Methods and Skin Friction

Tests with

Charts for Estimating Drag in Wind Tunnel Zero Heat Transfer

bv K. G. Smith

LONDON: HER MAJESTY’S STATIONERY OFFICE

1965

PRICE 7s 6d NET

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ILUISTRATIONS (Cont'd)

Laminar boundary layers: mean skin friction coefficient as a function of Reynolds nuniber based on momentum thictiess ofboundarylayer

Turbulent boundary layers: local skin friction coefficient as a function of Reynolds number based on distance from effective start of boundary layer

Turbulent baundasy layers: mean skin friction ooefficient'as a function of Reynolds nuniber based on distance from effective . start of boundary layer

Turbulent boundary layers: relationship between Reynolds numbers based on momentum thickness of boundary layer and distance from effective start of boundary layer

Turbulent'boundary layers: local skin friction coefficient as a functia of Reynolds nxmiber based on momentum thickness of boundary layer

Turbulent boundary layers: man skin friction coefficient as a function of Reynolds number based on momentum thickness of boundary layer

Fig.

5

6

9

10

m~aanloflowthefr?iotiaseldragnrrybe dedwedfranthebtnmdarylayw praemrr at the lzvuing edge, The praotioal use 09 Cooke's Jmthoda ia quite UffSoult ad tediau. Before badary layer oaloulati~ oan M made the ~amstrJraQt~~eznalnonneedstobelaramn~tsaoourafely. Atprermt it isnot pomaiblatooaloulatethefl~ fields BbQutgeneral~adiea inany detail,and~ 3nveatigatlon offlou fieldsiebothlaborlam ard lengthy. The variam asmaqtiona whioh different workers have nude about velooity pralmh ad SW stmes dietributians have been 0=iat3=a dn eamBdetailbyC~. Thoere ammptianrhave allbeenbaoadanthammll ammntdlcmapeeddataavailabl e,andthevaUditg of theirextona3.onto oaqrerielblefloubasnotbeen adquatolyoheohed. Asanexm@eat'the moertdntywhiahexbta, sllthomethodanhlohh~veb6enpmpoemdtalmfha variatiarof eurfaoe ahearbgcrt~ss alonga etreanillnetobe the mm aa in two dimmsiauXl flar. Emever memntr by Ashlcsnae ma BU~BU~ rrhcrned fhef therate Cp bamdarylqergrmth(inthe etreemdireotlaa) overaywedflat #ate was rrlightly hi@mr than the rate of grmth over an urqawd plate, 10 that thA8 allmqtation ia olrly a~toly oorxwot. In view af these ~brtt~ties it seam that, at least until a nuoh gmater body of eaprinmtal

vaiUble,ram aim$lermthodvf.Allgiveadequatemaults.

Cverthepaatf~yearseaperienoeinthe8ftx8ftw3ndtunnelat Bedfozd haa ahmn tbet flat plate fonnilae are oap&ble of givin& quite aooumto eatinmter of the skin fzdotion drag of mdele. FortuoslenderwSnga fao: rhioh the~dcee~~dfeeaseavailirble~~ely~a9~lrheRathrrt errtinmt3txk of friotim dmge by the methods givsn hera are oorreot tithin the expmiamtal aoouraoy d the other rims-ta. These wings had fairly mmll premare gmadients, but ewai for quite general typoa of flow tb errora are - relativelymmll. Forapartioulmoa&ereddeltawingwlthfeir

9 --=&I

prerarregradlcmtscudthree dinmaimaleffeotcrWinterBlld&aith~ hewe ehmn that the d3fferenoe b&men uverell dzq and warn drag agreed with e&Anu&es of rrldn frlotim within abmt lQ8 umr a range of &oh &era and a fairly nidemngeofinoidenw. Formreoa~@.extypeeofflanrfleld~~mdpamaure diatrlbuticmtlm u8eofflatplatefonmlaemaybe lees -te, but the predlotian at male effeot ppay still be mffioiently accumte formmy purpo6eh

The pressure drag of a model is obtained by subtraoting the estimated friction drag from the measured total drag. The total drag of the full scale aircraft is then obtained by adding the estimated full scale friction drag to the deduced pressure drag. An implicit assqtion here is that the pressure drag af a model does not change with Reynolds number. The possibility of small changes in pressure drag with change in scale should not be overlooked, but this effect is not ocnsidered in the present note.

