Local and average skin friction and heat transfer on flat plates in air - chart

Bertram, M. H.

Neal, L., Jr.

English

NASA Technical Reports Server

TURBULENT-SKIN-FRICTION A N D  
HEAT-TRANSFER CHARTS ADAPTED 
FROM THE SPALDING A N D  CHI METHOD 
by Lzctber Ne&& Jr., und MitcbeZ H,  Bertram 
Lungley Reseurcb Center 
LangZey Station, Humpton, Vu. 
N A T I O N A L  AERONAUTICS A N D  SPACE A D M I N I S T R A T I O N  W A S H I N G T O N ,  D. C. MAY 1967 
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i 
TURBULENT-SKIN-FRICTION AND HEAT -TRANSFER CHARTS 
ADAPTED FROM THE SPALDING AND CHI METHOD 
By Lu the r  Neal ,  Jr., and Mitchel  H. B e r t r a m  
Langley R e s e a r c h  Cen te r  
Langley Station, Hampton, Va. 
NATIONAL AERONAUT ICs AND SPACE ADMINISTRATION 
For sale by the Clearinghouse for Federal Scientific and Technical Infmmation 
Springfield, Virginia 22151 - CFSTI price $3.00 
TURBULENT-SKIN-FRICTION AND HEAT-TRANSFER CHARTS 
ADAPTED FROM THE SPALDING AND CHI METHOD 
By Luther Neal, Jr., and Mitchel H. Bertram 
Langley Research Center 
SUMMARY 
Charts a r e  presented which allow a rapid determination of local and average skin 
friction and heat transfer on flat plates in air. The charts cover a Mach number range 
5 9 from 0 to  20, a Reynolds number range from 10 to  10 in decade increments uf the expo- 
nent, and a wall-temperature-stagnation-temperature ratio range of 0.1 to the adiabatic- 
wall case. 
INTRODUCTION 
In the design of large hypersonic vehicles, the need for accurate estimates of turbu- 
lent skin friction and heat transfer is important. A number of prediction methods have 
accumulated over the years, but the results from these methods often differ widely and 
recourse to experiments must be made. In early 1964, Spalding and Chi (ref. 1) compared 
experimental skin friction with 20 different turbulent theories. A theory of Van Driest 
(ref. 2), employing the Karman mixing length, was found to  yield the lowest root-mean- 
square e r ro r  (11.0 percent). 
applicable to  a flat plate that gave an e r ro r  of 9.9 percent. In a paper by Bertram and 
Neal (ref. 3), a body of heat- transfer data in the Mach number range from about 5 to 9 and 
wall-temperature-stagnation-temperature ratios down to about 0.1 was found to  support 
the predicted effect of wall-temperature-stagnation-temperature ratio from the Spalding- 
Chi method. 
present, in convenient graphical form, the results of extensive calculations that were 
accumulated while preparing reference 3. 
However, Spalding and Chi were able to  formulate a method 
Because of current interest in the Spalding-Chi method, it was decided to  
SYMBOLS 
CF 
c, 
average skin-friction coefficient based on conditions at edge of boundary layer 
local skin-friction coefficient based on conditions at edge of boundary layer 
functions in the Spalding-Chi theory given by equations (3), (2), and (l), Fc 9 FRx, 
respectively 
M 
Npr Prandtl number 
Mach number at edge of boundary layer 
NSt local Stanton number based on conditions at edge of boundary layer 
R, Reynolds number based on distance x and conditions at edge of boundary 
layer 
T absulut e temperature 
t = T / T ~  
Subscripts : 
aw adiabatic wall 
i incompressible value 
W wall 
t total 
e edge of boundary layer 
DISCUSSION 
In developing their method, Spalding and Chi postulated that a unique relation exists 
between FcCf and FRR where Cf is the local skin-friction coefficient, R is the 
Reynolds number, and Fc and FR depend only on Mach number and wall-temperature- 
stream#-temperature ratio. The function Fc was  calculated by means of the mixing- 
