DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been...

66
AND NASA TECHNICAL NOTE- NASA TN D-7723 Z (NASA-TN-D-7723) BOUNDARY-LAYER N74-3275 1 TRANSITION AND DISPLACEMENT THICKNESS EFFECTS ON ZERO-LIFT DRAG OF A SERIES OF POWER-LAW BODIES AT MACH 6 (NASA) 66 p Unclas 'HC $3.75 CSCL 20D H1/12 4 896 5 _. ' BOUNDARY-LAYER TRANSITION AND DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF A SERIES OF POWER-LAW BODIES AT MACH 6 by George C. Ashby, Jr., and Julius E. Harris Langley Research Center Hampton, Va. 23665 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION * WASHINGTON, D. C. * SEPTEMBER 1974 https://ntrs.nasa.gov/search.jsp?R=19740024638 2020-06-28T10:03:11+00:00Z

Transcript of DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been...

Page 1: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

AND

NASA TECHNICAL NOTE- NASA TN D-7723Z

(NASA-TN-D-7723) BOUNDARY-LAYER N74-3275 1

TRANSITION AND DISPLACEMENT THICKNESS

EFFECTS ON ZERO-LIFT DRAG OF A SERIES OF

POWER-LAW BODIES AT MACH 6 (NASA) 66 p Unclas'HC $3.75 CSCL 20D H1/12 48 9 6 5 _. '

BOUNDARY-LAYER TRANSITION AND

DISPLACEMENT-THICKNESS EFFECTS

ON ZERO-LIFT DRAG OF A SERIES

OF POWER-LAW BODIES AT MACH 6

by George C. Ashby, Jr., and Julius E. Harris

Langley Research Center

Hampton, Va. 23665

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION * WASHINGTON, D. C. * SEPTEMBER 1974

https://ntrs.nasa.gov/search.jsp?R=19740024638 2020-06-28T10:03:11+00:00Z

Page 2: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

NASA TN D-77234. Title and Subtitle 5. Report Date

BOUNDARY-LAYER TRANSITION AND DISPLACEMENT- September 1974THICKNESS EFFECTS ON ZERO-LIFT DRAG OF A SERIES 6. Performing Organization Code

OF POWER-LAW BODIES AT MACH 67. Author(s) 8: Performing Organization Report No.

George C. Ashby, Jr., and Julius E. Harris L-9646

10. Work Unit No.9. Performing Organization Name and Address 502-37-01-04

NASA Langley Research Center 11. Contract or Grant No.

Hampton, Va. 2366513. Type of Report and Period Covered

12. Sponsoring Agency Name and Address Technical Note

National Aeronautics and Space Administration 14. Sponsoring Agency CodeWashington, D.C. 20546

15. Supplementary Notes

16. Abstract

Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and Reynolds numbers, based on body length, from1.5 x 106 to 9.5 x 106. Pressure distributions were computed on the nose by the inversemethod and on the body by the method of characteristics. These pressure distributionsand the measured locations of boundary-layer transition were used in a nonsimilar-boundary-layer program to determine viscous effects. A coupled iterative approach between theboundary-layer and pressure-distribution programs was used to account for boundary-layerdisplacement-thickness effects. The calculated-drag coefficients compared well with previ-ously obtained experimental data.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Hypersonic flow Unclassified - UnlimitedMinimum drag

Boundary-layer displacement thicknessStar Category 12

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price*

Unclassified Unclassified 64 $3.75

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

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BOUNDARY-LAYER TRANSITION AND DISPLACEMENT -THICKNESS

EFFECTS ON ZERO-LIFT DRAG OF A SERIES OF

POWER-LAW BODIES AT MACH 6

By George C. Ashby, Jr., and Julius E. Harris

Langley Research Center

SUMMARY

The influence of viscous effects on the zero-lift drag of a series of power-law bodies

has been examined analytically and experimentally at a Mach number of 6. Bodies con-

strained to length-diameter ratios of 6.63 with values of the power-law exponent in the

shape equation of 0.25, 0.50, 0.60, 0.667, 0.75, and 1.00 were studied. The pressure drag

of each body was computed by using the method of characteristics, and the skin friction

was computed by using a numerical solution of the compressible boundary layer with

the local flow properties predicted by the method of characteristics. The location of

boundary-layer transition on the models was determined from phase-change-coating heat-

transfer measurements, and the effects of transition and boundary-layer displacement

were included in the calculations. The calculated drags were compared with experimental

results obtained at a Mach number of 6 over a Reynolds number range, based on body

length, of 1.50 x 106 to 9.50 X 106. The analysis showed that the exponent for minimum

drag with either inviscid or fully laminar flow was approximately 0.667. However, if

boundary-layer transition existed on the bodies, the exponent for minimum drag was lower.

For the experimental results at the higher Reynolds numbers, boundary-layer transition

occurred, and the resulting exponent for minimum drag was approximately 0.6.

The effect of displacement thickness on total drag was significant but was nearly

constant for all the bodies exerting only a minor influence in the determination of the body

with minimum zero-lift drag. For the present tests, the boundary-layer displacement-

thickness effect on total drag was as high as 9 percent with the major contribution coming

from wave drag.

INTRODUCTION

Early studies, typified by references 1 and 2, which were conducted to determine the

body and/or wing profiles with minimum zero-lift drag considered only the inviscid drag

and used approximate pressure laws. Subsequent investigators included the effects of

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skin friction but in a simplified manner. (See, for example, ref. 3.) The results fromsuch basic investigations, however, have been useful design guides and have been shownto be generally applicable under lifting as well as zero-lift conditions (refs. 4 and 5).

More recently, with the focus of attention on the advanced space transportation sys-tems, the practical application of body shaping to improve hypersonic performance hasreceived greater impetus, and some refined results have been obtained. For example, inreference 6 a critical assessment of the pressure laws used in previous optimizationstudies (primarily the Newton pressure laws) revealed that more exact pressure laws,such as the method of characteristics, defined optimum bodies which had more volumeforward than previous studies had shown. Furthermore, experimental data for a series ofpower-law bodies obtained at a Mach number of 6 over a large Reynolds number range insupport of reference 6 indicated that a blunter body than the one predicted by the exactpressure laws would have the minimum zero-lift drag. This result was attributed to thevariation of boundary-layer transition between the bodies at the test flow conditions andpromoted the current study.