The charts given here were prduced in order to make esUmating fri&ion drags a routine matter, end it is felt worth while meking them more widely available* The curves have been calculated for zero heat transfer and a tunnel total temperature of 300C. They cover Hach rnmibers up to 5, and Re

Y olds n ers between 105 and 107 for laminar boundary layers, and between

90 P and IO for turbulent boundary layers. Variations of total tunperature between O°C and 60% have an insignificant effect on the calculated skin fric- tion coefficients, and even raising the total terqerature to 150°C, as may be necessary at a Maoh nmiber of 5 in order to avoid liquefaction of air, changes the skin friction coeffioient (at constant Reynolds nuder) by less than 35%.

The skin friction formulae used are given in Section 3. Section 4 describes the charts, and Seotion 5 describes methods of using the charts. The charts are strictly applicable only to wind tunnel tests within the above range of total temperatures. Little error wouldbe incurred, however, by using the charts for estimation of full scale friction drag at Meoh numbers up to about 2.5. Above this speed the higher temperatures encouuterod in flight would lead to significant errors, snd in any case heat transfer affects are likely to bcoome jmportant in this range of speeds.

2

cf

% M

Rex

Ree T

U

X

e

P

P

T

LIEC OF sY?vBOLs

local skin friction coefficient

mean skin friction coefficient

Machnumber

Reynolds number based on distance frcm effective start cxf boundary layer

Reynolds number based on mcxnentum thickness of boundary layer

temperature (degrees absclute)

velocity

distance from effective start of boundary layer

momentum thickness of boundary layer

viscosity

density

surface shearing stress

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1 free atm8m oamtw

3 aRmR!Rlm!IoN-

4 DESC'XFTIONCIti'(IZ%R!TS

4mI Lamjnar boundsry layers

Frcm equations (1), (2)

and similarly

!f?

49 Assuming a recovery factor'& of 0.85, with the viscosity given by Sutherlands f oxmula

and (4) we have

P = 3.Q45 x IO -' Cl ftoJ slug/ft set ,

it is found that the factor [(T, p*)/(T* p,)]; varies only between 0.967 and 1.014 for Mach nmbers less than 5. This factcr may Easonably be ignored, as any errors incurred thereby are likely to be less thsn errors arising from neglect uf' pressure gredients snd three dimcnsicmal effects. Thus we have

=f -$ = 0,664 Rex , ('51

shown in Fig.1, and

shown in Fig.2. Equations (12) and (16) give

Re6 = 0.664 Re$ ,

shown in Fig.3. Substituting (17) into (15) and (1G) gives

07)

cf z O.&I Rei' ,

d&him @tted iuFlgm7forthe mneHaohndmr6. From(l2) azd(2l)we

experimental results. Frm equations (7) snd (12) this corresponds to Rex = 0.9 x 10% Thus it seems reasonable to take Rez = 105 as a lower limit to the validity of equations (7) and (9). C urves are given on Figs.6 to IO for Rez = 105. It is worth pointing out that the values of Rex corresponding to Rez = IO5 (see for example Fig.6) agree quite closely with those that Van Driest and Blumer 19 found were needed for the establishment of a fully developed turbulent boundary layer downstream af three dimensional transition trips.

Measurements on wind tunnel models may be made with boundary layer transition either free or fixed, and the methcds of calculation will differ acooxdingly. If trensiticn is allowed to cccur naturally regions of laminar and tmbulent flow are normally detected by flow visualisation. On the other hand if transition is fixed (normally near the leading edge) supplementary flow visualisation tests are not requi-d. These two techniques each hsve ' advantages and disadvantages. If regions of laminar fluw are present on the model the external flow msy be different frcm that occurring at full scale, as separations and shock wave/boundary layer interactions may be different. On the other hand if trsnsition is fixed allowance must be made for the drag of the tripping device. Evans9 has shcwn that at moderate Reynolds numbers (5 - IO x 106 based on mean chord) it is possible to fix transition with only a small penalty which can be estimated sufficiently acourately.

