NASA CR-112147

Caisc_ '.,.

TECHNICALREPORT

AN EXPERIMENTAL INVESTIGA T/ON OF TURBULENTBOUNDA R Y LA YERS A T HIGH MACH NUMBER

AND REYNOLDS NUMBERS

By: Michael S, Holden

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CAL No. AB-5_I72-A-1

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Prepared for:

National Aeronautics and Space AdministrationLangley ResearchCenterLangley Field, Virginia

FINAL REPORT ." "_ _'_1_ _`

Contract No, NAB1-11242November1972 _ ____.'_ _'i, !.,, .

(NASA-CP-1121471 ._.N EXPERIMENTAL N7]-12290_;_VESTIC;A'_ION OF T[15B[ILENT BOUNDAEY LAYERS

,_ HIGH MACH NHMBEP AND REYNOLDS NIIMBEP_

'inal Rep,)[-t _.S. Holden (Calspan Corp... Unclas

_',]ffalo, N.Y.) No&'. 1972 85 p CSCL iOD (;3/12 48378

-CORNEI..L AERONAUTICAL LABORATGRY, INC.

_'_O OF CORNELL UNIVERSITY, BUFFALO, N.Y. 14.221

m

1973003563

https://ntrs.nasa.gov/search.jsp?R=19730003563 2020-05-16T06:20:29+00:00Z

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I FOREWORD

:. This report was prepared for the National Aeronautics and Spacet

..:. Administration by Cornell Aeronautical Laboratory, Inc. (CAL), Buffalo,

. New York in partial fulfillment of Contract NAS 1-11242 "Experimental

Investigation of Turbulent Boundary Layers at High Mach Numbers and

:: Reynolds Numbers", The program was performed by the Aerodynamic.7

_% Research Department of CAl. under the sponsorship of the NASA Langley

!. Research Center, Hampton, Virginia. Mr. Aubrey M. Cary, Jr. of NASA

4. Langley was the technical monitor. This report is also being published as

," CAL Report No. AB-5072-A-1. The author would like to acknowledge the

k" valuable discussions with Mr. Cary duri:g the course of this program. The

flat-plate model and instrumentation employed on this program was prepared

' and calibrated in conjunction with U.S. Air Force Contract F33615-70-C-1280.

The perfo, mance period for this contract was 11/23/71 to 10/2/72.

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1973003563-002

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AB STRAC T

Skin friction, heat transfer and pressure measurements were obtained

in laminar, transitional and turbulent boundary layers on flat plates at Mach

numbers from 7 to 13 at wail-to-free stream stagnation temperature ratios

from O. I to O. 3. Measurements in laminar flows were in excellent agreement

with the theory of Cheng. Correlations of the transition measurements with

measurements on flight vehicles and in ballistic ranges show surprisingly good

agreement. Our transition measurements do not correlate well with those of

_ Pate and Schueler. Comparisons have been made between the skin friction

,_ and heat transfer measurements and the theories of VanDriest, Eckert and

: Spalding and Chi. These comparisons reveal in general that at the high end

of our Mach number range (10-13) the theory of Van Driest is in best agree-

, ment with the data, whereas at lower Mach numbers (6.5-10) the Spalding

Chi theory is in better agreement with the measurements. However, the

'trelative agreement, between the theories and the data in the FcCf-FxRex ,_

plane is dependent upon the definition of the virtual origin of the turbulent

,: boundary layer for the low Reynolds number high Mach number measurements.

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1973003563-003

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e TABLE OF CONTENTS

Section Page

I INTRODUCTION 1 _

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II. EXPERIMENTAL FACILITIES AND iMEASUREMENT TECHNIQUES ............ 3

2.1 Experimental Facilities ............. 3

Z.Z Model and Instrumentation , ....... , . • . 4

Z.3 Data Recording and Processing .......... 5

IiI. DISCUSSION OF EXPERIMENTAL MEASUREMENTSAND COMPARISON WITH PREDICTION METHODS ..... 6

3. I Transition Correlation .............. 6

3. Z Comparisons of Heat and Skin FrictionMeasurements with Turbulent Theories ....... 8

• 12IV. SUMMARY AND CONCLUSIONS ............

, V. REFERENCES . . ............. . . . . 14

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1973003563-004

LIST OF ILLUSTRATIONS

_' 5 Figure Page

% 1 Wave Diagram for Tailored - Interface Shock Tube Z4

• _ Z Test Time Available for Tailored Interface Z4

• \ { Operation of the Shock Tunnel

_ 3 Flat Plate Model Z5

"_" i f

4 Drawing of Section Through Skin Friction Transducer 26 ,

., 5 High Frequency Pressure Mounting Z6 i

."! 6 TypicalFlatPlateDistributi°nModelof Heat Transfer Along the 27

: I 7(a-o) Schlieren Photographs of the Boundary Layer Z8i of the Flat Plates

8 Correlation of Laminar Heat Transfer 31

'. "'I 9 Correlation of the Scale Length of the Transition on 3ZFlat Plates and Cones t

i I0 Correlation of Transition Showing the Effects of 33 j_" Tunnel Size

11 Correlation of the Flat Plate Transition Data in Terms 34. of the Parameters Suggested by Pate and Schuelers _ ,

lZ Correlation of Transition Measurements in the CAL 35Shock Tunnels with Ballistic and Downrange Measurements

