Heat transfer and skin friction for turbulent boundary layers on heated or cooled surfaces at high speeds

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Material Information

Title:
Heat transfer and skin friction for turbulent boundary layers on heated or cooled surfaces at high speeds
Series Title:
NACA RM
Physical Description:
20 p. : ill. ; 28 cm.
Language:
English
Creator:
Donaldson, Coleman duP
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: The method presented in NACA TN 2692 for evaluating the skin friction of a turbulent boundary layer in compressible flow on an insulated surface is extended to evaluate the turbulent skin friction and heat transfer in compressible flow on a surface which is heated or cooled. The results of this analysis are in good agreement with the heat transfers measured in flight on the NACA RM-10 missile up to Mach number of 3.8.
Bibliography:
Includes bibliographic references (p. 16).
Statement of Responsibility:
by Coleman duP. Donaldson.
General Note:
"Report date October 2, 1952."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003809208
oclc - 133142839
sobekcm - AA00006185_00001
System ID:
AA00006185:00001

Full Text
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SEARCH MEMORANDUM





S r TRANSFER AND SKIN FRICTION FOR TURBULENT
i!i .. .
BOUNDARY LAYERS ON HEATED OR COOLED

SURFACES AT HIGH SPEEDS
. j : ."



By Coleman duP. Donaldson


:. Langley Field, Va.


UNIVERSITY OF FLORIDA
.. DOCUMENTS DEPARTMENT
::120 MARS.TON SCIENCE BRARY

P INESVILLE, F 32611-7011 USA


piON AL ADVISORY COMMITTEE
OR AE RONAUTICS
S WASHINGTON
"' .October 2, 1952
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NACA RM L52B04

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


RESEARCH MEMORANDUM


HEAT TRANSFER AND SKIN FRICTION FOR TURBULENT

BOUNDARY LAYERS ON HEATED OR COOLED

SURFACES AT HIGH SPEEDS

By Coleman duP. Donaldson


SUMMARY


The method presented in NACA TN 2692 for evaluating the skin fric-
tion of a turbulent boundary layer in compressible flow on an insulated
surface is extended to evaluate the turbulent skin friction and heat
transfer in compressible flow on a surface which is heated or cooled.

The results of this analysis are in good agreement with the heat
transfers measured in flight on the NACA RM-10 missile up to Mach num-
bers of 3.7.


INTRODUCTION


Within the last decade the tremendous increase in flight Mach num-
bers that has been achieved has placed the problem of aerodynamic heating
among the most important of the present time. In particular the problem
of the heat transfer through turbulent boundary layers at high speeds has
received considerable attention. It has long been known that the heat
transfer through turbulent boundary layers is so intimately connected with
the skin-friction force exerted by the boundary layer that the two problems
are almost one.

The analysis of the turbulent boundary layer presented in NACA
TN 2692 (ref. 1) permits an easy extension of the turbulent skin-friction
law from incompressible to compressible flow. The analysis led to expres-
sions for the extent of the effective laminar sublayer 6L and for the
velocity uL at the point 5L. These results permitted the skin-friction
law to be derived. If use is made of the quadratic dependence of temper-
ature on velocity derived by Crocco in reference 2, the results of ref-
erence 1 may be extended to obtain the heat transfer through the turbulent







2 NACA RM L52H04


boundary layer in both incompressible and compressible flows. It is
the purpose of the present short analysis to make this extension.


