A summary of available knowledge concerning skin friction and heat transfer and its application to the design of high-sp...

MISSING IMAGE

Material Information

Title:
A summary of available knowledge concerning skin friction and heat transfer and its application to the design of high-speed missiles
Series Title:
NACA RM
Physical Description:
17 p. : ill. ; 28 cm.
Language:
English
Creator:
Rubesin, Morris W
Rumsey, Charles B
Varga, Steven A
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics -- Research   ( lcsh )
Skin friction (Aerodynamics)   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: A review is made of the existing information concerning boundary-layer characteristics: the temperature recovery, the skin-friction coefficients, the heat-transfer coefficients of both the laminar and turbulent boundary layers and the position of the transition from laminar to turbulent flow. Comparison is made between existing flight data and results computed by the boundary-layer momentum-integral method in a preliminary attempt to establish some rational way of approaching the design of a missile whose Mach number range and body geometry are markedly different from those of existing data.
Bibliography:
Includes bibliographic references (p. 9).
Statement of Responsibility:
by Morris W. Rubesin, Charles B. Rumsey and Steven A. Varga.
General Note:
"Report date November 9, 1951."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003810325
oclc - 133153233
sobekcm - AA00006214_00001
System ID:
AA00006214:00001

Full Text
b X-: 1 : .; ; 1;~ ~ i~i: A r ; ,
wt i "' I T i;;i q
15.. "
S ... .., :" ". "" ."
a:.i
/ il i,: ,, :: ,:, ,,


1. M. LA ?5 25a

RM A51l25a


:i" t"h .,' ."". ."..: :* :
{ i;:' ":;- :!.. "
i i: i':' .. : "


J,
: ': ":': "t
I:..r. :.'" 1 '
ili
rC.i; ;.; R~ CH
' ?; .I" ,;: ,' "
,. ;
aii i*. .,) < ;.. .,


MEMORANDUM


iJMlOARY OF AVAILABLE KNOWLEDGE CONCERNING SKIN

FReITION AND HEAT TRANSFER AND ITS APPLICATION


II, '.'TO THE DESIGN OF HIGH-SPEED MISSILES

I. By Morris W. Rubesin
i i;i:.. Ames Aeronautical Laboratory
; :lf;:i'.: :: :
Charles B. Rumsey
,.; 'Langley Aeronautical Laboratory

i:~.~..a : and Steven A. Varga
.~~' Ames Aeronautical Laboratory
|i ". i i :: = *.'


UNIVERNOSfYS FLORIDA
SCM;E'l:.ENTS DEPARTMENT
: 20 MARSTON SCIENCE UBRARY
3P PO. BOX 117011
GAtt$VL FL 32611-7011 USA

AL ADVISORY COMMI

1... 4, FOR AERONAUTICS
S. i WASHINGTON
; ;November 9% 1951
i" C "", '.; ,.' % ;, ..] .!.,. ., ,= : :" ,


TTEE


.1 t.i

... ... ...: ..".. ....
:." ',".. ,::: !." :.' .. ,,4 .. .,
::. :;i::'~ ..' ..: .:,:!.".2 .:6i~ .


. ".. :..


.; : ':. /.* :*:;


I ~1 ___


___ _~


;r ~: .


































































































































.... .. ..







NACA RM A51J25a

A SUMMARY OF AVAILABLE KNOWLEDGE CONCERNING SKIN

FRICTION AND HEAT TRANSFER AND ITS APPLICATION

TO THE DESIGN OF HIGH-SPEED MISSILES1

By Morris W. Rubesin
Ames Aeronautical Laboratory

Charles B. Rumsey
Langley Aeronautical Laboratory

and Steven A. Varga
Ames Aeronautical Laboratory


To determine the skin friction and heat transfer on the surfaces of
high-speed missiles, it is necessary to know certain characteristics of
the boundary layers. These characteristics are: the temperature
recovery, the skin-friction coefficients and the heat-transfer coeffi-
cients of both the laminar and turbulent boundary layers and, also, the
position of the transition from laminar to turbulent flow. In this
paper a review is made of the existing information concerning these
characteristics. In addition, comparison is made between existing flight
data and results computed by the boundary-layer momentum-integral method
in a preliminary attempt to establish some rational way of approaching
the design of a missile whose Mach number range and body geometry are
markedly different than those of existing data.

The problem of determining the position of transition from a laminar
to a turbulent boundary layer is very important. For flight conditions
in which transition occurs in regions other than very near the nose of
the missile, the average skin friction and heat transfer may be influenced
more by the location of transition than by the absolute values of the
skin friction or heat transfer corresponding to either the laminar or
turbulent flow. At present there is no accurate method for determining
the location of transition.