13 Turbulent Boundary Layer Displacement and Momentmn 36 ,Thickness from Shock Tunnel Nozzle Data

14(a, b) Correlation of the Eddy Reynolds Number with 37Transition Measurements on the Flat Plates

15 Reynolds An_logy Factors for Turbulent 39Hypersonic Boundary Layers

16 Reynolds Analogy Factors for Turbulent 39Hypersonic Boundary Layers

17(a-d) Comparison Between Measured Heat Transfer and the 40

Theories in the F c ZC H versus Fr6 Re 8 Plane

• 18(a-d) Comparison Between the Measured Skin Friction and the 44

Theories in the F c Cf versus Fr6 Re 8 Plane

" 19(a-c) Comparison Between the Measured Heat Transfer and 48

the Theories for (O V = OB + OBE )

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1973003563-005

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7 i

_ LIST OF ILLUSTRATIONS (cont.) ._

Figure Page

7 20(a-c) Comparison Between the Measured Skin Friction 51_ and the Theories for (eV = e B + eBE) !_ 21(a-c) Comparison Between the Measured Heat Transfer 54

i and the The°ries f°r (eV= eB+eBE/2) IZZ(a-c) Comparison Between the Measured Skin Friction 57

and the Theories for (ev = e B + eBE/2 )23(a-c) Comparison Between the Measured Heat Transfer 60

and the Theories for (eV = e B + eBE )24(a-c) Comparison Between the Measured Skin Friction 63

and the Theories for (eV = e B + eBE Laminar)

ZS(a-c) Comparison Between the Measured Heat Transfer 66

and the Theories for (e V : eB) -_

. 26(a-c) Comparison Between the Measured Skin Friction 69 _

and the Theories for (eV = eB)

27(a-c) Comparison Between the Measured Heat Transfer 72 ,

and the Theories for (X v = 0 at e B + eBE/2)

28(a-c) Comparison Between the Measured Skin Friction 75 '

and the Theories for (X V = 0 at e B + eBE/2 )f

s

Table Page

I Test Conditions 16 _

Il(a) Heat Transfer 17

Il(b) Skin Friction 20

Il(c) Pressure 22

IlI Gage Positions on Flat Plate 23

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1973003563-006

_ NOMENC LAT URE

_" _ - Speedof sound !

.;.. • ft lbf

_, C " Specific heat s-T_F

C_e - Local skin friction coefficient %/_ _,, uL_z

CH - Stanton number _/_,, U,, (H o- H_,)

Cp " Pressure coefficient -_ / _ _e, _

- Compressibility transformation dr:/C4:

" Compressibility transformation ,_e,_/Re_

Fo - Compressibility transformation "Ree/R, e

i H " Total enthalpy ft Ibf

_ " Static enthalpy s_q&

H " Mach number _

" Pr - Prandtl number

_i, - Dynamic pressure (psia)

- Heat transfer rate BTU/ft2sec.

- Reynolds number ___ ) I

T " Temperature ("R)

Tx, _ " Reference temperatures

U - Velocity, ft/sec.

_t' - Velocity fluctuation level ( _' - O. 02 t_. )\

• _ - Distance along the surface of the flat plate

%v - Distance from the virtual origin

" Specific heat ratio

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1973003563-007

._IA ___.==..... l .ill -,_",

i '_rk°;''

d • _ - Boundary layer thickness

8 - Momentum thickness

• A - Turbulent macroscale _"8/I0

- Viscosity coefficient(slugs/ftsec) i

- Density slugs/ft 3 i,_ - Viscosity at point of maximum temperature in boundary layer i

}

- Shear stress

Subscripts

o - Nozzle supply conditions#

o - Stagnation conditionsbehind a normal shock

1 - Initialdriven gas condition

4 " Gas conditionsbehind reflectedshock

• A, " Incident shock in driven gas

_$ - Test sectioninitialconditions

- Initialconditionsat model surface

o._ " Adiabatic wall conditions

oo - Free stream or test section conditions

• " Local or edge conditions

v - Virtual origin

E " Eckert

. SC " Spalding and Chi

VD " Van Driest

8 " Beginning of tranIitions

£ " End of transition

ill " Beginning to end of transition

viii

1973003563-008

I. INTRODUCTION

o

• Methods of accurately predicting heat transfer and skin frictio._ are a

basic requirement in the design of vehicles, such as the space shuttle and the

, hypersonic transport, which are to be flown at hypersonic speeds. To main-

tain structural integrity for a sustained period of time at these speeds a vehicle

would probably have to be cooled internally or by transpiration techniques.

It is precisely in this hypersonic highly-cooled wall regime where the semi-

empirical prediction methods are in question. There is a dearth of simul-

taneous measurements of both skin friction and heat transfer in turbulent

boundary layers over highly cool.ed surfaces in high Reynolds number hyper-

sonic airflows.

