Similitude requirements for hypersonic, rarefied flow with nonequilibrium chemistry and vibration are presented. The full Navier-Stokes equations with catalytic or noncatalytic walls and with or without slip conditions are nondimensionalized. The heat transfer coefficient is written in terms of fourteen dimensionless parameters and reduced to four by making the binary scaling assumption. Duplication of blunt and sharp nose heat transfer requires the use of air over a geometrically similar model with the same free stream velocity, wall temperature and product of free stream density and characteristic length. Estimates of this heat transfer coefficient are also presented

Hendricks, W. L.

English

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NASA TECHNICAL
MEMORANDUM
NASA TM X-64859
SIMILITUDE REQUIREMENTS FOR HYPERSONIC,
RAREFIED, NONEQUILIBRIUM FLOW
By William L. Hendricks
Systems Dynamics Laboratory
July 1974
(NASA-TK-X-64859) SIOILITUDE REQUIREMENTS V74-27750
FOR HYPERSCNIC, RAREFIED, NONEQUILIBRIUM
FLOW (NASA) 14 p HC $3.00= CSCL 20D
Unclas
G3/12 43129
NASA
George C. Marshall Space Flight Center
Marshall Space Flight Center, Alabama
MSFC . Form 3190 (Rev June 1971)
https://ntrs.nasa.gov/search.jsp?R=19740019637 2020-03-23T07:11:05+00:00Z
TECHNICAL REPORT STANDARD TITLE PAGE
1. REPORT NO 12. GOVERNMENT ACCESSION NO. 3. RECIPIENT'S CATALOG NO.
NASA TMX-64859 I
4. TITLE AND SUBTITLE 5. REPORT DATE
SIMILITUDE REQUIREMENTS FOR HYPERSONIC, July 1974
6. PERFORMING ORGANIZATION CODE
RAREFIED, NONEQUILIBRIUM FLOW
7. AUTHOR(S) . PERFORMING ORGANIZATION REPORrT
William L. Hendricks O
9, PERFORMING ORGANIZATION NAME AND ADDRESS 10. WORK UNIT NO.
George C. Marshall Space Flight Center
Marshall Space Flight Center, Alabama 35812 1. CONTRACT OR GRANT NO.
3. TYPE OF REPORT & PERIOD COVERED
12. SPONSORING AGENCY NAME AND ADDRESS
Technical Memorandum
National Aeronautics and Space Administration
Washington, D. C. 20546 14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
Research Associate, National Research Council, National Academy of Sciences
16. ABSTRACT
Similitude requirements for hypersonic, rarefied flow with nonequilibrium
chemistry and vibration are presented. The full Navier-Stokes equations with
catalytic or noncatalytic walls and with or without slip conditions are
nondimensionalized. The heat transfer coefficient is written in terms
of fourteen dimensionless parameters and reduced to four by making the
binary scaling assumption. The duplication of blunt and sharp nose heat
transfer requires the use of air over a geometrically similar model with
the same free stream velocity, wall temperature and product of free stream
density and characteristic length. Estimates of this heat transfer
coefficient are also presented.
17. KEY WORDS 18. DISTRIBUTION STATEMENT
Heat Transfer Unclassified-unl imited
Space Shuttle
Transitional Flow Regime
William L. Hendricks
19. SECURITY CLASSIF. (of this trear1t 20. SECURITY CLASSIF. (of this page) 21. NO. OF PAGES 22. PRICE
Unclassified Unclassified 12 NTIS
MsFC - Form 3292 (Rev December 1972) For sale by National Technical Information Service, Springfield, Virginia 22151
TECHNICAL MEMORANDUM X-64859
SIMILITUDE REQUIREMENTS FOR HYPERSONIC,
RAREFIED, NONEQUILIBRIUM FLOW
I. INTRODUCTION
To determine the thermal protection system required for the Space
Shuttle, accurate predictions of the heat transfer must be made from wind
tunnel data. The heat transfer in the transitional flow regime, between
continuum and free molecular flow, with nonequilibrium chemistry and
vibration is an important part of the total heating to the Shuttle External
Tank. For the ascent of the External Tank using the Rockwell International
PRR Space Shuttle configuration, the preliminary estimate of the local
heating rate is largest during the transitional flow regime for points on
the ogival forebody. Thus, the prediction of aeroheating in the transi-
tional zone is of great interest. If the proper similarity requirements
are duplicated in the wind tunnel, the heat transfer coefficient on the
flight vehicle can be obtained.
Similitude requirements have been established for inviscid, hypersonic
small disturbance flows by Hayes and Probstein (Ref. 1), extended to blunted
bodies with the blast wave analogy by Cheng (Ref. 2), and further extended
to include real gas effects by Inger (Ref. 3). Using the thin shock layer
approximation, Cheng (Ref. 4) extended his original work to include viscous
effects. However, in the transitional flow regime a simple flow model is
not appropriate to describe the viscous, nonequilibrium phenomenon.
Jain (Ref. 5) shows that the thin layer approximation fails to give correct
results in the fully merged flow regime; but, that the entire Navier-Stokes