A compilation of data showing the beginning and end of boundary-
layer transition is shown in figure 1. The ordinate is the Reynolds
number based on the length along the body. The abscissa is the free-
stream Mach number. Open symbols designate the beginning of transition,
whereas the filled-in symbols designate the end of transition. Most of
these data, compiled by project Hermes, were obtained in the early stages
of V-2 flight and are, in effect, for a cooled surface.
1This is substantially a reprint of the paper by the same authors which
was presented at the NACA Conference on Aerodynamic Design Problems of Super-
sonic Guided Missiles at the Ames Aeronautical Laboratory on Oct. 2-3, 1951.







NACA HM A51J25a


Also shown are data obtained on unheated flat plates at Princeton (refer-
ence 1) and in the Ames 6-inch heat transfer tunnel (reference 2), on an
unheated RM-10 test body in the Langley 4- by 4-foot supersonic tunnel,
and on an unheated body of revolution at the Lewis Laboratory. It can
be seen that the scatter in the data is enormous; however, the general
trend of the data indicates that the Reynolds numbers of the beginning
and end of transition increase at the higher Mach numbers. That the
data shown do not correlate any better is expected. In these data no
control was made of such important quantities as surface roughness,
body shape, free-stream turbulence, and surface temperature.

One of these variables, the surface temperature, was isolated for
study in tests performed at the Ames Laboratory on a heated flat plate
at M = 2.4 (reference 2). The results of these tests are shown in
figure 2. The ordinate used is the Reynolds number based on the momentum
thickness whereas the abscissa is the ratio of the surface temperature
to the free-stream temperature. This form of Reynolds number was chosen
to localize conditions, thereby making the results applicable to bodies
of revolution with surface-pressure variations. The points at the
extreme left are for the unheated case. It can be seen that the Reynolds
numbers of the beginning and end of transition are reduced by about
50 percent from the unheated condition for a surface to free-stream
temperature ratio of 2.8. It is interesting to note that a determination
of the momentum thickness Reynolds number at the beginning of transition
on an unheated body of revolution tested at the Lewis Laboratory resulted
in a value identical to that shown for the unheated plate in this figure.
This agreement may have been fortuitous because the Mach number of the
body was 3.12 whereas that for the plate was 2.4.

Obviously, much more work needs to be done concerning transition
before an accurate means is available for predicting its position on a
missile. Because there is no alternative, it is recommended that until
more information is available the results of figure 1 be used as a guide
in design.

Before it is possible to determine the heat transfer, and often the
skin friction, it is necessary to know the recovery temperature. The
recovery temperature can be determined from the usual equation for
recovery factor shown in figure 3. In the equation at the left r is
the recovery factor, Tr is the recovery temperature of an unheated
body, To is the free-stream temperature, and Mo is the free-stream
Mach number. This figure shows a compilation of temperature recovery
factor as a function of Mach number obtained in wind tunnels at Mach
numbers below 4, and for two flight tests at Mach numbers approximately
equal to 2. These data apply to flat plates and bodies of revolution.
The length of the vertical bars, which represent wind-tunnel data, shows
the range of variation of the recovery factor with Reynolds number at a







NACA EM A51J25a


fixed Mach number. In general, the data points lie on two levels,
around r = 0.85 and r = 0.90. These levels agree with the usual
theoretical values of Pr1/2 for laminar flow and Prl/3 for turbulent
flow. The set of data for the laminar boundary layer on a flat plate
is not in agreement with the other laminar-boundary-layer data, or the
theoretical prediction.

In addition, two theoretical results showing the effect of Mach
number on the recovery factor are also indicated. The turbulent-
boundary-layer recovery factor determined by Tucker and Maslen (refer-
ence 3) by extending the approximate Squire analysis to include compres-
sibility shows a reduction with Mach number. Apparently the variation
of the recovery-factor data does not exhibit this change. It can be
concluded, therefore, that the usual theoretical value Pr1/3 be used
for the recovery factor in design work through the Mach number range,
neglecting the theoretical variation indicated. For the laminar boundary
layer, Klunker and McLean (reference 4) have shown that, under flight
conditions where extremely high air temperatures occur, the recovery
factor decreases with Mach number. These results were obtained from
the same basic boundary-layer theory which yields a recovery factor
of Prl/2 for the temperature levels occurring in wind tunnels. Thus,
the agreement of wind-tunnel data with Pr1/2 checks the basic theory.
The flight datum point shown is at too low a Mach number to indicate any
marked reduction. Since the experimental data agree with the basic theory,
the work of Klunker and McLean for flight conditions should yield satis-
factory results for design purposes at high Mach numbers.

