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NASA TECHNICAL NOTE NASA TN D-2054- ._ ­

c.a I

EFFECT OF CONTROLLED SURFACE ROUGHNESS O N BOUNDARY-LAYER TRANSITION A N D HEAT TRANSFER AT MACH NUMBERS OF 4.8 A N D 6.0

by Paul F. Holloway and James R. Stewett

Langley Research Center Langley Station, Hampton, Va.

N A T I O N A L AERONAUTICS A N D SPACE A D M I N I S T R A T I O N WASHINGTON, D. C. A P R I L 1964

TECH LIBRARY KAFB. "I

0154594

EFFECT OF CONTROLLED SURFACE ROUGHNESS

ON BOUNDARY-LAYER TRANSITION AND HEAT TRANSFER

AT MACH NUMBERS OF 4.8 AND 6.0

By Paul F. Holloway and James R. Sterrett

Langley Research Center Langley Station, Hampton, Va.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For saie by-the Office of Technical Services, Department of Commerce, Washington, D.C. 20230 -- Price $1.25

EFlFECT OF CONTROLLED SURFACE ROUGHNESS

ON BOUNDARY-LAYER TRANSITION AND HEAT TRANSFER

AT MACH NUMBERS OF 4.8 AND 6.0

By Paul F. Holloway and James R . S t e r r e t t

An invest igat ion has been conducted a t a free-stream Mach number of 6.0 t o determine the e f f ec t s of controlled three-dimensional surf ace roughness ( spheres ) on boundary-layer t r ans i t i on and neat t r ans fe r . Experimental data a re presented f o r a sharp-leading-edge two-dimensional f l a t -p l a t e model over a loca l un i t Reynolds number range of 1 . 2 x 106 t o 8 .2 x 106 per foot based on a loca l Mach number of 6.0 and 9.8 x 106 t o 13.9 x 106 per foot based on a l o c a l Mach number of 4.8.

The Reynolds number f o r na tura l t r ans i t i on has been found t o increase mark­edly with increasing Mach number above a Mach number of approximately 3.7. It w a s found t h a t surface roughness of height l e s s than the boundary-layer thick­ness can delay t r ans i t i on . The c r i t i c a l roughness Reynolds numbers determined experimentally i n t h i s invest igat ion a r e l a rge r than those found a t lower super­sonic Mach numbers. Roughness heights of approximately twice the calculated boundary-layer thickness a t t h e roughness posi t ion were required t o t r i p the boundary layer completely. Calculations of t he heat- t ransfer d i s t r ibu t ions based on simple f l a t -p l a t e theory a re shown t o agree reasonably well with the exper­imental r e su l t s .

INTRODUCTION

The nature of boundary-layer t r a n s i t i o n and i t s e f fec t on associated problem areas such as surface heat- t ransfer r a t e s and aerodynamic charac te r i s t ics has been the mbjec t of many studies, ye t there i s s t i l l much t o be learned about t he phenomena. The data and theor ies on boundary-layer t r a n s i t i o n current ly ava i l ­able f o r higher supersonic and hypersonic Mach numbers a re l imited a t bes t and contain a grea t deal of s ca t t e r . Indeed, t h e designer of a high-speed configura­t i o n f inds t h a t it i s d i f f i c u l t t o evaluate the meaning of the avai lable con­t rad ic tory experimental and theo re t i ca l results.

One of t h e most important fac tors affect ing t h e t r a n s i t i o n of t h e boundary layer from laminar t o turbulent i s the condition of t he body surface. Surface i r r e g u l a r i t i e s such as expansion s l o t s envisioned as necessary i n winged reentry

vehicles ( f o r example, t he X-20), connecting rivets,' o r d i s tor t ions due t o high-temperature buckling of t h e skin material may w e l l be prominent f ac to r s i n deter­mining the nature of t he l o c a l boundary layer . Since it i s possible t h a t surface i r r e g u l a r i t i e s may cause boundary-layer t rans i t ion , t h e designer of winged reentry configurations must be able t o estimate t h e extent t o which these i r reg­ularities may a f f ec t t h e locat ion and length of t r a n s i t i o n and, i n turn, t h e con­sequential e f fec t on t h e surface heat- t ransfer rates.

Surface condition i s also of importance from t h e experimental point of view since roughness may be u t i l i z e d as a boundary-layer t r i p i n wind-tunnel o r f ree-f l i g h t t e s t s . A guide i s needed f o r determining t h e s i ze of roughness required at high supersonic and hypersonic Mach numbers t o obtain f u l l y developed turbulent boundary layers which w i l l duplicate fu l l - sca le conditions.

Considerable study has been devoted t o boundary-layer t r ans i t i on and the fac tors affect ing t r a n s i t i o n at subsonic and supersonic speeds. (See, f o r example, refs. 1t o 7.) I n addition, recent work on t h e e f f ec t s of surface dis­to r t ions on the heat t r ans fe r t o a wing at hypersonic speeds has been published i n reference 8.

The purpose of t h i s report i s t o present t h e results of an experimental study of t he e f f ec t s of one type of surface d i s to r t ion of importance, controlled surface roughness (spheres), on boundary-layer t r ans i t i on and surface heat-t r ans fe r r a t e s a t Mach numbers of 4.8 and 6.0. The experimental invest igat ion w a s conducted on f l a t p l a t e s with sharp leading edges i n t h e Langley 20-inch Mach 6 tunnel and i n t h e Mach number 6.2 blowdown tunnel of t he Langley Research Center. The r e su l t s are compared with the avai lable data, empirical resu l t s , and theory f rom t h e l i t e r a t u r e f o r supersonic and l o w hypersonic Mach numbers.

*1?A2

C f

CF

CP,O

CW

d

h

k

coeff ic ients of T ' equation

loca l skin-fr ic t ion coeff ic ient

average skin-fr ic t ion coeff ic ient

spec i f ic heat a t outer edge of boundary layer

spec i f ic heat of w a l l material

diameter of roughness elements

heat- t ransfer coeff ic ient

v e r t i c a l height of roughness above p l a t e

2

length of t r ans i t i on region

Mach number

Prandtl number

Stanton number

experimental heating rate

recovery f ac to r

Reynolds number based on f l u i d conditions at top of roughness elements

%?kkand height of roughness, ­pk

PoU,un i t Reynolds number per foot a t outer edge of boundary layer, -PO

POUOtt r ans i t i on Reynolds number, -I-lO

l oca l free-stream origin, POUO+

P O

l o c a l free-stream

edge, po'ox

l o c a l free-stream

Reynolds number based on distance from v i r t u a l

Reynolds number based on distance from leading

Reynolds number based on distance from roughness l ocation, po'o*

I-lO

Reynolds number based on T ' conditions and distance from v i r t u a l origin, p 'UOXV

P '

la teral spacing of roughness

distance from leading edge t o t r ans i t i on ( e i t h e r at end of laminar flow or at beginning of turbulent flow)

temperature

reference temperature

veloci ty component of flow p a r a l l e l t o surface of p l a t e

3

X distance from leading edge

xk distance from leading edge t o roughness posi t ion

XV distance from v i r t u a l or ig in

H distance from roughness t o instrumentation locat ion

Y v e r t i c a l coordinate measured from p l a t e surface

P density

P leading-edge thickness

OW l o c a l wall thickness

7 time

6 calculated boundary-layer thickness at roughness posi t ion based on veloci ty

CL angle of a t tack

7 r a t i o of spec i f ic heats

P v i scosi t y

Subscripts:

1,2,3 denote r e l a t ive roughness height

c r c r i t i c a l

k conditions at top of roughness elements

l E U I l laminar conditions

0 l o c a l conditions a t outer edge of boundary layer

r recovery

T' based on reference temperature conditions

t r ans t r ans i t i ona l conditions

turb turbulent conditions

W wall

4

03 free-stream conditions

Primes denote parameters evaluated at reference temperature T ' .

APPARATUS AND TEST METHODS

Wind Tunnel

The t e s t program w a s conducted i n t h e Langley 20-inch Mach 6 tunnel and i n the Mach number 6.2 blowdown tunnel. The Langley 20-inch Mach 6 tunnel i s t h e intermit tent type exhausting t o t h e atmosphere through a d i f fuser augmented by an a i r ejector . Tests were run with tunnel stagnation pressures of 365, 440, and 515 pounds per square inch absolute with stagnation temperatures of 960° R t o 1,020° R. A more de ta i led descr ipt ion of t he tunnel i s given i n reference 9.