All the methods for predicting the skin friction and heat transfer to the

surface beneath a turbulent boundary layer depend to some extent upon experi-

mental measurement to determine semi-empirical constants contained in the

formulation. Thus the accuracy of these methods depends both on the precision

of the experimental measurements and the range of the basic parameter 4

covered in the experiments. Recently, the theories of Spalding and Chi 1,

Van Driest z' 3, Coles 4, and Sommers and Short 5 have been re-examined, with

emphasis on their ability to predict skin friction and heat transfer to highly

_ cooled surfaces in hypersonic flow. Whereas Bertram 6 and Cary 7 find best

agreement between experimental measurements and the Spalding and Chiq

method, Hopkins and Inouye claim the "Van Driest 11" theory is superior in

predicting the properties of flat-plate turbulent boundary layers under highly _ 0,:

cooled hypersonic conditions8. In part, the agreement, or lack of it, between

,_ theory and experiment results from the framework in which the experimental i

_ results have been evaluated and the assumptions made on the magnitude of the

Reynolds analogy factor. This situation highlights the need for further detailed i|

heat transfer and skin friction measurements in turbulent boundary layers

over cooled surfaces in the hypersonic high Reynolds number regime.

• In the experimental study described herein, heat transfer, skin friction

. and pressure measurements were made at Mach number from 7 to 13,

; Reynolds numbers from 6 to 180 x 106 and wall-to-freestream stagnation

temperature ratios from 0. I to 0.3. The test facilities_ model and instru-

_ mentation together with details of the model configurations and freestream

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1973003563-009

conditions at which the measurements were made are described in the following

, section; the results of the experimental program are then presented.

Comparisons between the measurements in laminar flow, the properties of

• transition and transitional flows, and the measurements in fully turbulent

boundary layers are made with existing measurements and semi-empirical

prediction methods. The conclusions from this study are then presented.

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1973003563-010

II. EXPERIMENTAL FACILITIES AND MEASUREMENT TECHNIQUES

2.1 Experimental Facilities|

The experimental program was conducted in Cornell Aeronauticalk,

Laboratory's 48-inch and 96-inch Hypersonic Shock Tunnels.' The operation

of these tunnels can be shown simply with the aid of the wave diagram shown

in Fig. I. The tunnel is started by rupturing a double diaphragm which

_ permits the high-pressure air in the driver section to expand into the driven

_- section, and in so doing generates a norreal shock which propagates through

the low-pressure air. A region of high-temperature, high-pressure air is

produced between this normal shock front and the gas interface between the

driver and driven gas, often referred to as the contact surface. When the

primary or incident shock strikes the end of the driven section, it is reflected

leaving a region of atmost stationary high-pressure heated air. This air is

then expanded through a nozzle to the desired freestream conditions in tLe ?test section, c)

_ The duration of the flow in the test section is controlled by the inter-

, actions between the reflected shock, the interface, and the leading expansion

:, wave generated by the nonstationary expansion process occurring in the

_" driver section. The initial conditions of the gases in the driver and driven

: sections are controlled so that the gas interface becomes transparent

to the reflected shock, as shown in Fig. l , thus there are no waves generated F

/. by interface-reflected shock interaction. This is known as operating under

",_ "Tailored-lnterface" conditions. Under this condition, the test time is control-

t led by the time taken ior the driver-driven interface to reach the throat_ or

the leading expansion wave to deplete the reservoir of pressur_ behind the

_ reflected shock; the flow duration is said to be either driver-gas limited or

t expansion limited, respectively. Figure 2 shows the flow duration in the

• test section as a function of the Mach number of the incident shock. Here it

can be seen that for operation at low Mi's , running times of over 2S milliseconds

. can be obtained with a long driver section. Measurements oi heat transfer,-/

_ skin friction and pressure were made on flat plate models at Mach numbers

_' from 7 to 13; a complete list of the test conditions are given in Table I.

1973003563-011

r12.2 Model and Instrumentation

The flat-plate model used in this experimental program is shown in

Fig. 3. The model, which_/as instrumented with heat transfer, skin friction

and pressure gages, was equipped with two interchangeable leading-edge sections

of 18 and 36 inches giving total model lengths of 39 and 52 inches, respectively.

Heat transfer measurements were made on these two leading-edge segments

to detect the position of transition as well as to provide information on the

properties of the transitional regions. Both leading-edge sections had leading

edges of less than I0 "3 inches in thickness.

Skin Frictiov. and Heat Transfer Instrumentation - A diagram of the

skin friction transducer which we used in the present studies is shown in

, Fig• 4• The transducer I0 consists of a diaphragm which is supported flush

• with the model surface by two piezo-ceramic beams, which develop a charge

" when placed in bending by a surface shear on the diaphragm• A third beam is

used to provide acceleration compensation; the beams are connected electrically

to eliminate thermal, normal, and transverse pressure effects. An FET * im-

pedance transform circuit is mounted internally to eliminate cable noise effects

at low levels of skin friction. The gage, which has been refined and developed

• over the past I0 years, has been used to measure very low levels of skin friction

encountered in separated regions in low Reynolds number hypersonic flow and

' more recently very high levels in regions of shock wave-turbulent boundary

layer interactions in hypersonic flow• Because of the very severe he¢tting

conditions encountered in the latter studies, special care was taken to minimize i

