COPY - NASA · h heat-transfer coefficient, (Taw k thermal conductivity K Bach's relaminarization...

NASA TECHNICAL NOTE \W '*j| NASA TN D-6965

cCOPY

EXPERIMENTAL STUDY OFNOZZLE WALL BOUNDARY LAYERSAT MACH NUMBERS 20 TO 47

by. Joseph H. Kemp, Jr., and F. Kevin Owen

Ames Research CenterA

Mo/eft Field, Calif. 94035<

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • OCTOBER 1972

https://ntrs.nasa.gov/search.jsp?R=19730002534 2020-03-21T23:55:20+00:00Z

1. Report No.

NASA TN D-69652. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle ^

EXPERIMENTAL STUDY OF NOZZLE WALL BOUNDARYLAYERS AT MACH NUMBERS 20 TO 47

5. Report DateOctober 1972

6. Performing Organization Code

7. Author(s)

Joseph H. Kemp, Jr., and F. Kevin Owen

8. Performing Organization Report No.

A-4359

9. Performing Organization Name and Address

NASA Ames Research CenterMoffett Field, Calif. 94035

10. Work Unit No.

136-13-01-16-00-21

11. Contract or Grant No.

12. Sponsoring Agency Name and Address

National Aeronautics and Space AdministrationWashington, D.C. 20546

13. Type of Report and Period Covered

Technical Note

14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

The nozzle wall boundary layer of the Ames M-50 helium tunnel has been thoroughly investigated with pitotpressure, total temperature, skin friction, and wall heat transfer measurements at five stations and hot wire measurementsat two stations. The results indicate that the boundary layer was turbulent with a thick viscous sublayer. Pressure gradientswere observed across the boundary layer; the effect of these gradients on the equations of motion are discussed in thispaper.

The direct skin friction measurements were higher than expected from empirical predictions; the Reynolds analogyfactors however, 2C^/Cy, were lower than expected. Hot wire measurements indicated mass flow fluctuations as large as80 percent of the local mean mass flow at the edge of the viscous sublayer with a maximum value relative to the edge massflow of about 15 percent at y/S « 0.8.

17. Key Words (Suggested by Author(s))Hypersonic DowTurbulent boundary layersNozzle wall

18. Distribution Statement

Unclassified - Unlimited

19. Security Ctassif. (of this report)

Unclassified

20. Security Classif..(of this

Unclassified

21. No. of Pages

106

22. Price"

S3.00

* For sale by the National Technical Information Service, Springfield, Virginia 22151

TABLE OF CONTENTS

Page

SYMBOLS vSUMMARY 1INTRODUCTION 1EXPERIMENTAL PROCEDURES 2

Facility 2Pressure Measurements 2Stagnation Temperature.Measurements 3Skin-Friction Measurements 3Heat-Transfer Measurements 3Boundary-layer Fluctuation Measurements 4

RESULTS AND DISCUSSION 5Throat Reynolds Number and Possible Relaminarization Effects 5Temperature Pressure Variations Along Nozzle 6Profile Measurements 7Velocity and Density Profiles 7Effects of Wall Temperature Ratio 8Effects of Nozzle Station 9Effects of Reservoir Pressure 10Correlations of the Velocity Profile 10Wall Measurements 11Boundary-Layer Fluctuations 12Flow Model 13

SUMMARY OF RESULTS 13APPENDIX A - DERIVATION OF PARAMETERS USED TO CORRELATE TOTAL-

TEMPERATURE-PROBE CALIBRATION DATA 15APPENDIX B - EFFECT OF DENSITY ON SUBLAYER THICKNESS 16REFERENCES - 17TABLES 23FIGURES 59

SYMBOLS

2 T,'wfriction coefficient,

Pe"e

Cu Stanton number,

Cp pressure coefficient

d diameter of thermocouple wire used in heat-transfer gage, cm

D diameter of wire in total temperature probe, cm

e voltage

/ frequency

Fc transformation function for Cf

Fr> transformation function for Rea

h heat-transfer coefficient,(Taw

k thermal conductivity

K Bach's relaminarization parameter (ref. 24),Peue dx

M Mach number

hdNu Nusselt number, —r-

K-

p pressure, atm

q wall heat-transfer rate, W/m2

Qr total radial heat conduction at radius r in the thin skin at the heat-transfer gage

r radius, cm

Ay recovery factor

^? Reynolds number

I 1/2

/ thickness of sensing element in heat transfer gage, cm

T temperature, °K

w velocity, m/sec

u+ transformed velocity

x longitudinal distance downstream from the nozzle throat

y distance normal to the surface, cm

y transformed distance normal to the surface

T 2 fd pu I T-T e \ I y\r energy thickness, rw - rw 1 / I - =r I 1 1 — — I i

L rw Jo P e u e \ l w ~ l e ' \ w'

5 boundary-layer thickness, cm

5 * displacement thickness, rw - rw 1 / (1 111 I dy\\_ rw o \ Peue/ \ rwl J

[ 2 r5 ou / u \ I v\ 11/2

I / —— (1 - 1 1 1 -—]dy?w J Pe^e \ MS I \ ^wl Io x ' ^ ' -i

p. coefficient of viscosity

v kinematic viscosity

p density, kg/m3

r shear stress

Subscripts

act actual value

aw adiabatic wall condition

c at disk center

e outer edge of the boundary layer

/ local value

VI

L outer edge of the viscous sublayer

m measured value

r at radius r

T at nozzle throat based on throat diameter

VD I Van Driest /

w wall or wire

o local stagnation condition

2 static value behind a normal shock

6 based on momentum thickness

Superscripts

' fluctuating part of dependent variable

- time-averaged part of dependent variable

VII

EXPERIMENTAL STUDY OF NOZZLE WALL BOUNDARY

LAYERS AT MACH NUMBERS 20 TO 47

Joseph H. Kemp, Jr. and F. Kevin Owen

Ames Research Center

SUMMARY

The nozzle wall boundary layer of the Ames M-50 helium tunnel has been thoroughly investi-gated using pitot pressure, total temperature, skin friction, and wall heat-transfer measurements atfive stations covering about 85 percent of the nozzle length and with hot wire measurements attwo stations near the nozzle exit. Data were obtained for temperature ratios varying from 0.35 to 1and Reynolds numbers based on momentum thickness varying from 900 to 5600. The resultsindicate that the boundary layer is turbulent with a thick viscous sublayer. Pressure gradientsobserved across the boundary layer are large relative to the static pressure but small relative to thedynamic pressure.

The direct skin-friction measurements were higher than expected from empirical predictions:the Reynolds analogy factors 2Cfj/Cf, however, were lower than expected. Hot-wire measurementsindicated mass flow fluctuations as large as 80 percent of the local mean mass flow at the edge ofthe viscous sublayer with a maximum value relative to the edge mass flow of about 15 percent aty/S = 0.8.

INTRODUCTION

The development of flow models and provision of test cases for hypersonic turbulentboundary-layer theories and prediction techniques require experimental data over the full rangeof conditions for which such theories and techniques may be used. Such data should includeindependent measurements of skin friction, wall heat transfer, and two flow property distributionsacross the boundary layer (usually pitot pressure and stagnation temperature).

Although more than 50 studies of hypersonic boundary layers have been reported, it is stilldifficult to make satisfactory tests of theories and prediction techniques because researchers have notmeasured all four quantities listed above except in the studies reported in references 1 through 4.Furthermore, the data are even more limited at the higher hypersonic Mach numbers where densityratios across the boundary layer become very large. For example, the only published data aboveMach number 15 appear to be those presented in references 4 through 8.

The present investigation was undertaken to augment available data by providing data onturbulent boundary layers with very high edge Mach numbers. The data include direct measure-ments of skin friction, wall heat-transfer rates, and mean and fluctuating measurements across the

boundary layer. The data obtained on the nozzle wall of the Ames M-50 helium tunnel cover abroad range of test conditions with measurements at five stations covering 85 percent of the nozzlelength, wall temperature ratios varying from 0.35 to 1, and Reynolds numbers based on momentumthickness varying from 900 to 5600. The boundary layer thickness varied from 2.4 cm to 29 cm,and the nozzle exit radius was 36 cm.

EXPERIMENTAL PROCEDURES

Facility

Tests were conducted on the nozzle wall of the Ames M-50 helium tunnel, which is a blow-down tunnel with an axisymmetric contoured nozzel and an open test section (ref. 9). It useshelium (heated by an electrical resistance heater) as a test gas and operates at nominal exit Machnumbers of 42 to 47 for test times up to 20 min.

Figure 1 shows the five survey stations where pitot pressure, total temperature, and walltemperature were measured, together with the ten wall pressure stations. Wall heat transfer andskin friction were measured at the last four survey stations; the hot-wire surveys were made at thelast two survey stations. The tunnel was operated at reservoir pressures ranging from 65 to 272 atmand total temperatures from 300° to 900° K.

Pressure Measurements

Primary pitot pressure surveys between x = 1.067 m and 3.56 m were obtained by traversing asingle 0.476-cm-dia probe through the boundary layers. In addition a 0.159-cm-dia probe was usedto check the effects of probe diameter on measured pitot pressure at these stations and to obtainsurvey data at x = 0.508m.

Pressures were measured by a capacitance type transducer system similar to one described inreference 10. With this system, one transducer provides accurate readings over the full pressure

.range encountered in the boundary layer (0.004 to 0.14 atm). Errors in pressure resulting fromnonlinearity and temperature effects on the transducer sensitivity were less than ± 1 percent of themeasured value. Additional errors due to variations in reference pressures were less than 4 percentof the minimum pressure measured.

No corrections were made for the various rarefaction effects on the pressure measurements.However, the possible variations attributable to these effects are summarized as follows. The effectsof thermal transpiration errors due to temperature variations along the tube connecting the orificeand the transducer are negligible (refs. 11,12). Errors in wall pressure resulting from orifice effectsdue to heat transfer to the wall are less than 3 percent (ref. 13). Errors due to viscous interaction andrarefaction effects on the pitot pressure measurements are less than 4 percent for the outer region ofthe boundary layer (ref. 14). Near the wall, these effects may introduce larger errors; however, thethe conclusions of this report do not depend on these data.

Stagnation Temperature Measurements

Stagnation temperatures were measured with a single shielded thermocouple probe (fig. 2)designed specifically to have low conduction losses at very low density flow conditions.

The probe was calibrated-in the free-jet facility described in reference 15, which producedflows at Mach numbers ranging from 5 to 40 and flow densities comparable to those encountered inthe boundary layer of the M-50 tunnel.

The probe calibration is shown in figure 3 where the data are correlated over a wide range oftemperatures and pressures. These data could not be correlated solely on the basis of conductionerror as proposed by Winkler (ref. 16). The present correlation parameters were derived using theassumption that radiation losses constituted the major source of error (appendix A).

Conduction effects probably cause some scatter in the data. Nevertheless, the correlation isreasonably good for a wide range of temperatures, and there is substantial agreement between datafrom the calibration in the free jet and the test points in the M-50, where differences in supporttemperatures should cause significant differences in the conduction effects.

Skin-Friction Measurements

Skin friction was measured using a floating element, magnetically nulling balance similar tothe one described in reference 17. At each station, the elements were contoured to the local nozzlesurface. The balance had an accuracy of ±2 percent.

The effects of floating element position, both above and below the surface, were investigated,and it was found that the measured skin friction was much less sensitive to the element position thanhas been observed previously at lower Mach numbers (refs. 18-20). Indeed, the indicated skinfriction increased by only 10 percent when the element was raised 0.0075 cm above the surface,and no differences were observed when the element was recessed the same distance below thesurface. The relative insensitivity of the data to element position is probably due to the combinationof large boundary-layer thickness and low wall density.

Heat-Transfer Measurements

Wall heat transfer was determined from the steady-state heat conduction in a thin-skin gagecontoured to the nozzle wall. The local temperatures were measured by chromel-constantan thermo-couples spot-welded to the thin skin at the locations shown in figure 4.

Since the temperature variation between the center of the thin skin and the wall was small,relative to the driving potential,

(rc"7w) < 0.06

constant heat transfer over the gage surface area was assumed. With this assumption, the equationgoverning the heat transfer in the gage is

/JTQR = -2-nRkt— = vR2q

** uK

Solving for the temperature, one obtains

9= — 4'*

In the present experiments, heat transfer rates corresponding to the temperature difference betweenthe center and each of the other three thermocouples (fig. 4) were calculated and the resultsaveraged to obtain the values presented in this paper. The individual measurements were generallywithin ±5 percent; however, in a few cases the variations were as high as ±10 percent.

Errors due to conduction down the thermocouple wires, convection from the back of thegage, and radiation losses have not been accounted for in the data reduction. However, conductionerrors were held low by using small diameter wires, d/t < 0.6. Convection losses were held down byevacuating the back of the gage to pressures less than or equal to the local wall pressure; and radia-tion errors were not large, since the temperature difference between the gage surface and the sur-rounding radiative media was small (Tc- Tw < 35° K). It is estimated that the combined error dueto these effects is less than 5 percent. Basic instrument accuracy and the accuracy of the normaliz-ing quantities (p, u, Taw) also introduce error in the heat-transfer coefficient. Consequently, theoverall accuracy of the heat-transfer coefficient is estimated to be about ±12 percent.

Boundary-Layer Fluctuation Measurements

The character of the fluctuations across the boundary layer at x = 3.56 m were obtained witha constant temperature anemometer system, which uses a water-cooled platinum film probe similarto the one described in reference 21. Water cooling permitted film operation at a temperature wellbelow the free-stream total temperature. The upper frequency limit (-3dB) of the system, as deter-mined by a standard square-wave technique, was found to be 60 kHz. Since the boundary layer isapproximately 29 cm thick at the survey station, this frequency response makes it possible tomeasure fluctuations with a length scale down to one-eighth the boundary layer thickness. In lightof the work by Kistler (ref. 22), over 90 percent of the energy should be contained within thisfrequency range. Mass flow and temperature fluctuation intensities were measured at x - 2.793 musing a constant temperature anemometer with a 0.00063-cm-dia by 0.317-cm-long platinum10 percent iridium wire.

A preliminary calibration of the probe was made in the free-jet facility (see ref. 15). In thiscalibration, the voltage across the wire ew was measured for various wire temperatures Tw, streamtotal temperatures To, and mass flows pu. The measurements were then plotted against pu andTo for constant values of Tw and TO or pu. From the calibration it was found that

dew2 \

3pH/_ and>«, Tw

were approximately constant for the flow conditions of the present tests. This is consistent with theindication that for this diameter wire the flow would be essentially free molecular. The fluctuationlevels were obtained at a number of points through the boundary layer using a technique similar tothat of Kistler. The wire was operated at six different over-heat values and a least-squares parabolicfit of the data was made to the equation

(Ae)2 =(~

where de/dTo and de/dpu were obtained from the calibration.

RESULTS AND DISCUSSION

Throat Reynolds Number and Possible Relaminarization Effects

The effects of Reynolds number based on throat diameter and relaminarization due to favorablepressure gradient have been examined through comparisons with the work of Bach et al. (refs. 23,24).In'figure 5 the relaminarization parameter K of Bach et al. is plotted against x. The data presentedare for the throat region of the nozzle only; farther downstream the data continue the trendestablished here and should be of little importance.

Presented are the limiting cases for the present tests. For the high pressure limit (po - 270 atm)with a throat Reynolds number of 4.8 million and a maximum value of K of about 2.6X106, thework of Bach etal. indicates the flow should definitely be turbulent at the throat with no discernibleeffects of relaminarization. For the lower pressure limit (po = 65 atm), the throat Reynolds numberof 0.5 million is on the borderline for turbulent flow (ref. 23), and the value of K is sufficientlyhigh that a significant region of relaminarization might occur (ref. 24). However, if relaminarizationdoes exist, its effects are not apparent in the data farther downstream as will be seen later.