SYMBOLS


parameter,


1

[n(r- l) 0 1
k2 uo6


2r,
skin-friction coefficient, 2Tw
pouo


r i .f.(7 1)M2 uL Tw Tadw
F = 1 + + +
S 2 u\ o To



Sr.f.( 1)M L Tw T


L0.56



U 0.448

(i_)


K constant (0.045)

k,m,n,r constants

M Mach number

qx
Nx Nusselt's number, --
K AT

N5 Nusselt's number, qb-
K LT

q local rate of heat transfer

r.f. recovery factor

Rx Reynolds number, u-
Vo

RE Reynolds number, u--
Vo
T absolute temperature







NACA RM L52HO4


u velocity in x-direction

x distance along surface

y distance normal to surface

7 ratio of specific heats

b boundary-layer thickness

e boundary-layer momentum thickness

K thermal conductivity

p viscosity

v kinematic viscosity

0 density

T shear stress

Subscripts:

av average value

adw adiabatic wall conditions

L conditions at edge of laminar sublayer

o free-stream conditions

w wall conditions


DERIVATION OF THE HEAT-TRANSFER LAW FOR TURBULENT BOUNDARY LAYERS


Before the present analysis is presented, a summary of the findings
of reference 1 is in order. It was found that for boundary layers of
the type



o n
U( ()







NACA RM L52H04


the dimensionless effective extent of the laminar sublayer


given by


Sn
L n(r 1) L n+l
& k2 ubt


and the velocity at the point 5L was given by


LL [ln(r- 1) Ln"1
uD k2 uo, 1


With these results, it was reasoned that the skin friction was


= "lL
5L


which, when the necessary substitutions had been made, resulted in
following skin-friction formula

l-n 2 2
Cf r(r n+1 o n1 L ( n1l
k2 O- ) Sie



Since


PL TL
0o TL


and if it is assumed that


was


the




(5)







(6)


PL TL
0 (To)L m


Tw w
w







NACA RM L52H04


then


1-n
cf = 2jn(r 1) n+l
k2 ]


2 n-2m-1
'n+ T n+l
RE 6 T


For a velocity profile having n = 7, it was found in reference 1 (from
skin-friction measurements in incompressible flows) that

n(r 1) = 158
k2


so that the local skin-friction coefficient for the compressible case
became


cf = 0.045 R5


-1/4 /(T)0.56
L


With these results it is now possible to pass on to the derivation of
the heat transfer through the turbulent boundary layer.

If a temperature distribution in the boundary layer similar to
that used by Crocco (ref. 2) for Prandtl number = 1 and zero pressure
gradient is adopted; namely:


T =A + B -+ C 2
the temperature in the bodary layer may then be written
the temperature in the boundary layer may then be written


T = (Tw Tad) (Tadw To) 2


(10)


It might be noted that, although the quadratic form of the tempera-
ture dependence on velocity was derived under conditions of zero pressure
gradient, this form appears also to be a very useful relationship for
most supersonic-missile shapes where the pressure gradients are generally
quite small.






NACA RM L52HOh


Differentiating equation (10) and evaluating it at the wall where
u- = 0 gives
uo

Tw TUo Ld (11)



Hence, the heat transfer at the wall is


-q= Y/= w(Tw Tadw (12)
Ww Uo w


From equation (4) it will be seen that


S=- L (13)



so that


q, L(T Tadw)L (14)
P w6L uo


If the values of 6L and UL that are obtained from equations (2)
uo
and (3) are substituted into equation (14) and the heat transfer is
expressed in terms of a Nusselt number based on the dimension 6, the
result is

l-n n-1
N q (r 1) n+lUo l n+ w
Ng ; 1)] (15)
T5 2 J/ o w

If it is assumed that

K- -- (i6)
K0 4o







NACA RM L52HO0


equation (15) may be written


1-n
n(r 1)n+
k2


Substituting the value


n-1
of (v)n+l
(TL0)


and IL
Io


obtained from the use of


equations (6) and (7) results in


1-n
S[n(r 1 n+l
N [ k2


n-1 n-2m-1
n+l n+1
R5 iT -


(18)


Writing the equations for dimensionless skin friction and heat


transfer together


where


1-n

and expressing 2 nr n+
L k2 J


2
cfF = Constant X Rg n+l

n-1
N8F = Constant R n+l
2



n-2m-1
F T- n+l
F T
Vo/


as a constant yields


(19)


(20)


(21)