_" the heat conduction through the flexures• The very large dynamic loads generatedt_

on the transducers during tunnel shutdown when run at the high dynamic pressure

. : conditions used in our studies caused the diaphragms to be torn from the sup-

porting beam• This p:oblem was overcome by careful design of the flexure and

by mounting the transducer in the seismic mass-rubber suspension system shown

in Fig. 4,

i Thin-film heat transfer gages were used in the present study. Thistechnique is based on sensing the transient surface temperature of a non-

conducting model by means of thin-film resistance thermometers. Because

_ the thermal capacity of the gage is negligible, the instantaneous surface

; i " temperature of the backing material is related to the heat transfer rate by the

i _Field Fffects Transistor

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1973003563-012

classical semi-infinite slab theory.9 Analog networks were used to convert

. the outputs of the gages, which are proportional to surface temperature, to

a voltage directly proportional to heat tran3fer. The gages are fabricated

on either small pyrex buttons or contoured inserts and mounted flush with

"+ the model surface.

Surface Pressure Measurements - We employed two types of surface

pressure transducers ir ou." earlier studies of shock wave-turbulent boundary

' layer interaction. The CAz.-designed and constructed lead zirconium titanate11

; piezoelectric pressure transducers were used to obtain essentially the mean

pressure distribution through the interaction region, though the transducer

and orifice combination could follow fluctuations up to 15 kHz. A second

flush-mounted transducer, especially designed for high frequency measure-

rnents by I=CB in Buffalo, was used to obtain surface pressure fluctuation

measurements from Z00Hz to 120 kHz. To prevent a resonance, a special

mounting system was developed (as shown in Fig. 5) to lock the gage firmly

into the model. A thin insulating barrier of aluminized mylar was attached

, to the diaphragm o_ the transducer to prevent thermal heating effects.

•_ Gage,.,,Locations - The distance of each piece of instrumentation from +

' " the leading edge of the flat plate is given + , Table II . Runs I through II were

conducted with model configuration A while runs 12 through 19 were conducted

_+ with the model in con/iguration B.

+,2.3 Data Recording and Processing ,_

, _- The outputs from the transducers were recorded on s. NAVCOR mag-

_ netic drum system and on a high frequency FM tape recorder, and also

monitored on oscilloscopes. The NAVCOR system, which holds 48 channels

in digital _orm, is essentially a low frequency system, whereas the 18 chan*

nel AMPEX FM recorder had a range of 0 to I MHz an_ was used to record

the fluctuation data. The fluctuation measurements were recorded in ana}og

form and subsequently processed by an ana?os-to-digital conversion/data

storage system and digital computer program using a fast Fourier transform

to yield the statistical properties of these measurements•

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1973003563-013

III. DISCUSSION OF EXPERIMENTAL MEASUREMENTS ANDCOMPARISON WITH PREDICTION METHODS

For each of the free stream conditions listed in Table I, the heat transfer,

skin friction and pressure which were obtained on the _at-plate model

are listed in Table III. In each case definitive measurements were made in the

laminar, transitional and fully turbulent regions on the flat plate. A typical

_, set of such measurements is shown in Fig. 6 . Figure 7d shows the particular

schlieren photograph which corresponds to these measurements. Well-defined

_ regions of laminar flow were observed upstream of transition and a correlation

of the heat transfer measurements made in these regions is shown in Figure

8. Here we have compared the measurements with Cheng's theory ;.ncluding

the effects (small) of boundary layer displacement. In its simplified form

Cheng's theory gives

•:. ,(,-,) [ r,., ] ,, ' where /_ = 2 To_CP,)+2Btp,)J

} •$ and _ = M_

:_ C is Chapman-Rubesin constant and A = 0.968, B = 0. 145. The measurementst

correlate well when plotted in this framework and are in good agreement with

"_ ChengWs zeroth to order theory. #

_i 3. 1 Transition Correlation

_ Downstream of the region of laminar heating, transition was first indi-

J cated by a series of spikes in the heat transfer traces. These sharp increases

i_ in heating may be associated with turbulent bursts created at the edge of the

. viscous sublayer. This observation is supported by the records from adjacentgages which indicated that the disturbance responsible for the temporally

. increased heating was convected downstream at a fraction of the free stream

velocity. At a short distance downstream of where the "spikes" were first

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1973003563-014

II

I

observed, the output of the gages exhibited strong fluctuations and the mean

heating and skin friction level increased above that for laminar flow. In this

region the outputs from the high frequency pressure transducers showed

fluctuation levels considerably in excess of the levels subsequently observed

downstream of transition. The heat transfer rises monotonically through the

transition region to reach a maximum near the end of transition, decaying

with downstream distance beyond this point to approach the one fifth power

law at approximately I 1 5% of the Reynolds number for maximum heating.

\ These studies and earlier measurements on flat plates and cones have indi-

cated that the beginning of the transition process occurs at approximately one

half of the distance to the point of maximum heating, as indicated in Fig. 9

" where we have plotted the ratio of the local Reynolds number based on the

:" beginning and end of transition versus Mach number. The beginning of trans=p

ition was defined in these studies as the point of departure of the heat transfer

from the laminar value.