In a hypersonic nozzle, the flow is similar to that shown in figure 6 (ref. 9). Downstream of thecontour point there are two separate regions of flow: the boundary layer and the uniform core. Inthis situation, the density distribution is like the one most commonly associated with boundarylayer flow — that is, the density decreases smoothly from a constant outer edge value to the wallvalue. Upstream of the contour point, however, there is a nonuniform flow region between theboundary layer and the uniform core. This region is one of expanding hypersonic flow with signifi-cant Mach number variations across it. These Mach number variations cause inviscid density varia-tions similar to those shown schematically in figure 7. In this situation, viscous effects act on avariable density inviscid flow field, rather than on the more common constant density flow field.The resulting density distribution is similar to that indicated by the dashed line in figure 7.

For nozzle flows, the inviscid density variation has not usually been considered, and the edgeof the boundary layer has been defined as the point where density (or pitot pressure) is maximum.However, from figure 7, it is apparent that the maximum density can occur a significant distancefrom the point where density first deviates from the inviscid variation — that is, where viscouseffects are first encountered. For most hypersonic nozzles, surveys are taken at or ahead of thecontour point; thus, this variation would be typical of much of the nozzle wall data.

In the present tests two measurements have substantiated the above model. First, totaltemperature measurements indicated the existence of temperature and velocity gradients fartherfrom the wall than the maximum density point. Second, hot wire surveys indicated the existence ofstrong intermittencies in the same area. Consequently, for the present tests .the edge of the boundarylayer was determined by using stagnation temperature, which is constant in the inviscid flowirrespective of variation in local Mach number. The data were plotted, a straight line was fairedthrough the free-stream values, and a curve was fitted to the remaining data. The point of junc-ture between the straight line and the faired curve was taken as the edge of the boundary layer.

Temperature Pressure Variations Along Nozzle

Figure 8 shows typical temperature and pressure distributions measured on the nozzle walland static pressure distributions for the outer edge of boundary layer. The static pressure wascalculated from pitot pressure and reservoir pressure measurements assuming isentropic flow. Thevalues are for nominal reservoir temperatures of 500° and 900° K. The nozzle has a hot throat,which is at very nearly recovery temperature during the tests. However, as can be seen, the tempera-ture drops off rapidly with distance down the nozzle, so that even for the higher temperature condi-tions the wall temperature is nearly room temperature at all survey stations.

The reservoir temperature has no measurable effect on either the wall pressure or the free-stream static pressure. However, there is a marked difference between these pressures, indicatingthe existence of a pressure gradient across the boundary layer. This pressure difference appearsto correlate with edge Mach number as shown in figure 9. Furthermore, from data of other investi-gations presented in figure 9, it appears that this phenomenon is not limited strictly to nozzle walltype flows.

The main reason for the high pressure ratios at the higher Mach numbers in figure 9 appears tobe the relatively low value of Peat these Mach numbers. If the pressure differences Pw~ Pe arenormalized using the edge dynamic pressure 1/2 PeUe2 , the resulting ratios vary from about1X1CT2 at the lower Mach numbers to about 1X10~3 at the higher Mach numbers. Thus, althoughthe pressure differences at the higher Mach numbers are very large relative to the static pressure,they are still small relative to dynamic pressure. An order-of-magnitude analysis of the time-averaged Navier-Stokes equations indicates that, as in the case of lower Mach numbers, the ymomentum is small relative to the x momentum. So from this point of view, the y momentumequation need not be considered in the solution of the problem. However, the large relative pres-sure variations result in large density variations. Since density, is involved directly in the continuity,x momentum, and energy equations, some way of accounting for these density variations is needed.Thus, it may be necessary to include the y momentum equation to properly describe the densityvariations in high Mach number flows.

Profile Measurements

The measured pitot pressure and uncorrected stagnation temperature for the five surveystations are presented in tables 1 through 5 with a representative plot presented in figure 10.The probe was traversed from near the wall to the free stream and back to the wall, with data beingobtained going both directions. As can be seen, the repeatability of the pressure data was excellent;however, the temperature data show some hysteresis effects particularly in the region where lowpressures exist. Since the hysteresis is undoubtedly due to the variations in support temperature,fairings of the data were weighted toward the measurements made on the outward traverse at thelower temperature ratios near the wall, and toward measurements on the inward traverse for highertemperature ratios farther from the wall. This procedure gives velocity values that vary as much as5 percent from the value given by unweighted fairing; however, it is believed to be more accuratesince it should minimize the effects of conduction losses into the supports.

Also shown on the temperature curve is the wall temperature slope as determined from heat-transfer measurements. As can be seen, within the accuracy of the data, the slope agrees withmeasured temperature profile. However, it is apparent that accurate determinations of the heattransfer from the temperature data alone would be very difficult. ,

Velocity and Density Profiles

Velocity and density profiles calculated using faired values from the pitot pressure and correctedstagnation temperature profiles are given in tables 6 to 10; table 11 is a summary of the morepertinent parameters associated with the data. In these tables, corrections have been made forreal-gas effects (ref. 25). The reservoir pressure stated is that associated with the pitot pressuresurvey, and the total temperature is that associated with the temperature surveys. The viscosity lawused in formulating some of the parameters was that of Akin (ref. 26). In some cases, there aredifferences between reservoir pressure and/or temperature for the pressure surveys and the cor-responding values for the temperature survey. However, it will be shown later that the crosscoupling between temperature and pressure is small; therefore, these differences should not signifi-cantly alter the accuracy of the data. The calculations for tables 6 through 10 required an assump-tion on static pressure p. Three possible assumptions were examined: p = pw , P - Pe andp - pw - (pw - pe) y/&. Figure 11 shows typical velocity and density profiles resulting fromeach of the assumptions. As can be seen, the assumption p - pe results in a value for u/ue near thewall that is too high. The other two assumptions, however, show little difference in either velocityor density ratio, thus it makes little difference which of these two assumptions are used. In thispaper, the linear variation is used because it provides more accurate absolute values for pressureratio and Mach number near the outer edge of the boundary layer. This assumption is not exact,but barring large excursions in p - values much larger than pw or much smaller than pe—it shouldnot substantially affect the conclusions of this paper.

From figure 11, it appears that the boundary layer has a relatively thick inner region withlinear velocity variations typical of those occurring in a viscous sublayer. In this region, the densityis nearly constant. The outer region has small velocity variations and fairly full density profilestypical of profiles normally associated with a turbulent boundary layer. Furthermore, skinfriction measurements suggest a velocity slope at the wall that is in reasonable agreement with themeasured velocity profile. The deviation of velocity very near the wall is probably the result of either

wall-probe interference effects or rarefaction effects on the pitot probe. The data at x = 3.56apparently have some end effects due to the close proximity of the open test section at x - 3.61,which affect the flow near the wall. Because of these effects these data are not included in thefollowing discussions.

Effects of Wall Temperature Ratio

Stagnation temperature and pitot pressure variations across the boundary layer are shown infigure 12 for various wall temperature ratios Tw/TOe. Differences in Tw/TOe were obtained byvarying the free-stream total temperature. These surveys were obtained at x = 1.067 where theedge Mach number was sufficiently low that wall temperature ratios near unity could be obtainedwithout liquefaction problems.

For Tw/Toe - 0.98 there is an overshoot in the total temperature near the outer edge of theviscous sublayer (y/8 ~ 0.2). Farther from the wall, the temperature drops to a value lower thanTw(y/8 * 0.4) and then rises uniformly to TOe at the edge of the boundary layer. The overshootin total temperature is similar to that previously observed at wall temperature ratios close to unityand is attributed to viscous dissipation. It is of interest to note that in the present investigation theseeffects persist to much lower wall temperatures and cause some perturbation of the profile forvalues of Tw/TOe as low as 0.535.

Another point of interest is that in the outer region of the boundary layer there are essentiallyno significant differences in temperature or pressure profiles for a wide range of wall temperatureratios.

The effect of variation in temperature on computed velocity, density, and Mach number pro-files is shown in figure 13. As might be expected from the variations in temperature and pressure,the temperature ratio affects the thickness of the viscous sublayer — the thickness increases as walltemperature ratio is decreased - but it has little effect on the computed values in the outer regionof the boundary layer.

From figure 13(c), it is apparent that most of the boundary layer is hypersonic. For the probemeasurements, one may obtain the energy relationship

{ 1 + [(7 - l)/2] Me*} T

\T 0' oer

The hypersonic approximation (M » 1) to this relationship gives

u \2 ToT0e

From this approximation

TO - Tw (u/iie)1 ~ (Tw/T0e)

TQS ~ TW 1 — (Tw/TOe)

Examination of this equation (referred to hereafter as the "hypersonic approximation") revealsthat as Tw/TOe approaches zero, the functional relationship between (To ~ Tw)/(TOe — Tw) andu/ue for the hypersonic portion of the boundary layer approaches the quadratic relationshiprecently noted by a number of investigators. However, if Tw/TOe is greater than zero the value ofthe temperature relationship will fall below the quadratic. Thus, it appears that as edge Machnumber is increased, conditions may be encountered in the outer region of the boundary layer forflows on all types of bodies where the temperature-velocity relationship could vary substantiallyfrom the familiar Crocco (linear) relationship. Consequently, the recently observed deviation froma Crocco velocity-temperature relationship may not be strictly a pressure gradient phenomenon ashas been suggested (refs. 27-28), but simply may be more evident in nozzle flows where the edgeMach number is generally higher and where the existence of a pressure gradient tends to augmentthe differences. Softley and Sullivan (ref. 29) have obtained data on a cone at M = 10.2 that agreevery well with the hypersonic approximation. Furthermore the existence of hypersonic effects mayexplain why the data of Jones and Feller (ref. 27) show very little tendency to "relax" to the linearprofile for as much as 100 boundary layer thicknesses downstream of the end of the nozzle.

Figure 14 replots data for two wall temperature ratios, 0.72 and 0.36 (fig. 12), on familiarCrocco energy type plots. Curves are given for the Crocco relationship, a quadratic relationship,and the hypersonic approximation just discussed. Excellent agreement is apparent between the dataand the hypersonic approximation in the outer region of the boundary layer. Although this regionappears small on this type of plot, the hypersonic data cover 60 to 75 percent of the boundary-layer thickness. In the viscous sublayer region the temperature-velocity relationship shows markeddifferences for the two temperature ratios presented in figure 14. The relationship for the higherwall temperature ratio is complicated by the temperature overshoot in the sublayer. The lower walltemperature data show a much simpler behavior. However, in either case it is evident that thetemperature-velocity relationship is more complex than either the Crocco or the quadratic relation-ship indicates.

References 27 and 30, indicate that the velocity variation near the wall may be linear, eventhough the variation in the outer region is significantly different. The earlier discussion of thevelocity profiles indicated that the measured u/ue may be high in the region near the wall; thus,it appears that the present data is in basic agreement with this indication.

Effects of Nozzle Station

Typical variations in velocity, density, and Mach number profiles with nozzle station areshown in figure 15. It is apparent that there are significant changes with nozzle station. .For thevelocity profile, the effect seems to be limited to the sublayer region, with the relative thickness ofthe sublayer increasing with distance from the nozzle throat. This increase in sublayer thicknessis probably associated with the increase in Mach number or, more exactly, the accompanyingdecrease "in wall density (appendix B). The density and Mach number profiles show differencesthroughout the boundary layer, but most of these differences are probably associated with adjust-ments required to account for the changes in relative thickness in the sublayer.

Effects of Reservoir Pressure

Typical variations in velocity, density and Mach number profile with reservoir pressure areshown in figure 16. Again, all differences seem to be associated with changes in the sublayer thick-ness. In this case, the sublayer is relatively thicker at lower pressures.

Correlations of the Velocity Profile

A correlation of the velocity profiles using measured wall density and shear stress in law-of-the-wall parameters is presented in figure 17. The four cases shown are typical of the results for allcases and represent considerable ranges in tunnel pressure, temperature, and station. It can be seenthat in the viscous sublayer the data are in excellent agreement with incompressible correlationvalues (ref. 31). However, above y+ = 10, the data do not agree with the incompressible correla-tion. For this region, changes in pressure and station affect the y+ value at the edge of the boundarylayer, while changes in total temperature affect the relative level of u+.

Correlations of the present data using the transformations of Coles (ref. 32) and Van Driest(ref. 33) are shown in figures 18 and 19. Coles' transformation stretches the y variable using afunction of the density variation across the boundary layer. As can be seen, with this transformation,the data retain the approximate agreement with the incompressible profile in the viscous sublayer,while in the logarithmic region, the y variable is stretched to values comparable to those obtainedin incompressible flows. Although this transformation seems to stretch the y+ variable to properproportions, it does not provide the changes in slope in the logarithmic region of the boundarylayer required for complete agreement with incompressible data. It appears that the need for a slopechange requires an operation on the velocity variable. Consequently, it is doubtful that othertransformations that operate only on the y variable (refs. 34-36) will be successful in transformingthe hypersonic turbulent boundary layer to the incompressible plane.

The transformation of Van Driest (ref. 33), which operates only on the u, variable, appears tobe very successful in obtaining a consistent correlation of the data (fig. 19). Two ways of applyingthe Van Driest transformation are shown in figure 19. The first uses the measured density profileto compute u+, while the second uses equation (54) from reference 33 to compute u+. Thisequation was obtained assuming a Crocco temperature-velocity relationship. As indicated previously,the Crocco relationship is somewhat different than the measured temperature-velocity relationship.Therefore, the substantial agreement of both curves indicates that at least for the present conditionsthis transformation is not strongly affected by the assumed temperature-velocity relationship. Ingeneral, the transformation provides reasonable agreement between the present data and incom-pressible data. However, the present data terminate at lower y+ values.

Velocity defect correlations using the Van Driest transformation on the data are shown infigure 20. The data below y/8 = 0.5 are in the viscous sublayer, and, as might be expected, theyare not correlated well. In the outer region, however, where flow conditions meet the requirementsfor correlation using the velocity defect law, the data agree very well with the incompressiblecorrelation, indicating that the mixing length concept may be applicable to very high Mach numberflows.

10

From the law of the wall correlations, it appears that the viscous sublayer thickness is adjust-ing very rapidly to the local density changes. Consequently, the sublayer is growing with distancedown the nozzle. However, it is possible that the high Mach number of the outer, turbulent regionof the boundary layer may restrict the rate of growth of turbulent bursts. As a consequence, therelationship between the sublayer thickness and the turbulent layer thickness may be affected. Itappears from figure 21 that if such effects occur they are correlated by variations in Me and RQ.Figure 21 is basically .the same as figure 19 of reference 37, except that additional data from testson flat plates and cylinders have been added. This figure also shows that the relationship between6^/6 and Mel\/R^, is the same for flat plates and cylinders, where there are no pressure gradienteffects, as it is for nozzle walls where such effects may be present. This suggests that locally appliedtheories, developed for use with flat plates, which obtain their solution from Me and Ro , mayprovide reasonable agreement with values obtained on nozzle walls.

Wall Measurements

Measured recovery factor— The recovery factor for the present tests was determined by inter-mittently reading the heat-transfer rate as the reservoir temperature was slowly varied from roomtemperature to about twice room temperature and back again. The heat transfer versus temperatureratio was then plotted and the recovery factor taken as the temperature for which q is zero. Therecovery factor determined by this procedure is about 0.8. This factor is somewhat lower than the0.9 values previously measured at lower Mach numbers (ref. 38); in fact, it is near the value forlaminar boundary layers. The reasons for this value are not understood, but they are probablyassociated with the fact that, as discussed later, the viscous sublayer is so thick that large turbulentfluctuations never reach the wall.

Skin-friction and heat-transfer measurements— Figure 22 shows the skin-friction and heat-transfer variations with RQ for the different data stations. The variation with RQ for each datastation is approximately the 1 /4 power relationship normally associated with a turbulent boundarylayer. However, there is a marked difference in level of the coefficients at the various data stations,primiarily because of Mach number variations as shown in figure 23, where data with RQ between2000 and 3500 are plotted against Mach number. It is apparent that both skin friction (fig. 23(a))and heat transfer (fig. 23(b)) show variations that agree in general with trends indicated by previousinvestigations at lower Mach numbers. In figure 23(a), the skin-friction data are also compared withvalues obtained from five commonly used turbulent skin-friction prediction techniques (refs. 32,33, 39-41) and for reference purposes with the value predicted by the T method (ref. 42) forlaminar boundary layers. (The program used to make these comparisons is that developed byHopkins and Inouye (ref. 43).) The turbulent prediction techniques were developed at lower Machnumbers (up to Mach number 10) for flows with no normal pressure gradient, and in this region theagreement with the data is reasonable. However, at the higher Mach numbers there is a very widedispersion in the predicted values with most of the predictions being much lower than the data.Only the Van Driest / prediction technique (ref. 33) give a value that is close to the data. This mightbe expected, since this technique also provides a good transformation of the-velocity profile datainto the incompressible plane.