The purpose of the present investigation was to determine the influence of viscouseffects on the minimum zero-lift drag. Accordingly, calculations were made of skin-friction and boundary-layer displacement effects on a series of power-law bodies whichwere tested at a Mach number of 6 and reported in references 6 and 7. In the referencetests the condition of the boundary layer was not established, but the range of Reynoldsnumbers was large enough for the boundary layer to vary from laminar over the entirebody to a laminar-transitional-turbulent state over the entire body. For the present study,the boundary-layer transition location was determined on reproductions of the force modelsby phase-change-coating heat-transfer measurements at zero angle of attack for Reynoldsnumbers corresponding to those of the reference test. (See refs. 6 and 7.) These meas-ured transition locations along with a pressure distribution calculated by the inversemethod for the nose and the method of characteristics for the body were used in anonsimilar-boundary-layer program to compute the skin-friction and wave-drag coeffi-cients on each body. An iterative approach was used to couple the boundary-layer and thepressure-distribution programs to account for boundary-layer displacement effects. Thecomputed results are presented and compared with experimental drag coefficients herein.

SYMBOLS

Measurements and calculations were made in the U.S. Customary Units. They arepresented herein in the International System of Units (SI) with the equivalent values givenin the U.S. Customary Units.

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A model base area, meters 2 (feet 2)

DragCD drag coefficient, qAqA

Total skin-friction drag

CD,f total skin-friction-drag coefficient, qA

Total drag

CD,T total-drag coefficient, sum of wave and skin-friction drag, qA

Wave drag

CD,W wave-drag coefficient, qA

Cf local skin-friction coefficient

Cp pressure coefficient, Pe - Po

1 model length, centimeters (feet)

n exponent in shape equation for power-law bodies, R (n (see fig. 1)

pe body surface pressure, newtons/meter 2 (pounds/foot 2)

p free-stream static pressure, newtons/meter 2 (pounds/foot 2 )

q free-stream dynamic pressure, newtons/meter 2 (pounds/foot 2 )

R base radius of power-law bodies, centimeters (feet)

R, ,l Reynolds number based on free-stream properties and model length

r,x coordinates on meridian curve of power-law body, centimeters (feet)

(see fig. 1)

S distance along body surface, centimeters (feet)

Uej velocity at edge of boundary layer, centimeters/second (feet/second)

xt,f end of boundary-layer transition

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xt, i beginning of boundary-layer transition

6 inclination of body surface relative to free stream, degrees

5* boundary-layer displacement thickness, centimeters (feet)

MEASURED LOCATIONS OF BOUNDARY-LAYER TRANSITION

Models

The heat-transfer models used in this investigation were cast reproductions of the

force models (ref. 7) and are shown in figure 1. The six power-law bodies, which are

defined by = with n = 0.25, 0.50, 0.60, 0.667, 0.75, and 1.00, were circular in

cross section, had a fineness ratio of 6.63, and were constrained for constant length and

base diameter. Each model had an aluminum nose and a high-temperature epoxy plastic

body. The measured coordinates of the heat-transfer models are within 0.040 cm of thosefor the force tests in reference 6. The noses of the heating models were individually

shaped to match those of the force models.

Wind Tunnel and Tests

The transition measurements were conducted in the same tunnel as the force tests

of reference 7, the Langley 20-inch Mach 6 tunnel. It is a blowdown type exhausting to

either atmosphere or vacuum with an operating range of stagnation pressures from 3 to

35 atmospheres (0.304 to 3.55 MN/m 2 ) and stagnation temperatures up to 589 K. The

general details of the tunnel, along with schematic drawings, are presented in reference 8.

The location of boundary-layer transition was measured on each body, by using aphase-change-coating technique described in reference 9, at five Reynolds numbers, from1.50 x 106 to 9.50 x 106 based on model length. Four of the Reynolds numbers matchedthose of the reference force tests. Since transition did not occur on any of the bodies atthe lowest Reynolds number, an additional intermediate Reynolds number was selected togive a more complete trend. All tests were conducted at zero angle of attack and sideslip.

Method of Determining Transition

Transition could be observed visually on these models because of heat-rate varia-tions between laminar, transitional, and turbulent boundary-layer regions. (See fig. 2.)Note that the light region in the photograph is the unmelted area. In the phase-changetechnique the heating rate is inversely proportional to the time required to melt the coat-ing. Progressing rearward from the apex of the model nose, the heating rate decreases

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through the laminar region and becomes a minimum at the start of transition. The rate

increases through the transition region until fully turbulent flow is established and then

decreases at a relatively low rate in the turbulent region. Under these conditions, a coat-

ing which melts at a specific temperature when applied uniformly to the model will begin

to melt at the nose and then at or near the rear, depending on whether or not transition is

completed on the body. The two melting regions approach each other and meet at the ini-

tial transition point, which is the last place the coating melts. The series of photographs

in figure 2 show an example of the melting sequence.

For each test, the model was coated and mounted on a tunnel injection strut in the

retracted position. After starting the tunnel and setting the flow conditions to the desired

Reynolds numbers, the model was injected and observed visually until the phase-change

process was complete. The time histories of the phase change of the coating were

recorded with motion-picture cameras on both the top and right side of the models. The

location of the beginning of transition (last melt point) and, where possible, the end of

transition (first rearward melt point) were scaled from those pictures.

Locations of Boundary-Layer Transition

Typical peripheral distribution of the beginning of transition obtained from the phase-

change-coating tests at a = 00 are presented in figure 3, and the values used in the

boundary-layer calculations are shown in figure 4. The average values were used because

transition occurred at a smaller value of xt i on top than on the side and, in general, was

not uniform about the body. Similar results were reported in references 10 and 11. For

the present paper, the variation was less than 20 percent of the model length and could

have been caused by any of several factors. For example, model irregularities could alter

the location of boundary-layer transition. In reference 11 a deliberate nose-tip offset of

0.1 percent of the tip length (0.005 cm) caused transition location to vary circumferentially

about the body by 10 to 35 percent of the length. Nonuniform distribution of the phase-

change coating or small flow angularity could also cause an asymmetric distribution of

transition location.

The variations in the location of the beginning of transition with Reynolds number

for the five bodies in figure 4 are reasonably well behaved and do not exhibit any large

anomalies. The apparent reversal of transition in the data for the bodies with n = 0.60

and 0.75 at the higher Reynolds number is believed to be associated with nonuniformity

of transition around the body. No reason is known for such transition reversal under these

flow variations; therefore, the curve presented in figure 4 for n = 0.60 was faired with-

out reversal. Calculations showed that, for the body with n = 0.60 at a Reynolds number

of 9.5 x 106 and a transition-location variation from 52 to 69 percent of the body length,the skin-friction drag coefficient would vary only about 0.001, which is within the accuracy

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of the measured drag coefficients. Thus, the difference between the curve as shown and

a fairing through all the points was found to have an insignificant effect on skin-friction-

drag coefficients.

The distance to transition from the nose varies inversely with the power-law expo-

nent (directly with nose bluntness, i.e., bluntness delays transition), which is in agreement

with the results presented in reference 12. As the Reynolds number increases, transition

moves forward, but the rate of forward movement is a direct function of n (inverse func-

tion of nose bluntness). At the higher Reynolds numbers of these tests, the rate of for-

ward movement approaches zero for the bodies with n = 0.60 to 0.75; whereas, the loca-

tion of transition on the body with n = 1.00 is more linear with Reynolds number. The

difference in the variation of transition location between the body with n = 1.00 and the

others at the high Reynolds numbers is not understood, but similar results were reported

for a slightly blunted 50 half-angle cone at a Mach number of 8 in reference 13.