I n order t o extend the t e s t Reynolds number range below that obtainable i n the Langley 20-inch Mach 6 tunnel, addi t ional t e s t s were conducted i n the Mach number 6.2 blowdown tunnel. The tunnel i s a l so of t he intermit tent type exhausting t o a 40,000-cubic-foot sphere which can be pumped t o pressures as low as, 1millimeter of mercury absolute. Tests were run with tunnel stagnation pres­sures of approximately 65, 165, and 265 pounds per square inch absolute with stagnation temperatures of 8400 t o 1,020° R. A more de ta i led description of t he tunnel i s given i n reference 10.

Models

The models t e s t ed consisted of f l a t p l a t e s with sharp leading edges con­structed from s t a in l e s s s t e e l . Each model assembly w a s 9 inches wide and approx­imetely 11 inches long f o r t he t e s t s i n the Langley 20-inch Mach 6 tunnel. P l a t e 1w a s a continuous plate , a sketch of which i s given i n f igure l ( a ) . The remaining models consisted of p l a t e 2 with interchangeable leading edges (one f o r each roughness he ight ) . (See f i g . l ( b ) . ) The instrumented p l a t e s and the leading edges were mounted on a support p l a t e as shown i n f igure 1. The smaller s i z e of t h e Mach number 6.2 blowdown tunnel required t h a t t h e models be smaller. For the t e s t s i n t h i s tunnel, p l a t e 2 and the leading edges were cut down so t h a t

t h e model assembly w a s 7-1 inches wide and 1C-1 2

inches long.2

For t h e roughness t e s t s , t h e leading-edge pieces were interchanged, each having a d i f f e ren t s i ze roughness mounted 2 inches from the leading edge. F l a t p l a t e 2 and the support p l a t e completed t h e assembly. (See f i g . l ( b ) . ) The spheres were glued in to s m a l l spherical indentations i n t h e leading-edge p l a t e . The location, spacing, height above t h e plate , and diameter of t he spheres are given i n f igure l ( b ) .

The leading edge of a l l t h e models w a s a 20° wedge t h a t tapered t o a cylin­d r i c a l leading edge with a radius of approximately 0.002 inch o r l e s s . Two models

5

of each p l a t e were constructed- one being instrumented with 0.050-inch I D pressure o r i f i c e s and t h e other with 30-gage iron-constantan thermocouples. The instnunen­t a t i o n w a s located chordwise along t h e center l i n e of t h e p la tes . The undersur­face of each p l a t e instrumented with thermocouples w a s s lo t t ed along t h e center l i n e t o a width of 0.6 inch and a surface skin thickness of approximately 0.020 inch t o minimize the l a t e r a l heat conduction i n t h e skin.

Test Methods and Techniques

For t h e t e s t s i n t h e Langley 20-inch Mach 6 tunnel, t h e free-stream Mach number outside t h e boundary layer of t h e models w a s varied by changing the angle of a t tack of t he p la tes . This method gave a l o c a l surface Mach number M, of 6.0 a t an angle of a t tack of Oo and o f 4.8 at an angle of a t tack of 8'. The resu l t ing un i t Reynolds numbers per foot based on conditions outside the boundary layer were 5.82 x 106, 7.02 x 106, and 8.22 x 106 f o r Mo = 6.0 and 9.84 x 106, 11.86 x 106, and 13.92 x 106 f o r Iv&, = 4.8 f o r t h e tunnel stagnation pressures of 363, 440, and 315 pounds per square inch absolute, respectively.

The t e s t s w e r e run i n the Mach number 6.2 blowdown tunnel with the p la tes at an angle of a t tack of Oo, a nominal l o c a l Mach number of 6.0 being obtained. The resu l t ing l o c a l un i t Reynolds numbers per foot were approximately 4 x 106, 2.6 x 106, and 1.2 x 106.

Pressure t e s t s . - Pressure d is t r ibu t ions along t h e center l i n e of the plates were obtained i n t h e Langley 20-inch Mach 6 tunnel for & = 6.0 and 4.8 with free-stream Reynolds numbers per foot of 5.82 x 106, 7.02 X 106, and 8.22 x 106. The loca l s t a t i c pressures on t h e p l a t e s were measured by connecting the o r i f i ce s t o pressure-switching devices which i n tu rn connected t h e o r i f i ce s i n sequence t o e l e c t r i c a l pressure transducers. The e l e c t r i c a l outputs from t h e transducers were recorded on a d i g i t a l readout recorder. Each pressure-switching device w a s con­nected t o two transducers with ranges of 1and 5 pounds per square inch absolute. The accuracy of t h e transducers i s approximately 1/2 percent of fu l l - sca le reading. All pressure t e s t s were run on the same support system as w a s used f o r t h e heat-transfer tests.

Heat-transfer t e s t s .-The aerodynamic heating w a s determined by t h e t rans ien t calorimetry technique by which t h e r a t e of heat storage i n the model skin i s meas­ured. The models, o r ig ina l ly at room temperature o r s l i g h t l y cooler, were sud­denly exposed t o t h e air flow by quick in jec t ion from a shel tered posi t ion beyond t h e tunnel w a l l . In jec t ion w a s accomplished i n l e s s than 0.23 second and t h e model remained i n t h e tunnel f o r only 4 seconds so t h a t t h e model w a s i n a nearly isothermal condition, lateral conduction i n the skin being kept t o a m i n i m .

t i c a l methods.- During t h e pressure and heat- t ransfer t e s t s i n the Langley shadowgraphs and schl ieren photographs were occasionally

taken t o a id i n determining the type of boundary l aye r ex is t ing on the p la tes .

6

DATA REDUCTION

The e l e c t r i c a l outputs from the thermocouples were recorded on a high-speed d i g i t a l readout recorder. The reading from each thermocouple w a s recorded a t 0.025-second intervals , converted t o a binary d i g i t a l system, and recorded on magnetic tape. The temperature-time data were f i t t e d t o a second-degree curve by the method of l e a s t squares, and the time derivative of temperature w a s computed on a card-programed computer.

The tunnel stagnation temperature range w a s 840' R t o 1,020° R and t h e w a l l temperature of t he p l a t e w a s approximately 550' R. Because of t he short time required f o r t he in jec t ion of t h e model, t he p la tes were considered t o have been subjected t o a s tep function i n t h e applied heat-transfer coeff ic ient . The thin-skin equation used t o calculate t h e l o c a l surface heating r a t e w a s

The measured l o c a l heat- t ransfer coeff ic ient w a s then calculated by the re la t ion

i n which conduction effects a re neglected, and where T r i s t h e calculated , recovery temperature defined as

Tw i s t h e measured w a l l temperature, and M, i s t h e l o c a l Mach number outside the boundary layer calculated from the measured pressure d is t r ibu t ion . For t h e t e s t s i n the Mach number 6.2 blowdown tunnel, t he pressure d is t r ibu t ion w a s assumed t o be t h e same as t h a t obtained i n the Langley 20-inch Mach 6 tunnel f o r a given configuration. The recovery temperature T r w a s calculated by assuming a recovery f ac to r equal t o 0.830 i n t h e laminar region and 0.883 i n the turbulent region. For t h e t r ans i t i ona l region, T r w a s calculated by assuming a l i n e a r var ia t ion given by

AxTr,trans = T r , l a m + i ( T r , t u r b - T r J l a m ) (31

where

ax l i n e a r distance from beginning of boundary-layer t r ans i t i on

2 length of t r a n s i t i o n region

Finally, t h e Stanton number, based on l o c a l conditions outside of t he boundary layer, w a s calculated by use of t h e equation

7

NSt = h (4)

po'ocp, 0

The experimental heat- t ransfer parameters 6, h, and N S t presented i n t h i s report were determined by reading the slope of t h e temperature-time curve at a time 0.05 second a f t e r t he model w a s i n posi t ion i n the tunnel. The maximum surface temperature increase a t the time f o r which t h e parameters were calculated w a s always l e s s than 25' and generally l e s s than l 5 O . This low temperature increase combined with t h e thin-skin thickness kept t h e conduction e r r o r at a minimum. The inaccuracy of t he d is t r ibu t ion i s thought t o be l e s s than -+lopercent.