The reasons for the disagreements between data and theory are not fully understood at thepresent time; however, some of the more obvious possibilities are discussed here. First, the existence

11

of pressure gradients may introduce some disagreement between the data and the various predic-tions. However, there are two indications that this may not be so. The first is the close relationshipbetween RQ and the relative sublayer thickness as discussed previously. The second is shown infigure 24 by the comparisons between data and theory. The present data and those of reference 8were obtained over a wide range of pressure gradients, and although both sets of data are somewhathigher than the predicted values, neither set shows trends associated with the variations in pressuregradient. Second, the actual temperature profiles differ from the Crocco profiles used in thetheories. However, in the critical region near the wall, the temperature gradients appear to be nearlyCrocco; therefore, the skin friction may not be strongly affected by the differences in temperature.Third, the transformed RQ values are somewhat lower than any RQ (fig. 24). Thus the value forincompressible flow at these Rg is not known directly but can be obtained only from extrapolationof available incompressible data; such extrapolations, of course, could be in error.

Reynolds analogy factors 2C^/CV determined from the measured skin friction and heattransfer data are shown in figure 25. It is apparent that the Reynolds analogy factors are somewhatbelow the values measured by prior investigators at lower Mach numbers. The reasons for this arenot clear at the present time. However, there is a tendency for the Reynolds analogy factor todecrease at the lower RQ (fig. 25(a)) and at the higher Mach numbers (fig. 25(b)).

Figure 26 shows the variation of Cf> Cfj, and 2Cfj/Cf with wall temperature ratio. The opensymbols are the data as measured, and the filled symbols represent data adjusted to a constant RQ(equal to 3786) by means of the 1/4 power relationship between RQ and Cf and Cfj. It isapparent that all three coefficients Cf, Cjj, and 2 C/f/Cy vary with wall temperature with the valueof the coefficients increasing as wall temperature ratio decreases.

Boundary-Layer Fluctuations

Figure 27 shows oscilloscope traces indicating the fluctuations in heat transfer to the cooledfilm probe. Of interest here is the intermittency at both the outer edge of the boundary layer andthe edge of the viscous sublayer. The inner intermittency is similar to that reported by Corrsin(ref. 44) in incompressible flow and is as predominant as that at the outer edge of the boundarylayer. The existence of this intermittency is not really surprising since such regions are commonat turbulent boundaries. It is apparent that turbulence is being dissipated in the viscous sublayer,and it is unlikely that large fluctuations ever reach the wall — probably because of the largethickness of the sublayer.

The magnitude of the mass flow and temperature fluctuations are shown in figures 28 and 29.Figure 28(a), where the mass flow fluctuations are normalized by local mean mass flow, shows avariation similar to that reported in reference 2. Very large amplitude fluctuations on the order of80 percent are observed near the edge of the viscous sublayer. (These fluctuations are sufficientlylarge that the linear assumptions used in deriving the equation for the data reduction may beviolated. As a result, the magnitude of the fluctuations near the edge of the viscous sublayer may bedifferent from those presented in this report. However, it is unlikely that the difference would besufficiently large to alter the conclusions drawn from these data.) This type of plot gives the falseimpression that fluctuation amplitude is increasing as the edge of the sublayer is approached, whenin fact there is a decrease in the amplitude as shown in figure 28(b), where ;he edge value of massflow is used as the normalizing factor. The data show a maximum of around 15 percent aty/8 — 0.8 with consistent decrease from this point to the wall. Thus, the main reason for the large

12

relative amplitude of fluctuations is the small local mean mass flow near the edge of the viscoussublayer. From figure 29 it is apparent that the temperature fluctuations are nearly constant overthe outer 70 percent of the boundary layer at a value between 1 percent and 2 percent, probably asa result of the very low temperature gradient throughout most of the turbulent region.

Flow Model

The present results suggest that a possible flow model for the origin and development ofhypersonic turbulent boundary layers is that presented in figure 30. This model is essentially thesame as the one presented by Maddalon and Henderson (ref. 45), except for the intermittent regionnear the outer edge of the viscous sublayer and the relative extent of each region. The main featuresincorporated into this model are: (1) breakdown to turbulence originates near the outer edge of thelaminar boundary layer at hypersonic speeds (refs. 45-47); (2) the rate of growth of the turbulentbursts normal to the wall is restricted by high Mach number; (3) the viscous sublayer thickens withincreasing Mach number due to the associated decreases in wall density; and (4) the existence of anintermittent region between the viscous sublayer, and the turbulent region, and at the outer edgeof the boundary layer.

SUMMARY OF RESULTS

The nozzle wall boundary layer of the Ames M-50 helium tunnel has been thoroughlyinvestigated with pitot pressure, total temperature, skin-friction, and wall heat-transfer measure-ments at five stations covering 85 percent of the nozzle length, and hot wire measurements at twostations near the exit of the nozzle. The resulting set of data is sufficiently complete and redundantthat it should provide satisfactory tests of most available theories and prediction techniques.

In general, tests results are as follows:

1. The velocity and density profiles, the skin friction and heat transfer variations withRQ, and the hot-wire measurements all indicate that the boundary layer is turbulent with a thickviscous sublayer.

2. The sublayer thickness, skin friction, and heat transfer vary substantially with nozzleposition, and these variations were shown to be consistent with Mach number and Reynolds numbertrends previously observed in turbulent boundary layers at lower Mach numbers.

3. Observed pressure differences across the boundary layer were large relative to the staticpressure but small relative to the dynamic pressure. In connection with these pressure differences,it was determined that the y momentum is still small relative to the x momentum. However, thevariations in density caused by the pressure variations may be sufficiently large to affect thex momentum equation, and therefore they should be considered in theories applicable to high Machnumber flows.

4. The temperature-velocity relationship in the outer region of the boundary layer wasfound to be accurately predicted by the hypersonic approximation obtained from a local energyrelationship rather than either the Crocco or quadratic relationship more commonly associatedwith turbulent boundary layers.

13

5. Mass flow fluctuations were found to be as large as 80 percent of the local mean massflow near the edge of the viscous sublayer with the maximum value relative to the edge mass flowbeing about 15 percent at y/5 « 0.8.

Ames Research CenterNational Aeronautics and Space Administration

Moffett Field, Calif., 94035, March 24, 1972

14

APPENDIX A

DERIVATION OF PARAMETERS USED TO CORRELATE

TOTAL-TEMPERATURE-PROBE CALIBRATION DATA

If it is assumed that radiation losses are the major source of error and that the conductionlosses can be neglected, then the relationship for determining recover factor of the probe is

*' W\T 4 IVo/

where

k_D

h = — NUw

Furthermore, since the wire diameter is very small and the densities very low, it is reasonable toassume that the Nusselt number variation may be approximated by the free molecular value

For helium

T -o.i47RW * Pw

To

and

k ex r0°-6470

thus

h Pw3-5

oc a

or

To* To To4'5

TV'5

15

APPENDIX B

EFFECT OF DENSITY ON SUBLAYER THICKNESS

To demonstrate the effect of density on the thickness of the sublayer one can take y+ fromthe correlations in figures 17 and 19 and the definition for shear stress TW -^ Mw(u£/5/,), and arrive at the following relation for the laminar sublayer thickness

§£ = const

With the additional assumption u^ « ue (an assumption that gives only a first-order approxi-mation but appears to be more accurate at Higher Mach numbers), one obtains

MW>6 j - constP\vue

It becomes apparent that for constant UQ and Tw, 8r is primarily affected by the density andmay become very large if the density is sufficiently low. Furthermore, in constant pressure andtemperature flows the viscous sublayer will be of nearly constant thickness.

16

REFERENCES

1. Lee, Roland E.; Yanta, William J.; and Leonas, A. C.: Velocity Profile Skin Friction Balanceand Heat Transfer Measurements of the Turbulent Layer at Mach 5. Proc. 1968 HeatTransfer and Fluid Mechanics Institute, Univ. of Washington, Seattle, Washington,June 17-18, 1968, Stanford Univ. Press, 1968, pp. 3-17.

2. Wallace, J. E.: Hypersonic Turbulent Boundary Layer Measurments Using an Electron Beam.Rep. AN-2112-Y-l, Cornell Aero. Lab., Buffalo, N.Y., Aug. 1968, NASA CR-96612.

3. Hopkins, Edward J.; Rubesin, Morris W.; Inouye, Mamoru; Keener, Earl; Mateer, George C.;Polek, Thomas E.: Summary and Correlation of Skin Friction and Heat Transfer Data fora Hypersonic Turbulent Boundary Layer on Simple Shapes. NASA SP-216, 1969,.pp. 319-344.

4. Kemp, Joseph H., Jr.; and Owen, F. K.: Nozzle Wall Boundary Layers at Mach Numbers20 to 47. AIAA Paper 71-161, 1971.

5. Bertram, M. H.; and Neal, L., Jr.: Recent Experiments in Hypersonic Turbulent BoundaryLayers. NASA TM X-56,335, 1965.

6. Burke, A. F.: Turbulent Boundary Layers on Highly Cooled Surfaces at High Mach Numbers.Rep. 118, Cornell Aero. Lab., Buffalo, N.Y., Nov. 1961.

7. Kemp, J. H., Jr.;and Sreekanth, A. K.: Preliminary Results From an Experimental Investigationof Nozzle Wall Boundary Layer at Mach Numbers Ranging from 27 to 47. AIAAPaper 69-686, 1969.

8. Fischer, Michel C.; Maddalon, Dal V.; Weinstein, Leonard M.; and Wagner, Richard D., Jr.:Boundary-Layer Surveys on a Nozzle Wall at M^ « 20 Including Hot-Wire Fluctua-tion Measurements. AIAA Paper 70-746, 1970.

9. Kemp, J. H., Jr.: The Ames M-50 Helium Tunnel. NASA TN D-5788, 1970.

10. Thurtell, G. W.; and Tanner, C. B.: Momentum Transport Measurement in the AtmosphericSurface Layer With the Anemoclinometer. Final Rep. 1962-65, Dept. of Soil Science,Univ. of Wisconsin, 1965.

11. Arney, G. D., Jr.; and Bailey, A. B.: An Investigation of the Equilibrium Pressure AlongUnequally Heated Tubes. AEDC-TDR-62-26, Feb. 1962.

12. Arney, G. D., Jr.; and Bailey, A. B.: Investigation of Equilibrium Pressures Along UnequallyHeated Tubes, Addendum. AEDC-TDR-62-188, Oct. 1962.

13. Potter, J. L.; Kinslow, M.; and Boylan, D. E.: An Influence of the Orifice on Measured Pres-sures in Rarefied Flow. AEDC-TDR-64-175, Sept. 1964.

17

14. Potter, J. L.; and Bailey, A. B.: Pressures in the Stagnation Regions of Blunt Bodies in theViscous-Layer to Merged-Layer Regimes of Rarefied Flow. AEDC-TDR-63-168,Sept. 1963.

15. Horstman, C. C.: Surface Pressures and Shock-Wave Shapes on Sharp Plates and Wedges inLow Density Hypersonic Flow. Proc. 6th International .Symposium on Rarefied GasDynamics, Vol. 1, New York, Academic Press, Inc., 1969, pp. 593-605.

16. Winkler, E. M.: Stagnation Temperature Probes for Use at High Supersonic Speeds and Ele-vated Temperatures. NAVORD Rep. 3834, Oct. 1954.

17. Spangler, J. E.: A Sensitive Magnetic Balance for the Direct Measurement of Skin FrictionDrag. LTV Rep. 0-71000/3R-39, Dec. 1963.

18. Coles, D.: Measurements in the Boundary Layer on * Smooth Flat Plate in Supersonic Flow.Ph.D. Thesis, Calif. Inst. of Tech., 1953.

19. Dhawan, Satish: Direct Measurements of Skin Friction. NACA Rep. 1121, 1953.

20. Fenter, Felix W.: Analyses and Direct Measurement of the Skin Friction of Uniformly RoughSurfaces at Supersonic Speeds. IAS Preprint 837, July 1958.

21. McCroskey, William J.: An Experimental Model for the Sharp Leading Edge Problem inRarefied Hypersonic Flow. Ph.D. Thesis, Princeton Univ., Jan. 1966.

22. Kistler, Alan L.: Fluctuation Measurements in a Supersonic Turbulent Boundary Layer. Phys.Fluids, vol. 2, no. 3, May-June 1959, pp. 290-296.

23. Bach, L. H.; Massier, P. F.; and Cuffel, R. F.: Some Observations on Reduction of TurbulentBoundary-Layer Heat Transfer in Nozzles. AIAA, J., vol. 4, no. 12, Dec. 1966,pp. 2226-2229.

24. Bach, L. H.; Cuffel, R. F.; and Massier, P. F.: Laminarization of a Turbulent Boundary Layerin Nozzle Flow. AIAA J., vol. 7, no. 4, April 1969, pp. 730-733.

25. Erickson, Wayne D.: An Extension of Estimated Hypersonic Flow Parameters for Helium asa Real Gas. NASA TN D-l 632, 1963.

26. Akin, S. W.: The Thermodynamic Properties of Helium. Trans. ASME, vol. 72, no. 6, Aug. 1950,pp.751-757.

27. Jones, Robert A.; and Feller, William J.: Preliminary Surveys of the Wall Boundary Layer in aMach 6 Axisymmetric Tunnel. NASA TN D-5620, 1970.

28. Bushnell, Dennis M.; Johnson, Charles B.; Harvey, William D.; and Feller, William V.: Com-parison of Prediction Methods and Studies of Relaxation in Hypersonic TurbulentBoundary Layers. NASA SP-216, 1969, pp. 345-376.

18

29. Softley, Eric J.; and Sullivan, Robert J.: Theory and Experiment for the Structure of SomeHypersonic Boundary Layers. AGARD CP-30, May 1968, pp. 3-1-3-18.

30. Fiore, Anthony W.: Turbulent Boundary Layer Measurements at Hypersonic Mach Numbers.ARL 70-0166, 1970.

31. Coles, D. E.: Measurements in the Boundary Layer on a Smooth Flat Plate in HypersonicFlow. Ill Measurements in a Flat Plate Boundary Layer at the Jet Propulsion Laboratory.JPL Rep. 20-71, 1953.

32. Coles, Donald: The Turbulent Boundary Layer in a Compressible Fluid. Phys. Fluids,vol. 7, no. 9, Sept. 1964, pp. 1403-1423.

33. Van Driest, E. R.: Turbulent Boundary Layers in Compressible Fluid. J. Aero. Sci., vol. 18,no. 3, March 1951, pp. 145-160,216..

34. Mager, Artur: Transformation of the Compressible Turbulent Boundary Layer. J. Aero. Sci.,vol. 25, no. 5, May 1958, pp. 305-311.

35. Baronti, Paslo O.; and Libby, Paul A.: Velocity Profile in Turbulent Compressible BoundaryLayers. AIAA J., vol. 4, no. 2, Feb. 1966, pp. 193-202.

36. Laufer, John: Turbulent Shear Flows of Variable Density. AIAA J., vol. 7, no. 4, April 1969,pp. 706-713.

37. Beckwith, I. E.; Harvey, W. D.; and Clark, F. L.: Comparisons of Turbulent Boundary-LayerMeasurements at Mach Number 19.5 With Theory and an Assessment of Probe Errors.NASA TND-6192, 1971.

38. Rudy, David H.; and Weinstein, Leonard M.: Investigation of Turbulent Recovery Factor inHypersonic Flow. AIAA J., vol. 8, no. 12, Dec. 1970, pp. 2286-2287.