No satisfactory method has been found to correlate measurements of

• the position of transition made in the present studies. Because of the high

Reynolds numbers and Mach numbers at which these tests were conducted, .

and the large scale of the tunnels involved, the radiated noise from the ".annel ' •

wall boundary layer was probably considerably smaller than in the studies of

Pate and Schueler. 12 The latter study indicated that decreasing the size oft" •

:, the tunnel, for the same free stream conditions, should decrease the transition/

._ Reynolds number. The transition measurements shown in Fig. 10, made in

: the A and D nozzles, which have exit diameters of 24 and 48 inches respectively,

+_ do not exhibit this scale effect, even though a unit Reynolds number variation +is evident in the correlation. Plotted in terms of Pate and Schueler's parameters,

- ._ in Fig. 11, the data falls below their correlation. For agreement, our tran-+_

_, sition Reynolds numbers would have to have been over Z00 million. In Fig. 12

we have compared our transition measurements on the flat plates with measure-

_ meats on flight vehicles and in the ballistic range. The agreement betweenb

these measurem:nts is surprisingly good. It is clear that in these tests the

_: . position of transition was influenced by factors which were different, and or

more complex than those governing transition in the tests conducted by Pate)

i - and Schueler. Whether the fluctuating pressure level of the free stream is themain factor influencing transition, as indicated by the measurements of

Stainback, Fischer and Wagner, 13 or whether this quantity merely follows more

I subtile parameters remains to be determined.

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1973003563-015

Looking at transition from another aspect, we can examine the lowest

" Reynolds number based on the turbulent macroscaLe at which fully turbulent14.

flow can exist. Following a similar reasoning to that used by Bradshaw m

* discussing transition reversal we can say a self-sustaining turbulent boundary

layer can exist only when the energy or shear stress producing eddy size and

dissipating range of eddy size just overlap, or where a viscosity independent%

region can be established. This criterion can be expressed as Re_ = constant

• where Re x and where Bradshaw 14 and Finson suggest that the

" constant is approximately 30. For a hypersonic turbulent flat plate boundary

layer the measurements of Demetriades 16 indicate that t_'= 0.02 u.,_ , A= _1o

Then evaluating _'Mn_ at the edge of the sublayer for turbulent boundary

layers with momentum thicknesses equal to those at the beginning and end of

transition (see Later) where 8 and _ are related through Fig. 13 we obtain

the correlation shown in Fig. 14. Here we see that for our measurements

transition is complete for Ren's of approximately 50, with transition begin-

ning at Reynolds numbers of approximately one half this value. These cot- _pv• relations suggest the hypersonic boundary layer will remain fully turbulent only •

after a certain Reynolds number based on a turbulent macroscale is exceeded.

: This idea is supported by experimental studies conducted at CAL and I_angLey 17

, where attempts have been made to cause premature transition in flows above

Mach 8. Although roughness was found to disturb the structure of the laminar

• boundary layer, causing an increase in the local skin friction and heat transfer,

the Reynolds number (based on distance to the end of transition) at which the

,'. boundary Layer exhibited the characteristics of a fully developed turbulent

: boundary layer was only slightly less than if "natural transition" had been ,

allowed to occur. Our studies indicate that Re h must be in excess of 50 for _"i

a fully developed turbulent flow to be maintained at hypersonic speeds,

"_ 3.2 Com parisions of Heat and Skin Friction Measurements with Turbulent'_ Theories

To make a comparison between the measurements beneath the turbulent

portion of the boundary layer and the theories of Eckert, Van Driest, and

_ Spaldlngand Chi, it is necessary to calculate the momentum thickness. In

Figs. 15 and 16 the measured Reynolds analogy factor (2St/¢_) is shown for both

transition and turbulent flows and from this correlation we have concluded

1973003563-016

.. that Z St/C¢ = 1 best represents the measurements on hypersonic highly-cooled

flows. We have calculated the momentum thickness from the heat transfer

:i: . distribution using the relationship 0 =forCed+ _'.:+

We have chosen to perform the comparison between the measurments

" and tt,e three theories within the framework of the Spalding-Chi method, where

the local skin friction coefficient Cf is related to the skin friction coefficient

in an equivalent incompressible flow Cf. through the relationshipl

.. c_ = _ (M_. r,,/ro) C_and the local Reynolds number based on momentum thickness 0 or distance

from a virtual origin _v" Reo and Re_ respectively are related to the

: equivalent quantities through the relationships

'q_-o:: F_R_oand

We have assumed that the Karman-Schoenherr relationship- 11z ¢

where the average skin friction CFI" is related to the local skin friction Cfz_. by

c+:= o2+2 % [o_4z +08_8__c,i"]++ accurately describes the skin friction distribution in the incompressible plane.

' " In Spalding-Chi analysis the transformation factors F , F 0 and

F_ were determined from Van Driest's analysts and correlations of experi- .,,i

. mental measurements. For this analysts the transformation factors are

:': (Fo)+_ r m, (.+/_-'oc + _ /3) .) -o._o_ -o.,a,.

_; ( F, ) sc .-. ( T. / T, ) ( Tw l T_W)

-I t

!+ . and F,. : F.Fo_d

;' where 0¢ (2 B)('t/q+B_) "'/'and /_ " B('_/_z+Dz)"lz• 'A =

' i% •

[ /_f'l" _; (r+f'-+;, and _ = f'_'letTt/ J and _ - (1 +Fro e- T. "_

] 973003563-0] 7

where a recovery factor (r) of 0.89 was used.