Several theories exist for determining the magnitude of the skin-
friction and heat-transfer coefficients. For the case of flight condi-
tions where extremely high temperatures occur, the previously mentioned
theory of Klunker and McLean also provides a means to calculate the
laminar-boundary-layer skin-friction and heat-transfer coefficients.
In addition, Van Driest (reference 5) has obtained similar results by
extending the work of Crocco to include flight conditions with the
resulting high air temperatures. Although the Crocco method is restricted
in that the Prandtl number is assumed constant and the viscosity is
expressed in Sutherland's equation in terms of enthalpy rather than
temperature, figure 4 indicates that the results of average skin friction
for the laminar boundary layer at Mach numbers below 10 are within 1 per-
cent of the more exact method of Klunker and McLean. As good agreement
is also obtained for the recovery temperature and the local heat transfer,
it can be concluded that for practical purposes the two theories give
equal results.







NACA RM A51J25a


The data with which these theories can be compared are relatively
meager. Published skin-friction data on unheated flat plates (refer-
ences 2, 6, and 7) represent the average skin friction from the leading
edge to the point of measurement of boundary-layer surveys. These
average skin-friction coefficients obtained at Mach numbers around 2
are about 30 percent higher than those given by Crocco's theory made to
apply to wind-tunnel conditions. Similar results were obtained at Lewis
from unpublished data on a hollow cylinder placed parallel to the air
stream. This discrepancy between theory and experiment has been attri-
buted to the momentum loss in the boundary layer caused by the bluntness
of the sharp leading edge. Unpublished data of average skin friction
obtained at the Langley Laboratory on a 60 wedge in a flow at a Mach
number of 6.9 exceeds by about 14 percent the estimated theoretical
value based on the Crocco method when the wedge is at a zero angle of
attack. Further unpublished tests at the Lewis Laboratory have indicated
that the laminar-boundary-layer theories compare favorably with the
experimental average skin-friction coefficients determined experimentally
on a cone-cylinder body at M = 3.85. Although no local skin-friction
data have as yet been correlated with the theory, local heat-transfer
data shown in figure 5 have been determined on a cone having approximately
a constant surface temperature (reference 8). The data are, on the
average, about 12 percent lower than those given by the Crocco theory,
corrected to a cone. In general, it can be concluded that the Crocco
theory predicted the skin friction and heat transfer within engineering
accuracy up to a Mach number of 7, for the wind-tunnel tests. It then
would be expected that the theories for the laminar boundary layer for
flight conditions are adequate for design.

For the turbulent boundary layer there are several theories from
which the skin friction and the heat transfer on flat plates can be
calculated (reference 9). Each of these theories indicates a marked
reduction in the skin-friction and heat-transfer coefficients with an
increase in Mach number or surface temperature. Because of the large
effects indicated by the theories and because they are of a semiempirical
nature it is important to compare them with existing data. This com-
parison is made in figure 6 for the case of an unheated flat plate in a
wind tunnel at a Mach number of 2.4 (reference 9). It is observed that
the average skin-friction coefficient is reduced from the values of the
incompressible case; however, the reduction estimated by Von Karman was
not realized. The compressible theories for turbulent flow on a flat
plate give good agreement with the data over the range of Reynolds numbers
below 6,000,000. In figure 7 are shown unpublished local skin-friction
data obtained on unheated cylinders with their axes placed parallel to
the air flow. The Mach number of these tests was 3.1. The abscissa used
in this figure is the Reynolds number based on the momentum thickness.
This characteristic dimension was used to avoid the necessity of knowing
the exact location of transition. In the lower figure there are shown







NACA RM A51J25a


the data with natural transition. These data agree approximately with
the compressible flat-plate theories at the lower Reynolds numbers.
Beyond a Reynolds number of 6000 the data drop off toward the Von Karmxn
estimation. The data with artificial transition shown in the upper
figure exhibit a reduction from the incompressible case; however, the
data have a different slope than any of the theories and give no insight
into which of the theories agree best with the physical phenomena.

Figure 8 is intended to show that a modified Reynolds analogy exists
at a Mach number of 2.4. This unpublished datum point was obtained on
a cooled flat plate in the Ames 6-inch heat transfer tunnel. The ordinate
is written in a fashion which permits comparing heat-transfer data with
theoretical skin-friction computations through a modified Reynolds
analogy. The abscissa used in this figure is the Reynolds number based
on the momentum thickness to avoid the necessity of knowing the location
of transition. The single datum point of heat transfer compares favor-
ably with the theories of Frankl and Voishel and of Van Driest.

In general, it can be concluded from the last three figures of
wind-tunnel data that the compressible-turbulent-boundary-layer theories
represent the available data of skin friction and heat transfer on flat
plates with an accuracy sufficient for design. The same cannot be said
from the data obtained on cylinders with their axes parallel to the air
stream, except for the data obtained with natural transition at Reynolds
numbers below 6000 when based on momentum thickness which did agree
fairly well with the theories.