5.1 Free transition

Transition fronts are usually detected by a sublimation technique 20 . Before calculations can be made the relationship between the sublimation pattern and the surface shear needs to be known. It is comonly assumed that the boundary shown on the sublimation pattern indicates the beginning of the fully turbulent boundary layer, and that the flow ahead of this point is laminar, with a sudden rise in surface shear at this point from the value appropriate to a 1 sminar boundary layer to that for a turbulent boundary layer at' the same momentum thickness. Tests. on a cone reported by YJinter, Scott-Wilson snd Da&l have shown that in general this assumption is incorreot. They show that the sublimation front indicates the end of wholly laminer flow and is followed by a transition region in which the shear grsdually rises to that chsracteristic of a turbulent boundary layer. This conclusion is consistent with the type of flow which Schubaucr and Klebanoff

22

have s Brillhart 3 9

to exist in the trsnsitim region. Recent tests by Pate and on swept wings show that the sublim&ticn front may occur closer to

the middle of the transiticn regicn, so that the sublimation pattern may need different interpretations in different types of flows.

The extent of the transition region and the variation of surface shear through the transition region still need to be known. Dhawan and Narasimha~ have collected and analysed most of the available information on flow in the transition regicn.. From their Fig.5, it can be seen that the extent of the transition region is between about 4 and + of the length of leminar flow

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(a) Wings

The wing issplit up into a mmiber of chordwise strips, the v3riaticn of tr3nsition Reynolds number and chord between neighbouring 3trips being 3m3Il.l. From the extent of wholly 13minar flow we obtain C14, the momentumthickne3s of the laminar boundary layer at the start of the tr3nsition region, from Fig.3. We also obta.in.the local skin friction coefficient at this point, , frcm . cf

Fig.1. Knowing (or a3suming) the extent of the transition rsgion,ARcx, ' (whioh may be 3ero),we obtain the momentum thichess 9+, and local skin friction coefficient cf at the end of the transition region by the following procedure.

t

(1) ~~~ A3sume a value of momentum thicknes3 et at the end of the tran3ition region.

(2) Find c$') from R&A: from lXg.9. t

(3) Assuming a linear growth of cf with di3tance the momentum equation may be integrated to give

or

(23)

(4) Rq=t steps (2) ELM (3) until two succe3sive values af Re* agree t

to the accuracy required.

From this value of ReO we obtain, from Fig.8, the value, Re of the t . xt'

effective Reynolds number of the turbulmt bcUndary layer at thi3 point. The effective Reynolds number of the boundary layer at the trailjng edge, Rcc, is obtained by adding the Reynolds number based on the length of' chord remaining between the point concerned and the trailing edge, and we obtain ReO fromRe

C c and Fig.8. The mean 3kin fricticm coefficient for this strip is then obtained from

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wheretheiatega7Mani8ti mrthewlao lnlrPaoa,andthe ocpeotialfaotor %EI tjbtabdby dividjngABbytho platxt'a~~~area.

Measurements of skin friction have also been made on a cambered delta wing10 with fairly strong pressure gradients and three dimcnsicnal effcots. The measurements covered a range of incidence and although the local skin friction coefficients were at some points on the surfaoc quite different from flat plate formulae, the integrated friction drag varied only a little with incidence and was within about lO$ of the flat plate estimates.