39. Sommer, Simon C.; and Short, Barbara J.: An Experimental Study of Nozzle Wall BoundaryLayers at Mach Numbers 20 to 47. NACA TN 3391, 1955 (also J. Aero. Sci., vol. 23,no. 6, June 1956, pp. 536-42).

40. Spalding, D. B.; and Chi, S. W.: The Drag of a Compressible Turbulent Boundary Layer on aSmooth Flat Plate With and Without Heat Transfer. J. Fluid Mech., vol. 18, pt. 1,Jan. 1964, pp. 117-143.

41. Van Driest, E. R.: Problem of Aerodynamic Heating. Aero Aspects Session, Nat'l SummerMeeting IAS, Los Angeles, 1956 (also Aeron. Engr. Rev., vol. 15, no. 10, Oct. 1956,pp. 26-41). . ' '

42. Rubesin, Morris W.; and Johnson, H. A.: A Critical Review of Skin Friction and Heat TransferSolutions of the Laminar Boundary Layer of a Flat Plate. Trans. ASME 71, no. 4,1949, 383-388.

19

43. Hopkins, E. J.; and Inouye, M.: An Evaluation of Theories for Predicting Turbulent SkinFriction and Heat Transfer on Flat Plates at Supersonic and Hypersonic Mach Numbers.AIAA J., vol. 9, no. 6, June 1971, pp. 993-1003.

44. Corrsin, S.: Some Current Problems in Turbulent Shear Flow. Proc. First Symp. NavalHydrology, Nat. Acad. Sci., 1951.

45. Maddalon, D. V.; and Henderson, A., Jr.: Boundary-Layer Transition on Sharp Cones atHypersonic Mach Numbers. AIAA J., vol. 6, no. 3, March 1968, pp. 424-431.

46. Nagamatsu, H. T.; Graber, B. C.; and Sheer, R. E.: Critical Layer Concept Relative to Hyper-sonic Boundary Layer Stability. GE Rep. 66-C-192, 1966.

47. Owen, F. K.: Fluctuation and Transition Measurements in Compressible Boundary Layers.AIAA Paper 70-745, 1970.

48. Scaggs, N. E.: Boundary Layer Profile Measurements in Hypersonic Nozzles. USAF ARL66-0141, Aerospace Res. Lab., July 1966.

49. Speaker, W. V.; and Ailman, C. M.: Spectra and Space Time Correlations of the FluctuatingPressures at a Wall Beneath a Supersonic Turbulent Boundary Layer Perturbed by Stepsand Shock Waves. NASA CR-486, 1966.

50. Wagner, R. D., Jr.; Maddalon, D. V.; Weinstein, L. M.; and Henderson, A., Jr.: Influence ofMeasured Free-Stream Disturbances on Hypersonic Boundary Layer Transition. AIAAPaper 69-704, June 1969.

51. Matting, Fred W.; Chapman, Dean R.; Nyholm, Jack R.; and Thomas Andrew G.: TurbulentSkin Friction at High Mach Numbers and Reynolds Numbers in Air and Helium. NASATRR-82, 1961.

52. Watson, Ralph D.; and Gary, Aubrey M., Jr.: The Transformation of Hypersonic TurbulentBoundary Layers to Incompressible Form. AIAA J. (Tech. Notes),; vol. 5, no. 6, June 1967,pp. 1202-1203.

53. Shall, Paul Joseph, Jr.: A Boundary Layer Study on Hypersonic Nozzles. MS Thesis, U.S.Air Force Inst. Tech., March 1968. (Available from DDC as AD, 833 236.)

54. Perry, J. H.; and East, R. A.: Experimental Measurements of Cold Wall Turbulent HypersonicBoundary Layers. AASU-275, Univ. Southampton, Feb. 1968.

55. Hill, F. K.: Turbulent Boundary Layer Measurements at Mach Numbers from 8 to 10. Phys.Fluids, vol. 2, no. 6, Nov. - Dec. 1959, pp. 668-680.

56. Lobbe, R. Kenneth; Winkler, Eva M.; and Persh, Jerome: Experimental Investigation of Turbu-lent Boundary Layers in Hypersonic Flow. J. Aero. Sci., vol. 22, no. 1, Jan. 1955,pp. 1-10.

20

57. Winkler, E. M.; and Cha, M. H.: Investigation of Flat Plate Hypersonic Turbulent BoundaryLayers with Heat Transfer at a Mach Number of 5.2. NAVORD Rep. 6631, Sept. 1959.

58. Samuels, R. D.; Peterson, J. B., Jr.; and Adcock, J. B.: Experimental Investigation of theTurbulent Boundary Layer at a Mach Number of 6 with Heat Transfer, at High ReynoldsNumbers. NASA TN D-3858, 1967.

59. Owen, F. K.; and Horstman, C. C.: A Study of Turbulence Generation in a HypersonicBoundary Layer. AIAA Paper 72-182, 1972.

60. Stalmach, Charles J., Jr.: Experimental Investigation of the Surface Impact Probe Method ofMeasuring Local Skin Friction at Supersonic Speeds. DRL-410 CF 2675, Univ. ofTexas Research Lab., Austin, Texas, Jan. 1958.

21

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40

TABLE 6. - COMPUTED PARAMETERS FOR x = 0.508

p0g = 66.l pw = 5.26X 10-4 pe = 3.74x 10"4 To = 499 Tw = 362 Te = 3940

ue = 2668 Me = 19.4 pp = 4.62 X 10'3 — = 8 . 5 2 x l 0 6 6=2 .87m

— = 0.520 —=0.00568 — = 0.0109 Ra = 13816 6 6 »

y5

1.00000.92440.88890.80000.71110.62220.53330.44440.35560.31110.26670.17780.08890.0000

po2

P°ie1.00001.03851.03210.95510.81410.6603.0.49360.33650.18270.09130^03210.00760.00450.0026

TOT0g

1.00000.99330.98660.97770.96990.96770.96990.97921.00651.01030.99120.88660.79600.7249

u"e

KOOOO0.99670.99320.98830.98340.98080.97940.97930.98040.95640.86520.57690.41100.0027

PPe1.00001.04541.04610.97780.84160.68590.51400.35020.18920.09880.04140.01900.01730.0154

TO-TWT T' Oe~ ' w

1 .00000.97570.95140.91900.89070.88260.89070.92451.02371.03730.96790.58790.25840.0000

TTe

1.01.01.01.11.31.72.33.56.7

13.031.570.479.591.7

MMe

0.99891.00250.99240.93850.85220.75510.64270.52250.37890.26520.15430.06880.04610.0003

p0 =66.2 pw = 5.26X 10-4 pg = 3.93X lO"4 ro = 820 Tw = 397 7> = 6.56

Rewe = 2908 M e = l 9 . 2 p = 2 . 9 2 X 1C'3 — = 4 . 8 9 X 1 0 * 6 = 2 . 8 7c e m

1.00000.90270.79650.70800.61950.53100.44250.35400.26550.17700.0885.0.0000

1.00001.06850.97510.82240.64800.44550.19630.04670.01400.00700.00440.0025

-11=0.544 -f-=0.00650 -£ = 0.0122 R f i = 9 \ 3o o o u

1 .00000.98980.97980.97020.95960.95680.95950.93460.85490.70670.58270.4836

1.00000.99500.98940.98360.97660.97180.95990.87970.69440.50510.35100.0075

1.00011.07940.99600.84980.67900.47110.21210.05910.02680.02280.02300.0224

.00000.98020.96080.94240.92180.91630.92150.87340.71900.43200.1918

+0.0000

1.01.01.11.31.72.55.7

20.947.156.757.660.5

1 .00001.01710.95510.86500.75750.61970.40550.19380.10180.06750.04650.0010

p 0 g =lQ7.6 pw = 8. 6 1 X 1 0 - " pe = 5 . 4 1 X l O - 4 ro = 3!5 Tw = 333 Te = 2.36

u e=1803 y W e = 1 9 . 9 P e = l . l l X l O - 2 -^-=2.27X10 7 6 = 2 . 7 2

A! = 0.5 19 4=0.00430 4=0.00838 Ra = 26620 0 0 "

1.00000^90650.8411 .0.74770.65420.56070.46730.37380.28040.18690.09350.0000

1.00001.0411L01370.87330.69180.50340.33220.21230.12670.04040.00610.0027

1.00000.99290.98770.98060.97880.97880.98590.99821.02821.09361.09041.0582

r.oooo0.99640.99350.98910.98660.98380.98200.97910.97670.92560.55870.0119

1.00001.04861.02690.8925-0.1103 .0.51960.34370.22060.12180.0459 -0.01480.0113

1.00001.1212

.2121

.3333

.3636

.3636

.2424

.03030.5152

-0.6074-0.5536-0.0000

1.01.01.11.31.72.43.86.3

10.932.5

104.3141.4

1.00000.99290.96200.87080.75670.6309-0.50120.39210.29630.16250.05480.0010

41

TABLE 6. - COMPUTED PARAMETERS FOR X = 0.508 - Continued

p0g= 107.6 pw = 8 .6 lX 10-4 pg = 5.09X 10-4 T0 = 393 Tw = 333' Te = 2.88«e

we = 2014 M e = \ 9 . 9 p = 8 . 6 1 X 1 0 - 3 _ = ] . 7 3 X 1 0 7 6 = 2 . 7 4c m

4r= 0-505 -|-= 0.00550 -£-=0-0 107 Ra = 26086 6 6 o

ys

1.00000.90740.83330.74070.64810.55560.46300.37040.27780.18520.09260.0000

POIP°*e1.00001.05671.03190.89360.71280.52840.34750.21630.11420.03330.00620.0028

ToToe

1.0000.0.99440.98870.97880.97180.96890.97030.97881.00421.02830.93800.8475

u'ue~

- 1.00000.99710.99400.98820.98310.97900.97450.96940.95980.87290.51440.0107

PPe

1.00001.06281 .04430.91490.73710.55060.36510.22920.12280.04220.01770.0147

T T1 O ~ 1 W

Toe - Tw

1.00000.96300.92590.86110.81480.79630.80560.86111.02781.18570.59370.0000

TTg1.01.01.11.31.72.4 •3.86.3

.12.237.192.2

115.7

MMe

1.00000.99650.96180.87030.75680.63530.50280.38730.27460.14340.05360.0010

PQ =108.1 pw = 8 .6 lX 1Q-4 pe = 5.62X 10"4 7"o = 481 7\v = 339 Te = 3.66

Ke = 2227 /We =19.8 p e = 7 . 4 9 X ! Q - 3 -^£.= 1.42X10' 5 = 2 . 7 2

AI=0.511 -f-= 0.005 68 4- =0.0 109 R a = 2 \ 9 66 6 6 o

1.00000.91590.84110.74770.65420.56070.46730.37380.28040.18690.09350.0000

1 .00001.03021.00340.87920.6980 ,0.53360.37920.24500.12180.01310.00430.0027

1 .00000.99080.98500.97800.97110.96880.97230.97921.00130.98580.83120.7052

1.00000.99530.99210.98790.98280.97940.97710.97310.96300.71820.39490.0076

1.00001.03991.01920.9007

. 0.72220.55570.39640.25780.13030.023 10.01670.0166

1.00000.96860.9490

- 0.92550.90200.89410.90590.92941.00430.95170.42740.0000 .

1.01.01.11.31.62.23.25.2

10.662.288.892.6

1.00000.99300.96180.88020.76750.65720.54300.42790.29550.09110.04190.0008

/70g=109.2 pw = S.6\ X 10-" pe = 5.51 X 10-4 To = 988 Tw = 400 Te = 1.39

«e = 3193 JW e =19.8 p. = 3.63 X 10'3 — = 6 . 1 5 X 1 0 6 6=2 .69e m

-y-=0.518 -|-= 0.00856 -j=0.0161 Re = 1420

1.00000.94340.89620.84910.75470.66040.56600.47170.4245

. 0.37740.28300.18870.09430.0000

1 .00001.03751.05461.04100.91130.72700.5358

. 0.36520.29010.18770.0379.0.00930.00450.0027

1 .00000.99280.98740.98220.97250.9621

. .0.9516 .0.94360.9352 '0.9156

. 0.7906' 0.6429

0.51140.4048

• i.oooo0.99630.99360.99090.98520.9784'0.9706

" 0.96200.95410.93490.78520.52060.31660.0052

1.00011.0452

'1.06821.06020.93870.75900.56820.39380.3178

.0.21370.05970.02960.02790.0294

1 .00000.98790.97890.97020.95380.93630.91870.90520.89110.85820.64810.40010.17920.0000

1.01.01.01.0 •1.2

. 1.62.23.3

. . 4.26.4

23.950.055.0

'54.1

1 .00001.00270.99810.97940.89460.78080.65570.52980.46730.37180.16190.07420.04300.0007 ,

42


P0 g=199 pw= 13.4X 10-4 pe = 9.63X 10" ro = 516 Tw = 352 re=3.80

-«e = 2308 Me = 20.1 p = 1 . 2 4 X 1 0 - 2 — = 2.38 X 107 8=2 .44"' m _.11=0.494 -^=0.00577 -£=0.0112 Ra = 33430 0 0 "

y8

1. 00000.9375•0.83330.72920.62500.52080.41670.31250.20830.10420.0000

POJ

~*°*e1.00001.03130.98960.84370.66320.48610.33330.21880.07360.00750.0023

ToToe

1.00000.99570.98820.9806 .0.97200.96660.96560.96880.98640.89790.6814

uue

1.00000.99790.99380.98910.98330.97800.97300.96720.93610.59090.0048

P"e

1 .00001.03571 .00200.86220.68560.50770.35140.23300.08290.01810.0151

T Tto- * wT0e - Tw

1 .00000.98650.96280.93920.91220.89530.89190.90200.95720.67950.0000

TTe

1.01.01.11.31.72.33.55.5

15.874.992.6

MMe

1 .00001.00330.96370.87320.76010.63930.52020.41430.23550.06830.0005

P0g=199.2 pw= 13.4X 10" pe = 9.73X 10" 7Oe = 942 Tw = 402 Te = 6.94

ue = 3118 A/e = 20.0 p =6.84X10' 3 — -= . 1 9 X 1 0 7 8 = 2 . 5 7*• nt

-11=0.490 -£.= 0.00882 -£=0.0170 Ra = 26886 0 0 "

1.00000.89110.79210.69310.59410.49500.39600.29700.24750.19800.14850.09900.04950.0000

1.00001.04700.97740.81740.62960.45910.30430.14260.06520.02780.009 10.00560.00370.0023

1.00000.98310.97010.96090.95330.94780.94380.93980.93370.90780.82070.69080.56310.4270

1.00000.99150.98450.97890.97330.96780.96070.94200.90450.81880.60840.45910.31090.0086

1.00011.06491.00820.85280.66410.48960.32900.15980.07860.04000.02170.02070.02170.0240

1 .00000.97060.94790.93180.91860.90880.90200.89490.88430.83910.68720.46050.23750.0000

1.01.01.11.31.82.53.88.0

16.532.961.565.463.458.0

0.99881.00160.95070.85460.73760.61980.49660.33420.22340.14330.07790.05700.03920.001 1

p0g = 270 pw= 18.1 X 10" p e = 1 2 . 8 X ! 0 " r0e = 503 7,,, = 318 Te = 3.65

RP«e = 2278 Me = 20.3 p = 1 . 7 1 X 1 0 - ' — ~ = 3 . 3 4 X 1 0 7 8=2.36e m

.11=0.492 -|-= 0.00689 -£=0.0134 Ra = 5430O 0 O v

1.00000.96770.91400.86020.75270.64520.53760.43010.32260.21510.16130.10750.05380.0000

1.00001.02101.03851.02100.87760.70980.51050.33920.23080.09650.02450.00660.00340.0024

1.00000.99560.99230.98670.97680.96910.96240.95910.95690.95470.95730.93520.84030.6298

1 .00000.99780.99610.99320.98740.98220.97630.97000.96230.93430.82380.57360.35030.0080

1 .00001.02551.04661.03500.90000.73540.53510.35970.24840.10950.03460.01640.01410.0163

1.00000.98800.97900.96400.93690.91590.89790.88890.88290.87690.88400.82400.5661

-0.0060

1.01.01.01.01.21.62.23.45.2

12.239.184.099.186.9

1 .00001.00371.00120.98210.89180.78600.65380.52290.42340.26830.13180.06270.03520.0009