Since the right-hand sides of equations (19) and (20) are the
incompressible expressions for cf and N5, respectively, the effect
of Mach number and temperature potential must be contained in the


n-l n-1
5 n+-- O n+1
R6 n+ I


(17)







NACA RM L52H04


factor F. This factor may be evaluated in the following manner.
equation (10)


From


TL = Tw_ (Tw Tadw)
To To To


(Tad U Lo 2
To V


which may be written


L 1+ r.f. (_ 1)M2 1- L2
1--+ -(y 1wl ii
To 2 u ou


The factor F therefore becomes


F = 1 + r.-- (7 1)M2 (u)
2 \uo)


T, Tadw 1
To
w o j


Tw Tadw 1
10


It may be seen that the second term within the brace represents
the effect of Mach number, and the third term the effect of temperature
potential on the skin-friction and heat-transfer coefficients. The
value of is usually of the magnitude 0.4 and although it depends
uo
Tw Tadw
upon M, Rb, and it generally does not have a first-order
To

effect upon the factor F. The value of LL, however, must be found
uo
in the following way:

Since

1
n+1
uL (r 1,
o k2 u6(3)


(22)


(23)


n-2m-1
- n+
uL
Ou


(24)







NAC


-~ -1)
h2


1 1
n+l Vn+1

R9-6 vo


1

where the value of --V
7O 1


has been expressed in terms of temperatures


by means of equations (6) and (7) and where


A n(r 1) 1
k2 R5



Now, if the value of TL given by equation (23) is substituted in
To
equation (25) there results


1
UL = A n+1 + r M2 -




uL
This equation for may be expressed as
uo


n+1
/u \m+l
I- l

m+1
1/
A 1


1
Am+l r.f.(y 1)
2
2


+Tw Tadw
To


1
m+l Tw Tadw
+A
To


M2UL 2
M -
\ uo


(26)


Sm+
u\ n+l



(27)


UL
uo


+ r.f.(-2 1) M2 + w Ta = 0
2 To /


UL
and solved graphically for for given values of Rg,


(28)


M, and


Tw Tadw
To


1
n+1
=A


m+1

(TLn+1
LTO!


(25)


A RM L52104






NACA RM L52H04


For the particular case of a turbulent boundary layer with a one-
seventh power velocity profile in air when = 1.4 and m may be
taken as 0.76 (an approximation which is usually adequate for calcula-
tions such as these), the equations necessary for the calculation of
heat transfer become


NF = 0.02 0.75
NKF = 0.0225R&


T Tadw
To
1


0.56
UL
- o
L"or


(29)


(30)


uL
and is found from the following equation
Uo


( 4.55 18 0.568
w, / (R r


r.f.M2 2UL
5 VUQI


0.568 Ty Tadw L
(& r T \uT -


158\0.568
RS


r.f.M2 + Tw Tadw o0
5 To


uL
Tables I and II give values of -UL
uo


S\2
and (-z}
\U


found by using


equation (31) for Mach numbers up to 5 for a range of values of Rg
T -T ri
and Tw Tadw that are useful in heat-transfer calculations.
To


COMPARISON WITH EXPERIMENT


In order to compare the results of this analysis with experiment,
it is necessary to have measurements of local rates of heat transfer
for conditions of turbulent flow when the local values of Rg, M, and


where


(31)