There may be regions on models where the flow is clearly three dtinsional in char b&ies). Bradfield 2

ter (e.g. conic&L or ogival noses or boat-tailed after- has made a ccnqrehensive survey of masurcments of turbu-

lent boundary layers on cones, and concludes that the local skin friction coefficient on a cone is 18:; higher than that on a flat plate at the same Mach number and Reynclds number. For the rsnge cf Reynolds numbers likely in wind tunnel tests this implies (see Appendix IV of Ref.27) that the mesn skin fric- tion coefficient on a cone is about 73 higher than that on a flat plate at the ssme Xach number snd Reynolds number, so that an approximate correction for nose effects should be applied. Usually this ccrrecticn will be found to be insignificant. The only measuremen ts in converging flow of the type which occurs on after-bodies seem to be those of Winter and Smithlo. The local skin friction coefficients they measured were as low as half the flat plate values and can be explained, at least qualitatively, by the combined effects of con- verging flow with an adverse pressure gradient.% In any case any body with a boat-tailed after-body is likely to have a ccnical or ogivsl nose. The two effects are of cpposite sign and would therefore at least partially cancel, SO that little error should be incurred by assuming that they cancel completely.

6 CONCLUSIONS

Forxmilae and methods for estimating the skin friotion drag of wind tunnel models at zero heat transfer have been discussed. Charts arc provided to make the estimation simple and rapid.

Estimates which have been made over the past few years suggest that if the flow is attaohed everywhere, and pressure gradients and three dimcnsicnal effects are small, then the estimates may bc accurate to 2 or j$.* On tne other hand if there are large regicns of separated flow, or strong pressurc~ gradients or three dimensional effects the estimates may be in error by 1%

?l?ecent tests on a waisted body of revolution 28 have also given low values of local skin friction coefficient. These results agree moderately well with theoretical calculations.

**Recent measurements at high Reynolds numbers 29 suggest that interYi&Liate texqerature hypothesis may only correlate compressible ard incozItpressible skin friction to within about 5,d.

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!title. do.

1 Cooke, J.Ci Bull., KG

2

3 Cooke, J.G

4 Cooke, J.C.

5 Cooke, J.C.

6 Cooke, J.C.

9

10

11

12

13

hake, J.C.

AdlkmK-, ‘& RiddelI., KR. .

hum, J.Y.G.

Winter, ILG SkKLth, ILG.

Eolcert, u.Gb

I4rm@an, R.J.

&udnrylayersintbreed~~i~~ Pirogress 5n Aeronaut~al SOiOmOS, V&2, Jtp.221-282.

A oaloulathm metho for three diunnsional ldtailent boudarywrs. A.R.C. R. &M. 319% ktobar, 1958,

Boudarylaywrsoverinfiniteya~~~ Aeroa (2ll-L fi(l960), 333-34x

Tkeedimensionaltuz%wlentb~layers. A.&C. (2.635. Ji-, Ml.

!Eurbulentboudary layers on&elt&wings at sero

i%k c.p.696. Mh, li63.

Theboundaqylayerdzqofbodiesw5thswept txaiUngedge~insuperso~~ A.&C. C.P. 69% ~ebru=-y, W3.

Investigdionof the tdbulent boudarywona yama flat plate. IUCA TeohnSoal Note 3383. April, 19%.

Uae~a~~~l~aet~~~~p~~e of sledor wings su5tabl.e for a supersonh trans- port airoraft. R.&X. !i!eo~al Note k. Aem 2084 -w.c. aJH3. w63. A.R.C. CLP.742. Mu-oh, 1963.

U@blished X.&A. Report. .

Survey on heat transfer at high speed. wi9D.G Teohnioal Repmt w70. 1gyk

. &mtu3aeardapproximtionsforaoxdynmda heating antes in high speed flight. A2.C. GP.360. Cotober, 1955

Boundaqlaymrl2wry,4ti~tio~ UoGzww-Bill, NewYokk. 1960.

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R3ItZRBlCdS (Co&d.)

&

14

15

16

17

Smith, K.G. Unpublished M.O.A. Report.

Smith, D.1:. Skin friction measuremnts in incompressible flow. Walker, J.H. N..A.S..L TR R-26. 1gy3.

18 Preston, J.H.

20 Main-Smith, J.D.

21

22

23

24

Author Title, etc.