The end of transition could not be positively identified for all cases (i.e., the begin-

ning of turbulent flow did not occur on the body); those for which it could be identified are

shown in figure 5. The ratio of the location of the end of transition to the location of the

beginning of transition varies from approximately 1.2 to 1.7. Because no systematic vari-

ation exists, a nominal value of 1.5, represented by the dashed line, was used for all cal-

culations. A value of 1.5 for the average extent of boundary-layer transition is at the

lower end of the range of values mentioned in reference 14 for a large amount of data.

VISCOUS EFFECTS ON POWER-LAW BODY FOR MINIMUM

ZERO-LIFT DRAG

Effect of Boundary-Layer Transition

The variation of the calculated skin-friction, wave, and total-drag coefficients with

power-law exponent for the bodies at the four test Reynolds numbers are presented in fig-

ure 6. The procedure used to compute the total-drag coefficients for the power-law

bodies, including the effects of boundary-layer transition and displacement thickness, is

presented in the appendix. Note that the measured locations of transition were used in the

calculations. By considering wave drag only, the body with minimum drag would have an

exponent n of approximately 0.667 (middle group of curves in fig. 6). This body is

blunter than that predicted by Eggers et al. and by Miele (n = 0.75) in references 1 and 3,

respectively, by using Newton's pressure law (Cp cc sin25). The difference in the bodies

for minimum drag between this investigation and those of Eggers and Miele is attributed

to the difference in pressure laws. This investigation used the method of characteristics,

and Eggers and Miele used Newton's pressure law. A comprehensive discussion of the

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effects of several pressure laws and methods used to predict minimum wave-drag shapes

is contained in reference 6.

When viscous effects are included (CD,T in fig. 6), the exponent for minimum drag

remains near 0.667 until the Reynolds number increases beyond 4.37 x 106, above which

it shifts towards a value of 0.60. This trend is caused by the variation of boundary-layer

transition with Reynolds number for each power-law body (fig. 4). The location of transi-

tion onset from the body nose xt i at a given Reynolds number varies inversely with the

power-law exponent n (directly with bluntness, i.e., bluntness delays transition), while

the rate of forward movement of transition location with Reynolds number dxt,i/dRw,l

varies directly with n. These divergent trends cause the increase in skin friction to be

larger for the higher exponent bodies as Reynolds number increases and therewith the

minimum total drag to shift towards the lower exponent, blunter bodies. Although these

calculations include displacement thickness, it has little effect on the power-law body for

minimum drag as is shown subsequently.

The results of figures 4 and 6 confirm and explain more completely the shift in

minimum drag towards a blunter body exhibited by the experimental data for the same

series of bodies at a Mach number of 6 and the higher Reynolds numbers (refs. 6 and 7).

In the references the shift was correctly attributed to the variation of the location

boundary-layer transition from body to body at a constant Reynolds number.

A comparison of the calculated values of total-drag coefficient with the experimental

data is presented in figure 7. The height of symbols in the figure is representative of the

experimental measuring accuracy, and the agreement between the measured and computed

data for the majority of the points is within this accuracy. Of those points for which the

measured and computed data do not agree, all but two are for the sharper bodies at the

higher Reynolds numbers. If the location of transition for the bodies with n = 0.667

and 0.75 had been the linearly extrapolated value (21 percent of length in fig. 4) and for

the body with n = 1.00 had been at 50 percent of the body length, then the calculated val-

ues of CD,T for Ro, 1 = 9.5 x 106 would have been represented by the dashed-line

curves in figure 7. Under these assumptions the agreement between the calculated values

and the measured data is good. At R, l = 6.42 x 106, the calculated-drag-coefficient

value for the body with n = 1.00 can be adjusted to agree with the measured value by

assuming that the location and the nondimensionalized extent of transition are 58 per-

cent of body length and 2.0, respectively, instead of the measured values of 46 percent

and 1.5. Similar assumptions would bring the data for the body with n = 0.75 at the

same Reynolds number into agreement with experiment. This exercise indicates that

the calculated data agree with the experimental data for reasonable assumptions of

boundary-layer transition.

7

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Effect of Boundary-Layer Displacement Thickness

Boundary-layer displacement thicknesses for each body are presented in figure 8 asa function of S/l at each of the test Reynolds numbers. The location of transition usedin the calculations is indicated on each curve if transition occurred for that particularReynolds number. When transition begins somewhat forward of the body base, a markeddip in 6* is followed by a rapid increase. However, the maximum thickness neverexceeds that for the completely laminar-flow case at the lowest Reynolds number. Thescale at the right (fig. 8) is a measure of the displacement thickness at the base in percentof the base radius. For the conditions of this investigation, the displacement thickness atthe base varies from approximately 4 to 12 percent of the base radius.

Effect on surface pressure.- The computed pressure distributions on each body withand without boundary-layer displacement effects at the highest and lowest Reynolds num-bers are presented in figure 9. The plots arbitrarily terminate at the body station wherethe two pressure distributions appear to converge. These results indicate that the pri-mary effect of displacement thickness on surface pressure is generally confined to thefirst 4 percent of the body length.

Effect on skin friction.- Skin-friction coefficients for each body with and withoutboundary-layer displacement effects are presented in figure 10 for the maximum andminimum test Reynolds numbers. These comparison plots were terminated, in general,prior to 1 percent of the body length because the effects were negligible beyond this region.This small effect is reflected in the less than 3-percent difference in the integrated valuesof viscous drag with and without displacement-thickness effects, as discussed in the nextsection. The distributions of the skin-friction coefficient, including the effect of boundary-layer displacement thickness are shown in figure 11. The location of transition used inthe calculations is also indicated in figure 11 where appropriate.

Effect on drag.- The pressure and skin-friction distributions were integrated overthe entire wetted surface to get the total drag for each body and each test Reynolds number.The change in drag coefficient due to boundary-layer displacement effects is observed tobe nearly constant with power-law displacement at all Reynolds numbers (fig. 12), therebyindicating that boundary-layer displacement does not have a strong influence in determin-ing the power-law body with the minimum zero-lift drag. The primary effect of displace-ment thickness is on the wave-drag coefficient (fig. 12(b)). This effect is more clearlyevident in figure 13 where the changes in the total-drag-coefficient components due todisplacement effects are plotted against the power-law exponent for the four Reynolds num-bers. The increase in drag due to boundary-layer displacement ranges from 2 to 9 per-cent for total drag, 3 to 16 percent for wave drag, and below 3 percent for skin-frictiondrag. The level of the effect is, in general, an inverse function of Reynolds number.