length theory, and the function FR was  deduced semi-empirically. Thus, values of skin 
friction could be obtained but heat transfer was  not considered. 
Tables and charts for  Fc and FR to be used in the calculation of skin friction 
were presented; however, for  extensive use of the theory, it was found more convenient to  
2 
compute values of Fc and F R  When the Reynolds number is based on momentum 
thickness, FR = FRO and is given by the empirical expression (eq. (52) of ref. 1). 
-1.474 0.772 
= b taw 
or  when based on length of turbulent flow x, FR = F R ~  and is given by equation (25) of 
reference 1. 
FRO F e = -  
F C  
Only the Reynolds number based on distance x is considered in this paper. To compute 
Fc, equation (39) of reference 1 was integrated to yield the expression 
The proper quadrant in which to evaluate the inverse sine is shown in the following sketch: 
4 8 12 16 20 
U 
M 
In the limit as M approaches 0, equation (3) reduces to 
1+G2 
F c = (  2 ) 
By using these expressions and the variation of F,Cf and F c C ~  with FmRx,  
as given in table 7 of reference 1, and extended where necessary by equations (28) to (30) 
3 
of reference 1*, values of Cf and CF were computed. Stanton numbers were calcu- 
lated by the modified Spalding-Chi method described in appendix A of reference 3. This 
approach consists of applying the Karman form of the Reynolds analogy in the incom- 
pressible plane so that the Karman equation is modified to be 
6 
- = [1+ 5 j T ( N p r  FcCf - 1 + loge 
Cf 
(5) 
When the Karman factor was applied in this paper, Npr was taken as 0.725. 
Although directly applicable to a flat plate, the results for local skin friction and 
heat transfer may be applied to a cone with a known or assumed location of transition by 
using the transformation given in appendix B of reference 3. Results given in reference 4 
(pp. 25 to 30) may aid.in the application of the results to obtain the average skin friction 
on a cone o r  tapered wing. 
PRESENTATION OF RESULTS 
Charts of Cf/Cf,i, NSt/NSt,i, and CF/cF,i are presented in figure 1 as a func- 
tion of Mach number for  Rx = l o5  to lo9 and Tw/Tt = 0.1 to the adiabatic-wall value. 
The recovery factor was taken to be 0.89. The charts are normalized by using the incom- 
pressible values of skin-friction coefficient given by the Karman-Schoenherr formulas: 
J 
.- 
Equation (29) of reference 1 actually contains two typographical errors ;  the correct * 
form which was determined by referring back to the paper in which the expression was 
originally presented is 
Rx,i = -&(u&)~ + (K3E)-l {[6 - 4(Ku+G) +(KuL)lexp(Ku&) - 6 - ~(Ku;) 
-'(KG) 12 --KuG 20 ( +)5 - ~ ( K u ; ) ~  60 - L(Ku&T} 252 
where u& = ($'2 , K =  0.4, and E = 12. 
4 
The normalizing values of N s , i  are those obtained from the unmodified Karman equa- 
tion (eq. (A4) of ref. 3) in conjunction with equation (6) and are presented in figure 2. 
In figure 1 for  the adiabatic-wall cases at M = 0, the value of unity is not neces- 
sarily obtained because the Karman-Schoenherr formulas were used for normalizing 
rather than the incompressible values of Spalding and Chi. Actually, the incompressible 
values of skin-friction coefficients of Spalding and Chi do not differ significantly from 
those of Karman-Schoenherr as seen in figures 3 and 4, but because of the widespread 
use of the Karman-Schoenherr formulas in normalizing experimental data in past publica- 
tions, they were chosen for  use here. For some cases, the skin-friction or heat-transfer 
ratio drops below the value 0.1 before Mach number 20 is attained. Extrapolation is 