Data of skin friction and heat transfer have been measured in flight
on the RM-10 missile, the earliest of which are included in references 10
and 11. Figure 9 shows time histories of the flight characteristics for
a typical boosted PJ~-l0 flight during which average skin-friction coef-
ficients were obtained from boundary-layer rake measurements to a maximum
Mach number of 3.7. The characteristics shown are a surface temperature
parameter, the Reynolds number based on the length to the rake location
just ahead of the fins, the Mach number, and the average skin-friction
coefficient.

The surface temperature parameter shown was used since its numerical
value indicates the magnitude of the heating regardless of Mach number
and indicates cooling and heating of the boundary layer by negative and
positive values, respectively. The experimental skin-friction coefficients
are from 20 to 30 percent higher than Van Driest's theoretical prediction
for a flat plate at the test conditions, except near peak Mach number.
During the first part of the test which is after booster separation but
prior to firing of the sustainer rocket, transition would be well forward
on the pointed nose of the missile so that close to 100 percent of the skin
area would have turbulent boundary layer and the measured values would be







NACA RM A51J25a


average turbulent coefficients. It is expected, however, that at the
high Mach numbers during sustainer firing, the strong cooling of the
boundary layer indicated by the surface temperature parameter would
stabilize the laminar boundary layer and cause transition to move back
on the body. The measurements during this part of the flight would
thus be lower than average turbulent coefficients. During the period
after sustainer firing, the heating parameter became less stabilizing,
and transition would be expected to move forward causing a relative
rise in the average coefficient. These trends are shown by the data.

At a time of about 23 seconds, the heating parameter became positive,
or destabilizing, and nearly all of the skin area would again be covered
by turbulent flow. The 20 to 30 percent difference shown between turbulent
flat-plate theory and the data for times of nearly complete turbulent
boundary layer is attributed to the missile geometry and to the pressure
distribution at the flight Mach numbers.

Figure 10 shows unpublished flight conditions and results from a
cylindrical body with an ogive nose. This configuration more closely
approximates a flat plate. The measured values of average skin-friction
coefficient are relatively lower than the RM-10 results and are in close
agreement with Van Driest's flat-plate theory. The extent of laminar
flow on this model is believed to have been small because of the values
of Mach number and Reynolds number, at least during the first half of
the test.

Presented in figure 11 are values of average skin-friction coef-
ficient at the condition of zero heat transfer which have been obtained
at four points in the skin-friction tests, all occurring at a Reynolds
number of about 60 x 10 but at different Mach numbers from 1.1 to 3.
Also shown is a value at zero Mach number and 60 x 106 Reynolds number
which was recently obtained from rake measurements in under-water tests
performed on an RM-10 body in the Langley tank no. 1. A flat-plate
theory is also included to show its variation with Mach number. Below
Mach number 1 no reduction is shown by the data. From Mach number 1
to 3, the data show a reduction of about 30 percent whereas the flat-
plate theory shows a reduction of 35 percent.

Local heat-transfer coefficients measured in flight on the
RM-10 missile are shown in figure 12. The data are plotted as NuPr-/3
against Reynolds number with the velocity and air properties based on
free-stream conditions. Above a Reynolds number of about 6 x 106 the
heat-transfer coefficients are for turbulent flow. Below approximately
2 x 106, the coefficients measured on the nose of model C show a
decrease from the turbulent correlation indicative of laminar flow. The
values are, however, considerably higher than the laminar theory for a
cone. Plotted in the present manner the data lie midway between the
laminar-boundary-layer theory for a cone and the measured turbulent data.







NACA RM A51J25a


The heat-transfer data for turbulent boundary layer can be repre-
sented to t7 percent by a line having the equation indicated. These
data were obtained over a Mach number range from 1 to 2.8 and at several
stations along the body as indicated in the legend. It is interesting
to note that, for three models, almost all of the data for all stations
along the length of the body and over the complete Mach number range
agree to within 7 percent.

It is concluded from the flight-test data that for missiles not
greatly different in shape from the RM-10, and for conditions similar
to the test range, heat-transfer characteristics for turbulent flow can
be obtained from the RM-10 equation for design purposes. It should be
emphasized that the heat-transfer data do not explicitly show a Mach
number effect in the range of Mach numbers tested. The test data
further indicate that the skin-friction coefficients can be obtained by
reference to the flat-plate theory in the following manner. For ogive-
cylinder bodies practically no modification to the theory is necessary.
For bodies of higher fineness ratio than the RM-10 it would be expected
that the values of skin friction are between those of the RM-10 and the
flat plate.

In view of the conclusions drawn from the flight-test data, it is
apparent that some rational method is necessary for extrapolating the
known data to blunter bodies or to bodies flying at flight conditions
much different than those of the available tests. As the flat-plate
theories including compressibility agreed well with the skin-friction
data obtained with the ogive cylinder, it was believed that some method
accounting for body shape might bring the theories in line with the data
obtained on the RM-lO, thereby extending the scope of the data. There-
fore, computations were made of heat transfer and skin friction for the
RM-10 shape and flight conditions by means of the well-known momentum-
integral method using the Frankl and Voishel flat-plate theory.