43

TABLE 6. - COMPUTED PARAMETERS FOR X = 0.508 - Concluded

P0 =268 pw= 18.2 X 1Q-" pe =12.6X10-" To = 882 Tw = 361 Te = 6.40

ue = 3018 Me = 20.2 pg = 9 . 5 8 x l O - 3 — -=1 .71X10 1 6 = 2 . 4 1

•^=0.500 4-= 0.00809 -£-=0.0157 #,,=33375 6 6 g

y6

1 .00000.94740.90530.84210.73680.63160.52630.42110.31580.26320.21050.15790.10530.0000

P02

P°'e1 .00001.02131 .02841.00000.87230.66310.47160.32620.18790.13050.06210.02590.00580.0024

ToToe

1.00000.99460.98860.98290.97170.96020.95080.94260.93630.93350.92310.88680.80630.4060

u"e

1.00000.99720.99420.99110.98470.97720.96960.96090.94720.93520.89440.79710.49860.0094

P

1.00001.02691 .04041.01790.89950.69410.50110.35260.20860.14820.07640.03910.01820.0259

T0-TW

T0e - Tw

1 .00000.99080.98060.97110.95210.93260.91680.90280.89210.88740.86990.80840.6723

-0.0053

TTe

1.01.01.01.11.21.72.43.66.39.0

17.835.377.156.0

MMe

0.99880.99780.99230.96560.88280.75390.62330.50840.37850.31220.21250.13440.05680.0013

44

TABLE 7. - COMPUTED PARAMETERS FOR X = 1.067

p0 = 6 5 3 p w = 1 . 4 5 X 1 0 - " pe = 0.682 X 10-" To = 500 Tw = 315 re = 2.01

Reue = 2276 A/e = 273 pe = 1 .65 X 1 0° — - = 4 . 5 9 X 1 0 6 6=7 .98

-^ = 0587 4-= 0.00384 4-= 0.007 19 ~Ra = 1440O O O a

Cf= 3.62 X 10-" C H = 1 . 2 1 X 1 0 - 4 -^=0683 -^- = 5.961 " cf \/R9

ys

1 00000.93950.87580.79620.716606369055730.47770.39810.31850.23890.15920.07960.0000

po2

P°*e1 .00001.03190.99470.88480.74110.56560.39890.23580.07540.01330.00620.0039000270.0020

To7^7

1 .00000.99730.98840.97920.97050.96360.96500.97270.99231.01000.87850.7508065950.6306

uue

1 .00000.998609939098880.98360.97870.9769097500.95570.80380.60610.44980.29540.0000

ppe1.00001.03481.00690.90480.76580.59010.41750.24750.08180.01940.01460.01430.01430.0135

T0 - Tw

T0e - Tw

1.00010.99280.96870.9437092020.90150.90530.92620.97901.02710.67100.32520.07830.0000

TTe

0.0.1.4.7

2.43.66.4

20.691.2

127.5136.6142.4156.9

MMe

1.00000.98300.93440.84860.74980.63380.51600.3852021120.08440.05380.03860.02480.0000

nn =65 p, =1.45X10-" p,= 0.784 X 10"4 To = 933 7"lt, = 358 7> = 3.9I( 0£ r VV * C " v c

! Ue = 3109 Me = 26 4 pe = 9 .79X10-" - = 2 . 4 I X 1 0 6 5 = 7 . 6 7

-^-=0629 4-= 0.00496 -£- = 000909 K f l = 9 l 66 5 o f

C>=3.80X10-" C,,= 1 4 1 X I O - " —^=0.742 -^- = 0.875-> " -Cr ^Ka

J O

1 .00000.94040.82780.74500.66230.57950.49670.41390.33110.24830.16560.08280.0000

1.00001 .02290.91000.74470.55320.34860.11870.02390.01150.00690.00430.00280.0018

1. 00000.99470.97300.95980.95030.94770.96490.97900.77130.62080.51300.43690.3826

1.00000.99720.98580.97830.97200.96750.95980.87760.69010.53810.4025027530.0000

1.00011 .02860.93630.77790.58510.37200.12820.03020.02260.02110.02080.02100.0202

1.00000.99140.95620.93470.91940.91520.94300.96600.62880.38470.20960.0861

-0.0019

1.01.01.31.62.23.7

11.450.771.079.584.186.491.5

1.00000.98690.89130.78270.65590.50690.28790.12470.08290.06110.04440.03000.0000

p0 =107.7 pH, = 2.08X 10-" pe= 1 . 1 2 X 10"" To = 305 Tw = 299 7" e=l .22

«e=1777 Me = 21. 3 pe = 4 . 5 0 X l O - 3 -^1=139X10 7 6 = 7 1 1

-^-=0.557 4=000383 -^=0.00745 Rg = 37860 O 0 u

2CH MCy=2.33x 10-" % = 0.866 X JO"1 -^- = 0.746 -^- = 0443

1 .00000.93570.8S7I0.80360.71430.62500.53570.44640.35710.26790.22320.17860.13390.08930.0000

1.00001.03060.97990.90200.74540.56170.38850.24760.14790.06040.01890.00660.00420.00300.0017

1.00000.99450.98720.98000.97090.96720.96720.9727099271.03391.05481.06161.04081.01670.9891

1.00000.99720.99330.98930.98400.98080.97840.97660.97790.96740.88040.70280.58050.46770.0061

1.00001.03640.99310.92140.76960.58350.4054025910.15400.06380.02350.01190.00990.00890.0075

1 .00000.727303636

-0.0000-0.4545-0.6364-0.6364-0.3636

0.63642.69293.73694.07593.03581.83580.4545

1.01.01.11.31.62.33.45.7

10.125.570.8

142.6176.5200.0247.6

0.99850.98710.93340.87790.77330.65160.52680.40940.30820.19170.10460.05890.04370.03310.0004

45


p0e= 107.7 pw = 2.08 X 10-1 p g = 1 . 0 9 X 1 0 J 1 T0 = 418 7\v = 301 r e=1.66

ue = 2080 /W<. = 27.5 p, = 3.23 X 10'3 — = 9.57 X |Q« 5 = 7 . 1e m

4f= 0.567 4 = 0.00408 -£ = 0.00780 /?„ = 27675 5 6 8

2CH A<C Y = 2 . 5 3 X I O - " CH = 0.967 X W* _fl = 0.765 J2_ = 0.523' i cf >/*»

y6

1.00000.93210.85710.80360.71430.62500.53570.44640.35710.26790.22320.17860.08930.0000

po,Po,e1.00001.02760.98040.90290.73110.55030.38020.24130.12470.01960.00690.00480.00280.0017

ToT^e

1 .00000.99340.98540.97740.97210.96940.96680.96730.98800.99650.96750.8959076200.7194

uue

1.00000.99660.9924.0:98800.98450.98180.97790.97350.9716

,0.86100.67890.57380.37830.0052

P"e1.00001 .03460.99550.92480.75400.57060.39710.25410.13140.02550.01330.01220.01170.0105

TO - TWT T'oe- lw

1.00000.97630.94790.91940.9005 '0.8910088150.88330.95730.98740.88400.62910.1518

-0.0002

TTe

1.01.01.11.31.72.43.65.9

12.065.2

128.4143.5156.5181.7

MMe

0.99850.98270.93070.87470.76140.64040.51690.40040.28000.10670.05990.04790.03020.0004

p =107.7 p,,, = 2.08X 10-1 p = 1 . 0 2 X 1 0 - * To = 577 7\v = 577 Te = 2.22I-Qe r-\v e

Rfue = 2444 Me = 21.9 p. = 2.23 X IO'3 — - = 6.42 6 = 7 . 1 6e m

-4^ = 0.578 4= 0.00392 -£= 0.00737 K f l = I 8 0 16 5 5 "2CH M

C /-=3.02X 10-4 CH= 1.161 X 10^ -^=0792 -^—=0.655/ v^0

1.00000.91130.85110.79790.70920.62060.53190.44330.35460.26600.17730.08870.0000

1.00001.03820.99530.91230.72110.5233034510.19590.06620.01190.00500.00290.0018

1.00000.99560.98800.98180.97380.97020.96780.96690.97200.92010.72950.60210.5342

1 .00000.99770.99370.99020.98530.98180.97750.97010.94230.75730.51720.33530.0045

1 .00001.04301.00800.93030.74260.54250.36070.20760.07390.01960.01550.01540.0148

1.00010.99060.97410.96070.94350.93570.93060.92870.93960.82760.4168

'0.1422-0.0042

1.01.11.21.31.82.64.27.7

22.890.8

120.6127.7139.1

0.99850.97330.92660.86650.74240.61.100.48030.35090.19760.07960.04720.02970.0004

p0e=109.8 pw = 2 0 4 X 10-4 pf = 1 . 1 2 X 1 0 - * To = 920 Tw = 328 7V = 361

»e = 3087 Me = 21A p ^ = 1 . 5 1 X l Q - 3 — = 3 . 9 2 X 1 0 ' 6 = 7 . 2 4e ' m4^ = 0.582 4=0.00594 4-= 001 1 1 Ra = 16855 6 6 *

Cy=3.22X 10-4 C W = 1 . 2 2 X I O - 4 -^- = 0.780 _f!L=0.671/ V«9

1.00000.92980.89470.84210.78950.73680.68420.63160.57890.5263047370.42110.36840.31580.26320.21050.15790.07020.0000

1.00000.03531.01930.96360.88240.79040.6898

. 0.57540.45990.33690.21390.10700.04010.01900.00800.00500.00330.00220.0017

1.00000.99400.98850.97870.96860.95860.95070.9439093950.94020.94180.93810.89560.82240.68090.56690.48930.39430.3560

1 .00000.99690.99400.98890.98350.97800.97340.96910.96550.96360.95950.94450.88260.78480.60330.47260.36040.22310.0000

1.00011.04171.03150.98530.91210.82620.72780.6124

.0.492903624

- 0.23180.1193005070.02990.0201001910.0188002040.0201

1 .00000.99080.98210.96700.95120.93570.92350.91290.90610.90720.90960.90390.83790.72410.50450.32740.20700.05940.0000

.0

.0

.1

.2

.3

.5

.82.22.83.96.3

12.630.653.381.388.092.088.190.9

1.00000.98930.96850.92350.8672080600.73990.66430.58420.49210.38590.26850.16120.10850.06760.05090.0379002400.0000

46


P0i?=189 pw = 3 . \ 2 X ID- p^ 1 3 9 X 1 0 - To = 534 7W = 319 7>=1.85

«e = 23S2 Me = 29.4 pe = 3.68X 10° — - = 1 1 5 x 1 0 ' 6 = 6.96

11=0.531 -1=000465 -£=0.00894 «,, = 37100 O 0 °

2Cu v,0=2.73X10-" Cr,= I . I 1 3 X 10- -^=0.842 -^- = 0.479

/ ^Ke

yT

1.00000.90510.82120.72990.63870.54740.45620.36500.27370.18250.13690.07300.0000

PO:

P^e

1.00001.05220.98880.82460.64180.46640.31720.16790.05600.00460.00300.00260.0018

ToTo7

1.00000.99480.98650.97610.96570.95700.95400.96050.97470.96590.88260.75260.5963

u•

1.00000.99730.99280.98690.98050.97430.9696096400.93470.58390.43540.34590.0000

p"e

1.0000, 1.0580

1.00310.84650.66720.49100.33690.18010.06330.01120.01040.01180.0130

TQ-T WT0e - Tw

1.00000.98700.96630.94040.91450.89290.88560.90160.93700.91510.70770.3842

-0.0052

T~T7~

1.01.11.21.62.23.25.09.9

30.1' 180.9

200.3182.9172.2

MMe

1 .00000.97040.89980.78620.66590.54670.43520.30620.17070.04350.03080.02560.0000

P0 e=189 pw = 3.!2x 10-4 pe= 1.48X 10- 7\> = 877 7\v = 348 TV = 3.09

1^ = 3015 /We = 29.0 p- = 2.33 X 10° — - = 6 5 6 X l O 6 6 = 6.99e tnx* a r•V= 0.562 4-= 0.00520 -f= 0.00989 /?„ = 2380

0 0 O °

C y = 3 . 0 1 X 1 0 - 4 CH= 1.02X 10- -j^ = 0.690 ^.= 0594

1.00000.96000.91640.81820.72730.63640.54550.45450.36360.31820.27270.24550.18180.09090.0000

1.00001.02171.03260.94570.77900.59420.42390.26090.10140.04710.01850.01120.00510.00240.0017

1 .00000.99460.98990.97860.96720.95830.95330.95340.95870.95230.92230.85160.73930.49280.3925

1 00000.99720.99480.98870.98230.97650.97190.96750.95240.91740.82380.72510.53730.26410.0000

1.00001.02741.04330.96720.80720.62280.44830.27820.11120.05520.02630.02000.0151

. 0.01700.0187

1.00000.99110.98330.96460.94570.93100.92270.92280.93160.92100.87140.75410.56820.1598

-0.0063

.0

.0

.1

.3

.62.33.45.9

15.632.369.993.1

128.3120.2111.5

1.00000.98910.97210.88710.77330.65060.53070.40280.24320.16270.09930.07580.04780.02430.0000

47

TABLE 7. - COMPUTED PARAMETERS FOR X = 1.067 Concluded

p0(? = 270 pw = 3.86 X 10- /> e= 1.86X10- To = 502 • 7"M, = 318 Te = \ 68

we = 2281 Me = 29.9 pe = 5.40 X 10'3 -^- = 1 .74 X 1 0'1 6 = 6 4 7

-^-=0528 -|-= 0.00423 -^=0.00820 R0 = 4756O O 0 p

Cy=2.03X 10- CH = 0.957 X 10- -1 -= 0.938 -^- = 0.439

y6

1.00000.96080.90980.88240.78430.68630.58820.49020.39220.29410.26470.19610.17650.14710.09800.0000

P0;

P0ie

1.00001.04321.05951.04860.91890.72700.53510.36760.22430.12160.08380.91570.00810.00420.00260.0016

1°.Toe

1. 00000.99780.99450.99230.98450.97680.96900.96130.96010.96710.97370.92620.98840.9351

. 0.79190.6338

u~ue

1.00000.99890.99720.99600.99170.98690.98160.97520.96950.96180.95480.80980.73140.57210.40130.0000

ppe

1.00001.04561.06551.05710.93430.74620.55510.38610.23810.13090.09130.02290.01390.01060.01050.0109

TO - Tw

T0e - Tw

1 .00000.99400.98490.97890.95770.93660.91540.89430.89100.91000.92810.79850.96820.82260.43160.0000

JL

.0

.0

.0

.1

.3

.82.64.07.0

13.519.681.4

135.9181 7188.4189.2

MMe

0.99840.99910.98140.96360.86270.73660.60840.48670.36770.26220.21550.08980.06270.04240.02920.0000

p0(j = 265 pw = 38Sx 10- pg= 189X10- T0 = 876 Tw=339 Te = 2.96n

</<, = 3013 Me = 29.6 Pf = 3.09 X 1Q-3 — = 8.96X106 6=6.60c tn

4-= 0557 -f- = 0.00484 -£= 0.00924 R0 = 28706 6 6 "2CW M

Cc=2.58X 10- C H = I . 0 2 X 1 0 - -^=0.813 -^-=0552/ Vg

1.00000.96150.92310.86540.76920.67310.57690.48080.38460.33650.28850.24040.19230.09620.0000

1.00001.03281.04371.01640.83330.67210.48360.31420.16560.09560.04150.01580.00710.00270.0016

1.0000• 0.9972

0.99450.9900

• 0.97940.96810.95750.94900.94980.94950.94600.91240.79000.51300.3855

1 .00000.99860.99720.99480.98880.98220.97520.96760.95990.94740.90980.80680.63090.32780.0000

1 .00001.03581 .04971.02710.85220.69650.50820.33510.17910.10590.04950.02340.01610.01650.0173

1.00000.99540.99100.98380.96630.94790.93060.91680.91810.91770.91200.85720.65750.2057