REVIEW OF HEAT-TRANSFER EQUATIONS

There a re many methods available for t he theo re t i ca l calculat ion of the Stanton number. (See, for example, r e f s . 11t o 16.) I n t h i s analysis, t he T ' method of Monaghan ( r e f s . 13 and 14) has been employed. From reference 16, the T I equation may be wri t ten as

L ' J

The Stanton number may be determined from t h e modified Reynolds analogy

n

where t h e Prandtl number Npr i s based on T I conditions.

Laminar Boundary Layer

For t h e case of a laminar boundary layer, t h e coeff ic ients of equation ( 5 ) (see r e f . 12) become .

By using the Blasius equation f o r laminar flow, equation (6) becomes

8

I :

1

11 where NSt i s based on free-stream conditions f o r d i r ec t comparison w i t h data, , and the conversion parameter from T I reference conditions t o free-stream con­

d i t ions C ' i s given by

Turbulent Boundary Layer

For the case of a turbulent boundary layer, t he coef f ic ien ts of equation ( 5 ) ( see r e f . 13) become

A1 = 0.54 \ A2 = 0.149

The Karman-Schoenherr equations were used t o determine the l o c a l sk in- f r ic t ion coeff ic ient as follows. (See r e f . 15 f o r a p lo t of these parameters and fu r the r discussion.)

0.242

JCF,T' = loglO(Rx,T')(%,T1)

where

and C f Y T l may be converted t o free-stream conditions as follows:

Cf = ' f ,T '

T '/TO

and f i n a l l y Stanton number i s determined by modifying equation (6), basing NSt on free-stream conditions so t h a t

where Cf i s determined by equation (14 ) . The dis tance xv of equation (12) i s defined as the dis tance from t h e v i r t u a l o r ig in . I n t h i s paper, t he v i r t u a l o r ig in f o r t he f l a t p l a t e with undisturbed flow w a s defined as the point a t which laminar flow ends, as determined by t h e surface heat- t ransfer rates. (See r e f . 15 f o r f u r t h e r discussion.) For t h e f la t p l a t e with roughness, t he v i r t u a l o r ig in w a s found by determining t h e length of t r a n s i t i o n 2 on t h e smooth p l a t e f o r a given free-stream Reynolds number, assuming ful ly developed turbulent flow t o begin at the t r i p posit ion, and then defining t h e v i r t u a l o r ig in t o be located at a dis tance 2 forward of the t r i p posi t ion.

For the lower Reynolds number tests, f u l l y developed turbulent flow w a s not obtained on t h e smooth f l a t plate; therefore, t h e length of t r ans i t i on w a s not known. I n order t o compare t h e experimental r e s u l t s with theory f o r these t e s t s , t w o theore t ica l d i s t r ibu t ions w e r e calculated by:

Assuming t h e v i r t u a l or igin t o be l ocat ed at the roughness posi t ion p o , v = RO,*)

( 2 ) Assuming t h e Reynolds number based on t h e v i r t u a l or igin t o be R0,v = Ro,* + 2.7 x 106, where 2.7 x 106 w a s based on t h e length of t r a n s i t i o n f o r t he higher Reynolds number t e s t s .

Transi t ional Boundary Layer

A simple semi-empirical method of predicting t h e heat t r ans fe r i n the t r ans i ­t i o n a l boundary-layer region has been presented. This method i s ' based on the near l i n e a r increase i n heat t r ans fe r which begins at t h e end of laminar flow and i s presented only f o r general comparison purposes. I n t h i s method, it i s assumed t h a t the Stanton number increases l i nea r ly from t h e end of laminar flow t o the point a t which f u l l y developed turbulent flow i s f i rs t obtained. The d i f f i c u l t y with the re la t ion i s t h a t t h e beginning and end of t r ans i t i on of t h e boundary layer must be determined before it can be applied.

RESULTS AND DISCUSSION

Determination of Transit ion

A s has been discussed by Probstein and Lin i n reference 2, many f ac to r s a re known t o influence t h e select ion of t h e so-called " t rans i t ion point" from laminar t o turbulent flow i n wind-tunnel t e s t s . For instance, t he t r ans i t i on process occurs over a f i n i t e distance covering a s igni f icant range of Reynolds numbers; and the type of instrumentation and method used t o detect t r ans i t i on influences t h e select ion of t h e t r a n s i t i o n point.

An accurate method of detecting t r ans i t i on i s needed f o r t he proper evalua­t i o n of many wind-tunnel t es t resu l t s . I n t h i s report, t he locat ion of t r ans i ­t i o n has been determined primarily by noting a change i n t h e l o c a l surface heating rate along t h e p la tes . A t yp ica l example of t h i s e f fec t i s shown i n f igure 2 which gives t h e d i s t r ibu t ion of t he l o c a l heating r a t e along the p l a t e a t two l o c a l Mach numbers and several un i t Reynolds numbers. A s t he laminar boundary layer on t h e forward end of the p l a t e thickens, t h e l o c a l heating r a t e decreases u n t i l t r ans i t i on occurs. Once t r ans i t i on of t h e boundary layer begins, t h e l o c a l heating rate increases rapidly. When f u l l y developed turbulent f l o w i s obtained, t he heating rate peaks and begins t o decrease with increasing distance from t h e leading edge. The general shape of t h e surface-heating-rate d i s t r ibu t ion is, of course, similar t o t h a t of t h e usual shear d i s t r ibu t ions f o r a f l a t p la te . (See, f o r example, r e f . 17.) The locations defined as t h e beginning and end of t he

10

transition region are sketched on the in se r t of f igure 2 (a ) . The beginning of t r a n s i t ion (end of laminar flow) w a s taken as tha t region where the heating r a t e begins t o increase rapidly. The end of t r ans i t i on (beginning of f u l l y developed turbulent flow) w a s taken as t h a t region where t h e l o c a l heating r a t e s begin t o l e v e l off rapidly. An objection t o the surface-heating-rate method of deter­mining t r ans i t i on i s t h e e r ro r introduced by l a t e r a l thermal conduction i n the skin. However, i n these t e s t s great care has been taken t o minimize the thermal conduction i n t h e skin so t h a t t h e t r ans i t i on data presented herein are thought t o be accurate.

Mach number, Reynolds number, surface roughness, leading-edge thickness, temperature of model, free-stream turbulence, and so for th , all a f f ec t t he loca­t i o n of t r ans i t i on . Also, t h e method of detecting t r ans i t i on influences t h e apparent experimental locat ion of t r ans i t i on . Therefore, as might be expected, a compilation of t he avai lable t r ans i t i on data shows considerable sca t t e r . These data were obtained from references 18 t o 25 and t h e present invest igat ion and a re presented i n f igure 3 with t r ans i t i on Reynolds number p lo t ted as a function of Mach number. Reference 25 has indicated t h a t two very important parameters f o r correlat ing the data from the various in s t a l l a t ions are free-stream un i t Reynolds number and leading-edge thickness. However, a fu r the r correlat ion of t h e data i n f igure 3 i s d i f f i c u l t and out of t he scope of t h i s paper; ra ther f igure 3 i s meant t o show t h a t t r ans i t i on data of t h i s invest igat ion agree reasonably well w i t h those found i n the l i t e r a t u r e and t h a t the general t rends of t he available data indicate t h a t t he s t a b i l i t y of t h e boundary layer increases with increasing Mach number above Mach numbers of approximately 3.5 t o 4. Similar r e su l t s have been found by Pot te r and Whitfield i n reference 26 where f o r a given un i t Reynolds number and leading-edge thickness, t h e t r ans i t i on Reynolds number w a s found t o increase markedly above a free-stream Mach number of approximately 4.0. The t r ans i t i on Reynolds number i s shown ( i n r e f . 26) t o increase as much as 500 per­cent i n going from a free-stream Mach number of 4.0 t o a free-stream Mach number of 8.0.