The momentum-integral method consisted of solving the equation
shown in figure 13. This equation relates the rate of growth of the
boundary layer with the compressibility effect, the acceleration of the
air outside the boundary layer, the geometry of the body, and the local
skin-friction coefficient. The solution of this equation is obtained
through the use of the flat-plate relationships of the skin-friction
coefficient and the Reynolds number based on the momentum thickness.
The solution yields the distribution of the momentum thickness, from
which the skin-friction coefficient can be determined. The local heat-
transfer coefficient is obtained from the local skin-friction coefficient
through a modified Reynolds analogy.

In figure 14 there is shown a comparison of some preliminary results
of the momentum-integral method with the local heat-transfer coefficients







NACA RM A51J25a


measured on the RM-10. For the theoretical computations the local skin-
friction coefficient was expressed in terms of the Reynolds numbers based
on momentum thickness according to the flat-plate theory of Franki and
Voishel. The ordinate shown is the local heat-transfer coefficient.
The abscissa is the dimensionless length along the body. Two sets of
data are shown; the upper set is for a Mach number of 2.3, whereas the
lower is for a Mach number of 1.02. It should be noted that the Reynolds
numbers of these data are roughly in proportion to the Mach numbers. The
solid lines represent the distributions given by the equation representing
the bulk of the RM-10 data. The dashed line represents the results
obtained from the momentum-integral method. At the lower Mach number,
and consequently the lower Reynolds number, the momentum-integral method
agrees well with the data and the RM-10 equation. The results of the
Van Driest flat-plate theory for these conditions were about 10 percent
lower than the data along the entire body. At the higher Mach number
the momentum-integral method gave results which are about 15 percent
higher than the data on the front of the missile and about 3 percent
higher than the data towards the rear of the missile. The data apparently
do not show the geometry effect expected from the momentum-integral
method. In fact, the Van Driest flat-plate theory gives results which
pass through the data near the front of the missile and then drop to
values about 3 percent low in the rear portions of the missile. From
the latter results it can be concluded that, for slender bodies such
as the RM-10, the RM-10 equation or flat-plate theory represents the
data as well as does the more tedious momentum-integral method at a
Mach number of 2.3. The momentum-integral method may become necessary
for blunter bodies.

The momentum-integral method is evaluated further in figure 15.
The ordinate shown is the average skin-friction coefficient and the
analogous heat-transfer parameter. The abscissa is the Mach number. The
average skin-friction data shown are for the RM-10 at recovery temperature.
These data were shown previously in figure 11. The average heat-transfer
parameter shown was evaluated from the RM-10 equation and it is noted
there is no Mach number effect in this equation representing the bulk of
the RM-10 data. For comparison, two flat-plate theories corresponding
to the recovery temperature are included. These theories are for condi-
tions comparable with the curve of the RM-10 heat-transfer parameter
because the same value of the parameter was obtained under conditions
of both cooling and heating.

It is noted that the momentum-integral method does not reconcile
the flat-plate theories with the characteristics of the RM-10 data in
that first, the momentum-integral method does not sufficiently increase
the values obtained using flat-plate theory to agree with the RM-10 skin-
friction data, and second, the Mach number effect remains in the
theories even when based on the momentum-integral method thereby resulting







NACA RM A51J25a


in a lack of agreement with the RM-10 heat-transfer equation. It is
apparent that no conclusive method for extrapolating existing data to
greatly different conditions can be given at present.

REFERENCES


1. Ladenburg, R., and Bershader, D.: Interferometric Studies on Laminar
and Turbulent Boundary Layers along a Plane Surface at Supersonic
Velocities. Symposium on Experimental Compressible Flow. U. S.
Naval Ordnance Lab. (White Oak, Md.), June 29, 1949, pp. 67-86.
2. Higgins, Robert W., and Pappas, Constantine C.: An Experimental Inves-
tigation of the Effect of Surface Heating on Boundary-Layer Tran-
sition on a Flat Plate in Supersonic Flow. NACA TN 2351, 1951.

3. Tucker, Maurice, and Maslen, Stephen H.: Turbulent Boundary-Layer
Temperature Recovery Factors in Two-Dimensional Supersonic Flow.
NACA TN 2296, 1951.
4. Klunker, E. B., and McLean, F. Edward: Laminar Friction and Heat
Transfer at Mach Numbers from 1 to 10. NACA TN 2499, 1951.

5. Van Driest, E. R.: Investigation of the Laminar Boundary Layer in
Compressible Fluids Using the Crocco Method. Rep. AL-1183, North
American Aviation, Inc., Jan. 9, 1951.
6. Blue, Robert E.: Interferometer Corrections and Measurements of
Laminar Boundary Layers in Supersonic Stream. NACA TN 2110, 1950.