The Van Driest analysis used Prandtl-Karman mixing length theory to

describe the compressible turbulent boundary layer over a fiat plate. Thee

transformation factors which can be derived from this analysis are

-1 -1//_

(Fo),,o = p.,/ U..and _,

The function F c is identical in the Van Driest and Spatding-Chi analyses.

From Fckert's reference enthalpy method we can deduce the trans-

formation factors

(F_)_ _- T*/ Te _.1

". (F e ), : Fie / I_"

• and F, = Fo Fc-' '_'

where we have evaluated these expressions for two definitions of the reference

temperature T*

T_ : o.s T_ + o.zz TAw, o.ZST,

T;: 0.5T.+ T.+o.333Tt

The viscosity for the transformations was calculated from the Sutherland c

relationship for temperature above 500°R with the Hirschfelder, Cur __ss and i I

:_" Bird 19 values used below this temperature. !i:

:_ In order _,,)compare the experimental measurements with the theories

L- in the Fc Cf - FRxRe x plane it is necessary to define a virtual origin from

which the turbulentboundary layer is assumed to grow. We assumed th_.t

' the mon,entum thickness of the turbulent boundary layer at the end of .

tran_:ttion ( _)v ) was given by

(i) 8_ ,, d.x + C,,oZ_ -: 8s, es_

" (;i) a v : as + (8,. /Z)

(iii) 0_, : 0s + d Z

(iv} O_ - e,10

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1973003563-018

t

The first assumption gives a comparison in the FcC f - FRx Rex plane which is

directly equivalent to the FcC f - FoRce correlation. We make the second

assumption %r a direct comparison with the method for determining the

. virtual origin suggested by Bertram and Carry to be discussed later.

The third and fourth assumptions are based on pragmatic situations where no

measurements exist of transitional heating and the momentum thickness is

\ calculated by assuming the boundary layer is laminar to the end and beginning

of transition respectively. In all cases the distance from the virtual origin

":: to the end of transition was calculated using the respective theories of each

of assumption for momentum thickness. Finally we have used the method

suggested by Bertram and Garry who based on the best correlation of their

data with the Spalding-Chi theory place the virtual origin at the point where

; I_ee = R_s "1-(-_)

Comparison between the three prediction methods and the heat transfer

and skin friction measurements in the FeC f - FxRex framework are contained

in Figs. 19 through Z6. Using our first assumption ( 8v = 8at0se )we obtain "r

• equivalent correlations on the FxRex plane (Figs. 19 and 20) to the FoRce%

described in the previous two figures. We again observe that Van Driest

theory is in closer agreement with the measurements than the Spalding-Chi

,,. analysis which consistently underpredicts the heat transfer and skin friction. ':

',". With decreasing 8v the agreement between the measurements and the'

_ Spalding-Chi theory improved however even for e v = 8s Spalding-Chi under ,_

.. predicts the measurements at the highest Mach numbers whereas causing the :

. ._ Van Driest method to underpredict in this region, Employing Bertram and

;. Carry's approach t o calculating the virtual origin we obtain good agreement

between the measurements and the Spalding-Chi method for Mach numbers i!

_: below I0. This approach underpredicts the measurements above Mach I0.., Van Driest's theory overpredicts the measurement below Mach 10 using the

_'i' later method for specifying the virtual origin.

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1973003563-019

IV. SUMMARY AND CONCLUSIONS

.. Heat transfer, skin friction and pressure measurements have been

• obtained in laminar, transitional and turbulent boundary layers over flat platemodels at Mach numbers from 7 to 13, Reynolds numbers from 6 x 106 to

200 x 106 and wall-to-free stream stagnation temperature ratios from 0. 1 to

0.3. The measurements of heat transfer beneath the laminar boundary

,: layers were in good agreement with the zeroth order theory of Cheng.

Transition was first observed as "spikes" in the heat transfer records; and,,

records from adjacent gages suggest that these "spikes" result from turbulent

: bursts which grow in size as they accelerate downstream. Surprisingly, our ,

_: measurements of the Reynolds number based on momentum thickness at tran-

sition were in good agreement with results from ballistic and down range tests.

These measurements did not correlate within the framework of Pate and Schueler's

studies which suggest that in these high Mach numbers, high Reynolds number

flows, noise radiated from the tunnel walls is not the dominant factor influencing

' transition. It was found that the end of transition did not occur (or a turbulentu.' A

boundary layer could not be supported) until the eddy Reynolds number Re;_ = _---_

exceeded a value of 50; and, in fact, the end of transition was correlated well

with Be _ = 50. The beginning of transition occurred at approximately one

half this value, and, in fact, the transition region had approximately the same

length as the laminar run.