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Boundary-layer displacement effects on total drag reach a maximum at n near

0.5 (fig. 13(a)); whereas for wave drag they peak near a value of 0.6 (fig. 13(b)). The

reduction in displacement effects beyond this peak is more abrupt for wave drag than for

total drag. In both cases, however, the drop is tempered somewhat by increasing Reynolds

number. Cross plots of the drag coefficients against Reynolds number (fig. 14) tend to

obscure these anomalies and, in general, show an inverse variation of boundary-layer

displacement effects with Reynolds number. The large increases in displacement effects

for n = 1.00 at the higher Reynolds numbers is attributed to the earlier transition on

that body.

CONCLUDING REMARKS

Wave and skin-friction-drag coefficients have been numerically calculated for a

series of power-law bodies at a Mach number of 6 and Reynolds numbers, based on body

length, from 1.5 x 106 to 9.5 x 106. These calculations were used to evaluate the impact

of boundary-layer transition and displacement thickness in the determination of the power-

law body with minimum drag. The calculated-drag results were compared with previously

obtained experimental data.

The results of this investigation indicate that, when boundary-layer transition occurs,

the body for minimum zero-lift drag can be blunter than when the boundary layer is fully

laminar. This condition occurs because the location of transition at a given Reynolds num-

ber varies directly with nose bluntness, thereby causing the skin friction to decrease with

nose bluntness and the minimum total drag to shift to a blunter body. The calculated val-

ues agree well with the experimental data.

The effect of displacement thickness on drag is significant but is nearly constant for

all the bodies and therefore has very little influence in the determination of the body with

minimum zero-lift drag. For the present tests, boundary-layer displacement-thickness

effect is as high as 9 percent of the total drag.

The results from this investigation are applicable to the design of high-performance

missiles and cruise and entry configurations at hypersonic speeds. For example, if at

zero-lift, conditions for boundary-layer transition are encountered in flight, a blunter nose

should be considered as a trade-off for improving aerodynamic performance as well as

reducing aerodynamic heating. Furthermore, the boundary-layer displacement thickness

is a sizable percentage of the body radius, varying from 4 to 12 percent at the base. The

effect of boundary-layer displacement thickness on drag can also be sizable (up to 9 per-

cent from this investigation) and should be included in performance calculations.

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These results are also useful as a guide in the design of hypersonic missiles andcruise and entry vehicles for lifting conditions. Previous work has shown that for somecases body shapes for minimum wave drag at zero lift have the best performance underlifting conditions at hypersonic speeds. Analogously, the identification, in this paper, of

blunt bodies for minimum drag at zero lift when the boundary layer is transitional shouldbe applicable to the design of lifting cruise vehicles and entry configurations. For such

applications, blunting for practical packaging and aerodynamic stability requirements

might be less penalizing than indicated by previous studies.

Langley Research Center,

National Aeronautics and Space Administration,Hampton, Va., June 25, 1974.

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APPENDIX

DETERMINATION OF BOUNDARY-LAYER DISPLACEMENT EFFECTS

ON WAVE AND SKIN-FRICTION DRAG

Procedure

The effect of boundary-layer displacement thickness on drag was determined by an

iterative procedure diagramed in figure 15. Seven different computer programs were

utilized with the iteration requiring three passes through five of the programs to obtain a

solution.

To start the calculations, the inverse method for blunt-nose bodies at hypersonic

speeds (ref. 15) was used to compute the surface pressures on the nose of a sphere and a

paraboloid and the flow properties in the shock layer along a line normal to each of these

nose shapes in the transonic flow region. This line is called a field data line in the refer-

ence. The sphere and paraboloid were fitted to the particular power-law body (n < 1.00)

at the body slope of the field data line. The flow-field properties of the nose shape, which

best matched that of the body nose shape, were used in the method-of-characteristics pro-

gram of reference 16 to compute the pressure distribution over the remainder of the body.

A curve-fit program using cubic spline functions was then applied to the complete pressure

distribution. The coefficients of the equation from this pressure curve fit, the body coordi-

nates from a body geometry program, and the measured location of boundary-layer transi-

tion were input to the nonsimilar-boundary-layer program of reference 17, which used the

numerical technique reported in reference 18. In the latter program, the integrated wave

and skin-friction drags as well as the boundary-layer properties were determined, and the

coordinates for the body plus the displacement thickness were obtained.

The iteration procedure for displacement effects began with a program which deter-

mined an effective power-law body for the basic body enlarged by the boundary-layer dis-

placement. This effective body was required because the boundary-layer displacement

was somewhat nonuniform with transition present, and the method-of-characteristics

program is difficult to use without a continuous surface slope. From this point, the cal-

culation procedure looped back through the sphere or paraboloid nose match, the method-

of-characteristics, the pressure curve-fit, and the boundary-layer programs. This loop

was repeated until the velocity at the edge of the boundary layer agreed with that of the

previous pass to within 1 percent. Agreement was accomplished in all cases after the

third pass through the boundary-layer program.

For the body with n = 1.00 (cone), the initial pass used a constant surface pressure

from the cone tables (ref. 19), the geometric parameters, and the measured transition

11

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APPENDIX - Continued

location in the boundary-layer program. In the subsequent iterations, the pressure distri-

butions were obtained from a method-of-characteristics program, developed by Lillian R.

Boney of NASA Langley Research Center, for the effective body with a conical nose.

Program Applicability

Blunt-body pressure.- The only exact nose match for the pressure calculations was

the paraboloid to the body with n = 0.50. For all bodies with 0.50 _ n < 1.00, the parab-

oloid was the closest match; and the properties for the field data line from it were used

in the method-of-characteristics calculation. The power-law body with the exact nose

match to a sphere is n = 0.413; therefore, the spherical nose properties were used only

for the pressure calculations for n = 0.25. The following table shows the maximum axial

distances (normalized by body length) between the body nose and the nose of the parab-

oloid or sphere, which was matched to the body at the initial point of the method-of-

characteristics solution for the initial pass, first iteration, and the second iteration:

Ax/1 for -n

Initial pass First iteration Second iteration

0.25 -0.00330900 -0.00323000 -0.00323600

.50 .00000004 .00003130 .00003220

.60 .00000841 .00010470 .00014150

.667 .00003770 .00005440 .00005340

.75 .00000432 .00001010 .00000973

In all cases, the differences between the actual and calculated nose shapes isextremely small. All of the shapes fitted to the power-law bodies lie inside of the bodyexcept for the sphere fitted to the body with n = 0.25, which is somewhat blunter than asphere.

Method-of-characteristics solution.- The method-of- characteristics solution extendedthe full body length for all cases except the body with n = 0.75, which covered only 5 per-cent of the body. However, comparison of this calculation with the pressure distributionobtained with the method-of-characteristics program of Lillian R. Boney, using a 450 start-ing cone, showed that the two became coincident at 0.007 percent of the body length fromthe nose; therefore, the two solutions were joined at this point to obtain the completedistribution.

