A theoretical study of heat transfer for zero pressure gradient hypersonic laminar boundary layers for various gases with particular application to the flows produced in an expansion tube facility was conducted. A correlation based on results obtained from solutions to the governing equations for five gases was formulated. Particular attention was directed toward the laminar boundary layer shock tube splitter plates in carbon dioxide flows generated by high speed shock waves. Computer analysis of the splitter plate boundary layer flow provided information that is useful in interpreting experimental data obtained in shock tube gas radiation studies

Cook, W. J.

English

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CORRELATION OF HEAT TRANSFER FOR
THE ZERO PRESSURE GRADIENT
HYPERSONIC LAMINAR BOUNDARY
LAYER FOR SEVERAL GASES
A TECHNICAL REPORT
By
William J. Cook
Associate Professor of Mechanical Engineering
Iowa State University of Science and Technology
Ames, Iowa 50010
Prepared for the
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Langley Research Center
Hampton, Virginia 23665
Under
Master Contract Agreement No. NAS1-11707
Task Authorization No. 21
Submitted by the
Old Dominion University Research Foundation
owl P. 0. Box 6173
Norfolk, Virginia 23508
November 1973
https://ntrs.nasa.gov/search.jsp?R=19740007863 2020-03-23T12:40:16+00:00Z
REPORT OF RESEARCH ACTIVITIES - SUMMER, 1973
William J. Cook
Department of Mechanical Engineering
Iowa State University
Ames, Iowa
Introduction
This report covers the author's research activities at the NASA Langley
Research Center during the ten-week period June 11 to August 17, 1973. This
work was supported under NASA Contract No. NASl-11707-21, and was carried out
in the Hypervelocity Impulse Facilities Section of the Advanced Entry Analysis
Branch, Space Systems Division. Mr. Jim Jones served as immediate supervisor.
The effort involved work in the three areas described briefly below.
Correlation of Aerodynamic Heating Rates
This work involved a theoretical study of heat transfer for zero pressure
gradient hypersonic laminar boundary layers for various gases with particular
application to the flows produced in Langley expansion tube facility. A,
simple correlation based on results obtained from solutions to the governing
equations for five gases was formulated. See the attached first draft of a
paper entitled, "Correlation of Heat Transfer for the Zero Pressure Gradient
Hypersonic Laminar Boundary Layer for Several Gases," which will be submitted
to the AIAA Journal.
Heat Transfer Measurement in Short-Duration Flows
The author served as a consultant on data reduction techniques in relation
to the measurement of heat transfer rates in the expansion tube facility.
Techniques in use by the Langley scientists which were previously developed
by the author for reduction of data obtained from thin-film resistance thermo-
teter heat-flux gages were extended for use in conjunction with thin-film
thermocouple gages.
Shock Tube Boundary Layer Analysis
The author served as a consultant on the subject of high temperature
laminar boundary layer flows in shock tubes. Particular attention was directed
toward the laminar boundary layer on shock tube splitter plates in carbon
B-ii
oxide flows generated by high speed shock waves. Computer analysis of the
plitter plate boundary layer flow provided information that is useful in
nterpreting experimental data obtained in shock tube gas radiation studies.