-0.0021

.0

.0

.0

.1

.52.02.94.79.3

16.335.978.2

116.6120.4114.3

1.00000.99620.98250.94330.81850.70670.57790.45010.31630.23640.15290.09190.05880.03010.0000

48


P0 =66.9 pw = 0.698 X 1Q-* pe = 0.372 X 10"4 ro = 515 rw = 301 T e = \ . 6 Q

Reue = 23\0 Me = 30.9 p = 1 . 1 3 X 1 Q - 3 — £-= 3.78 X 106 8 = 13.34e • m

4^-= 0.563 4"= 0.00434 -£ = 0.00820 Rfi =21900 0 0 p

y6

1.00000.97140.92380.90480.85710.80950.76190.71430.66670.61900.57140.52380.47620.42860.38100.33330.28570.23810.19050.14290.09520.04760.0000

/° — "?Cf 2

Pp2

P°*e1.00001.01921.03211.02440.99490.94620.87820.79870.70770.60770.50000.38460.26540.12780.03170.01330.00810.00560.00410.00300.00220.00170.0013

.66X 10-4 CH = 0.758 X 1

TOT0e1.00000.99310.98740.98520.97760.97010.96510.96020.95660.95650.95780.95930.96250.96820.96200.90260.80670.73850.68820.65200.62160.60030.5827

it"e

1.00000.99650.99360.99250.98850.98460.98180.97900.97680.97620.97600.97530.97420.96780.91390.80760.69450.60320.52020.43750.34960.23230.0083

H AA0-4 " - r) cci m - r

f ^Rsp

f>e1.00001.02631.04541.03991.01810.97600.91100.83320.74150.63750.52460.40400.27920.13600.03730.01970.01560.01390.01280.01190.01120.01050.0101

TO~T WTpe ~ TW

1.00010.98340.96960.96430.94600.92800.91590.90410.89550.89530.89830.90190.90960.92350.90850.76540.53440.37020.24890.16170.08860.0372

-0.0052

.660

TTe1.0.0.0.1.1.2.3.5.8

2.12.63.55.3

11.141.681.1

104.7120.8134.5148.2160.6175.6187.3

MMe

1.00000.99720.98370.97250.94050.90070.85270.79950.74030.67520.60310.52100.42650.29150.14230.09000.06810.05510.04500.03610.02770.01760.0006

p0 =65.3 pw = 0.698 X IQ-4 pf = 0.379 X 10"4 To = 890 7W = 321 Te = 2.77

ue = 3036 Me = 30.6 pe = 6.66 X 1Q-4 -^ = 1.99 X 106 6 = 13.46

-^ = 0.593 -r= 0.00484 4-= 0.00885 R,. = 1296o d o o2C(/ JM

Cy=3.00X 10-4 C#=1 .00X 10

1 .00000.94340.90570.84910.80190.75470.70750.66040.61320.56600.51890.47170.42450.37740.33020.28300.23580.18870.14150.09430.04720.0000

1.00001.03761.04391.01000.94360.85590.75310.63910.5125038100.23870.09960.03260.01720.01070.00800.00580.00430.00320.00230.00170.0013

1.00000.99350.98810.97910.97040.96140.95280.94600.94220.94210.94210.94610.89110.74570.65480.57010.51040.46030.42210.39180.369903'609

1.00000.99670.99390.98930.98470.97980.97510.97100.96820.96660.96310.95220.88230.76080.66320.58120.50570.43390.36320.28460.18730.0078

-4 " n ££a "' n

/ v'/fy

1.00001.04451.05661.03200.97310.89130.79200.67760.54650.40740.25700.10940.04120.02880.02310.02190.02050.01970.01890.01810.01720.0163

1.00010.98990.98140.96730.95370.93970.92620.91560.90960.90940.90940.91560.82960.60210.45980.32740.23400.15550.09580.04830.01400.0000

839

1.01.01.01.11.2.4

1.61.92.53.45.6

13.636.954.269.575.082.187.693399.9

107.6115.9

1.00000.99520.98350.94670.89940.84230.77750.70500.62180.52810.41200.26210.14710.10470.08060.06800.05660.04700.03810.02880.01830.0007

49


P0g = 107.3 pw = 1 .05 X 10" pg = 0.432 X 10" To = 495 Tw = 306 Te = } .36

ue = 2876 M e =33. pp = 1.55 X 10'3 — = 5.70X 106 8 = 12.57e m~= 0.582 -j-= 0.00390 4~= 0.00739 Ra = 28005 6 8 8

Cy=2.38X 10" C

•y«

1.00000.97980.93940.90910.85860.80810.75760.70710.65660.60610.55560.50510.45450.40400.35350.30300.25250.20200.15150.10100.05050.0000

P0-i

P°*e1.00001.0311

• 1.03881.02950.99070.92550.84780.73600.61960.50160.38660.27640.17330.08450.03200.01260.00660.00440.00320.00230.00180.0015

ToTo,1.00000.99500.99020.98680.97890.97260.96880.96550.96240.96060.96120.96330.96700.97370.98220.97140.90640.79780.71760.66300.62900.6155

— n 7Q7 v in-4 ** — o ^cn "* — ff ^Re

u"e

1.00000.99750.99500.99320.98910.98570.98350.98140.97910.97720.97610.97450.97100.95930.91800.81710.68190.55500.45030.34080.20500.0099

P"e

. 1.00001.03631.04921.04361.01250.95240.87640.76410.64600.52490.40550.29060.18330.09130.03730.01800.01290.01200.01180.01150.01110.0109

TO - Tw

T0e - Tw

1.00010.98690.97440.96550.9451

. 0.9286

. 0.91870.91010.90190.89720.89890.90420.91390.93130.95370.92530.75590.47260.26340.12090.0322

-0.0029

).625

rTe

1.0.0.0.1.2.3.5.9

2.33.04.15.99.8

20.451.9

111.8162.0179.5188.6200.2214.8225.2

MMe

1 .00001.00100.97750.95420.90730.8516

. 0.79290.71960.64390.56570.48550.40160.31130.21270.12770.07750.05370.04150.03290.02410.0140

i 0.0007

pn =107.6 p =1 .04X10" p =0.471 X 10" To = 886 7\v = 317 Te = 2.47Og W C

Rf!/e = 3032 Me = 32.5 p. = 9.29 X 10" — - = 3.00X 106 5 = 12.70e m

— = 0.606 -|-= 0.00440 -£ = 0.00808 Ra = 16808 & & "

C/-=2.68X 10" C

1 .00000.93000.90000.85000.80000.75000.70000.65000.60000.55000.50000.44500.40000.35000.30000.25000.20000.15000.10000.05000.0000

1 .00001.02521.01480.96000.88150.78810.68150.56300.43700.31110.19700.10220.03290.01630.00960.00680.00490.00340.00250.00180.0014

1.00000.99250.98920.98240.97590.96760.96040.95170.'94530.94330.94520.94780.92490.82410.70740.60360.52150.45930.41030.37510.3570

^f

u 093 IX 10" " 0695 ^ (" -- •- £y «• '-> ^JR^

1.00000.99610.99440.99080.98730.98270.97860.97340.96890.96580.96240.95290.89730.79030.67000.56870.47450.38460.29720.19890.0058

1 .00001.03321.02620.97780.90430.81590.71150.59390.46520.33320.21230.11210.04020.02530.02020.01930.01890.01850.01840.01820.0176

1 .00000.98830.98330.9727 '0.96250.94970.93840.92490.91490.91180.91480.91890.88330.72650.54500.38360.25590.15910.08300.02830.0000

).792

1.01.11.1 .1.21.41.62.02.53.34.77.7

15.243.872.393.4

101.0106.7112.0115.8120.6128.2

1 .00000.97220.95150.90150.84250.77800.70720.62890.54260.44870.35010.24680.13710.09400.07010.05720.04650.03680.02790.01830.0005

50


p0g=198 p w = 1 . 4 5 X l O " pe = 0.655 X 10" ro = 515 7\v = 304 7e=1.29

ue = 2312 ;We = 34.5 p. = 2.47 X 10'3 . — - = 9 .52X10 6 6 = 11.42e m

4r= 0.561 —=0.00376 4-=0-00722 Ra = 40960 0 0 V

2CH MCo< v i n-4 /^ n £70 v i ri~^ — n^\7° ~ o ^ ^ or— .VO X, 1U Cry — U.O / J X 1U — U.O /Z - . \J.JJ7

y0

1.00000.92220.88890.83330.77780.72220.66670.61110.55560.50000.44440.38890.33330.27780.22220.16670.11110.05560.0000

P02

P°te1 .00001.05031.04160.98440.86480.79380.67940.55460.42980.31540.21660.13340.05720.01460.00590.00350.00230.00170.0013

Toroe

1 .00000.99350.99030.98490.97840.97090.96340.96200.96100.96110.96460.96830.97560.98090.95820.82640.70950.63340.5894

uue

1 .00000.99670.99500.99210.98860.984S0.98020.97880.97710.97530.97390.96940.95140.85790.70730.54740.41070.28590.0058

Ppe

1.00001.05721.05201.00000.88480.81880.70690.57860.44990.33120.22800.14160.06270.01910.01070.00960.00970.00980.0094

TO " TW

Toe ~ Tw

1 .00000.98430.97640.96330.94750.92910.91080.90760.90490.90520.91370.92280.94050.95350.89820.57720.29240.10700.0000

TTe

1.0.0.1.2.4.6

2.02.53.44.97.4

12.328.998.3

182.8210.0

'215.5219.7234.7

MMe

1 .00000.97980.95820.90510.82560.77080.69580.61410.52860.44310.35970.27660.17730.08670.05240.03780.02800.01930.0004

p0 =200 p w = 1 . 4 5 X 1 0 " pe = 0.618 X 10" To = 888 7\v = 321 re = 2.15

Reue = 3036 Me = 34.9 p_ = 1.40 X 10'3 — -=4 .99X10 6 6 = 11.68K m

-4r=3036 4=0.00501 -r= 0.00945 «fl = 2916o o o "

— ^ «i c v l C\~Q ' /""* — O QT^ v 1 O*^ O T ^ O — C\ f-iAft/ • " jC . tOA IU L-t/ — W . 7 Z U A 1U — TT U. / J Z — ' VJ.OHD/ ^Rg

1.00000.97830.92390.86960.81520.76090.70650.65220.59780.54350.48910.43480.38040.32610.27170.21740.16300.10870.05430.0000

1.00001.03271.06721.03630.96910.87840.76230.63340.50090.36840.25050.14880.07150.02290.01090.00630.00400.00270.00190.0013

1 .00000.99650.98910.98270.97530.96620.95860.95280.94760.94470.94400.94700.94530.92040.83510.67070.53890.46090.40240.3613

1 .00000.99820.99440.99110.98720.98230.97810.97450.97090.96790.96480.96040.94380.87320.75450.5988 '0.46210.35320.23700.0080

1 .00001 .03631.07911.05490.99440.91020.79660.66670.53110.39290.26870.16090.07980.02940.01820.01610.01610.01610.01610.0160

1 .00000.99450.98290.97290.96130.94700.93530.92620.91790.91330.91240.91700.91430.87540.74180.48440.27810.15590.06430.0000

.0

.0

.0

.1

.3

.5

.82.23.04.26.4

II. 123.466.1

110.5129.4134.8139.1143.2149.4

1 .00001.00170.98400.93910.88130.81550.73950.65710.57030.47780.38530.29060.19700.10850.07250.05310.04020.03020.02000.0007

51


P0e = 268 p w = l . 8 8 X W A ^ = 0 .742X10^ 7-o = 514 7^ = 302 7>=1.20

ue = 2310 Me = 35.8 p, = 3.03X1Q-3 — = 1 2 3 X 1 0 ' 5 = 11.04e m

-^=0.521 -f-=0-QQ4U 4-= 0.00794 RH = 5582o o o *

Cf~ 1 88 X 10"4 Crj ~ 0 794 X 10"4 0 8^7 ~ 0 478

ys

1.00000.94250.87360.80460.74710.68970.63220.57470.51720.45980.40230.34480.28740.22990.17240.11490.05750.0000

Pp7

P°*e1 .00001.07291.0979

• 1.03450.92900.81190.67370.52780.39540.28410.20150.12090.05120.01190.00450.00260.00170.0013

TQ

T0e

1.00000.99460.98920.98380.97730.97080.96650.96260.96200.96020.96020.96190.96650.96920.91700.77760.64250.5864

uue

1 .00000.99730.99450.99160.98800.98440.98160.97870.97690.97370.97030.96330.94000.81890.62540.44940.26840.0026

Pf>e1.00001.07881.11011.05220.95150.83770.69900.55080.41390.29920.21360.12980.05750.01700.01010.00960.01000.0101

T0-TW

Toe ~ Tw

1 .00000.98690.97390.96080.94520.92950.91910.90950.90820.90380.90380.90790.91910.92560.79930.46220.13560.0000

rTe

1.01.01.11.21.51.82.23.04.26.19.0

15.536.6

129.2226.6247.9245.5252.3

MMe1.00000.99290.95880.89200.81790.74140.65600.56490.47630.39380.32380.24500.15570.07220.04160.02860.01720.0002

p =263 p, = 1 . 8 7 X 1 0 - " p =0.794X 1Q-" To = 893 rw = 314 Te = 2.14rOe w c

ue = 3211 Me = 35. p = 1 . 8 1 X 1 0 - 3 — -=6 .50X10 6 5 = 11.17e m

-^= 0.570 -4-= 0.00465 -4~-= 0.00879 Rfi = 33785 8 6 v

C y = 2 . 1 7 X 10-4 Cfj = 0.782 X 1

1 .00000.92050.85230.79550.73860.68180.62500.56820.51140.45450.39770.34090.28410.22730.17050.11360.05680.0000

1.00001.04220.99820.91010.80550.69170.56700.44040.31380.21280.12620.04940.01580.00710.00400.00260.00170.0013

1.00000.99130.98440.98180.96700.96020.95450.95130.94920.94920.94840.92440.85430.73920.61240.47820.40230.3517

1 .00000.99550.99190.99040.98250.97860.97500.97220.96920.96580.95830.92020.80720.64850.49530.35180.22160.0000

0-< 2 C H_ Q ^ } M _Q

^/ V^g

1.00001.05151.01440.92780.83430.72210.59620.46560.33370.22780.13700.05780.02350.01560.01400.01510.01570.0164

1 1.00000.98660.97600.97190.94910.93860.92980.92480.92170.92160.92040.88340.77520.59770.40220.19500.07800.0000

.603

1.01.11.21.41.72.02.63.55.17.8

13.533.385.4

133.6153.8148.3147.7147.1

0.99830.96840.91060.84290.77010.69420.61220.52620.43370.34910.26290.16050.08800.05650.04020.02910.01840.0000

52


P0g= 108.2 pw = 0.395 X 10" pe = 0.212 X 10" To = 523 7^ = 300 Te - 1 .07

Reue = 233\ JWe = 38.1 P,= 9.68X10" — - = 4 . 2 2 X 1 0 6 6 = 2 5 . 0e m

-^=0.616 4-= 0.00196 -4-= 0.00368 «„ = 20740 0 O D

r° — i £/i Y i fi-4 c — n dO'* Y i o4 — n ^oo — 0 9 ? ^^ /" — I .OH A 1 U O fj — \J .^ls J A l\J ,-, — V/. (_)(-/,' V7.O *5 J

.y8

1.00000.95430.90360.86290.81220.76140.71070.65990.60910.55840.50760.45690.40610.35530.30460.25380.20300.15230.10150.05080.0000

P0j

7T°^1.00001.00961.01440.98560.90670.79900.73920.57420.44020.30860.19860.09710.03010.01010.00630.00430.00300.00220.00150.0010

L 0.0009

ToTJ Og

1.00000.99890.99440.98910.98420.98060.98000.97830.97980.98300.98600.99150.99840.99100.93280.81860.72380.66060.62090.58770.5730

uue

1 .00000.99940.99710.99440.99180.98970.98930.98780.98780.98790.98670.98150.95170.86460.77480.66500.55980.46640.35430.18000.0036

Pfe

1.00001.01081.02020.99680.92180.81560.75530.58830.45100.31600.20370.10050.03290.01310.00990.00900.00850.00800.00740.00670.0067

To - Tw

TOg - Tw

1.00010.99730.98690.97450.96290.95460.95320.94920.95280.96030.96720.98010.99620.97900.8426.0.57510.35300.20500.11210.03430.0000

TTe

1.01.01.11.11.31.51.72.23.04.47.1

14.846.4

120.3163.9185.2201.8217.7243.4272.7281.3

MMe

0.99820.98380.96600.93710.88170.81250.76770.66500.57260.47170.37240.25630.14020.07910.06070.04900.03950.03170.02280.01090.0035

p0 e=110.2 pw = 0.395 X 10" pg = 0.224 X 10" To = 967 Tw = 300 r e=1.96

ue = 3\61 Me = 31.9 p =5.57x10" — - = 2 . 1 4 X 1 0 6 8 = 2 4 . 1e c re m

-^=0.604 -4-= 0.00280 -^-= 0.00508 Ra = 14500 0 0 °

2Cu MCy = .90 X 10" C

1 .00000.94740.86840.84210.78950.73680.68420.63160.57890.52630.47370.42110.36840.31580.26320.21050.15790.10530.05260.0000

1 .00001.00280.98040.95800.88240.76750.66950.52100.39220.25700.14500.03290.01270.00800.00530.00410.00280.00180.00110.0008

1.00010.99750.99020.98800.98170.97580.96930.96450.96300.95960.93510.86670.75420.63860.54680.47600.42350.37870.33650.3102

— n ^&n Y iLJ — U.OOVJ A 1

1.00000.99870.99490.99380.99050.98720.98360.98060.97890.97510.95830.8914 -0.77740.67190.57460.49970.41780.32240.20580.0000

0" — ^-=0.