r.f.M2 (uL
F= 1 +1- U







NACA RM L52HOh 11


T, Tadw
are known. This type of information for a range of Mach
To
numbers, Reynolds number, and temperature potentials can be obtained
from the heat-transfer and skin-friction measurements made on the NACA
RM-10 missile in flight and reported in part in references 3 and 4. For
two of the body stations at which local heat transfers were measured in
reference 3 (stations 85 and 122 inches from the missile nose) the bound-
ary layer had been surveyed under similar conditions for the skin-friction
study reported in reference 4, so that the boundary-layer thickness 5
and hence RE were known. The pertinent measured quantities at these
stations for several flight conditions are given in table III. The first
six points are taken from data published in references 3 and 4. The last
four points are computed from heat-transfer data not yet published. The
correlation of these data by the present method is shown in figure 1
wherein the measured and theoretical results are plotted in the form NBF
versus RS. A reasonably good correlation of the data is obtained for
Tw Tadw
Mach numbers ranging from 1.6 to 3.7 and temperature potentials
T
ranging from 0.15 to -1.8. In comparing these experiments with the
theory, it was assumed that the boundary-layer profile had a one-seventh
power profile even though the surveys show that the power of the boundary-
layer profile varied somewhat from test to test. This is not considered
to be a serious matter in making a comparison between the theory and
experiment, as experience has shown that small variations of profile power
from the value of seven do not materially affect the magnitude of the heat
transfers involved.

In general, heat-transfer data are not plotted as has been done in
figure 1, in terms of local correlations, but in terms of Nx and Rx.
This usual practice is generally permissible in the case of a flat plate
having no pressure gradient, but it may be of interest to see how the
results of the present analysis appear when integrated along such a flat
plate so as to be presented in more conventional form. The integration
necessary (carried out in detail in the appendix) result in the following
relations for incompressible flow with n = 7


-1/5
Cf = 0.0578Rx



Nx = 0.0289Rx


Thus the normal minus one-fifth and four-fifth power variations of
cf and Nx with Rx are found.







NACA RM L52HO


For the case of compressible flow, the solution is not quite so
Tw Tadw
simple and only in the case of a flat plate where is con-
To
stant can the following useful approximations be made. For n = 7,


cfG = 0.0906Rx 1


NxG = 0.0453Rx


where


G = /


+ r.f.M2 2
5 Vo/av


It is, however, desirable for the sake of accuracy and generality
to retain the relations given in equations (29), (30), and (31) and
correlate turbulent-boundary-layer heat transfer and skin friction on
the basis of the length 5 rather than x.


CONCLUSIONS


The method presented in NACA TN 2692 for evaluating the skin friction
of a turbulent boundary layer in compressible flow on an insulated surface
is extended to evaluate the turbulent skin friction and heat transfer in
compressible flow on a surface which is heated or cooled.

The results of this analysis are in good agreement with the heat
transfers measured in flight on the NACA RM-10 missile up to Mach numbers
of 3.8.

Langley Aeronautical Laboratory,
National Advisory Committee for Aeron3atics,
Langley Field, Va.


Tw Tadw
To


v L 0.f8
\"% j







NACA RM L52H04 13


APPENDIX


DERIVATION OF DEPENDENCE OF SKIN FRICTION AND

HEAT TRANSFER ON Rx FOR A FLAT PLATE


The momentum equation for the boundary layer on a flat plate may
be written


de Cf
dx 2


e
Since
b


is a constant on a flat plate at a given Mach number


(Al)


(A2)


dx c
dx 2 f


Substituting the expression


2
K n+l
cf =

into (A2) there results


2
2 K (F.Vn
2e F OF


(A4)


When M = 0


T -T
and w Tadw
To


O, F = 1 so that equation (A4) may be


integrated to obtain


(A3)







NACA RM L52H04


n+l n+1 2 n+l
8 = Kn+3 + 3 n n+ n+3
U-o


(A5)


The local skin friction and Nusselt
M = 0 and Tw = Tadw as

n+l

Cf = K n+3


n+l
Kn+3_ + 1
llx-i-n--J


number may then be expressed for


2 2
2 n+3 ,,)n3


2 n+1
n+3 n+3
2) (uox
B (vor


(A6)




(A7)


For n = 7, K = 0.045, and I = -. so that
5 72


cf = 0.0578Rx-


Nx = 0.0289Rx /


(A8)


(A9)


These formulas are the more conventional expressions for local skin-
friction coefficient and Nusselt numbers as a function of Reynolds
number.