To complement t h e data determined by the heat- t ransfer method, shadowgraph and schl ieren photographs (see, f o r example, re fs . 9 and 27) were a l s o employed t o detect t he locat ion of t r ans i t i on . A typ ica l shadowgraph f o r t h e f la t p l a t e i s shown i n f igure 4 ( a ) . The beginning of a change i n t h e slope of t he th ick white band represents t h e beginning of t rans i t ion . The end of t r ans i t i on i s indicated when the band converges t o the apparent p l a t e surface. A t yp ica l schl ieren photograph of t he flow over t h e f la t p l a t e i s shown i n f igure 4 ( b ) . The beginning of t r ans i t i on i s assumed t o occur a t t h e point on t h e schl ieren photograph where the boundary layer begins t o thicken more rapidly. (See r e f . 18.) A comparison of t h e t r a n s i t i o n data from t h e shadowgraph and schl ieren photographs with no roughness with t h a t from heat- t ransfer measurements ( f i g . 2) as given i n f igure 3 shows reasonable agreement i n t h e loca t ion of t r a n s i t i o n by t h e several methods when it i s remembered t h a t t h e beginning of t r ans i t i on i s not a point but ra ther a s m a l l region which may s h i f t somewhat with time.

A s w a s pointed out i n reference 2, t h e work of Coles ( r e f . 20) shows t h a t t he Reynolds nbmber f o r t he end of t r ans i t i on d i f f e r s from t h e Reynolds number f o r t h e beginning of t r ans i t i on by a f ac to r of 1.5 f o r supersonic Mach numbers. It i s in te res t ing t o note (from f i g . 3) t h a t t h e data of t h i s invest igat ion f i t

11

approximately t h i s empirical r e l a t ion f o r a given free-stream Reynolds number a t l o c a l free-stream Mach numbers of 4.8 and 6.0.

Ef fec ts of Roughness on Boundary-Layer Transit ion

One of t h e governing var iables i n producing t r ans i t i on by three-dimensional roughness elements i n supersonic flow i s t h e height of t h e roughness. Figure 5 presents a sketch t y p i c a l of t h e t r a n s i t i o n posi t ions f o r various height spheres taken from t h e data of references 5 and 6 f o r supersonic flow with zero heat t ransfer . The d i f f e ren t regions of t r a n s i t i o n as indicated by the curves f o r various s ized roughness are labeled 1, 2, and 3. I n zone 1, t h e free-stream disturbances are predominant i n es tabl ishing t rans i t ion , whereas t h e disturb­ances from t h e roughness play t h e predominant ro l e i n zone 3 and a fu r the r increase i n sphere height has l i t t l e e f f ec t on t h e locat ion of t r ans i t i on . I n zone 2, both t h e free-stream disturbances and t h e disturbances created by the roughness have an e f f ec t on the pos i t ion of t rans i t ion , and a fu r the r increase i n t h e height of t h e roughness w i l l cause t h e t r a n s i t i o n locat ion t o move forward. The data of reference 5 a l s o ind ica te t h a t t he l a t e r a l spacing of a single row of spheres has l i t t l e e f f ec t on boundary-layer t r a n s i t i o n provided t h e spheres a re not so close together t h a t they a c t as a two-dimensional roughness element. The e f f ec t of sphere spacing w a s not examined i n t h i s test program; however, a roughness element w a s always located on t h e center l i n e of t he model, forward of t h e thermocouples ( f i g . 1)i n order t o minimize any e f f ec t of sphere spacing.

I n t h i s paper, a roughness t r i p i s defined as ef fec t ive (or c r i t i c a l ) when a fur ther increase i n roughness height causes l i t t l e change i n the forward move­ment of t h e loca t ion of t h e end of t h e boundary-layer t r ans i t i on .

I n f igures 6 and 7, t h e d is t r ibu t ions of t h e l o c a l heating rates along the p l a t e f o r various height roughness are presented f o r l o c a l free-stream Mach num­bers of 6.0 and 4.8, respectively. I n these figures, t h e data given by the c i r ­cular symbol represent t h e heating r a t e s obtained on p l a t e 2 w i t h a sharp leading edge and no roughness. The dashed l i n e represents t he f a i r e d data from f igure 2 f o r t he continuous p l a t e 1. The discrepancy between t h e two s e t s of data i s thought t o result not only from s m a l l var ia t ions i n t h e leading-edge thickness (see, f o r example, r e f s . 25 and 26), but a l s o from a s m a l l angle-of-attack varia­t i o n (less than 1/20) between t h e two assemblies. However, t h e angle of a t tack f o r each s e r i e s of roughness tests w a s invar iant since t h e data represent a group of t e s t s made without changing t h e mounting.

Examination of f igu res 6(a) and 6(c) shows t h a t t he smaller diameter rough­ness elements (k = 0.0018 foa t and O.OO3O foo t ) were ac tua l ly found t o delay t r ans i t i on beyond t h e t r a n s i t i o n point found with p l a t e 2 and no roughness. Since these data indicated t h a t s m a l l roughness heights may delay t rans i t ion , fur ther l imited roughness tests w e r e conducted with t h e model assembly having t h e same leading edge so t h a t possible e f f ec t s of slight var ia t ions i n nose bluntness would be eliminated. (This work w a s stimulated by the comments of Po t t e r and Whitfield i n ref. 28 i n regard t o t h e summary of t h i s study published i n r e f . 29.) Figure 6 ( f ) presents t h e heat flow rate d is t r ibu t ions f o r t h e model with t h e same leading edge, and w i t h various height roughness elements at a u n i t Reynolds number similar t o t h a t of f igure 6(a). A schematic of t h e model

12

i

f shown i n the f igure i l l u s t r a t e s t h e new mounting technique, t he roughness s t r i p s! being interchangeable and with t h e same leading edge (thickness diameter Of

0.0025 inch), being used f o r a l l da ta of t h i s f igure. Comparison of t he da t a in figures 6(a) and 6 ( f ) ind ica tes t h a t although some of t h e var ia t ion i n t h e loca­t i o n of t r ans i t i on with s m a l l roughness and no roughness w a s due t o t h e slight var ia t ions i n t h e leading-edge thickness, under ce r t a in conditions t r a n s i t i o n i s apparently s l i g h t l y delayed when t h e surface roughness i s l e s s than t h e boundary-layer thickness.

The physical reasons f o r these phenomena are not ful ly understood; however, reference 19 has indicated t h a t t h e s t a b i l i t y of a separated laminar mixing l aye r increases with Mach number more rapidly than the s t a b i l i t y of an attached laminar boundary layer . This f a c t has l e d t o t h e speculation t h a t t he laminar separation t h a t must ex i s t near t he s m a l l roughness elements i s a t l e a s t p a r t i a l l y respon­s i b l e f o r a delay i n t h e t r a n s i t i o n a t t h e present Mach number. However, t h e pressure l o s s at t h e edge of t h e boundary layer caused by t h e roughness elements may be a contributing fac tor . These phenomena appear t o warrant fu r the r invest igat ion.

For the higher Reynolds number tests at = 6.0 ( f i g s . 6(a), 6(b), and 6 (c ) ) as the roughness i s increased above k = 0.003 foot, t r ans i t i on moves for­ward u n t i l a fur ther increase i n height i s inef fec t ive . For example, note t h a t i n f igure 6(a) t h e heating-rate d i s t r ibu t ions f o r t h e roughness of height k 2 0.0054 foot a r e approximately the same, and k = 0.0054 foot i s defined as t h e roughness height s l i g h t l y grea te r than t h e c r i t i c a l roughness height f o r t h i s free-stream Reynolds number.

Similar t rends a re evident i n f igure 7 f o r M, = 4.8, t h e smallest s i ze roughness (k = 0.0018 foo t ) now causing a considerably grea te r forward movement of t h e beginning of t r a n s i t i o n than w a s found f o r Mo = 6.0, since the l o c a l Reynolds number i s increased. However, t h e smallest s i z e roughness i s s t i l l below t h e c r i t i c a l height as previously defined.

Previous work has es tabl ished a c r i t e r ion f o r determining t h e c r i t i c a l height of three-dimensional roughness elements f o r subsonic and supersonic speeds. (See, f o r example, r e f s . 4 and 30 . ) This c r i t e r i o n s t a t e s t h a t t r ans i ­t i o n w i l l occur when t h e roughness Reynolds number exceeds a cer ta in value which i s approximately constant with the Mach number. This c r i t i c a l roughness Reynolds number is, by def ini t ion, based on t h e f l u i d propert ies a t t h e top of t h e raugh­ness element and t h e height of t he roughness element. The value of t h e c r i t i c a l roughness Reynolds number w a s not found t o be affected appreciably by moderate surface cooling. (See ref. 30.)