B-iii
FIRST DRAFT
CORRELATION OF HEAT TRANSFER FOR THE ZERO PRESSURE GRADIENT
HYPERSONIC LAMINAR BOUNDARY LAYER FOR SEVERAL GASES
WILLIAM J. COOK
AUGUST 1973
This work was supported under NASA Contract No. NASI-11707-21
B-iv
CORRELATION OF HEAT TRANSFER FOR THE ZERO PRESSURE GRADIENT
HYPERSONIC LAMINAR BOUNDARY LAYER FOR SEVERAL GASES
By William J. Cook1
NOMENCLATURE
C Chapman-Rubesin coefficinet, pp/pe"e
h Static enthalpy, J/kg
hr Recovery enthalpy, he + ru2/2
H Dimensionless enthalpy, (h - h )/he
k Thermal conductivity, W/(m K)
P Pressure, atm
q Heat transfer rate per unit area, (heat flux), W/m2
r Recovery factor
Re Reynolds number, XUePe/ e
M Mach number
T Temperature, K
s Exponent: cone, s = 1; wedge, plate, s = 0
St Stanton number, 
-qw/Peue (hr - h )
u x component of velocity, m/s
V y component of velocity, m/s
x Space coordinate parallel to wall, m
Y Space coordinate perpendicular to wall, m
8 u/ue
1 Associate Professor of Mechanical Engineering, Department of MechanicalEngineering, Iowa State University, Ames,Iowa 50010
2-I
y Specific heat ratio, C p/Cv
SDimensionless shear stress, p(au/ay)/peu2
S Viscosity coefficient, kg/(m s)
p Density, kg/m3
f Shear stress variable, Eq. (3)
a Effective Prandtl number
Subscripts
e Boundary layer edge
w Wall
INTRODUCTION
Current research in the area of convective heating in high speed flows
includes studies using gases and gas mixtures to simulate aerodynamic heating
in planetary atmospheres. Test facilities such as the shock tunnel and the
expansion tube are useful for generating the required high-energy flows for
investigating aerodynamic heating over a range of high Mach numbers. Simple
configurations such as the sharp flat plate, the wedge, and the sharp cone
that avoid boundary layer pressure gradients are frequently used as basic
body shapes in such studies. This note is concerned with correlation of
hypersonic aerodynamic heating parameters for these body shapes for several
gases under test conditions encountered in these facilities and for conditions
under which the boundary layer is laminar.
Testing in the above-mentioned facilities characteristically involves
models with cold surfaces (relative to the recovery temperature) since, due
to short testing times involved, little change from the initial room temperature
of the model surface occurs. Hence, considerations here will be directed mainly
toward flows with the cold wall condition. Further, only the boundary layer
flow downstream of the region of interaction of any leading edge shock wave
and the boundary layer that occurs near the leading edge will be treated.
To a good approximation the self-similar laminar boundary layer equations
for flows with zero pressure gradient are then applicable.
B-2
In order to assess the influence of the characteristics of various real
gases, hypersonic heating rates in air, argon, carbon dioxide, helium, and
hydrogen have been theoretically studied. These gases as ideal gases range
in molecular structure from monatomic to triatomic, the molecular weights
range from 2 to 44 and values of y vary from 1.28 to 1.67. In high Mach
number flows real gas effects can play an important role since, because of
pronounced dissipation effects, the temperature in the boundary layer can
reach large values. Real gas effects in hypersonic flows may be present
to varying degrees depending on the gas considered, e.g., carbon dioxide
dissociates at relatively low temperatures while helium exhibits ideal
gas behavior over a wide range of temperature. In view of these facts and
the range of gas characteristics considered, there is no assurance that a
precise universal correlation for heat transfer exists for a wide range of
flow conditions. However, insight for the present problem can be gained
by examining the boundary layer equations and solutions thereto for various
gases.
ANALYSIS
The method employed here to solve the governing boundary layer equations
is that of Crocco (e.g., see Ref. 1), in which the boundary layer equations
in terms of x and y as the independent variables are transformed to
equations that have x and u as the independent variables. For real gas
flows in thermochemical equilibrium, the laminar boundary layer equations
with B = u/u and x as the independent variables and t = V(au/ay)/p 2
and H = (h - he)/he as dependent variables are:
Momentum,
S
Q a [ [ (1)
Energy,
U2
H = et + Hjj - P x (2)
8-3
where s = 1 for the case of the cone and s = 0 for the plate and the wedge.
In Eq. (2) a = a(h,p) is the effective Prandtl number that serves to incorp-
orate real gas effects with regard to energy transport. Continuity is
implicitly satisfied in Eqs. (1) and (2). The boundary conditions for these
equations are,
OT/ B = 0, H = Hw , at a = 0
r = 0, H = 0, at B = 1
The boundary condition on t at B = 0 results from u = v = 0 at the wall.
By employing a separation of variables solution of the form r(x,8) = 0(B)/X(x),
the shear stress becomes,
T(x,B) = 0(B)/(Re)1/2  (3)
After Crocco, the term Hx  in Eq. (2) is taken as zero. Substituting Eq. (3)
into Eqs. (1) and (2),
00 + 3SCa/2 = 0 (4)
000H = (ue/he)02  + 0(A Hg) (5)
where C = p/p e e . The boundary conditions become
0 = O0, H = Hw , at B = 0
0= 0, H = 0, at B = 1
In integrated form Eqs. (4) and (5) can be written as,
1 P 11/2
B-4
and
1
H(s) = -H ( 0) d - dP
1 P
+ H n (x)d ad d dP
where O(8) = [0(0)]1-a(B). Equations (6) and (7) are coupled since C = C(h,p).
