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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
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
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
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.
2
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
3
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
4
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
5
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
6
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
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.
8
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.
9
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.
10
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
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
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
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
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
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
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
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.
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
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
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
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
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.
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.
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
- 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.
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.
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.
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.
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.
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
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.
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.
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
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
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
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
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.
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.
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
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
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
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
-- 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
-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
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
-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
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
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
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
-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
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
-- 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
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.
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.
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
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
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.
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
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.
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.
.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
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
.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