,_The Reynolds analogy factor was found to be close to ,,nity in the turbulent(

and transitional boundary layers we studied and this value has been used in the#

' data reduction and correlation. Plotted in the F'0df - F# _¢$ plane, the skin

! friction and heat transfer measurements were in best overall agreement with

_, the theories of Van Driest and Eckert, though both of these theories tended to

overpredict the levels in the Mach 7 to I0 range. Plotted in the Fc C$ - Fz Rez

plane we see the same general result whether the virtual origin is defined from

the momentum defect at the beginning or end of transition. While the theory of

} Van Driest is in best overall agreement, itoverpredicts the heat transfer and

_ skin friction in the low hypersonic Mach number regime_ where the agreement

! between the measurements and the Spalding-Chi theory is best. Because only

i 12

] 973003563-020

surface measurements were made, there remains a question on the magnitude

of the distance downstream of the maximum heating point, where all the

remnants of the transition process are dissipated. While this is unimportant

in our studies at the lower Mach numbers,where we have measurements at

many hundreds of boundary layer thicknesses downstream of the point of

, maximum heating, at Mach 11 and 13 one is forced to ask the question: Is the

disagreement between the measurements and the Spalding-Chi theory results

"_ from the persistence of turbulent remnants in the flow a low Reynolds number

_, effect or an inadequacy of the theory? Only boundary layer surveys will

- provide a complete answer on the relaxation distance of a transitional boundary7"

layer from the maximum heating point to the point where the boundary layer

": is fully turbulent.

I

i-!

V. REFERENCES

1. Spalding, D.B. and Chi, S.W., The Drag of a Compressible TurbulentBoundary Layer on a Smooth Flat Plate with and without Heat Transfer,Journal of Fluid Mechanics, January 1964, pp. 117-143.

2. Van Driest, E.R., Turbulent Boundary Layer in Compressible Fluids,J. Aero. Sci., 1951, 18, No. 3, 145-160.

5. Van Driest, E.R., Problem of Aerodynamic Heating, Aeronautical

Engineering Review, Vol. 15, No. I0, October 1956.

4. Coles, D. E,, The Turbulent Boundary Layer in a Compressible Fluid,

The Physics of Fluids, Vol. 7, No. 9, September 1964, pp. 1403-14Z3;also Report R-403-PR, 196Z, Rand Corp.

5. Sommer, S.C. and Short, B.J., Free-Flight Measurements of Turbu-

lent Boundary-Layer Skin Friction in the Presence of Severe Aerody-

namic Heating at Mach Numbers from 2.8 to 7.0, TN 3391, 1955, ?NACA; also Journal of the Aeronautical Sciences, Vol. 23, No. 6,

June 1956, pp. 536-542..

6. Bertram, M.H., Cary, A.M., Jr., and Whitehead, A.H., Jr., ,

. Experiments with Hypersonic Turbulent Boundary Layers on Flat Platesand Delta Wings, AGARD Specialists Meeting on Hypersonic Boundary

Layers and Flow Fields, London, England, May I-3, 1968.

7. Cary, A. M., Jr., Turbulent Boundary-Layer Heat Transfer andTransition Measurements for Cold-Wall Conditions at Mach 6, AIAA

Journal, Vol. 6, No. 5, May 1968, pp. 958-959.

r

8. Hopkins, E.J. and Inouye, M., An Evaluation of Theories for Pre-dicting Turbulent Skin Friction and Heat Transfer on Flat Plates at _ t

Supersonic and Hypersonic Much Numbers, ALAA Journal, Vol. 9, i

No. 6, June 1971, pp. 993-1003. i

9. "Description and Capabilities of the CAL 48-1nch Hypersonic ShockTunnel, " Cornell Aeronautical Laboratory, Inc., Buffalo, New York

(April 1968).

10. MacArthur, R.C., Contoured Skin Friction Transducer, CAL Report

No. AN-2.403-Y-I (August 1967).

1I. Martin, J.F., Duryea, G.R., and Stevenson, L.M., Instrumentation

for Force and Pressure Measurements in a Hypersonic Shock Tunnel,

Advances in Hypervelocity Techniques, CAL Report No. If3 (PlenumPress 1962).

14

%

1973003563-022

lZ. Pate, S.R., Schueler, C.J., An Investigation of Radiated Aerodynamic• Noise Effects on Boundary-Layer Transition in Supersonic and Hyper-

sonic Wind Tunnels, AIAA Paper 68-375, April 1968.

13. Stainback, P.C., Fischer, M.C. and Wagner, R.D., Effects of WindTunnel Disturbances on Hypersonic Boundary Layer Transition,AIAA Paper 72-181, January 1972.

\ 14. Bradshaw, P., A Note on Reverse T:ansition, JFM (1969), Vol. 35,Part Z, pp. 387-390.

15. Finson, M., Private Communication

16. Demetriades, A. and Laderman, A.J., Measurements of the Mean

and Turbulent Flow in a Cooled Wall Boundary Layer at Mach 9.37,

ALAA Paper 7Z-73, January 197Z.

17. Morrisette, E.L., Stone, D.R., and Cary, A.M., Downstream Effects

of Boundary Layer Trips in Hypersonic Flow, Langley Syposium on

Compressible Turbulent Boundary Layers, NASA SP Z16, Dec. 1968•

18. Eckert, E.R.G., Engineering Relations for Friction and Heat Transfer

to Surfaces in High Velocity Flow, J. Aeronautical Sciences, 1956,z3(8). s8s-587. )'

, S

19. Hirschfelder, J.O. ; Curtiss, C.F. ; and Bird, R.B., Molecular

Theory of Gases and Liquids, J. Wiley & Sons, 1954. :I

Z0. Cheng, H.K., Hall, J.G., Golian, T.C., and Hertzberg, A. ,_ Boundary Layer Displacement and Leading-Edge Bluntness Effects

:" in High-Temperature Hypersonic Flow, J. Aero/Space Sci., Vol. Z8,No. 5, pp. 353-381 (1961). ,;f

.

f

?