The var ia t ion of roughness Reynolds number Rk with various assumed rough­ness heights i s presented i n f igures 8 and 9 f o r l o c a l free-stream Mach numbers of 6.0 and 4.8, respectively. These f igures show the value of Rk f o r both adiabat ic w a l l conditions (Tw/Tr = 1.0) and t h e ac tua l w a l l temperature (Tw/Tr = 0.66) of t h e present t e s t s . The roughness Reynolds numbers presented were determined from boundary-layer temperature and veloci ty p ro f i l e s calculated by the Chapman-Rubesin method f o r an x-distance of 2 inches from t h e leading edge ( the roughness loca t ion) . A complete discussion of t h e va l id i ty and assumptions of t h i s theory i s given i n reference 31. me veloc i ty p ro f i l e s calculated by

t h i s method are a l so given i n f igures 8 and 9. I n addition, these f igures show t h a t t he e f f ec t of cooling the boundary layer i s t o cause a thinning of t h e boundary layer and thereby increase t h e Rk values f o r a given height roughness provided t h e roughness height i s l e s s than the boundary-layer thickness.

The c r i t i c a l roughness Reynolds number has been obtained experimentally f o r lower supersonic Mach numbers by several invest igators . A summary of t h e experi­mental r e su l t s from reference 4 (which included da ta from other sources, including r e f . 5 ) i s presented i n f igure 10 i n t h e form of t h e var ia t ions ofp G with Mach number. Also plot ted i n t h i s figure are the resu l t s of t h e

present invest igat ion where the open symbol ind ica tes t h a t t he roughness height i s s l i g h t l y less than t h e c r i t i c a l height and the s o l i d symbol indicates t h a t t h e roughness height i s s l i g h t l y grea te r than t h e c r i t i c a l height. This figure shows c lear ly t h a t t h e c r i t i c a l roughness Reynolds numbers determined experi­mentally f o r M, = 4.8 and & = 6.0 a re much l a r g e r than those observed i n reference 4 f o r lower supersonic ve loc i t ies . The c r i t i c a l roughness height of t h e present tests i s a l so much l a rge r than t h a t predicted by t h e semi-empirical equation of reference 6, which w a s determined from experiments conducted a t lower supersonic speeds.

It should be mentioned t h a t the def in i t ion and determination of t he c r i t i c a l Reynolds number f o r t h e previous data shown i n figure 10 from reference 4 are not exactly t h e same as t h a t used i n t h e present report . Also, t h e data of refer­ence 4 were taken under zero heat- t ransfer conditions. Some possible values based on t h e end of t r a n s i t i o n from t h e data of references 26 and 32 are a l so shown i n f igure 10. I n reference 32, t h e end of t r a n s i t i o n w a s determined by the shadowgraph method which i s similar t o t h e present t r ans i t i on detect ion method since both methods detect permanent changes i n t h e boundary-layer pro­f i les . The data i n f igu re 10 show t h a t basing t h e c r i t i c a l Reynolds number on t h e end of t r a n s i t i o n gives values which are higher than those given i n reference 4.

A modification of t h e c r i t i c a l roughness cor re la t ion parameter Rk has been given by Po t t e r and Whitfield i n references 26 and 33. The results of t h e present invest igat ion agree well with the extrapolated values given i n these references where it i s predicted t h a t f o r l o c a l hypersonic Mach number and w a l l s a t temperatures corresponding t o the conditions of t h e present investigation, t he Rk value needed t o br ing about t r ans i t i on a t t h e roughness elements would be approximately 2.4 X lo4. The values of Rk of t h e present invest igat ion were approximately 2 x 104 t o 4 x 104.

The experimental var ia t ion of t he c r i t i c a l roughness Reynolds number with free-stream un i t Reynolds number f o r a Mach number of 6.0 i s shown i n f igu re 11. Included i n t h i s figure a re both the data taken with one leading edge with rough­ness elements mounted on an interchangeable s t r i p and t h e data taken with in t e r ­changeable leading edges. The two s e t s of da ta do not coincide; however, t h i s condition may be due i n pa r t t o a d i f fe ren t loca t ion Q of t h e roughness ele­ments. A n increase i n free-stream un i t Reynolds number by a f ac to r of approxi­mately 8 w a s found t o result i n an increase i n c r i t i c a l roughness Reynolds num­ber by a f ac to r of approximately 4. Hence at a Mach number of 6.0, Rk,cr has been found t o be sens i t ive t o t h e free-stream un i t Reynolds number.

1 4

For l o c a l free-stream Mach numbers of 4.8 and 6.0, it has been shown herein tha t t he roughness parameter k/6 must be approximately 2 or grea ter t o move t h e beginning of f u l l y developed turbulent flow t o t h e region of t h e roughness. A s i s pointed out i n reference 26, t h i s requirement of a roughness height of twice the boundary-layer thickness may lead t o l imi ta t ions i n t h e use of three-dimensional roughness t o obtain turbulent flow i n wind-tunnel t e s t s because of t he flow d i s to r t ions created by t h e roughness extending outside the boundary layer . O f course, t h e requirement of k/6 values of 2 or grea ter i s p a r t i a l l y responsible f o r t h e l a rge var ia t ion i n Rk,cr with varying free-stream Reynolds number.

Further work i s warranted t o give a b e t t e r ins ight i n t o t h e importance of t he defining parameters i n boundary-layer t r a n s i t i o n work. It i s in te res t ing t o note t h a t t h e recent t heo re t i ca l invest igat ion o f Lees and Reshotho i n refer­ence 34 has a l so indicated t h a t t h e minimum c r i t i c a l Reynolds number f o r t h e boundary layer of an insulated p l a t e would rise sharply with increasing Mach num­ber above a Mach number of 3 and indicates a need f o r t h e reexamination of t he basic assumption of t h e theory of s t a b i l i t y of t h e laminar boundary layer at high supersonic and hypersonic Mach numbers.

Comparison of Heat-Transfer Distr ibut ions With Theory

Distr ibut ions over smooth continuous p la te . - The experimental heat- t ransfer-dis t r ibu t ions i n t h e form of t h e var ia t ion of Stanton number with distance from the leading edge are shown i n f igure 12. Also, given i n t h i s f igure a r e t h e cal­culated var ia t ions of Stanton number with distance from t h e leading edge f o r t he laminar, t rans i t iona l , and turbulent regions of t h e boundary layer obtained by the methods presented i n t h e sect ion e n t i t l e d "Review of Heat-Transfer Equations." This figure c l ea r ly shows t h e r e l a t i v e values of heat t r a n s f e r t o be expected f o r t h e various types of l o c a l boundary layers with na tura l t r ans i t i on . The experimental Stanton number i s seen t o increase by a f a c t o r of approximately 3 from t h e beginning of t r a n s i t i o n t o t h e end of t r ans i t i on . The calculations predict an increase by a f a c t o r of approximately 4; however, the level ing of t he experimental heat- t ransfer values i n t h e rearward port ion of t n e laminar boundary-layer region and t h e choice of t he v i r t u a l o r ig in may eas i ly account f o r t h e differences between t h e calculated and experimental values.

Comparison of t h e theory with t h e experimental da ta shows t h a t t he theory gives a reasonably good predict ion of t h e magnitude of heat t r ans fe r i n the lam­i n a r region, generally within 15 percent of t h e experimental value except f o r t h e rearward portion of t he laminar flow where t h e experimental heat t r ans fe r w a s found t o l e v e l of f p r i o r t o t h e rapid increase i n Stanton number associated with t h e beginning of t r ans i t i on .

The simple assumption of t h e v i r t u a l o r ig in f o r t h e turbulent flow being located a t t h e beginning of t r a n s i t i o n y ie lds a fair predict ion of t h e heat t r ans fe r i n t h e f u l l y developed turbulent boundary-layer region, t he maximum deviation between theory and experiment being approximately 30 percent. Note pa r t i cu la r ly i n f igure l2 (b ) t h a t t h e rate of decrease of turbulent Stanton num­ber with increasing Reynolds number follows very closely t h e r a t e of decrease predicted by theory.