7. Low, G. M., and Blue, R. E.: Measurements to Evaluate Leading Edge
Effects on Laminar Boundary Layer on Flat Plate in Supersonic Stream.
(Prospective NACA paper)
8. Scherrer, Richard, Wimbrow, William R., and Gowen, Forrest E.: Heat-
Transfer and Boundary-Layer Transition on a Heated 200 Cone at a
Mach Number of 1.53. NACA RM A8L28, 1949.
9. Rubesin, Morris W., Maydew, Randall C., and Varga, Steven A.: An
Analytical and Experimental Investigation of the Skin Friction of
the Turbulent Boundary Layer on a Flat Plate at Supersonic Speeds.
NACA TN 2305, 1951.

10. Rumsey, Charles B., and Loposer, J. Dan: Average Skin-Friction Coef-
ficients from Boundary-Layer Measurements in Flight on a Parabolic
Body of Revolution (NACA RM-10) at Supersonic Speeds and at Large
Reynolds Numbers. NACA RM L51B12, 1951.
11. Chauvin, Leo T., and deMoraes, Carlos A.: Correlation of Supersonic
Convective Heat-Transfer Coefficients from Measurements of the Skin
Temperature of a Parabolic Body of Revolution (NACA RM-10). NACA
RM L51A18, 1951.








NACA RM A51J25a


-TREND OF END
OF TRANSITION


-TREND OF
BEGINNING
OF TRANSITION


TRANSITION
BEGINS ENDS
o a HERMES (V-2)
A4 PRINCETON U.
(FLAT PLATE)
h L AMES (")
a LEWIS
(BODY OF REV.)
0 LANGLEY(RM-IO)


I I I I I
I 2 3 4 5
MACH NUMBER, M.


Figure 1.- Location


of boundary-layer transition on different
flight and in wind tunnels.


bodies in


FLAT PLATE
Me = 2.4







- END OF TRANSITION


HEATING START OF


TRANSITION


22 24 2.6 2.8 3.0


Figure 2.- Effect of heating on boundary-layer transition.


5x106 -


XUwPc
Rx .


2x106 -


5x1O5
O


2800 r


EATING


2400


2000


1200


800


400
2.C


I I I f I


:)








NACA RM A51J25a


RECOVERY FACTOR


.90 -

.88 'lI


TII(I(rFR MBA~IIP F3


(TURBULENT) 2
S- ---= /PR

i TURBULENT LAMINAR
== FLIGHT O
S WIND TUNNEL '
KLUNKER 8 McLEAN '-..,
LAMINARR)


N.>


MACH NUMBER, M,


Figure 3.-


Compilation of temperature recovery factors and comparison
with theories.


PREDICTED SKIN FRICTION IN FLIGHT


T.
a


KLUNKER
McLEAN


/ISULAED I


iKcn a McLEA 1
IN,
VAN DRIEST INSULATED-1
(CROCCO'S METHOD)


NO DISSOCIATION


MA 2 3 4 5 6
MACH NUMBER, M,


7 8 9


Figure 4.- Comparison of the theories of Van Driest (Crocco) and of
Klunker and McLean for flight conditions.


11

a.
I-
1.


1.4



1.3


1.2-


1.1


1.0



.9


r\LUn


-S


I I I


,,








NACA RM A51J25a


HEAT-TRANSFER
PARAMETER
Nux
Rx!.


CROCCO THEORY
(WITH CONE CORRECTION)



CONE DATA


I I I I J
.2 .4 .6 .8 1.(
X
DIMENSIONLESS LENGTH, L
L


Figure 5.- Comparison of Scherrer's cone data with Crocco's theory.


UNHEATED PLATE
M,=2.4


INCOMPRESSIBLE


VAN DREIST
'""'7


--..


VON KARMAN-'.


3 4 5 6 7 8 9 10I


Figure 6.- Comparison with theories of experimental average skin-friction
coefficients on a flat plate.


.006

.005


CF .004


.003



.002


nnmrI


FRANKLY
AND
VOISHEL-


1.5x 10 2


a I I I I I








NACA RM A51J25a




.0015r


.001
Cf

000
2

.0005


.00151


DATA OF BRINICH AND DIACONIS
ARTIFICIAL TRANSITION
-- a V IA


I reT


--FRANKL AND
VOISHEL


WILSON-

VON I
I ,


I I I I I I I I I


NATURAL TRANSITION


Figure 7.- Comparison with theories of experimental average skin-friction
coefficients on a cylinder in axial flow.


M, 2.42

= 2.00 (COOLING)
T.