1973003563-023

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1973003563-031

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Figure 4 DRAWING OF SECTION THROUGH SKIN FRICTION TRANSDUCER

, HEAT SHIELD (ALUMINIZED MYLAR .002 THICK)

MYLAR 1.00025THICK)pcbTRANSDUCER / /

T X_ ( BORONNITRIDE WASHER ',

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_ Figure 5 HIGH FREQUENCY PRESSURE MOUNTING

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1973003563-034

..t I I i I l,ltl,t, III I I illltl I .......

Figure7a RUN 5 M = 10.4,Re/FT= 5.2 x 106,TW/T0 = 0.16

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Figure7b RUN6 M=12, RelFT=6.3x106 ,Tw/T 0=0.16

Figure7c RUN7 M-11, RelFT-11x106 ,TWIT 0 0.19all

l

Figure7d RUN 8 M - 10.5, RelFT- 5.7 x 108, TW/T0 - 0.21

Figure7e RUN 9 M - 10.6 Re/FT - 13.6 x 108, Tw/T0 - 0.19

28

1973003563-036

<!

I IIIII

7f RUN 10 M = 10.6, Re/FT= 10.6 x 106, Tw/T0 = 0.21Figure

II................................................ __........... - __:_ _;'

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i

Figure7h RUN 12 M - 10.6, Re/FT - 13.3 x 106, Tw/T0 ,, 0.19

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Figure7i RUN 13 M -, 8, Re/FT n 16.0 x 106, Tw/T0 " 0.31

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:,i J

] 97:3003563-037

............................ .. ,..- .... ,._ * .... r_--T-_ _-'----f,"__IPP_m

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1973003563-038

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! Figure 8 CORRELATION OF LAMINAR HEAT TRANSFER_ o

1973003563-039

,I

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AT MAXIMUM HEATING

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i a •

1973003563-041

I

. Figure11 CORRELATIONOF THE FLAT PLATETRANSITIONDATA IN TERMSOFTHE PARAMETERSSUGGESTEDBY PATEAND SCHUELERS

34

1973003563-042

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35

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1973003563-043

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Figure 13 TURBULENTBOUNDARYLAYERDISPLACEMENTANDMOMENTUMTHICKNESSESFROMSHOCKTUNNELNOZZLEDATA

36

1973003563-044

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r Z_ TRANSITIONMEASUREMENTSON THEFLATPLATES

3?

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1973003563-045

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Figure 15 REYNOLDS ANALOGY FACTORS FOR TURBULENTHYPERSONIC BOUNDARY LAYERS

1.6 ......... , ........ ,- ........r- ........ r ......... -r ..........ii i !

I t I I o

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• Figure 16 REYNOLDS ANALOGY FACTORS FOR TURBULENT, HYPERSONIC BOUNDARY LAYERS

38

1973003563-046

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|r" I$ 2 25 $ 4 S 6 7 J $ IO w IS 2 25 $ 4 $ i I 8 $ !0

Figure 20a COMPARISON BETWEEN THE MEASURED SKIN FRICTION AND THE

t . THEORY OF ECKERT (e V = e B + eBE)

5O

1973003563-058

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Figure 20b COMPARISON BETWEEN THE MEASURED SKIN FRICTION AND

THE THEORY OF VAN DRIEST (O V = OB + eBE)

51

1973003563-059

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't

_,_Lt_} FA(.;E BLANK NOT FILMED

53

1973003563-060

Fc 2C H

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, Figure21b COMPARISONBETWEENTHE MEASUREDHEAT TRANSFERAND

THE THEORYOF VAN DRIEST (e V = OB + eBE/2)

54

r

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1973003563-061

Figure21c COMPARISONBETWEENTHE MEASUREDHEAT TRANSFERAND

THE SPALDING-CHITHEORY (9 V = eB + OBE/2)

55

1973003563-062

56

1973003563-063

f

rI

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ql

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Figure 22b COMPARISON BETWEEN THE MEASURED SKIN FRICTION AND

• _ THE THEORY OF VAN DRIEST 10 V = e B + OBE/2)

57

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1973003563-064

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} Figure22c COMPARISONBETWEENTHE MEASUREDSKIN FRICTION AND_': THE SPALDING-CHITHEORY (eV - e B + GBE/2)

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1973003563-065

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Figure24a COMPARISONBETWEENTHE MEASUREDSKIMFRICTIONANDTHE THEORYOF ECKERT (e V - OB + OBELAMINAR)

r

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1973003563-069

t

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1973003563-070

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_. Figure 24c COMPARISON BETWEEN THE MEASURED SKIN FRICTION AND

f THE SPALDING-CHI THEORY (OV = eB+ OBE LAMINAR)

1973003563-071

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1973003563-072

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1973003563-074

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1973003563-080

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