The simple empirical r e l a t ion f o r t h e var ia t ion of Stanton number i n t h e t r ans i t i on region (given i n eq. (15)) a lso leads t o a reasonably good predict ion of t h e experimental r e su l t s . (Note t h a t t h e end of t r a n s i t i o n f o r t h e Ro = 5.82 x 106 and & = 6.0 data has been assumed t o occur at x = 11.00 inches based on t h e heat-transfer-ra%e d i s t r ibu t ion of f i g . 2 i n the calculat ion of t h e Stanton number i n t h e t r a n s i t i o n region.) Therefore, t h e di f f icuxty i n considering a region i n which t h e boundary layer i s t r ans i t i ona l i s not i n determining how t h e heat t r ans fe r associated with t h e region w i l l vary, but i n determining where t h e laminar flow will end and where the f u l l y developed turbulent flow w i l l begin.

The var ia t ion of heat t r ans fe r with l o c a l Reynolds number f o r t h e case of pure laminar flow i s shown i n f igure 13 f o r t h e f u l l tes t range of free-stream un i t Reynolds number. The comparison of t h e experimental data with the theoret­i c a l d i s t r ibu t ion shows t h a t theory gives a good prediction, both of t h e magni­tudes of t he da ta and the slope of t he d i s t r ibu t ion (up t o the rearward region of laminar f l o w where t h e experimental da ta are observed t o l e v e l off p r i o r t o t r a n s i t i o n ) .

-__Variation of Stanton -n-gnber with R g o x s rt-umberfor turbulent flow.- The var ia t ion of t he heat t r ans fe r with Reynolds number (based on t h e distance from the v i r t u a l o r ig in f o r t h e higher Reynolds number tests of t he Langley 20-inch Mach 6 tunnel) f o r t h e case of f u l l y developed turbulent flow i s compared with theory i n f igure 14. A l s o i n t h i s f igure, t h e experimental turbulent heat-t r ans fe r data f o r t h e p l a t e with roughness ( f o r values of k equal t o o r grea te r than t h e c r i t i c a l height) are compared with theo re t i ca l predictions. The theory i s seen t o predict reasonably well both t h e magnitudes and t h e slopes of data f o r t h e p l a t e with and without roughness.

Inspection of f igure 14 shows t h a t t h e highest experimental Stanton number values occur near t h e roughness posit ion. These peak values of NSt decrease s l i g h t l y as the roughness height i s increased above t h e c r i t i c a l height. Increasing the roughness height apparently has t h e e f f ec t of increasing s l i g h t l y the e f fec t ive Reynolds number at any given posi t ion. However, t he overa l l reduc­t i o n i n Stanton number with increasing roughness height at any posi t ion i s ra ther s m a l l s ince t h e Stanton number var ia t ion i s a weak function of Reynolds number f o r turbulent flow. Therefore, it appears t h a t roughness can be used as a boundary-layer t r i p f o r high-speed wind-tunnel t e s t s designed t o study heat t rans­f e r i n a turbulent boundary l aye r without ser iously a f fec t ing t h e heating d is t r ibu t ion .

The turbulent Stanton number var ia t ion with Reynolds number 'based on t h e distance from t h e roughness locat ion f o r t h e complete t es t range of free-stream uni t Reynolds number at a Mach number of 6.0 and roughness of c r i t i c a l height o r grea te r i s given i n f igure 15. A s explained previously, f o r t h e low Reynolds number t e s t s , f u l l y developed turbulent flow w a s not obtained so t h a t the length of t r a n s i t i o n 2 could not be determined. The assumptions leading t o t h e two theo re t i ca l d i s t r ibu t ions of Stanton number i n f igure 15 a r e discussed i n t h e section e n t i t l e d "Review of Heat-Transfer Equations." Examination of f igure 15 shows t h a t the assumption of the v i r t u a l o r ig in t o be located at the roughness posi t ion gives the best agreement with the experimental data over t he complete t e s t Reynolds number range.

16

CONCLUSIONS

An investigation has been conducted i n t h e Langley 20-inch Mach 6 tunnel and i n the Mach number 6.2 blowdown tunnel of t h e Langley Research Center t o determine the e f f ec t s of controlled surface roughness on boundary-layer t rans i ­t i o n and heat t ransfer f o r l o c a l free-stream Mach numbers of 6.0 and 4.8. Anal­ys i s of t he experimental r e su l t s and comparison with theory and previous r e s u l t s from the l i t e r a t u r e have l e d t o the following conclusions:

1. A compilation of t h e current data and the da ta available from t h e l i tera­t u r e indicates t h a t t he Reynolds number required t o bring about natural t r ans i ­t i o n increases markedly with increasing Mach number above a Mach number of approximately 3.7.

2. For a Mach number of 6.0, surface roughness t h a t . i s l e s s than the calcu­l a t ed boundary-layer veloci ty thickness a t the roughness posi t ion can under cer­t a i n conditions delay t r ans i t i on .

3. The c r i t i c a l roughness Reynolds number required t o move the beginning of turbulent f l o w t o t he region of t he roughness i s grea te r f o r higher supersonic and hypersonic Mach numbers than has been found a t lower supersonic speeds.

4. The high values of t he required c r i t i c a l roughness Reynolds numbers f o r t he Mach numbers of t h i s invest igat ion l ed t o required roughness heights of approximately twice the calculated boundary-layer thickness at the roughness posit ion. The e f f ec t of the la rge required roughness on t h e heat-transfer dis­t r ibu t ion downstream of t h e roughness has been shown t o be small.

5 . Theoretical calculations of t h e heat- t ransfer d i s t r ibu t ions agree reason­ably well with the experimental data obtained with the smooth f l a t p la te . Also, theore t ica l calculations of t he turbulent heat- t ransfer data on the p l a t e with roughness of c r i t i c a l height o r greater agreed reasonably well when it i s assumed tha t f u l l y developed turbulent f l o w begins a t t h e roughness posit ion.

Langley Research Center, National Aeronautics and Space Administration,

Langley Station, Hmpton, Va., October 9, 1963.

I

REFIERENCES

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3 . Rubesin, Morris W., Rumsey, Charles B., and Varga, Steven A.: A Summary of Available Knowledge Concerning Skin Fr ic t ion and Heat Transfer and I t s Application t o the Design of High-speed Missiles. NACA RM A51J25a, 1951.

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I

26. Potter , J. Leith, and Whitfield, Jack D.: Ef fec ts of U n i t Reynolds Number, Nose Bluntness, and Roughness on Boundary Layer Transition. AEDC-TR-60-5 (Contract No. AF 40(600)-800), Arnold Eng. Dev. Center, Mar. 1960.

27. Pearcey, H. H.: The Indication of Boundary-Layer Transit ion on Aerofoils i n the N.P.L. 20 in . x 8 in. High Speed Wind Tunnel. C.P. No. 10, Br i t i sh A.R.c., 1950.

28. Potter, J. Leith, and Whitfield, Jack D.: Comment on "Effects of Controlled Roughness on Boundary-Layer Transit ion at a Mach Number of 6.0." A m Jour. (Tech. Comments), vol. 2, no. 2, Feb. 1964, pp. 407-408.

29. Ste r re t t , James R., and Holloway, Paul F.: Effects of Controlled Roughness on Boundary-Layer Transit ion at a Mach Number of 6.0. AIM Jour. (Tech. Notes and Comments), vol. 1, no. 8, Aug. 1963, pp. 1951-1953.

30. Braslow, Albert L.: Effect of Distributed Granular-Type Roughness on Boundary-Layer Transit ion a t Supersonic Speeds With and Without Surface Cooling. NACA RM ~ 5 8 ~ 7 ,1958.

31. Chapman, Dean R., and Rubesin, Morris W.: Temperature and Velocity Prof i les i n the Compressible Laminar Boundary Layer With Arbitrary Distr ibut ion of Surface Temperature. Jour. Aero. Sci., vol. 16, no. 9, Sept. 1949, pp. 547-3650

32. Jones, Robert A. : An Experimental Study a t a Mach Number of 3 of the Effect of Turbulence Level and Sandpaper-Type Roughness on Transit ion on a F l a t Plate . NASA MEMO 2-9-59L, 1959.

33. Potter, J. Leith, and Whitfield, Jack D.: The Relation Between Wall Temper­a ture and the Effect of Roughness on Boundary-Layer Transit ion. Jour. Aerospace Sei . (Readers' Forum), vol. 28, no. 8, AM. 1961, pp. 663-664.