aided by noting that the logarithm of these ratios is essentially linear with.the logarithm 
of M between Mach numbers of 10 and 20 at the high Reynolds numbers. Thus, extrap- 
olation may be simply done on log-log paper. 
' 
It should be noted that at the higher Mach numbers and lowest Reynolds numbers, 
the equivalent incompressible Reynolds numbers are much lower than those expected for 
turbulent flow to exist. (See ref. 5.) 
CONCLUDING REMARKS 
Utilizing the Spalding-Chi turbulent-skin-f riction theory, charts a r e  presented which 
allow a rapid determination of the local and average skin friction on flat plates in air. 
An extension was made to allow the determination of heat transfer by modifying Spalding- 
Chi's skin-friction results according to  the Karman form of the Reynolds analogy factor. 
5 The charts cover a Mach number range from 0 to  20, a Reynolds number range from 10 
to  10 in decade increments of the exponent, and a wall-temperature-stagnation- 
temperature ratio range of 0.1 to the adiabatic-wall case. 
9 
Langley Research Center, 
National Aeronautics and Space Admini st r at ion, 
Langley Station, Hampton, Va., January 13, 1967, 
129-01-09-14-23. 
5 
REFERENCES 
1. Spalding, D. B.; and Chi, S. W.: The Drag d a Compressible Turbulent Boundary 
Layer on a Smooth Flat Plate With and Without Heat Transfer. J. Fluid Mech., 
V O ~ .  18, pt. 1, Jan. 1964, pp. 117-143. 
2. Van Driest, E. R.: The Problem of Aerodynamic Heating. Aeron. Eng. Rev., vol. 15, 
no. 10, Oct. 1956, pp. 26-41. 
3. Bertram, Mitchel H.; and Neal, Luther, Jr.: Recent Experiments in Hypersonic Tur- 
bulent Boundary Layers. Presented to  the AGARD Specialists Meeting on Recent 
Developments in Boundary-Layer Research (Naples, Italy), May 10-14,1965. 
4. Bertram, Mitchel H. : Calculations of Compressible Average Turbulent Skin Friction. 
NASA TR R-123, 1962. 
5. Preston, J. H.: The Minimum Reynolds Number for a Turbulent Boundary Layer and 
the Selection of a Transition Device. J. Fluid Mech., vol 3, pt. 4, Jan. 1958, 
pp. 373-384. 
6 
M 
(a) Rx = 16. 
Figure 1.- Variation of Cf/Cf,i, Nst/Nst,i, and C F / C F , ~  as  a function of Mach number for various values of wall-temperature-stagnation-temperature ratio. 
(b) Rx = 106. 
Figure 1.- Continued. 
'F/'f 
._ . . . .. - 
.6 
I I 
.4 
.2 
.1 
.2 .2 
.1 .1 
0 2 4 6 8 10 12 14 16 18 20 
M 
(c) R, = 107. 
Figure 1.- Continued. 
m 
(d) R, = 108. 
Figure 1.- Continued. 
2 
M 
(e) Rx = 109. 
Figure 1.- Concluded. 
.6 
6 x  
*t- I I I I 
1 2 4 5 6 7 8 9 1 0  
Rx,i (Scale range shown on individual curves) 
Figure 2.- Variation of incompressible Stanton number with Reynolds number based on the Karman form of the Reynolds analogy equation 
and the Karman-Schoenherr skin-friction equation. Npr = 0.725. 
12 
10 x 
9 
8 
7 
6 
5 
4 
'f,i 
3 
2 
1 
1 
Spalding 
Karma 
---- 
I I I I I I I I I I lilll 
2 
Chi I 
Schoenherr I 
I 
3 
i 
i 
T 
I 
- 
t I 
1 T 
1 
4 
Rx,i (Scale range shown on individual curves) 
5 6 7 ' 8  9 10 
Figure 3.- Variation of incompressible local skin-friction coefficient with Reynolds number. 
13 
9 
8 
7 
6 
5 
4 
‘F,i 
3 
2 
1 
M1-l ! I  
tt - Spalding - Chi Karman - Schoenherr 
1 2 3 4 5 6 7 8 9 1 0  
Rx,i (Scale range shown on individual curves) 
Figure 4.- Variation of incompressible average skin-friction coefficient with Reynolds numbers. 
14 NASA-Langley, 1967 - 33 L- 5369 
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Turbulent skin friction and heat-transfer charts adapted from the Spalding and Chi method

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