Monaghan, R.J. A review and assessment of various formulae for turbulent skin friction in compressible flow. A.R.C. C.P.142. August, 1952.

Peterson, J.&

Van Driest, R.R. Blumcr, C.B.

Winter, K.G. Scott-Wilson, J.B. Davies, P.V.

Schubauer, G.B. Klehnoff, P.S.

Pate, S.R. Brillhart, R.E.

D&wan, S. Narasimha, R.

Smith, K.G. The u5e o f surface pitot tubes as skin friction Gaudet, L. meters at supersonic speeds. Winter, K.G. A.R.C. R. & M. 3351. June, 1962.

A comparison of experimental and theoretical results for the compressible turbulent boudary layer skin friction with zero pressure gradient. N.A.S.A. T.N. D-1W.j. March, 1963.

The minimum Reynolds nunibsr for a turbulent bouuy layer and the selection of a transition device. J. Pluid Mech. ~(lg!ji'-58), 373-384.

Effect of roughness on transition in supersonic flow.

A.G.A.R.D. E2eport 2%. 1960.

.Chemical solids as diffusible coating films for visual indication of bourdary layer transition in air an3 water. A.R. C. R. 8: ?L 2755. February, 1950.

Methods of determination and of fixing boundary layer transition on wilti tunnel models at super- sonic speeds. A.R.C. C.P. 212. Scptcmbcr, l%L

Contributions on the mechanics of boundary layer transition. Boundary layer effects in Aerodynamics. Proceedings of a Symposia held at N.P.L. H.K S. 0. , 1955.

Investigation of boundary layer transition on swept wings at Mach numbers 2.5 to 5. Amc-TDR-63-109. 19 63.

Some properties of boundary layer flow during the transition from lsminar to turbulent motion. J. Fluid h%zch. ~(l%i'-58), 418-436.

-17-O

i&k Author

26 BmdtQld, KS.

28 Tinter, LG. Sdth, %G Rotta, J.C.

29 Winter, K.G. Smith, KS. Galldeli, L

Title, e-to.

Researohonlaminaradiadulent bazdqy~a atauperaonio Speeds, Find.8~~~ Minne8ota university, Rosemount Aero. bib. Report No.131. 1957.

iPi?-- masuremnts on 10’ ad 20' ooms

z 2.45 ad 8ero h0attransfer. A.R.C. C.P.264. W-d=, 1954.

bkmremnt~ of -hmbulentdd.nfkiotAonathigh Reynom nudmra atkch w&era of a2 ad 2.2. Prooeediw of Agad spooiall8t8 lbehing "Reoent Develoganents in Boudary 4er RcmadP, Naples, I-xv, -14*w, w&.

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FIG. I. LAMINAR BOUNDARY LAYERS : LOCAL SKIN FRICTION COEFFICIENT AS A FUNCTION OF REYNOLDS NUMBER BASED ON DISTANCE FROM START

OF BOUNDARY LAYER-

’ 005

‘004

-003

-002

l OOl l 0009 -0008

l OQ7

a0006

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FlG.2. LAMINAR BOUNDARY LAYERS: MEAN SKIN FRICTION COEFFICIENT A S A FUNCTION OF REYNOLDS NUMBER BASED ON DISTANCE FROM START

OF BOUNDARY LAYER.

4 5 6 7 * g,xu3

FIGa 3. LAMINAR BOUNDARY LAYERS: RELATIONSHIP BETWEEN REYNOLDS NUMBERS BASE0 ON MOMENTUM THICKNESS OF BOUNDARY LAYER

AND DISTANCE FROM START OF BOUNDARY LAYER.

n 003

0002

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l 0009 l oooEl

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me--~--

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2X102 3 4 5 6 709

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FfG.4. LAMINAR BOUNDARY LAYERSXOCAL SKIN FRICTION COEFFlClE.NT AS A FUNCTION OF REYNOLDS NUMBER

BASED ON MOMENTUM THlCKN/ESS OF BOUNDARY LAYER.

I A.R.C. COP. No&& I A&C. C.P. Nod&

C.P. No. 824

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