34. Lees, Lester, and Reshotko, E l i : S t a b i l i t y of t he Compressible Laminar Boundary Layer. Jour. Fluid Mech., vol . 12, p t . 4, 1962, pp. 555-590.

20

I ,

of instrumentation

L ~ u p p o r tplate

Plate with thermocouples or pressure orifices

( a ) continuous f l a t p l a t e 1.

Characteristics of spheres

I I I I X k l

A

S u p p o r t plate

Plate with thermocouples or pressure orifices

( b ) F l a t p l a t e 2 with roughness elements.

Figure 1.- Sketch of model assembly. A l l dimensions are i n f e e t .

21

10

I I

I

Beginning Of fully turbulent flow

10-1 0 1 2 3 4 5 6 7 8 9 10 ll

x, in.

( a ) M, = 6.0.

Figure 2.- Heating-rate d i s t r i b u t i o n on t h e continuous f l a t p la te .

2

0 9.04 x 1 0 11.86

13.52

8 6

0 0 5 0

0

0

0

C 00 0

0

0 0

1

1

2

1 1

1

10-1 1 11

0 1 2 3 Ir 5 6 7 9 10 11

x, in.

1 1 1

(b) M, = 4.80.

Figure 2.- Concluded.

23

107

8

6

4

0r\u

2 1 -F. d

R0,t d t IO6 1

Y 8 I 0

6

4

2

105 0 1 2 3 Ir 5 6 7

Mach number

Figure 3.- Variation of transition Reynolds number with Mach number.

24

I -

M 2, i n .

1.57 1.005 1.57 .DO5 1.96 ,005 1.96 ,005 2.44 .OD5 2.44 ,005 2.90 ,005 2.90 ,005 3.34 ,005 3.34 ,005

1.97 .001 1.97 ,001 2.57 ,001 2.57 .001 3.70 ,001 3.70 ,0014.54 ,001 4.54 .001 2.41 ,002

3.05 ,002 to 0.006

3.12 ,005

5.50 .w9 5.75 .w9 5.75 ,009 5.90 ,009 6.00 ,009 6.00 ,009 6.35 ,009 7.00 ,009

5.80

6.80 ,003 to ,005 6.80 ,001 to ,003 6.80 ,003 to ,005 6.80 ,002

4.80 .002 4.80 .002 4.80 .w2 4.80 .w2 4.80 . w 2 4.80 .w 2 6.00 ,002 6.00 ,002 6.00 ,002 6.00 ,002 6.00 ,002 6.00 .w2 6.00 .002 6.00 ,002

Ro/in. Model

2.2 x 105 to 6.9 x 1 9 Flat plate 4.7 to 8.1 Flat plate2.6 to 9.7 Flat plate 3.5 to 9.7 Flat plate2.6 to 7.5 Flat plate 3.1 to 7.5 Flat plate 2.3 to 6.8 Flat plate 2.8 to 6.8 Flat plate 2.4 to 5.5 Flat plate 3.0 to 5.5 Flat plate

1.5 3.0 .591.6

to 2.5

Flat plateFlat plateFlat plateFlat Dlate

1.4 1.0 1.6 1.2

Flat plateFlat plateFlat plateFlat plate

2.5 to 9.3 lollov cylinder

I92 to 5.75 lollov cylinder

1.0 to 6.5 Flat plate

2.0 5.0 6.0

Flat plateFlat plateFlat plate

5.0 LI.0 5.0 4.0 4.0

Flat plateFlat plateFlat plateFlat plateFlat plate

3.0 Flat plate

2.5, 4.0 lollov cylinder 2.5 lo l lov cylinder 1.4 lollov cylinder 1.55 to 1.75 1 0 l l O " cylinder

8.19 Flat plate9.88 Flat plate1.6 Flat plate8.19 Flat plate9.88 Flat plate-1.6 Flat plate4.85 Flat plate 5.85 Flat plate6.85 Flat plate 5.85 Flat plate6 .8 '~ Flat plate6.8j Flat plate 6.85 Flat plate6.85 Flat plate

Method

Shadowgraph -transition begins transition ends transition begins t T i l " S i t i 0 " ends transition begins transition end� transition beginstransition enas transition beginstransition ends

Minimum shearing GtreSP Maximum shearing atless Minimum shearing stress Maximum shearing stress M i n i m shearing stress Maximum shearing etress Mini" shearing stress Maximum shearing stress

Schlieren

Schlieren

Surface temperature

Surface temperatureSurface temperatureSurface trmperature Surface trmperature Surface temper*tllreSurface temperature Surface tempePaLuPe Surface trn,pcratuTe

L.lminescent1iicquer

Sh?duvgraphSchlieren

Shadowgraph Shadowgraph, echlieren

Reference

chapman, et a1.j ref. 19 chapman, et ai.; ref. 19 chapman, et ai.; ref. 19 chapman, et ai.; rei. 19 chapman, et ai.; ref. 19 chapman, et ai.; ref. 19 chapman, et ai.; ref. 19 Chapman, et el.; ref. l9 Chapman, et al,; ref. l9 chapman, et ai.; ref. 19

Coles; ref. XI Coles; ref. 20 Coles; ref. XI Coles; ref. XI Coles; ref. 20 Coles; ref. 20 Coles; ref. 20 Coles; ref. 20

0 'Dx" l ; ref. 18

Brinich and Dioconis; ref. 21

Brinich; ref. 22

k n a i s , et ai.; ref. 23 landis, et al.; ref. 23 mais is, et ai.; ref. 23 m a s , et ai.; ref. 23 Landis , et al.; ref. 23 l a n b i s , et ai.; ret-. 23 Landis, et al.; ref. 23 m b i s , et ai.; ref. 23

Korkegi; r e f . 24

Bertram; ref. 25 Bertram; ref. 25 Bertram; r e f . 25 Bertram; ref. 25

Heat flow rate, end of laminar f l o w current investigation Heat flow rate, end of l d n a r flow current investigation Heat flow rate, end of laminar flow current investigation

Heat flow rate, beginning of fully turbulent flow current investigation Heat flow rate, beginning of fully turbulent flow current investigetion Heat flow rate, beginning of filly turbulent f l o w current investieat1on

Heat flowrate, end of laminar flow Current investigation Heat flow rate, end of laminar flow current investlgatlon Heat f l a w rate, end of laminar flow current inYesti goti0"

Heat flow rote, beginning of fully turbulent flow current i"YestigRti0n Heat flow rate, beginning of fully turbulent f l o w current investigation

Schlieren current invcstlgntia" Shnduvgraph, end or lminar flow current invrstigntion

Shiduvgraph, beginning of f u l l y turbulent flow current i">rstigatio"~-

Figure 3.- Concluded.

25

End of t ransi t ionI rBeginning of transition

r

(a) Shadowgraph

Boundary layer thickensrI

( b ) Schlieren photograph.

Figure 4.- Shadowgraph and schl ieren photograph of flow over Ro = 8.22 X lo6.

Shock From lea4ing e4ge

White band indicates laminar flow

-Flow

the f l a t p l a t e . & = 6.0;

I

- - -

--

I I 4

-. - - >^ _ _ _ _ - - - ~

Smooth plate Roughness height equals k l

Roughness height equals kz, (kz > kl)

Roughness height equals k3, (k3 > k2)

T -.

Roughness pos i t ion 1­- - - - - ­

- __.

Reynolds number per inch

Figure 5.- Sketch showing variation of transition location with Reynolds number for various size surface roughness elements.

27

l111l11111l111111111111lllIII I1 II

\ \ ' -L.

Position of roughness

0 1 2 3 4

0 0 0 U .0018 .76 0 .0030 1.26

,0044 1.85 h .0054 2.27 b .0067 2.82 0 .0091 3.82

_ _ _ _ From fig

_D­-/

,, ,

5 6 7 8

x, in.

( a ) Interchangeable leading edges; Ro = 8.2 x lo6.

Figure 6.- Heating-rate d i s t r i b u t i o n on p l a t e 2 f o r var ious s i z e spheres. & = 6.0.

20

9

--

I

10

8

6

4

2

6

.' ''. 4

2

Position of roughness

10-1 Y0 1 2 3 4 5

x, in.

(b) Interchangeable leading edges; Ro

Figure 6.- Continued.