1 .00001.00550.99040.97000.89940.78740.69180.54170.40910.27000.15760.04100.02060.01700.01510.01480.01380.01280.01230.0115

715 - f

1.00010.99630.98580.98270.97350.96490.95550.94850.94640.94140.90600.80670.64370.47610.34290.24030.16410.09920.03800.0000

.975

1.01.11.11.21.31.61.92.43.35.29.2

36.374.592.8

107.2111.9123.0135.6145.1153.0

1 .00000.98190.94390.92460.87180.79940.73440.63760.54460.43430.32130.15040.09160.07090.05640.04800.03830.02810.01740.0000

53


P0g = 200 pw = 0.553 X 10-4 pe = 0.242 X W4 To = 498 Tw = 297 Fe = 0.83

we = 2273 Me = 42.2 p = 1.41 X 10'3 — = 7. 1 X 106 6 = 2 1 . 3e 777

4^=0.600 -1= 0.002 18 4^=0.00415 tffl = 3308o o o ^

^ >O

Cy=1 .47X 10^ C# = 0.418 X 1

yS

1.00000.95240.91070.8929

. 0.83330.77380.71430.65480.59520.53570.47620.41670.35710.29760.23810.17860.11900.05950.0000

PO,P°*e1.00001.05221.06651.05700.99050.87030.72780.57280.41460.27690.16460.06580.01520.00630.00390.00260.00170.00110.0009

ToT0g

1.00000.99480.99120.99010.98140.97570.97380.97460.97680.98040.98570.99641.02351.02760.94750.76780.67640.62610.5971

uue

1.00000.99740.99550.99500.99050.98740.98610.98600.98610.98620.98530.97730.92160.81730.70000.54620.41200.26570.0094

0-4 - n s^i — n '

P. pe

1.00001.05771.07601.06771.00960.89260.74840.58910.42610.28450.16930.06860.01750.00890.00730.00740.00710.00670.0064

T T'o- 'wT0e - Tw

1.00010.98720.97810.97540.95390.93970.93500.93690.94250.95130.96460.99121.05821.06850.86960.42370.19680.07180.0000

30

TTe

1.0. 1.0

1.01.11.21.51.82.53.65.69.9

25.6104.8215.3273.5280.5302.6331.6356.1

MMe

1 .00000.99580.97820.96400.90340.82130.72980.63000.52230.41640.31350.19360.09030.05590.04250.03270.02380.01460.0005

p0g = 201 pw = 0.553 X 10-4 pe = 0.244 X 10-4 To = 958 7^ = 300 7^=1.58

Re«e = 3154 ytfe = 42.1 p = 7 . 5 2 X 1 0 ^ — -=3 .34X10 6 5 = 2 1 . 8e 777

-^=0.632 -f = 0.00226 -^-= 0.00410 Ra = 16456 5 5 °

2CW MCf= 1.79 X 10-" Cw = 0.576 X 10-4 -^1 = 0.658 -51=1.030/ " 9 V*fl

1.00000.94190.90700.87210.81400.75580.69770.63950.58140.52330.46510.40700.34880.29070.23260.17440.11630.05810.0000

1.00001.02871.01590.99040.89170.75800.6146

. 0.46500.31210.17830.07770.02320.01090.00670.00440.00310.00190.00120.0009

1.00000.99460.99040.98880.98580.97860.97610.97330.97320.97430.98470.99930.87300.71460.58500.48880.41810.36110.3132

1.00000.99730.99510.99420.99250.98860.98690.98470.98320.98040.97540.94130.82230.68840.56680.46420.34750.22700.0021

1.0000• 1.0343

1.02601.00190.90510.77540.63090.47930.32260.18530.08140.02590.01560.01330.01270.01270.01210.01200.0123

1.00010.99220.98600.98380.97930.96880.96520.96110.96100.96250.97780.99890.81510.58460.39600.25590.15300.07000.0000

1.01.11.11.21.41.72.33.14.98.9

21.269.6

120.1146.5160.3166.0180.5187.8189.9

1.00000.97880.95330.92320.84940.76080.66660.56490.45140.33320.21480.11450.07610.05770.04540.03650.02620.01680.0002

54


p =270 pw = 0.658 X 10-4 pg = 0.251 X 10-

4 To = 523 Tw = 300 Te = 0.78

Reue = 2330 Me = 44.6 pe=1.60XlQ-

3 — -=8.49X106 6=21.3

—=0.549 -i= 0.00248 -£-=0.00476' «fl=44120 o o v

/~l

Cy = 1.44X 10"4 Cf

y

s1.00000.92860.86900.83330.77380.71430.65480.59520.53570.47620.41670.35710.29760.23810.17860.11900.05950.0000

po2P°2e1.00001.10331.12921.11441.00370.84130.68630.52400.36160.23250.13650.05720.01110.00410.00280.00180.00110.0009

ToT0e1.00000.99350.98880.98220.97890.97610.97450.97430.97550.97810.98020.98771.00650.97460.85960.73740.63900.5734

n ^ni Y i O"4 — n Mf\ — n,-0.,01X 10 -Cf 0.696 ^ 0.

uue1.00000.99670.99430.99100.98910.98740.98620.98540.98480.98380.98030.96870.88020.7139'0.59070.44360.24420.0096

P. "e

.0000

.1106. .1421

.1348

.02590.86280.70550.53940.37260.23990.14180.06060.01390.00720.00690.00680.00660.0069

T TJo ~ ' wT0e - Tw1.00010.98470.97370.95840.95050.94400.94020.93970.94270.94860.95360.97131.01530.94040.67090.38440.15380.0000

672 ;

TTe1.01.01.11.11.31.72.23.14.77.813.833.9155.5310.8341.2361.0386.7382.6

MMe1.00000.99430.96490.93640.85670.75800.66300.56210.45390.35420.26450.16680.07080.04060.03210.02340.01250.0005

P0g = 271 pw = 0.658 X 10-4 pg = 0.281 X 10-

4 . r0 = 913 Tw = 297 re=1.42

ue=3078 Me = 43.5 p =9.67X10-* — -=4.54X106 6=21.6

£ /ft

-V-= 0.606 -f = 0.00238 4- = 0.00432 tf „ = 23306 • 6 6 "

Cy=1.65X lO"4 CH = 0.666 X 10"

1.00000.94120.89410.85880.82350.76470.70590.64710.58820.52940.47060.41180.35290.29410.23530.17650.11760.05880.0000

1. 00001.0348

' 1.03831.02090.97910.85020.71080.55050.37980.23340.13070.04490.01390.00750.00460.00290.00180.00130.0008

1.00000.99710.99320.98880.98730.98150.97820.97280.96960.97060.97400.98180.96160.80180.52480.44180.39320.35460.3287

1.00000.99850.99650.99420.99340.99030.98830.98500.98230.98050.97750.96150.88930.74890.54690.43570.33070.24330.0067

2Cl7 M-4 n ~ n o i n ' l — n

/ ^Re1.0000

' 1.03801.0456.03270.99210.86690.72760.56730.39340.24260.13650.04830.01720.01280.01420.01330.01220.01220.0113

1.00010.99570.98990.98340.98110.97260.96770.95970.95490.95650.96150.97310.94310.70600.29530.17220.10010.04280.0045

395

.0

.1

.1

.2

.3

.62.02.74.06.912.838.0

111.1156.1

• 146.0162.8183.3190.7212.1

1 .00000.97930.95340.92630.88950.80360.71360.61090.49430.37790.27600.15790.08540.06070.04580.03460.02470.01780.0005

55


p0 =201 pw = 0.504 X 10"4 pe = 0.203 X 1Q-* ro = 538 Tw = 297 Te = 0.83

ue = 2363 Me = 43.8 p = 1 . 1 8 X 1 Q - 3 — = 6 . 1 4 X 1 0 6 6 = 2 9 . 0e m

-^=0.601 -f-= 0.002 10 4-= 0.00395 flfl = 3732 Q=0.92xlO' 46 6 8 o f

ys

1.00000.95610.92110.87720.83330.78950.74560.70180.65790.61400.57020.52630.48250.43860.39470.35090.30700.26320.21930.17540.13160.08770.04390.0000

po,P°*e1.00001 .00420.99220.96250.92260.88120.82320.74220.63650.50920.38120.25500.15310.07530.02740.00940.00540.00360.00250.00180.00140.00110.00090.0009

TOT0e

1 .00000.99460.99150.98740.98350.98060.97680.97530.97310.97130.97080.97210.97450.96790.95850.92360.86580.80500.74940.69880.64820.60650.57450.5522

uue

1.00000.99730.99560.99350.99140.98980.98780.98690.98540.98390.98280.98160.97910.96620.93000.83150.72890.63220.53920.44450.35080.24690.14350.0080

P"e

1.00001 .00981.00090.97510.93860.89930.84360.76200.65530.52580.39450.26440.15950.08040.03130.01320.00950.00810.00740.00690.00680.00680.00680.0071

TO - Tw

Toe ~ rvv

1.00010.98800.98100.97190.96300.95660.94810.94490.94000.93590.93480.93770.94300.92840.90740.82930.70020.56450.44040.32730.21440.12110.04990.0000

TTe

1.0.1.1.2.3.5.6.9

2.33.04.26.5

11.223.061.2

150.2216.0261.5295.8323.1338.4351.5

• 356.9355.7

MMe

1 .00000.97100.94240.90230.86000.81940.77300.71720.64970.56890.48230.38670.29400.20230.11950.06820.04980.03930.03150.02480.01920.01320.00760.0004

p0e = 21Q pw = 0.645 X 10-4 pg = 0.218 X 10"4 ro = 519 Tw = 297 Te = 0.73

we = 2321 Me = 45.9 p. = 1.44 X 10'3 — -=8.03X106 8=27 .2e m

-^-=0.599 -^-=0.00227 4-= 0-00427 /?„ = 4944 C>=0.56X 1Q-"O O O " /

1.00000.97200.94860.92520.88790.84110.79440.74770.70090.65420.60750.56070.51400.46730.42060.37380.32710.28040.23360.18690.14020.09350.04670.0000

1.00001.01041.01481.01120.99760.96910.92940.86000.75010.61690.48170.34540.21100.10630.04850.02160.00820.00470.00310.00220.00160.00130.00100.0010

1.00000.99770.99540.99320.99000.98470.98060.97790.97460.97170.97110.96980.97040.97260.97320.96420.92200.86000.80010.74330.69010.63970.60150.5727

1.00000.99880.99760.99650.99480.99200.98990.98840.98640.98450.98360.98170.97930.97360.95740.91660.80630.69400.58600.48910.39280.28640.16460.0075

1 .00001.01281.01961.01831.00800.98470.94850.88030.77080.63630.49780.35810.21970.11180.05260.02540.01210.00910.00780.00730.00710.00710.00710.0074

1.00010.99460.98930.98410.97650.96410.95450.94830.94060.93370.93240.92940.93080.93600.93740.91620.81750.67240.53200.39910.27460.15660.06720.0000

r .01.01.1

.11.2 •1.31.5.7

2.12.73.65.28.9

18.440.988.4

192.6267.7322.8356.1378.4393.8405.6404.4

1.00000.97870.96020.93920.90440.85970.81400.75860.68770.60640.52180.43070.32860.22780.15030.09780.05830.04260.03270.02600.02030.01450.00820.0004

56

TABLE 10. - COMPUTED PARAMETERS FOR -X = 3.56 - Concluded

,p0e = 270 pw = 0.645 X JO-4 pe = 0.233 X JO"4 To = 892 Tw = 297 TV =1.29 !

Me = 3044 Me = 45.D

p =8.84X 10-4 — = 4.37X 106 5 = 2 7 . 7K m

-^•=0.621 -4-=0-00309 -r=0-0°572 #,, = 3731 Q= 0.61 X 10'40 0 0 u 3

ys

1 .00000.96330.94040.91470.87160.80280.73390.68810.64220.59630.55050.50460.45870.41280.36700.32110.27520.22940.18350.13760.09170.04590.0000

po2

^e1 .00001.0217

' 1.02711.02330.99570.92710.78760.64690.50390.36200.22400.12790.06320.02450.01270.00730.00490.00330.00230.00170.00130.00100.0009

ToToe

1 .00000.99720.99510.99320.98880.97620.96590.95930.95160.94530.94060.93170.91290.87920.80810.72840.64850.57460.50930.45300.40120.36260.3329

u"e

1 .00000.99850.99750.99650.99420.98770.98220.97840.97390.96960.96480.95560.93490.88490.80390.70510.60950.51320.42130.33470.24480.14620.0037

_£.Pe

1.00001.02471.0323

• 1.03051.00740.95030.8164

• 0.67560.53110.38490.24040.13980.07200.03090.01920.01400.01210.0112 .0.01080.01090.01120.01160.0122

TO ~ T\vToe ~ T\v

1.00010.99580.99270.98980.98320.96440.94890.93900.92740.91810.91090.89760.86940.81890.7123

.0.59280.47310.36240.26440.18010.10240.0445

-0.0000

rTe

.0

.1

.1

.1

.21.51.82.33.14.67.6

13.727.867.6

113.0160.9192.6216.4230.5236.8237.0236.9231.1

MMe

1 .00000.97960.96410.94510.90110.82940.73220.64600.55580.45980.35340.26110.17940.10900.07650.05630.04440.03530.02810.02200.01610.00960.0002

57

TABLE 11.- TEST PARAMETERS

X

0.508

1.067

1.625

2.793

Me

19.419.219.919.919.819.820.120.020.320.2

27.326.4

27.327.527.927.429.429.029.929.6

30.930.633.132.534.534.935.8

" 35.1

38.137.942.242.144.643.5

Poe

66.166.2107.6107.6108.1109.2199199270268

65.365

107.7107.7107.7109.8189189270265

66.965.3107.3107.6198200268263

108.2110.2200201270271

.T0e

499820315393481988516942503882

500933

305418577920534877502876

515890495886515888514893

523967498958523913

6

2.872.872.722.742.722.692.442.572.362.41

7.987.67

7.117.117.167.246.966.996.476.60

13.3413.4612.5712.7011.4211.6811.0411.17

2524.121.321.821.321.6

PWX 10'

5.265.268.618.618.61•8.6113.413.4

, 18.118.2

1.451.45

2.08•2.082.08•2.043.123.123.863.85

0.6980.698.05.04.45.45.88.87

0.3950.3950.5530.5530.6580.658

Tw

362397333333339400352402318361

315358

299301309328319348318339

301321306317304321302314

300300297300300297

cyx io4

4.874.84*

4.06*4.28*3.78*4.38*3.37*3.71*2.39*3.42*

3.62 .3.80

2.332.533.023.222.733.012.03

, 2.58

2.663.002.382.681.962.451.882.17

.64

.90

.47

.79

.44

.65

CHK 104

1.211.41

0.8660.9671.161.221.111.020.9571.02

0.7581.000.7970.9310.6790.9200.7940.782

0.4930.6800.4180.5760.5010.666

'w*10'

7.46.5412.5512.6512.4310.6718.7316.4027.924.8

2.231.98

3.373.393.303.044.784.365.895.35

1.141.0851.721.6342.322/243.062.97

0.6580.6400.9020.9251.1031.092

*L6

0.30.390.190.200.250.290.170.280.180.21

0.390.16

0.250.310.330.380.280.320.230.29

0.400.460.360.420.310.350.260.34

0.400.460.350.410.330.38

UL

"e

0.950.940.94

0.930.940.910.900.880.920.91

0.930.93

0.940.940.920.920.940.930.930.92

0.940.940.930.930.930.930.920.92

0.940.940.920.940.940.94

MLMe

0.240.280.160.160.130.180.090.180.180.21

0.200.22

0.180.180.170.190.180.170.150.16

0.200.210.140.180.140.150.120.17

0.140.250.090.120.13().12

*•

13819132662260821961420

3343268854303337

1440916

37862767•180116853710238047562870

2190

129628001680

40962916

55823378

20741450

33081645

4412

2330

*These values estimated from velocity profiles.