Cf =2Nu
Rx PrI


FRANKL


.002


FLAT PLATE
EXPERIMENTAL POINT


.001J-----
0013 R 104


Figure 8.- Comparison with theories of experimental local heat-transfer
coefficient on a flat plate.








NACA RM A51J25a


-%5
Tw-Tr 0
Tr-Ts -5
-1.0
150x0- 6 RX BASED 01
100 RAKE STATIC
50
0 --1-I----------
3.5
3.0-
M. 2.5-
2.0-
1.51-
1.0-
S -EXPERIMENTAL-
.002- ---

VAN DRIEST
CF .001 FLAT PLATE THEORY-'

0 --- ----------------1
0 4 8 12 16 20 24 28 ~
TIME, SEC


Figure 9.- Flight data from the RM-10 missile and
mental average skin-friction coefficients


comparison of experi-
with theory.


TIME HISTORIES OF OGIVE-CYLINDER BODY TEST


Tw -T OT f
T,-T 0
Tr -T8 -1.0
150 x 106
R 100
50
01


2.5
2.0
Me 1.5
1.0
.002
CF
.001

0
0


Rx BASED ON LENGTH
TO RAKE STATION -10.42 FT

I I I I _


__--- EXPERIMENT

AN DRIEST FLAT
PLATE THEORY
I l I I I
I 2 3 4 5 6
TIME, SEC


Figure 10.- Flight data from ogive-cylinder body and comparison of
average skin-friction coefficients with theory.


N LENGTH TO
N. 10.42 FT


1 I


z









NACA RM A51J25a


R~ 60x106


RM-IO RESULTS

--VAN DRIEST FLAT-PLATE THEORY

VAN DRIEST FLAT-PLATE THEORY


Figure 11.- Comparison with theory of experimental average skin-friction
coefficients on the RM-10 in flight.





106
MODEL STATION < M<2.8
A, B, C, +
o 8.9 +
a 9.2
O 14.3 1 .
X 17.7 NuPr 3-.0296 R
17.8 NuPr 3=.571R2
o 18.3
104 o 36.2 CROCCO'S LAMINAR
D 49.9 THEORY CORRECTED
o 85.3 FOR A CONE
A 86.1
+ 87.3
Nu Pr 3 123.5 / y


REYNOLDS NUMBER


10


Figure 12.- Heat-transfer results from RM-10 flight tests.


003


002

CF


.001 F







NACA RM A51J25a


d8+ H+2-M2 dMI dr Cf

dx IMI (I+ Mij) dx r dx 2








Figure 13.- Integral-momentum equation for bodies of revolution in
compressible flow.


FLIGHT DATA
o M = 2.3
o M,= 1.02


.04 F


h .03
BTU
FT2 SEC "F
.02


RM-IO0 EQUATION-- o
Z -MOMENTUM INTEGRAL BASED-
/m ON FRANKL a VOISHEL THEORY
EL E U L -


0 .1 .2 .3 .4 .5 .6 .7 .8
XL NAC


Figure 14.- Comparison of experimental and theoretical heat-transfer
coefficients on RM-10.







NACA RM A51J25a


FRANKL-VOISHEL AVERAGE HEAT-TRANSFER
(FLAT PLATE) PARAMETER EVALUATED
FROM RM-IO EQUATION

.002 -
CF VAN DRIEST
S(FLAT PLATE)
or
2Nuav
Rx Pr3 MOMENTUM-INTEGRAL METHOD
.001 USING FRANKL-VOISHEL THEORY-


oAVERAGE SKIN FRICTION
ON RM-IO IN FLIGHT
0 I
0 I 2 3 4
M,


Figure 15.- Comparison with theories of average skin-friction coeffi-
cients on RM-10 in flight.


NACA-Langley 1-31-55 150











to TO,5
4 0 S ar-4C6
-I~~ -3 i5- 6-r.S



e g g"
^l-itr 7r i4~ir


0C. a 4)0.
x >M 4 ;5wI>



v 7- P

896=05wwo Mil ll. 0 43) :b iiMOOMN
02 m 5d
iiS|-is lil^Bp l l
a m t ,S LO F. 0~

74 11-9 1 I 11E! 1
Illdl 50 0: tol iE;i
9:o? ECoo 2 M 0
S 0 v a 0 ..O` 0o c a
6', 0j W O Ell 0- 'u] ) bg) E09 IOU -tl 0 o





--40M Q g? >s ~ e sa I r. -i g Q ? 0 C) Oc 0 0) 0
ew1U 1 ES 1Fo
CO CC a 'a 0

"i 1 o E) Cl o I T 3
wo CC E,
0 to L tka bll


1 0 cd 0. Z kl0l4~Ui~



I1 Si MQ C ^1 S^ k
ze o wE <0 LO k.' <
0 W m ~ D ~ ~,.