The heat flux qw = -k(6T/ay)w for the gas becomes
1/2
S= -Ow (H )wPeuehe/w (Re) (8)
where w and (H ) are obtained from simultaneous solution of Eqs. (6) and
(7). Substitution of Eq. (8) into the Stanton number expression yields,
St(Re)1/ 2  w(HB)w (9)
aw [(w)1/ 2  ue2 H
where the recovery factor r has been taken as (aw )1/2
It is useful to examine the equations that lead to the terms in Eq. (9)
in order to determine how the various flow quantities and gas properties
influence St(Re)1/2. To solve Eqs. (6) and (7) for Ow and (H )w  certain
quantities must be specified. For a given gas the variations of the properties
a, T, and pp with h at the specified pressure are required. Specification
of Tw and Te  then establish hw  and he and therefore H w. In addition,
C = pP/pe1e and a can then be determined as functions of H.
B-5
Finally, with specification of u , the term u2/he  in Eq. (7) is
[established. Thus, if Tw , Te, u , and the gas properties as a function
of h are specified, sufficient information is available to solve Eqs. (6)
and (7) for OW and (H )w . Therefore, if Tw , Te, u ,e and the pressure
p are specified, and various gases are considered, any variation that occurs
in 0w and (H )w  will arise from differences in the relations for T, pp,
and a vs h for the various gases. It should be noted however that fixed
values for Tw and Te do not yield the equal values of Hw for all gases
except for the case Tw = Te. Therefore, if comparisons are to be made between
the solutions to Eqs. (6) and (7) for various gases at conditions other than
Tw= Te, they should be made at the same values of Tw and Hw rather than
at the same Tw and Te. On this basis, specification of fixed values of
T , p, u2/he, and Hw  would permit the variations of pp and a vs h
on Ow and (H)w  for various gases to be assessed. The terms u2/he and
Hw also appear in the denominator of Eq. (9). The term u2/he  is related
to the Mach number. For ideal gases u2/he = M2(y -.1). Thus for large
Mach numbers u2/h >>H . The Prandtl number aw in the denominator of
Eq. (9) for the cold walls considered here is related to y. (According to
the theory of Eucken2 the Prandtl number for gases can be closely approximated
by 4y/(9y - 5).) Therefore, for high Mach numbers the denominator of Eq. (9)
is related to the Mach number and y. Computations performed to examine the
influence of variations of gas properties on Eq. (9) and the terms therein
are presented in the next section.
RESULTS
Figure la shows solutions to Eqs. (6) and (7) for a plate (s = o) in terms
of *(B) and H(s) obtained for carbon dioxide and helium for the values of
Tw,  p, u/he, and Hw noted in the figure. The numerical method employed
to solve Eqs. (6) and (7) is outlined in ref. 3. Figure lb shows C(B) and a(B)
for each of the gases. Property values for carbon dioxide were obtained from
refs. 4 and 5. Thermodynamic properties for helium were computed from ideal
gas equations. Also the relation v a T0O 647 and a constant Prandtl number
(o = 0.688) were employed for helium.
It is noted that the influence of the difference between the curves for
B-6
C(B) and o(B) on the profiles for *(B) and H(B) is small, but that
the values for w (H )w noted in the figure differ by about 20 percent.
However, the values of St(Re)1 /2 computed by Eq. (9) for the two gases
differ by less than one percent. This suggests that St(Re)I/2 and u2/h
are the correlating variables. Figure 2 presents results in terms of St(Re)1/ 2/y4 W
vs u2/he for air, argon, carbon dioxide, helium, and hydrogen. The term
v-s permits the results to be applied to the cone or the plate and wedge.