Cubic spline curve fit of pressure distribution.- The cubic spline curve fit was per-formed on a cathode ray tube and involved shifting spline curve junction points until the

12

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APPENDIX - Concluded

complete curve system was observed to conform to the pressure distribution. In the curve

fits, from 9 to 16 junction points were utilized, depending on the slenderness of the body.

Figure 16 shows some examples of the curve fit with the junction points identified by the

vertical tick marks.

Boundary-layer program.- The numerical technique presented in reference 18 was

used to solve the nonsimilar equations governing the laminar, transitional, and turbulent

compressible flow over the power-law bodies. The turbulent boundary layer was repre-

sented by a two-layer concept with appropriate eddy-viscosity models used for each layer

to replace the Reynolds stress term. A static turbulent Prandtl number model was used

to relate the turbulent-heat-flux term in the energy equation to the Reynolds stress model.

The mean properties in the transitional boundary layer were calculated by multiplying the

eddy viscosity by an intermittency function based on the statistical production and growth

of the turbulent spot in the transitional region of flow. The computer code and flow charts

are presented in reference 17.

Effective-body program (Body + 6*).- Before the procedure repeats, the last program

determines an effective power-law body for the basic body with the boundary-layer dis-

placement thickness superimposed. Several examples of this curve fit are shown in fig-

ure 17. The accuracy of representation varies inversely with nose bluntness; nonetheless,

the representation in all cases is good.

13

Page 16: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

REFERENCES

1. Eggers, A. J., Jr.; Resnikoff, Meyer M.; and DIennis, David H.: Bodies of Revolution

Having Minimum Drag at High Supersonic Airspeeds. NACA Rep. 1306, 1957.

(Supersedes NACA TN 3666.)

2. Drougge, Georg: Wing Sections With Minimum Drag at Supersonic Speeds. Rep.

No. 26, Aero. Res. Inst. of Sweden (Stockholm), 1949.

3. Miele, Angelo, ed.: Theory of Optimum Aerodynamic Shapes. Academic Press, Inc.,c.1965.

4. Spencer, Bernard, Jr.; and Fox, Charles H., Jr.: Hypersonic Aerodynamic Perform-

ance of Minimum-Wave-Drag Bodies. NASA TR R-250, 1966.

5. Spencer, Bernard, Jr.: Hypersonic Aerodynamic Characteristics of Minimum-Wave-

Drag Bodies Having Variations in Cross-Sectional Shape. NASA TN D-4079, 1967.

6. Love, E. S.: Design of Bodies for Low Drag and High Performance in Practical Hyper-

sonic Flight. Performance and Dynamics of Aerospace Vehicles, NASA SP-258,1971, pp. 103-174.

7. Ashby, George C., Jr.: Longitudinal Aerodynamic Performance of a Series of Power-

Law and Minimum-Wave-Drag Bodies at Mach 6 and Several Reynolds Numbers.

NASA TM X-2713, 1974.

8. Goldberg, Theodore J.; and Hefner, Jerry N. (With appendix by James C. Emery):

Starting Phenomena for Hypersonic Inlets With Thick Turbulent Boundary Layers at

Mach 6. NASA TN D-6280, 1971.

9. Jones, Robert A.; and Hunt, James L.: Use of Fusible Temperature Indicators for

Obtaining Quantitative Aerodynamic Heat-Transfer Data. NASA TR R-230, 1966.

10. Mockapertris, L. J.; and Chadwick, G. A.: Three-Dimensional Hypersonic Flow

Boundary Layer Transition Distribution on Sharp 8-Degree Cone With Ablating and

Non-Ablating Walls at Small Angles of Attack. BSD TR-67-197, U.S. Air Force,July 1967. (Available from DDC as AD 819 957.)

11. Kendall, J. M.: Wind Tunnel Experiments Relating to Supersonic and Hypersonic

Boundary Layer Transition. AIAA Paper No. 74-133, Jan.-Feb. 1974.

12. Moeckel, W. E.: Some Effects of Bluntness on Boundary-Layer Transition and Heat

Transfer at Supersonic Speeds. NACA Rep. 1312, 1957. (Supersedes NACA

TN 3653.)

14

Page 17: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

13. Stainback, P. Calvin (With appendix by P. Calvin Stainback and Kathleen C. Wicker):

Effect of Unit Reynolds Number, Nose Bluntness, Angle of Attack, and Roughness on

Transition on a 50 Half-Angle Cone at Mach 8. NASA TN D-4961, 1969.

14. Henderson, Arthur, Jr.: Hypersonic Viscous Flows. Modern Developments in Gas

Dynamics, W. H. T. Loh, ed., Plenum Press, 1969, pp. 83-129.

15. Lomax, Harvard; and Inouye, Mamoru: Numerical Analysis of Flow Properties About

Blunt Bodies Moving at Supersonic Speeds in an Equilibrium Gas. NASA TR R-204,

1964.

16. Inouye, Mamoru; and Lomax, Harvard: Comparison of Experimental and Numerical

Results for the Flow of a Perfect Gas About Blunt-Nosed Bodies. NASA TN D-1426,

1962.

17. Price, Joseph M.; and Harris, Julius E.: Computer Program for Solving Compressible

Nonsimilar-Boundary-Layer Equations for Laminar, Transitional, or Turbulent

Flows of a Perfect Gas. NASA TM X-2458, 1972.

18. Harris, Julius E.: Numerical Solution of the Equations for Compressible Laminar,

Transitional and Turbulent Boundary Layers and Comparisons With Experimental

Data. NASA TR R-368, 1971.

19. Sims, Joseph L.: Tables for Supersonic Flow Around Right Circular Cones at Small

Angle of Attack. NASA SP-3004, 1964.

15

Page 18: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

n

0.25.50,60.667,75

1,.00

n = 0.25 n = 0.667

n = 0.50 n = 0.75

r -____

-

n = 0.60 R 2.68 n = 1.00

L = 35.56 I

Figure 1.- Power-law bodies studied. (All dimensions are in cm.) Model shape equation: =)n

Page 19: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

FIRST TIME OF MELT--

- ..... INCREASINGTIME

FLOW

LAST TIME OF MELT-- TURBULENTTURBULENT

xt, f

TRANSITIONLAMINAR

HEATINGRATE

xt, i

xl/

L-74-1137

Figure 2.- Typical time history of phase change relative

to local conditions of boundary layer.

17

Page 20: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

Top

n = 0.60 n = 0.60 n = 0.60

n = 0.667 n = 0.667 n = 0.667

n = 0.75 n = 0.75 n = 0.75

n = 1.00 n = 1.00 n = 1.00

R I = 4.37 x 10 R, = 6.42 106 R = 9.5 x 106

Figure 3.- Typical peripheral distribution of boundary-layer transition.