-0 0 0 0c! .mi8 .a 26

-<? .0030 1.06 131A .0044 1.56 161 C. .0054 1.91 178

- c_\ ,0067 2.38 197 ~

IJ ,0091 3.23 230 - _ _ From figuure

! $ E ! 0

0

0

//

//,

,, ,, ,,

1 0 0

0

6 7 8

= 5.8 x lo6

29

I .- . - .. - ..._.. . .... .. ... ... .. - . - . . ... ..

9

I 111111111111lI1lII Ill I

1 , _­~

8 I k'ft 5 %CRk Ro

0 0 0 0 4.37 x lo' u .0018 .57 18.0 4.09

6 0 *W30 *93 78.7 4'04 A .0044 1-37 131.1 3.89 h .oO% 1.68 144.7 3.86 b .OW57 2.08 162.5 3.96 0 .oogi 2.88 19.0 4.14

4

2

h, Btu/ft2-sec

10-1

8

6

h

2

S S

10-2 . ._. . 0 1 2 3 4 5

x, In.

Qe;

A

0

0

0

. . . .

6 7 a

( c ) Interchangeable leading edges; R, = 4 x lo6.

Figure 6.- Continued.

9

I

1 k , f t

a 0 0 0 .0018 0 .W30

6 ~ . o o 4 4 h .OO% L . 6 7 D .W9l

4

2

2h,Btu/f't -aec

2

10-2­1 2 3 4 5 6 7 a

x, in.

(a) Interchangeable leading edges; R, = 2.6 x 106 . Figure 6.- Continued.

9

IIl1111111111l11111llIIl I I I Ill1Ill

0 0 0 0 1.26 X 10 0 .0018 - - ­0 .OOp .50 10.7 .1.32 A .W .70 19.9 1.13n .OO% - - ­n .0067 1.08 89.3 1.19 D .Owl 1.45 105.8 1.23

f Posit

0 1 2 3 4 5 6 7 a x, in.

( e ) Interchangeable leading edges; R, = 1.2 X lo6.

Figure 6.- Continued.

32

9

II I

'618 Rk Ro

0 0 8.36 .y."5 $0 8.43

1.10 15x3 8.37 1.6.1 189 8.10 1.96 214 8.44 2.47 236 8.34 3.32 271 8.10

16

D

b

n [1

0

2

1

8

6

4

2

10-1 0

$ 6 b B

0

0

Roughness strip

6 7 8 9

Roughness position

2 3 4 5

x, in.

( f ) One leading edge, interchangeable roughness s t r ip , Ro = 8.2 X lo6.

Figure 6.- Concluded.

33

10

a

6

2

6

2

10-1

0 1 2 3 4 5 6 7 a x, in

(a) Interchangeable leading edges; R, = 13.9 x lo6.

Figure 7.- Heating-rate d i s t r i b u t i o n s on p l a t e 2 for various s i z e spheres. M , = 4.8.

34

9

--

I

2

;I,Btu/ft2-aec

1

8

6

4

2

10-1

0 1

. k , f t k/6 ?

0 0 0 c3 .0018 1.13 CJ .QOjO 1.90 A ,0044 2.78 b .0054 3.41 h.0067 4.24 - - _ .From f igur i

-rX7!

i1

I

r P o s i t i o n of roughness

l l 2 3 4 5 6 7 0 9

x, in.

(b) Interchangeable leading edges; R, = 9.8 x lo6.

Figure 7.- Concluded.

35

- - -

111111ll1ll1ll1 111111111lII111111I I1 Ill IIIIIIII Ill

. . . ~ TWIT, = 1.00

- TWITr = 0.66

0 .w1 .om .003 .a -007

Y, ft

(a) Velocity p r o f i l e s .

c

/n -c-­/’ I

/ I_

0 .w1 .om .006 ,007

(b1 Rk values .

Figure 8.- Calculated boundary-layer ve loc i ty p r o f i l e s and corresponding Rk values f o r M, = 6.0. x = 0.167 f e e t .

36

.w1

280

240 I 200

160

.J­120

Bo

40

0

O,X = 2.32 x 106

3,x - 1.64 x 106

" I

.m .w3 .a .w5 .OM .w Y, fi

(a) Velocity profiles.

/

/ /

i /'

/

o,x = 1.64 x lo6

.w3 H

.a .w5 .OM .04

Y, rt

Figure 9.- Calculated boundary-layer velocity profiles and corresponding Rk Values for &, = 4.8. x = 0.167 feet.

37

280 1I 1

00.0030 1.9 .66 1.64 x lo6 .0&4 2.78 .66 1.64

240 U .0018 1.35 -66 2.32 .0030 2.26 .66 2.32

H­200

n .oo% 1.91 .66 .m h . a 7 2.38 .66 .fl 0 .0&4 1.85 .66 1.37 e .OO$ 2.27 e66 1.37

160 III-- I i Possible cr i t ical roughness Reynolds‘d Rk,cr tnumbers from refeRnce 26,.

120

Possible c r i t i ca l roughness Reynolds80 numbers based on end of transition

f r o m data of reference 30.­

1 2 3 4

Mach number

it

roughness Reynolds number

Tference 4 .

1 1 1 5 6

Figure 10.- Cr i t i ca l roughness Reynolds number a t several Mach numbers. Open symbols ind ica te k s l i gh t ly less than t h e c r i t i c a l value and so l id symbols ind ica te k s l igh t ly greater than the c r i t i c a l value.

38

7

-----------

----------

----------

0,. Interchangeable leading edge, Xk = 2.00 in. 0,. One leading edge, interchweable

roughness strip, Xk = 2.87 in.

240 0

x 106

Figure 11.-Variation of c r i t i c a l roughness Reynolds number with free-stream uni t Reynolds number f o r a Mach number of 6 . (Open symbols indicate roughness height l ess than the c r i t i c a l value. Closed symbols indicate roughness height i s s l i g h t l y greater than the c r i t i c a l value.)

w v)

.0010

.m 9

.oom

.OOW

.on06

." 5

.n&

Nst

.OOO)

.nncn

.on01 0

I I

- _ _ Theoret ical ; t u rbu len t

Fmpirical, t rnns i t i o n a l

-R, = 5.82 x 106

R, = 8.22 x IO&-

I 1 L

2 3 4 9 6

x, in.

(a) & = 6.0.

Figure 12.- Heating-transfer d i s t r i b u t i o n on

- 1I=-'

/' I,/'

0 o o I /

3

( / '

4/'

7 a JO

continuous f l a t p l a t e .

40

11

.OOlC

.ooog

. O M

.0 0 4 /

,0006 i / o

,0005 / .0004

84 x 106 Nst

.om', 0 0

/

Ro = 13.92 x lo6 A' 0002

.0001 0 1 2 3 4 5 6

x, in.

(b) % = 4.8.

Figure 12.- Concluded.

3 " 0 i"? 0

irical,transitional

7 8 9 10 ll

41

0 1.26 X lo6 0 ri 2.73 0 0 4.37 0 A 5.73 0e C b 5.82 0 I3 8.22 0Q P.I -Theory1

C

4

% @ n \

I

105 2 4 6 8 106 2 Ir 6 8 0 1"7

%: x

(a) M, = 6.0.

10-3 Ro k,f

8 0 9.84 x 106 0 0 13.92 -Theory

6

NSt 4

2

10-4

105 2 4 6 8 2 4 6 1.07

(b) Mo = 4.8.

Figure 13.- Comparison of laminar heat-transfer distributions as a function of local Reynolds number with theory.

42

2

~ ---, 2

10-4 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 8.OX IO6

( a ) &, = 6.0.

4

"s t I i

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0~10~

%. "

(b) Q = 4.8.

Figure 14.- Comparison of turbulent heat-transfer dis t r ibut ions with theory f o r both natural t r a n s i t i o n and roughness.

~ _. I I I 1 1 1 1 1 1 1 1 Ro k,f l

a ~- I 0 2.73 x loG ,0091 - .-_i 0 2.57 ,0067

,009 1 6 -___ ,0067

.w91

2

10-4 0 6 8 13 1? 16 16 18 20 2 1 24 ?E 2a 30 32 34 36 38 x 105

2 Ro. X'

2;P

Figure 15.- Comparison of experimental turbulent heat- t ransfer d i s t r i b u t i o n (with roughness) with d i s t r i b u t i o n predicted by theory f o r complete t e s t Reynolds number range.

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