TABLE 11

58

Noov-iI

oo4-tCO

(L>

OC/D

.U.

59

.ooo

o

2a,<D>H

a<u

nj

00rt•»-»00

CMO

•sCN

2OX)

60

<DO

OCO

E

—

o>

O)O)

Omi

I I I I I Io o o o o oX

00

CM

1int

X

r-r-i

i

X

o<j-i

?

X

o—1

10I

X

r--—i

101^^

X

<d-ini

iO O O O O o

X X X X X Xro t CM — in inoo CM — OJ ro in

00

roroIOJ

c>

inCVJI

(

CM — cn CM o00 O) 00CM oj cn

oo(O

o o o o o

CJ<i-•Ir -oo

OJr-OJ

OJ

(5

^ u>?• m oo CM CM oj cn

^ m cn CM CM io co - r~ i - i Q

om

O D O Q Q O

o

JO

2o,S

sID

miO

e0

OJQ.

in*o

in«

o1-

+3"o<DX•4—1

£

•oc

_0v^2

43

o_o%-»C3

1

6CO

mo

oo

m moo ooo

m

Eo

61

-*J<*-!oo

•*-*d3

I

o

o

8.?E $

m in to tof f>. CM CVJO O O Oo q q qo 6 o o

g,

(O(A

.b c Eo

•k

o ^j-

cvj N- m_' O Oq q q qd 6 o d

I 6^ x"*»en

Is- m mto CM <r> coq co N m— —' CM to".

62

IO-4,-

i Possible relaminarization

No relaminarization

P0 'a t m T°e°K "T65 861 0.5XI06

270 528 4.8X10

Figure 5.— Variation of the relaminarization parameter, K, along the nozzle.

63

\\x\\\\\\\\\\\\\\\\\\

VI

(a

o

•ocKS

(1)

oo<4-H

oo

IOon

r<u

IE

65

400 r-

350

T °Kiw, r\

300

250

2.4x10

2.0 -

-3

p.atm 1.2 -

.4 -

To,.°K

O 900

D 500

3.5

Figure 8.— Pressure and temperature variations along the nozzle.

66

Ak.ftOD

A

Q

O

A

O

SolidOpen

x = 0.508

x = 1.067x = 1.625 J Present data

x =2.793x = 3.56Fischer etol.8

Fiore30

Jones 8 Feller27

Harvey ft Clark (unpub.) Nit.

Scaggs48

Speaker 8 Ailman49

Creel (unpub.) Nit.

Coles31 flat plate

Wagner etal.50 wedge

Cary (unpub.) flat plate

Maddalon 8 Henderson 45cone

symbols -Heliumsymbols -Air

Nozzle wall data

A

Figure 9.— Variations in wall pressure ratio with edge Mach number.

67

(a) Pressure profile.

Figure 10.- Typical pilot-pressure and stagnation-temperature profiles;x = 2.793 m,po = 201 atm,TO = 958° K.

68

1.0

.8

.6

.4.

.2

Slope fromheat transfermeasurements

0

(b) Temperature profile.

Figure 10.— Concluded.

.8 .0

69

1.0 i-

.8

O p = pw-(y/S)(pw-pe)D p = pw

O P = Pe

Slope fromfriction balance

ID

E>

O

.4 .6 .8 1.0

(a) Velocity profile.

Figure 11.- Typical velocity and density profiles; A- = 2.793 m, po = 201 atm, To = 958° K

A-4359

.0

.8

.6

.4

O p = Pw-(y/S).(pw-pe)

a p = pw

o P = P.

D 4

D 4

3 4

a •

7TX I°

o .4 .6 1.0

(b) Density profiles.


71

1.2

1.0

.8

-!- .6S

.4

.2

.2

O 0.981D 0.720O 0.535A 0.357

A

A

AA

£A o

AA ^<$K.OA *»°^GP

* a° ^°0 0O

A I I(P

_rd_.4 .6

T0 /Tn

.8 1.0 1.2

(a) Total temperature ratio.

Figure 1 2.— Variations in stagnation-temperature and pitot-pressure profiles with wall temperature;p0= \Q8atm,M e = 2 7 , x = 1.067m.

72

1.0

.8

.6

.4 AA

O 0.981D 0.720O 0.535A 0.357

\ ' '.2 .4

" 1.6

Po2

1.8

1

1.0

(b) Pitot-pressure ratio.

Figure 12.- Concluded.

73

I.U

.8

.6

y8

.4

.2

0

— WJ

Po.a tm Tw/T0f Me

O 108 0.981 .27.3 g

D 108 0.720 27.5 D23

O 108 0.535 27.9 n%

A HO 0.357 27.4

A

A

A

A"

A O Q O

A a oA 0 Q 0

OA O D O

1 1 I I 1.2 .4 .6 .8 1.0

(a) Velocity.

Figure 13.— Variations in velocity, density, and Mach number profiles with wall temperatix = 1.067 m.

74

1.0

.8

.6

_y_8

A

®3

p O I atm

O 108

D 108

O 108

A HO

A

A0 00

0O*-

A0[D

0.981

0.720

0.535

0.357

M e

27.3

27.5

27.9

27.4

A

.6 .8

_|

1.0

(b) Density.

Figure 13.—Continued.

75

1.0

.8

.6

Po . Qtrn

O 108

D 108

O 108

A 110

A

O

0.981

0.720

0.535

0.357

A

O DO

AO ID

AM « 10

Me

27.3

27.5

27.9

27.4

O

A

1 1

.4 .6M

Me

(c) Mach number.


1.0

76

H o

o_£Xok_Q.CLO

.r c••- oo 2 S"« -5 «o ° £o o x

OJ

£VIjo

£SOo,23

4_t173

(D

Oo

co

l-03

CO

tt.

77

I.U

.8

.6

y8

.4

.2

0

*,m p0,atm Toe-°K Me

O 0.508 199 576 20.1 &

D 1.067 189 534 29.4 Q

O 1.625 198 515 34.5 g>

A 2.793 200 498 42.2 />

|^1

A O a

^a °D

O O

i i i i I

.2 .4 .6 .8 1.0uUe

(a) Velocity.

Figure 15.- Variations in velocity, density, and Mach number profiles with nozzle station.

78

x,m Po.a tm Toe,°K Me

1.0

.8

.6

y8

.4

.2

i

£

A0

O 0.508 199 576 20.1 Q

D 1-067 189 534 29.4

O 1.625 198 515 34.5

A 2.793 200 498 42.2

<*A o on

A 0

A 0

AO D

A 0

A QO

i>ai

I

Q 1 1 1 1 1

.2 .4 .6 .8 1.0P

(b) Density.

15.— Continued.

79

x,m P 0» a f m T0e,°K ^e

1.0

.8

.6

ys

.4

.2

0

O 0.508 199 576 20.1

D 1.067 189 534 29.4" .

O 1.625 198 515 34.5 &

A 2.793 200 498 42.2& o

4>

A <n °A ^ O

A ° °

A *° 0" 4

0 o

O D

^

1 1 1 1 1

.2 .4 .6 .8 1.M

Me(c) Mach number.


80

I.U

.8

.6

y8

.4

.2

r

P 0 , a tm T0e,°K Me J

O 265 876 29.6 KL-J

D 189 877 29.0 ^

O MO 920 27.4 fa

A 65 933 26.4 £>

<b

IA 0

0 8

A ° o°A ° a 0°

o* ° ° °

O ^

\ 1 1 1 10 .2 .4 .6 .8 1.0

(a) Velocity.

Figure 16.— Variations in velocity, density, and Mach number profiles with pressure;x = 1.067 m.

81

1.0

.8

.6

y8

.4

.2

0

P o ,atm • T0e,°K Me Q

O 265 876 29.6 |

D 189 877 29.0

O M O 9 2 0 27.4 (

A 65 933 26.4 AQ

A 0°

A <£>

O

A 0

0 Q °

rtt ' ' ' ' '.2 .4 .6 .8 1.0

••. P

(b) Density.

Figure 16.- Continued.

82

1.0

.8

.6

.4

.2

P 0 , a tm

O 265

D 189

O 110

A 65

A

Axv

D

A

T0e,°K

876

877

920

933

O

aao

Me

29.6

29.0

27.4

26.4.

OO

A 00

A

.4 .6 .8 1.0MMe

(c) Mach number.


83

01 fo f^ ^~c\j r>- oo ooto in ro ro6 d o o

•j; 03 10 r . ma CD ID CO CD

- CVJ CJ COo

a.

ro fO h- roCT> CD CD CDP>- h- O t^-

c\j c\j — cvj

o a o

ooo

oo

1 1 I I 1 1CVJ C\J — —

*

K-s

1c

ea>

ca

W)

'«3

I

<u6

* x

a,

co•*3J2"3

oo

IDJ3-t^CnO

I

n

3

84

- O» en

- so W

Q.

OJ

ooo

oo

oo

CMoCVJ

cvj 00

co

VI

03

CL>

"o

c

IcaID

s>>

•*-*'o

>-4—»

C

l-iOH

03

1OO

(U

o

«2i _

06

i

in

—oO

85

OOO

OO

OCM

C\J CO CO CVI 00

T3

«.»

S

<fr tOm to

o

co

en

rt

enU

efl

60

4>

EIcrt

S

>.•4-*

'o

cCD

O

o+-»03

oo

-4_»

ItX

O

I

Ov

8II

3

86

-4

-8

J"1"- I

-12

-16

-20

0.329

0.5730.387

0.384

Incompressible (Coles)

.6 .8

V8

(a) u+=-P/Pw

1.0

Figure 20.— Velocity-detect law correlations of the present velocity measurements using VanDriest's transformation.

87

-4

-8

-12

-16

-20

O

0.329

0.573

0.387

0.384

Incompressible (Coles!

.4 .6 .8

(b) u+ from equation 54 (ref. 33).


88

1000 i—

100 —

10 —

-

-

~ oD

- OA

- £

aQ

- oCi13)

~ oov

x = 0.508 \x = 1.067 1

, ___ > Presentx = 1.625 jx = 2..793 1Matting etal.51

Wagner etal.50

Watson 8 Cory52

Fischer et al.8

Shall53

Scaggs48

Burke6

Perry 8 East54

Hill55

Matting etol.51

Wallace2

Jones 8 Feller27

Lobb etal.56

Unpub AEOCBeckwith etal.37

Kist ler2 2

Winkler 8 Cha57 Flat

tests

plate^ Samuels et al.58 Hollow cylind«3

aColes3 1 Flat plateOwen 8 Horstman59 Cone cylind

Nozzle wall

Note: These data were obtained from high mochnumber tests with an assumed temperaturedistribution through the boundary layer

See note

.01.001 .01

Figure 21.- Variations in sublayer thickness

89

1/4

A

2.793 37.9-44.6

A

J I II03

(a) Skin friction.

Figure 22.— Variation in skin friction and heat transfer90

10'

I5x|0-5

10

I I

10

o

o.625 30.6-35.8

I I I

8r

1/4

(b) Heat transfer.


91

to-2 .

icr3

cf

10-4

2XIO-5

O x = 0.508D x = 1.067O x = 1.625A x =2.793O Coles31

b Stalmoch60

A Hopkins et at.3

O Wallace2

0 Winkler aCha57

Prediction techniques

Sommer & Short39

'oe

Present tests 0.35 -

Coles32

Spalding a Chi40

Van Driest I41

Van Driest I33

Laminar T'42

II

0.29 -0.460.120.6-0.88

0.338

10 20 30 40 50 60 70

(a) Skin friction.

Figure 23.— Variations in skin friction and heat transfer with Mach number.

92

icr3

O x = 0.508

D x= 1.067O x = 1.625A x = 2.793O Coles31

b Stalmach60

^J Hopkins et al.3

O Wallace2

0 Winkler a Cha57

Toe

Present tests 0.35-1

10.29-0.460.120.6-0.88

CH

2X10"5

a

1

A

A

A

110 20 30 40

Me

50 60 70

(b) Heat transfer.


93

I I I 11 I I IiO

toiO

l CD roCM tO Sl-

I I Io o\ °°N." 'fr 10CM to to

r--c>

oCVJ

E

o

1 -CDO

—

inCM10—

10<T>r>-C\J

o(D

<nc

_^Q.OX

(D0a

o$

0

<ol_a>c-OU)

il

c0)o:o

C/}

Ic

o O

_ o

roO

CDCM (K

2 CDo:

u.

co

c

T3(DUiCX

T3

rtTJ23c«w

C

ooJSO•sC8CX

soo

au-

CVJIO

roIO

O

0.°

94

o-—, O

m.ro (6 O» II

0) 0>o o

is. to ro(O CJ 0>O U) Is-.

— —' OJ

D O <

D

D

o o

Ic "a<B —Q. OO <S)

CDo:

C3VI

T3

"o

_cc

03

(N

10O

IOCM

10 i-

A Hopkins etal.3

O Wallace2

0 Winkler 8 Cha57

•oe

iO.35

«0.7

0.29-0.460.12

0.6-0.88

2CH

Cf

10 20 30 40 50Me

(b)M


96

^o

0>1-H

*.»

03

*

o•«_>

u

OB_O13c03

to

T3C

tn

2

<D

^o-*-Jo

00I x -

<o in10

q10

incJ

OCJ

oio

O

C3

&0£

97

y, cm

5 -

0 .2

Figure 27.— Profile of heat-transfer fluctuations to cooled film probe.

98

o-0 0 0

o o oo

CM

IOo

°0o

OOo

x>

, O rt

h o •§™

£ CMo

;§ oa ~ 0

K) 0E CD

*• N-X .

CM

—

~ o O 8 @ • Ooo 9 Q oQ °

1 1 1 1 1

— 3O

X>lU

-*— »

CO bb3O

-*— »onC

^ , .^ -2fe «3 3 3

US Q. tS> ^ c<•»— t

*. oCJVIC/5

C^

^06<N

2?n

o co . ID ^r CM o . E

99

1.0

.8

.6

ys

.4

.2

0

r OO 2.793 1

O

00

O<DO

©O

O OOG

O OO

GO G

O

i i i.02 .04 .06

v^Tf

Toe

P0 iatm

108.2 0.574

Figure 29.— Temperature fluctuations through the boundary layer; Af = 38.1,T0e - 500° K.

100

0>>»

_oJ3

<n3Ooin

Eo

\\\\SS\\\\\\XX\XXX\\

\\\\\\S\\\S,\S\\\\

S\S\SSS\\\S\S

S\S

\S

S

S

T3OE£ocUi

T3

O

o'SoS20>

T3%oa2a

<4-oo

'•4-tCO

E<uo

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