It may be seen from the differential equation (A3) that the extremely
simple expressions just derived are not valid for the case when M or
Tw Tadw
T- is other than zero. However, F is not a very sensitive
T1
function of R5 and, if an average value of R5 for a given problem
is used to evaluate an F = Fay, the resulting approximate equations
are extremely useful and fairly accurate. Thus, for a flat plate with
a constant surface temperature







NACA RM L52H04


n+l n+l 2
S K n+3n n+3 V n+3
Fav n+ \n+ 3 jj


n+l 2 2
S= K \n+3/n+1 2en+3vo n-+3
cf iav n + 3-6" uox


n+1
N = K n+3 1
2\av/ \n + 3


For n = 7 these expressions become


2 n+l
2\n+3 uon+3
T/ \,,-g-


-1/5
= 0.0906Rx


= 0.o453Rx


These equations may be written


cfG = 0.0906Rx-1/5


NxG = 0.0453R,/5


where


5 o2 To
r..M(A2 )
(A17)


In equation (A17)


o) av


is an average value of L
Uo


along the plate.


and


n+1
n+3


(A10)


(All)


(A12)


(A13)


(A14)


(A15)


(Al6)


/1/5 )4/5
Cfx-) (Fav


Nx 1/5 4/5
Nx (w F a







NACA RM L52H04


REFEREE NCES


1. Donaldson, Coleman duP.: On the Form of the Turbulent Skin-Friction
Law and Its Extension to Compressible Flows. NACA TN 2692, 1952.

2. Crocco, Luigi: Transmission of Heat From a Flat Plate to a Fluid
Flowing at a High Velocity. NACA TM 690, 1932.

3. Chauvin, Leo T., and deMoraes, Carlos A.: Correlation of Supersonic
Heat-Transfer Coefficients From Measurements of the Skin Temperature
of a Parabolic Body of Revolution (NACA RM-10). NACA RM L51A18,
1951.

4. Rumsey, Charles B., and Loposer, J. Dan: Average Skin Friction
Coefficients 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.








NACA RM L52H04


TABLE I

VALUES OF uL FOR n = 7
uo

ULIUo for n = 7 and Rg = -

2 x 104 6 x 10 1 x 105 5 x 105 1 x 106 1.5 x 106
Tw Tadw 0
To

0 0.5465 0.4900 0.4463 0.3655 0.3350 0.3192
1 .5603 .5038 .4593 .3778 .3457 .3295
2 .5943 -5365 .4911 .4060 .3752 .3555
3 .6318 .5725 .5285 .4392 .4140 .3852
4 .6702 .6108 .5660 .4734 .4352 .4171
5 .7085 .6493 .6032 .5076 .4663 .4463

Tu Tadw = 0.5
To

0 0.5704 0.5140 0.4679 0.3875 0.3560 0.33985
1 .5817 .5250 .4805 .3970 .3650 .3482
2 .6100 .5524 .5067 .4200 .3870 .3691
3 .6432 .5868 .5400 .4492 .4150 .39627
4 .6787 .6220 .5742 .48oo .4430 .42418
5 .7150 .6573 .6089 .5110 .4720 .4525

Tw Tadw = -
To

0 0.5140 0.4544 0.4135 0.3375 0.3055 0.2896
1 .5334 .4760 .4320 .3515 .3205 .3052
2 .5759 .5180 .4725 .3885 .3532 .3383
3 .6185 .5606 .5130 .4275 .3900 .3731
4 .6630 .6045 .5554 .4650 .4283 .4095
5 .7070 .6492 .5983 .5058 .4660 .4460

TW Tadw
= 1.0
To

0 0.5890 0.5330 0.4885 0.4045 0.3720 0.3561
1 .5990 .5422 .4972 .4120 .3783 .3629
2 .6230 .5651 .5200 .4330 .3970 .3808
3 .6539 .5963 .5499 .4590 .4240 .4052
4 .6860 .6281 .5815 .4870 .4500 .4308
5 .7185 .6610 .6135 .5160 .4775 .4565