The properties for argon and .hydrogen were obtained from ref. 4. Those for
air were taken from ref. 6. The Mach number and ue/he ranges applicable for
each gas are shown in the figure. Results obtained for Hw = 0 and Hw = -0.68
are indicated by the open and filled symbols respectively. These values of
Hw correspond to T e/Tw = 0 and T e/Tw 3, and cover the range of T e/Tw
commonly encountered in shock tunnel and expansion tube flows. With the
exception of carbon dioxide the results are relatively insensitive to the
value of Hw . The results in Fig. 2 were obtained for p = 0.01 atm. However,
computations at p = 1 atm for carbon dioxide (for which variation in pressure
was expected to have the most influence) yielded results that are within one
percent of those shown in Fig. 2, indicating that there is no significant
influence of pressure for the conditions considered here.
For simplicity a straight line has been fitted through the points in
Fig. 2. The computed results deviate by at most five percent from this
curve. Thus for most engineering purposes this simple relation should be
sufficiently accurate for the gases considered for Fig. 2. In view of the
wide range of gas characteristics considered here it is reasonable to
expect this relation would be applicable with the same accuracy to other
gases and gas mixtures provided the cold wall condition is maintained and
Hw is within the range noted in Fig. 2.
B-7
REFERENCES
i. Van Driest, E.R., "The Laminar Boundary Layer with Variable Fluid
Properties," Preprints of Papers, Heat Transfer and Fluid Mechanics
Institute, 7, 127-141, 1954.
2. Kennard, E.H., Kinetic Theory of Gases. New York: McGraw-Hill Book Company,
Inc., 1938, pp. 180-182.
3. Cook, William J. and Chapman, Gary T., "Characteristics of the Unsteady
Shock-Induced Laminar Boundary Layer on a Flat Plate," The Physics of
Fluids, 15, 2129-2139, (1972).
4. Marvin, Joseph G. and Deiwert, George S., "Convective Heat Transfer in
Planetary Gases," NASA Technical Report TR R-224, 1965.
5. Hilsenrath, J.; Beckett, C.W.; et al.: Tables of Thermodynamic and
Transport Properties of Air, Argon, Carbon Dioxide, Carbon Monoxide,
Hydrogen, Nitrogen, Oxygen and Steam. Pergamon Press, New York,
Oxford, 1960.
6. Hansen, C. Frederick, "Approximation for the Thermodynamic and Transport
Properties of High Temperature Air," NASA Technical Report TR R-50, 1959.
B-8
i ... H, Carbon dioxide
0.4 .- 0, Carbon dioxid -...-i. ---- 4.0
- . . -H- Helium -.
- ~ -'N 3.0
0.2 - - 2.0Hel i um
0.1 1.0
-r
-1.0
0 0.2 0.4 0.6 0.8 1.0
a. 0 and H vs B
1.4- .- 2 ..
--------- /h = 45 p = 0.01 ratm
1.3 H = -0.68 T = 300 K
.... --. " i . - + W • 'i.--.+ .. ::-
,2..
1.2. Gas Me Ow(HB)w St (Re)1/2
1 J CO2  11.8 5.291 0.3323
He 8.20 4.443 0.3339
1.0-
S0.9T- 0.9C, Carbon dioxide '
, Carbon dioxide
. .i I . : , Hel um
0.7 .__
0.6
,-:-: i -"C. Helium
0.5
0 0.2 0.4 0.6 0.8 1.0
b. C and o vs B
Fig. 1 Comparison of results for carbon dioxfde and helium.
B-9
St(Re) /2/(3) 1/ 2  0.385 - 0.0011(u2/he) :
0.4 . e e ..
0.3 -4
T 300 K .
RA E .. .. .. .. .
Open symbols, H 0.8 - I .I I 
Filled symbols, Hw = -0.68 RAIGE .
0.2 Gas Me
0 AIR . AIR 6.0-11.1 13.6-68.1 ..
6 HYDROGEN ARGON 6.0-10.0 20.0-66.6
0 ' ARGON CARBON DIOXIDE 6.4-15.8 14.0-58.2
E HL , .. HELIUM 5.7-9.9 . 21.6-65.0
. ARGON IO :i'i. ; '  :: CAHYDROGEN 6.2-11.1 16.0-51.0
0 .. .. . ..
0 IQ 20 30 2 40 50 60 70 80o HELIUoe ot f f r g .
Fig. 2 Correlation of heat transfer for several gases.
0


Correlation of heat transfer for the zero pressure gradient hypersonic laminar boundary layer for several gases

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