Page 21: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

n

0 0.50LI .600 .667S,.75 &-

(4. 330 cone) . 1, 00 ILinear extrapolation

oo 0

- 80r

,...

60

0 40

-- 20

I I I I 64 5 6 7 8 9 10x10

RCO,

Figure 4.- Average measured location of initial boundary-layer transition on

n-powered bodies at Mach number of 6 using phase-change coatings.

19

Page 22: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

n

O 0.667n ,75

2.0- I 1.00

1.8 -

xt, f 1.6- Nominal value

xt, i 1.4-

1.2 -

1.0 I 64 5 6 7 8 9 10 x 10Ro,

Figure 5.- Ratio of average measured location of end of transition to initiationof transition on n-powered bodies at Mach number of 6.

20

Page 23: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

Calculated

O 9. 5 x 10 05 iii !i it o5' " -, !i i~l 6. 42i ii

:i ill ii-i i~ i ii ..: , O 4. i37:' A I. 5 -7 :. -,01

:::: ~ ~ ll .. .. . i ... '; i L i iii ', i:'. .i i iii !iilim 040c. 5...1..... i : i ill i

, 177

::: ... .. ll i . i !ii il jl ! l t' i I ;" q

i l tl t t

.. ' iiii

04.37 1i tlr 1II l i T T ! r ,t

.... i- TtTt i:i

;:: : :: : ;ii ii iJ i ... ''

1.: 5 , HI CD, W CD,T

0

ti .6 HT .04

:::::i . N ',.. " r :: ' "jiii i ' : 1; : ::l l ;'i

;; '': +'i " * 1L :: 1: . . . .. .. ..

:e ::t' : i ; 11 ' : i i.!T . :~ I:" ; 17. ..... .: ::; Ir' ::: :: :::: : !! l! I : : :l i : ii : ! : i{I : i = : ' : ' : i

,g2 6.f V o :cua sin- ricton waveantota l cic

power-law:: exonn in moe shap eqio =il n.e r Meaue tasiin oa

c D ::: :: :: ::: :::::: :::::::::::::::::: :: illi i i : ! !i. .. ; , :i i "' i

.T . 0 3

.. .. 'I" " .. . ... ... .. .. .. .. . . .. .

'"17 7" ' . . . . . .

. . .6 .7 .8 .9 1.D

Figure 6.- Variation of calculated skin-friction, wave, and total-drag coefficients with

power-law exponent in model shape equation = esrdtasto oa

tions and boundary-layer displacement-thickness effects are included.

21

Page 24: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

14 -1 Experiment (Ref. 7)

.16 - . R

.12 0 1 9.5 x106S 6,42

.14 -0 1.37

1.5.1 4 -

Calculated, measured transition.12 - ,rlocation, x xt, i = 1.5

.08 - Calculated, extrapolated transitionii itII location, xtf/xt i 1.5or2.0

.10 -

lais l I it .;0 i, F R .ili t t I tIi 1I1 1, i lt t i il liil i

. 08 4 :4 W I ! t ii ifli i ii I i III I tI h !ill tillt It

_.02. 04 !il lifi iii iii ii i t 12

11 I Ill li ' N li ifi lil ll l I ill i!

I III!I Ii N il!i lit k il l i fl i ! ! i i ill "iiiii! i! i l i

"i tiii, i, f RI li Ii 11 111 i fi O w

.2 .3 .5 .6 .7 .8 .9 1.0

Figure 7.- Comparison of experimental and calculated total-drag coefficients forsix power-law bodies at Mach number of 6 and four Reynolds numbers.

22

Figurelliliil 49.- 11omp0111risol*n1 of eIeim n~ an VIalcu late It t I li!l] -dra IH cofi ie t o

si o e -a o ie t MacNh11H 1 number o11111111I 1fii It an four .Reynol 111111mbers.1iI il

2 2ll;:i1 11 l l!;! 1 IfI ti

Page 25: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

15.40 V R

1.5 x 106.012 4.37

.35 - 6.429.5

.30 - .010

10

.25 .008

.20

.006 -

.15 -

.004 -

.10

L 0.05 02

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

s/.

(a) n = 0.25.

Figure 8.- Variation of boundary-layer displacement thickness along body surface

for four Reynolds numbers. Tick marks locate beginning of transition.

Page 26: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

15

RO, 1

1.5 x 1064.37

.30 - .010 - 6.429.5

10

.25 .008.008

.20 -

.006

.15

. 5

.004

.10

.05 . -

S 0 I0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

S/i

(b) n = 0.50.

Figure 8.- Continued.

Page 27: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

RO~,l

1.5 x1064.376.42

9.5

10

.25.008

.20 I

.006

.15

- --

.004

.10o.10 00 I- I ---- -

.002.05 -- xti

0o 0 I I I I 00 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

S/I

(c) n = 0.60.Figure 8.- Continued.

IN

Page 28: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

- 15

R

1.5 x 1064.37

6.429.5

- 10

.25.008

.20

.15

- _i_- 5.004

.10 -

----

.002 -.05 -

0 o 0 00 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

S/i

(d) n = 0.667.

Figure 8.- Continued.

Page 29: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

15

R ,

1.5 x 1064.376.42

- 9.5

10

.25.008

.20

5.004

.10

.002 - - -----.002

.05

0 0 I I0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

S/i

(e) n = 0.75.

Figure 8.- Continued.

Page 30: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

15

1.5 x 1064.376.42

9.5

10

.25.008

.20 -

.006

.15 -

.10 --

.002 --

.05 -t i

--- x

0o I I I I I I 0

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

S/i

(f) n = 1.00.

Figure 8.- Concluded.

Page 31: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

18 x 103

360

16

320 -

14280

12 -240

Without boundary-layer displacement effects

10 ------ With boundary-layer displacement effects200

~ 60

6 120

4 80

2- 40 -

0 0I0 .001 .002 .003 .004 .005 .006 .007 .008 .009

I I I I { } {

.01 .02 .03 .04 .05 .06 .07 .08

s/1

(a) n = 0.25; Roo,1 = 1.5 x 106.

Figure 9.- Effect of boundary-layer displacement thickness on pressure distribution

of each body at minimum and maximum Reynolds numbers.

Page 32: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

100 rx 103

2000

90

1800

80-

1600

701400

Without boundary-layer displacement effects

60 ----- With boundary-layer displacement effects1200

5 1000 -

40 800

30 600-

20 400

10L 200

0 0 J t I I I I , I I I I . 0 I .0 .001 .002 .003 .004 .005 .006 .007 .008 . 09 .010 .011 .012 .013 .014 .015

.01 .02 .03 .04 .05 .06 .07 .08

s/1

(b) n=0.25; R,, 1 =9.5x 106.

Figure 9.- Continued.