0-0









Ual 14J -0 v. 2 |6 d P4 :8
.^6- Sa-: 5 g j1-" *<; / e -^'SS-- a<
V2 mcm Bn ioa g i i U g 3 S






-. Z0 0 0:-
E0 E
















i. .. I1 i; i
D 0 .2 e -X ;h:0 -





=0o ho '0 r.6M 1 4 W
0 a









cu.= E -1 I:


IDa^ Vg~ l'a -- il -UgS tCi^ 58. L,










i~ i
03~~~P k o014 0 k&
kIso |s| 0 i I ||
um 1 90 : E: c



























sC ComS" S- ,C go^ 14o >>2 'S S ~ "- 5
H'i~3 u2 ;5 2 5 2
~.cc 0 14g T; ~ C ~ o 41 w2
04 i Mqo 00 b DCi Ci
k 0sS s s w &s S p, 1,s1 C
4) 0.4 0 9 5Cw 0=lR 0











o~~C~Co C4 Co 4 o~o~
%%s, Ld k w 2 0o 1 a)1 Q EC
ZZ.EU t CZ OM oa S
$~~2~ gC) 11:5 0 'Q.) v rn""" r
LO 0 k~
4, 0 i .IH 8V z 7,b
,0 E-0 Vo 0, OWZ~Ovli8"B~e0'"m
g u 0 0" ti 0 *g
.4 0 m 0 = ;. t rl 0 0 0
pq~ D 5 P-0Z' to 8iA

Op 4 --01 4-.
w .- ics
bo g -.9A bD
-5 4 o O. 21: m4 p )
ul m0 l
.0 UM'a-P-0a
cqF > I 0e kP
M-m kk 4 U3 :5104
~~Q0 w C: kv -0 0m~


(L a t!l
0 : .4 4) U5~m~ P *L
cc k M t 0 pl, C
W, 0 w w 0 a iii
z C c -w -4 v 4 --(au 16











C44 m; ot-:

*s: 0w^ Tt :a"t
IZ
^:. ke2.< ta



11" Ii ul u n li
to.. 0 z rn ~ E

fc t 2 o 13 o t? iV 1 a- o to f. t hor (
0 3 8 CO to m


cz w
^Nc w Q) 5 ^< *;Mr





VD b0r.E

R Egz I; r 0 0
2r~~~~c toca~~to




cl ~ g .- | OZ0ll2-
|l^s^ ~~~ *sii|0 ,|ij ca~~j1;
M,.o' kt 0L'. -0 cd bau~ D0~-~t
~~~~ m- D 0 tb 0


ilga^40 i^3l^ & & i i ii^ Ii




01I I I E4~f.'

$' I01 ^o2 u 55 TT S
0 f^ ~ 4) 4) 0 V Z) -,06I
<0~~ Q 0'








*I tl01 \\
bO~W k -~oI tog2' 80





?| ,2 i b., I~ g ^ ^z
z >~











S~fc- I2 toao^- hsa iaHh
E-4. 0~c e-, 0 LO E
ul u










to 9: 0oc V) 06 to 0a
C4 s. 4 F. ~.


Ell
1s ..'




01

001V





|)|U
Ow to s.




0AD w


o ca-=
g|b^ t
S.. l.-q
0 01 01e





< 0 M J0


I 0 M t0 >




tbD UC

0 zB 2 4&.^
9a co2 f
o440* *'^


to w~
03
Cp~t
QS|
LV Go ^ as


0 (D 4
94.

01
01^
S be a
'i o1-
to- C
2. t A
5. fc, _






o _3 o g
O) ^. 01~


01 0 .0
4..,.t
01... 01
U to AD

















,v t
S5". c5 ^ i .





o-.0 OF. olbm
2|


- 0 0 .4 0 m
Fit mC W.C >
.gS Suvi4 4 ;


0 to



<0
f f, 0


E. I.. 0 1i
o wi 1, *



H 0 MQ









.4 t
cq ) M. >
*Sm M n&I

s z^-s
MQW ~


c'E;xa .4 a
z u |M


0
kt4


S0~
r- 0






CD
V 02
" .. LI'*
. -o 0







ho Q" ) M>
QL rI

u2'C
o.| o
0""'
*^l
o 0 a *2




0 ..> -.0

L) C L. <


E-4 't w





o"! S bO
4: m a




.4^ C4 i 4 tO ai rlbr
.IhoaM.fi




0-I .n *<- S Cu 4


0- (D 0IC
-" -? a-.-



9 f 0 l< 0 -





C.) !CO F4 W t
;a a) O3;s Cu4:<


~.>
E. ? .; h ;<




0g0!




(9 u




LO .. j 0
V 0 M
< _1< 43*

za

&0 s


Cd

I
ho



o. 00






EG U





*- 0 I
4 ) a


0 ;*, .1.o
0 .. 0













































































































































it.. ...




IU,!;