Tw Tad 1.0
V0
To

0 0.461 0.401 0.356 0.2755 0.2465 0.232
1 .494 .435 .3905 .310 .2814 .266
2 .553 .496 .449 .3645 .3335 .3165
3 .6085 .5495 .530 .4138 .3794 .362
4 .6561 .5965 .5495 .4558 .4202 .4015
5 .6968 .6385 .5878 .4925 .4555 .435








18 NACA RM L52H04




TABLE II


VALUES OF ( 2 FOR n = 7



(,L /ui 2 cfor n = 7 and R. -
M
2 x 10 6 10 1 x 10 5 x 10" 1 10 1.5 x 106

T, Tadw ,.3
To

0 0.2'87 0.2401 0.1992 0.1343 0.1122 0.1019
1 .319 .2.38 .21O .1427 .1191 .1086
2 .3 2 .2878 .2412 .1681 .1408 .L264
3 .4C0, 3310 .2793 .1955 .16i50 .192
4 .4492 768 .3204 .2241 .1903 .1740
.50X20 .42'1 .3022 .2528 .2174 .1992

T% Ta, i
To

0 O. i325 0.2642. 0.2189 0.1502 0.1267 0.1155
1 .3384 .2757 .2309 .1576 .1332 .1222
2 .3721 .30.1 .2567 .176. .1498 .1362
3 .1i59 .3443 .291' .2034 .1722 .1570
4 .462. .380'. 3297 .2314 .1971 .1799
5 .5084 .8290 .368- .2611 .2228 .2048

Tw Tad = -.
To

0 0.26442 0.2005 0.1710 0.1139 0.0933 0.0839
1 .285 .2266 .1866 .1236 .1027 .0931
2 .3317 .2683 .2233 .1502 .1262 .1144
3 .3851 .3160 .2669 .1828 .152. .1407
4 .4421 .?652 .311j .2151 .1834 .1677
5 .4965 .4i 3544 .2490 .2125 .1945

T, a 1.0
To

0 0.3469 0.2841 0.2386 0.1636 0.1399 0.1268
S .3588 .2940 .2472 .1697 .1441 .1317
2 .3831 .3193 .2704 .1875 .1576 .1450
3 .4280 .3556 .3024 .2107 .1798 .1624
4 .4701 .39' .3 01 .2372 .2025 .1856
5 .fi.2 .4 369 .3838 .2663 .2280 .2084

Tw Tadw = -1.0
'1.0

0 0.2125 O.1608 0.1267 0.0759 0.06076 0.05382
i .24L4 .1892 .1525 .0961 .07919 .07076
2 .3058 .2'ta .2016 .1329 .1113 .1002
3 .3703 .3020 .2515 .1712 .1439 .1310
4 .4305 .3558 .3020 .2078 .1766 .1612
S .4855 .4077 .3455 .2426 .2075 .1892







NACA RM L52H04


TABLE III

MEASURED QUANTITIES USED IN COMPARISON


Point M T Tadw, Tw Tadw Rg N5
(See fig. 1) OF To

1 1.59 67 0.145 0.385 x 106 270
2 1.61 58 .126 .697 422
3 2.15 ---153 -.320 .585 388
4 2.19 --171 -.365 1.39 799
5 2.52 --376 -.776 .7 450
6 2.58 -394 -.837 1.973 1130
7 2.60 49 .135 .208 180
8 3.12 -90 -.258 .457 331
9 3.60 -509 -1.312 .921 561
10 3.69 -745 -1.842 1.07 634







. / ,,, -%fL)tlA


> NACA RM L52H04


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i 'v lIhl4 Iit---- + # 1 +f1". 1 11 111


10,000
8,000
6.000

4,000



2,000



1,000
800
600


LIiS1


*ii -I- i. t n i ff


iilx Iij:q:
ilill:1ll~ii;:
-[^ti- f:^ *^ =


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UN IVRlTY I T v FLUIBUA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE UBRARY
RO. BOX 117011
GAINESVILLE, FU 32611-7011 USA