Page 33: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

18 x 103

360

16

320S-- - Without boundary-layer displacement effects

14 ---- - With boundary-layer displacement effects

280

12

10200

160

6120

4 - 80

2 40

0 0 001 .002 .003 .004 .005 .006 .007 .008.01 .02 .03 .04 .05 .06 .07 .08

s/I

(c) n =0.50; Roo, = 1.5 X 106.

Figure 9.- Continued.

CIO

Page 34: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

CO

100 -x 103

2000

90 -

1800

80

1600

Without boundary-layer displacement effects70 - - - With boundary-layer displacement effects

1400

601200

50 ',,S 1000

40 800

30 600 -

20 400

10 200

I I I _I _ _ II0 0 0 .001 .002 .003 .004 .005 .006 .007 .008.01 .02 .03 .04 .05 .06 .07 .08

s/1

(d) n = 0.50; R ,1 = 9.5 x 106.

Figure 9.- Continued.

Page 35: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

18 -x 10

360

16 -320

14 -280

--- Without boundary-layer displacement effects12 --- With boundary-layer displacement effects

240

10 -200

8 [ 160

6 - 120

4 80 -

2- 40

0i 00 .001 .002 .003 .004 .005 .006 .007 .008

.01 .02 .03 .04 .05 .06 .07 .08

5/ 1

(e) n = 0.60; Rol = 1.5 x 106

Figure 9.- Continued.

Page 36: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

100 -x 103

2000

90

1800

80

1600 -

70 -1400 \

SWithout boundary-layer displacement effects60 \ - - - With boundary-layer displacement effects

1200

S50 \E 50 1000

40800 i

30 600 -

20 - 400 "

10 - 200

0 0- I I I I I0 .001 .002 .003 .004 .005 .006 .007 .008 .009 .010

s/i

(f) n = 0.60; Roo, = 9.5 x 106.

Figure 9.- Continued.

34

Page 37: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

18 -x 103

360

16

320

14

280

Without boundary-layer displacement effects12 -- - With boundary-layer displacement effects

240

10E 200

160

6N120 . \

4 - 80 -

2 - 40

0 - 0 I I0 .001 002 .003 .004 .005 .006 .007 .008 _

.01 .02 .03 .04 .05 06 .07 .08

S/I

(g) n = 0.667; Ro,l = 1.5 x 106.

Figure 9.- Continued.

cn3

Page 38: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

100 x 10

2000

90

1800

80

1600

701400

-- Without boundary-layer displacement effects60 -- - With boundary-layer displacement effects

1200

50 t50 1000

z -

800 -

30

20 20 - 400

10 200

0 00 .001 .002 .003 .004 .005 .006 .007 .008 .009 .010

s/1

(h) n = 0.667; R l = 9.5 x 106.

Figure 9.- Continued.

36

Page 39: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

360

16

320

14-

280

Without boundary-layer displacement effects12 - - With boundary-layer displacement effects

240

S200

160

120

80 -

2- 40 -

0 0 0 .0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008.01 .02 .03 .04 .05 .06 .07 .08

s/I

(i) n = 0.75; Roo = 1.5 X 106

Figure 9.- Continued.

-C3

Page 40: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

C-O

100 -x 103

2000

90

1800

80

1600

70

1400

-- Without boundary-layer displacement effects60 ----- With boundary-layer displacement effects

1200

50--. o o1000

40800 -

30 600 -

10- 200 -

0 0 I I I0 .0001 .0002 .0003 .0004 .0005 .0006 .0007 .008

.01 .02 .03 .04 .05 .06 .07 .08

s/I

(j) n = 0.75; Roo,l = 9.5 x 106.

Figure 9.- Continued.

Page 41: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

Without boundary-layer displacement effects- -- With boundary-layer displacement effects

3500 r 70 R = 9. 5 x 10670 o

3000 I-60

250050 R 6. 42 x 106

----------------- -- -- -- - __ _____- ------ - -2000 40

_ R =4. 37 x 106

1500 3030

R 20 1.5 x 106

500 10

0L

00 .0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008

.01 .02 .03 .04 .05 .06 .07 .08

s/L

(k) n = 1.00.

Figure 9.- Concluded.

Page 42: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

1Without boundary-layer displacement effects

With boundary-layer displacement effects

10-1

R = 1.5 x 106

Cf

, 9.5 x 106

10-2

103

0 .004 .008 .012 .016 .020 .024 .028 .032 .036

S/I

(a) n = 0.25.Figure 10.- Effect of boundary-layer displacement thickness on skin-friction

distribution of each body at minimum and maximum Reynolds numbers.

40

Page 43: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

Without boundary-layer displacement effects

- With boundary-layer displacement effects

10

= 1.5 x 106

Cf

R = 9.5 x 106

-210 2

l -3 I I I I I10 0 .001 .002 .003 .004 .005 .006 .007 .008 .009

S/I

(b) n = 0.50.

Figure 10.- Continued.

41

Page 44: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

Without boundary-layer displacement effectsWith boundary-layer displacement effects

10-1

1.5 x 106

RR =1.5 x 10

10-2

10- 3 I I I I I I

.0 .001 .002 .003 .004 .005 .006 .007 .008 .009

S/I

(c) n = 0.60.

Figure 10.- Continued.

42

Page 45: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

1Without boundary-layer displacement effects

With boundary-layer displacement effects

10-i

R = 1.5 x 106

C

R = 9.5 x 106

10-210-

0 .001 .002 .003 .004 .005 .006 .007 .008 .009

S/I

(d) n = 0.667.

Figure 10.- Continued.

43

Page 46: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

-- Without boundary-layer displacement effects- -- With boundary-layer displacement effects

Ri = 1.5 x 106

Cf

0-110

S= 9.5 x 10

10- 2

0 .0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008

S/I

(e) n = 0.75.

Figure 10.- Continued.

44

Page 47: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

-110 Without boundary-layer displacement effects

With boundary-layer displacement effects

R 1.5 x 106

-210

10-3 I I I I

0 .001 .002 .003 .004 .005 .006 .007 .008 .009

S/I

(f) n = 1.00.

Figure 10.- Concluded.

45

Page 48: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

10 -1

1.5 x 106

4.37

i6.429.5

10-2

, ..

10- 3

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

S/I

(a) n = 0.25.

Figure 11.- Variation of local skin-friction coefficient along body surface for

four Reynolds numbers. Tick marks locate beginning of transition.

46

Page 49: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

-110

1.5 x 1064.37

6.42

9.5

-210

10-3

I-I

10 4

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

S/I

(b) n = 0.50.

Figure 11.- Continued.

47

Page 50: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

R_

1.5 x 1064.376.429.5

10- 1

Cf

10-2

-- i-----I10-I

0 .2 .3 .4 .5 .6 .7 .8 .9 1.0

S/

(c) n = 0.60.

Figure 11.- Continued.

48

Page 51: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

Rm

1.5 x 1064.376.42

- - 9.5

10-1

Cf

10-2

-- -

- 3 I I I I I

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

sh/

(d) n = 0.667.

Figure 11.- Continued.

49

Page 52: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

R

1.5 x 1064.37

6.429.5

10-1

Cf

10-2

10-30.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

s/i

(e) n = 0.75.

Figure 11.- Continued.

50

Page 53: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

-110

R

1.5 x 106

4.37- - - 6.42

9.5

10- 2

Cf

10-3

10

O-4 I

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

S/I

(f) n = 1.00.

Figure 11.- Concluded.

51

Page 54: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

With boundary-layer displacement effectsWithout boundary-layer displacement effects

.05

R .04

SI CD,T9.5 x 106

.03

.03

.02

.04

C .03D,T

.02

.04

4.37 x 1 .03 CD,T

I I .02

.05

CDT .04

D,T

.03

.02 I I.5 .6 .7 .8 .9 1.0

n

(a) Total drag.

Figure 12.- Comparison of drag coefficients computed with

and without boundary-layer displacement effects.

52

Page 55: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

-- With boundary-layer displacement effectsWithout boundary-layer displacement effects

R .03

00,.029.5 x 106 --. 02

CD,W

.01

.03 I I 0

.02 6.42 x 10CD,W

.01 -

0 I .03

4.37 x 106 -. 02

CD,W

.01

.03 I0

.02 5 x 10

CD,W

.01 -

05 .6 .7 .8 .9 1.0

n

(b) Wave drag.

Figure 12.- Continued.

53

Page 56: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

C

With boundary-layer displacement effectsWithout boundary-layer displacement effects

.03

.01

.03

.02

C .0 6.42 x 106D,f

.01

0 --1 .03

.02

4.37 x 106 CDf

x-0-.01

.03

.02 1.5 x 106

CD,f

.01

I I i I II I I0

.2 .3 .4 .5 .6 .7 .8 .9 1.0

n

(c) Skin friction.

Figure 12.- Concluded.

Page 57: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

R

0 9 . 5 x 10 6

E 6.424.371.5

8

C,

2

0 I I I I I I I

.2 .3 .4 .5 .6 .7 .8 .9 1.0

n

(a) Total drag.Figure 13.- Percent increase in calculated-drag coefficients due'to boundary-layer displacement effects.

Page 58: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

R ,l

'j 9.5 x 106[ 6.42

16 - > 4.37A 1.5

14 - ,C D,

12

10

8

2--

/.- '- -----

-

Q .L..2 .3 .4 .5 .6 .7 .8 .9 1.0

n

(b) Wave and skin-friction drag.

Figure 13.- Concluded.

56

Page 59: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

n

O i.oo000 .750 .667n .60

o10 - .50

8-

6

U 4

2

0I I I I I I I0 1 2 3 4 5 6 7 8 9 10 x10

6

Ro,

(a) Total drag.

Figure 14.- Percentage increase in drag coefficient for six bodies due to boundary-

layer displacement thickness as function of Reynolds number.

-zI

Page 60: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

0 16 -n

O 1.00l .75

.66714 A- .60

N .50L} .25

CD,W

12 - CD,f

10

8

0)

' 6 -

24

III I Ill II0 1 2 3 4 5 6 7 8 9 10 x 10

R

(b) Wave and skin-friction drag.

Figure 14.- Concluded.

Page 61: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

R [FREE-STREAMjCONDITIONS

SLOPE AT FIELD

DATA LINE BLUNT--BODY

-- PRESSURE

PROGRAM

SPHERE ORPARABOLOID FIELD DATA LINE

I MTCH TO AT MATCH POINTMATCH TO NOSE

BODY PRESSUREDATA

CHARACTERISTIC

PROGRAM FORPRESSURE AFT

OF MATCH POINT

[FREE-STREAM R PRESSURE -LCONDITIONS J DISTRIBUTION

CURVE-FIT

PROGRAM

[LOCATION OF COORDINATEO COORDINATE [ATION -- - PROGRAM

TRANSITION & Sx, r, & S

BOUNDARY-

LAYER I BODY: (

PROGRAM R

POWER-LAW NO CONVERGENCEBODY (r +6 U . - U . < 0.01 STOP

PROGRAM

Figure 15.- Calculation procedure for iteration of boundary-layer displacement-thickness effects on wave and skin-friction drag for power-law bodies.

59

Page 62: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

48 24

44- 20

40 16

Junction point 12

PeI P,o 36 12

32 - 8

28 - 4------ I

24 00 .002 .004 .006 .008 .010 .012 .01 .02 .03 .04 .05 .06 .07 .08 .09

S/I SAL

Ri = 1.5 x 106

0 Computed pressure2 Curve fit

0 I I l I I0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

s/1

(a) n = 0.25.

Figure 16.- Typical spline function curve fits of computed pressure distributions.Tick marks identify curve-fit junction points.

Page 63: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

48 24

44 R =9.5 x 106

40 -o Computed pressure 16

4 Curve fit

Pe/ Po 36 - 12

32 - 8

Junction point

28 - 4

24 I I 0 I I I I0 .000002 .000004 .000006 .000008 .000010 .000012 0 .0004 .00008 .00012 .00016 .00020 .00024 .00028 .00032

s/I s/1

12 - 3

8 2

4 - 1

0 I I I 0.0004 .0008 .0012 .0016 .0020 .0024 .0028 .2 .3 .4 .5 .6 .7 .8 .9 1. 0

s/I s/A

(b) n = 0.75.

Figure 16.- Concluded.

Page 64: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

.103.0

R , 1.5 x 106

.09 --

Body plus 6*

- Curve fit

2.5

.08

.02

Body

.06 A0

.5

.04

1.0

.03 -

.02

.01

0 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2

x, ft

0 10 15 20 25 30 35x, cm

(a) n = 0.25Figure 17.- Examples of effective power-law body representations of

basic body plus boundary-layer displacement thickness.

62

Page 65: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

S103.0

R. = 9.5 x 106

.09

'Body plus 6Curve fit

2.5

.08 -

.07 -

2.0

.06

1.5 .05

Body

.04

.02

.5

.01

0 00 .2 .3 .4 5 .6 .7 8 1.2

x, ft

I I I ' I I

0 5 10 15 20 25 30 35x, cm

(b) n = 0.667.

Figure 17.- Continued.

63

Page 66: DISPLACEMENT-THICKNESS EFFECTS ON ZERO-LIFT DRAG OF … · Wave and skin-friction drag have been numerically calculated for a series of power-law bodies at a Mach number of 6 and

.103.0

.09 -

2.5

.08 R, - 4.37 x 106

Curve fit

.07

2.0

.06 /

u 1.5 .05.04

.04/ .K ,

Body

1.0 /'-Z ,

.02

.5

.01

0 L 00 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2

x, ft

05 10 15 20 25 30 35x, cm

(c) n = 1.00.

Figure 17.- Concluded.

64 NASA-Langley, 1974 L-9646