equations are appropriate from the surface of the body to the free stream
for flow from the continuum regime through the fully merged regime. Thus,
the entire Navier-Stokes equations will be nondimensionalized to indicate
the similarity parameters.
II. SIMILARITY REQUIREMENTS
For locations near the stagnation point the basic equations for a
multicomponent mixture of chemically reacting gases made up of N, 0, N2,
02, NO, NO+, and e are written in spherical coordinates. By neglecting
external body forces and the radiative heat flux in comparison to the
convective heat flux (good for small nose radii, see Ref. 6) and using
Fick's law, the equations are nondimensionalized. The boundary con-
ditions are also nondimensionalized assuming a catalytic wall with no
slip conditions. For a diatomic gas the slip conditions have little
effect on the heat transfer, see Ref. 5.
The heat transfer coefficient, written in a functional relationship,
for the stagnation region is
CH = CH (8; Reo, Pr, Le, 7, M. , T o , Tw, XRj' KRj, Kcj, Kcj ,vi'
v2i' K .i) Species i = 1, .... N (1)
Reaction j = 1, .... N
where strict similarity requires the duplication of the following:
(1) Angle from the stagnation point, 0; (2) Reynolds number, Reo =
p0 It% R/go, based on the free stream density and velocity, body radius,
and viscosity corresponding to the free stream total temperature; (3)
Prandtl number, Pr; (4) Lewis number, Le; (5) ratio of specific heats,
Y; (6) free stream Mach number, Mo ; (7) free stream total temperature,
T ; (8) wall temperature, T ; (9) ratio of activation temperature for
chemical reactions to total temperature, XRj; (10) ratio of residence
time to chemical relaxation time, KRj = (R/U.)/(I/p Cfj To 7fJ), where
Cfj is the reaction rate constant in the forward direction and nfj is the
reaction rate temperature exponent; (11) ratio of forward to backward
reaction rate for two, K j, and three body reactions, Kcj = K cj/P,
(12) ratio of activation temperature for vibration: and molecular constant to
To , i.e. ,Avi v2 i; and (13) ratio of residence time to vibrational
relaxation time, Kvi = (R/U. )/(Klvi T5/6/P U2 ), where Kv i is a
constant which depends on the physical properties of the molecule.
2
If air is used for the test gas; y, Pr, and Le will be essentially
the same for wind tunnel and flight conditions, due to the relative in-
variance of these quantities with small differences in temperature and
pressure. By using a similar gas the molecular properties and the reaction
constants are identical for wind tunnel and flight conditions. The re-
maining parameters to be duplicated are
CH = CH [9; Re0 (p U, R/Io), M. (U/T. ), To, Tw, XRj(To),
KRj (0,R/U, T0 f j ) , Kcj (T 5 / 6 ) ]  Species i =  ,.... N (2)
Reaction j = 1,.... N
It is obvious from Equation (2) that, if the free stream thermodynamic
state and free stream velocity are duplicated, the wind tunnel model must
be the same size as the actual vehicle.
Using the free stream velocity and the stagnation temperature as
similitude requirements, the free stream temperature is, ideally, also
simulated and Equation (2) reduces to
C1H = CH [9; U,,, T , Tw, Re & KRj & K. (P.R), cj ()]* (3)
By simulating the free stream velocity for hypersonic, rarefied flight,
one has, in essence, simulated the total temperature. The free stream
temperature may not be that of flight, but for many existing wind tunnels
these low temperatures cannot be obtained. Matching the free stream
temperature is relatively unimportant (Ref. 7). Then Equation (3)
simplifies to
CH = CH [9; U,, Tw, Reo & KRj & Kvi (DPR), Kcj 0.)]. (4)
Once again similitude is impossible unless further relaxation of similarity
requirements are made.
At high altitudes the binary scaling assumption can be made with
confidence. Assuming that the nonequilibrium process is described by
two body reactions is equivalent to assuming that Kcj is large and
independent of density, which yields the final similarity relation
CH = CH (@; U., Tw, Pm R). (5)
The last parameter in Equation (5) is proportional to the number of
molecular collisions in a characteristic distance R. To duplicate a
3
nonequilibrium relaxation the number of collisions in a distance R
vehicle
in flight must be the same as the number of collisions in a distance
R in the wind tunnel. Hence, the parameter pR is characteristic of
model
the nonequilibrium vibration, chemistry and even radiative heat transfer
(see Ref. 8).
A similar approach to determine the nonequilibrium similitude param-
eters for the heat transfer on a sharp cone yields essentially the same
results as Equation (5), i.e., at a point L down the body
CH = CH (c', U., Tw, P. L) (6)
where 9 is the cone half angle.
To include effects of oxygen predissociation in the nozzle, the use
of an oxygen rich flow can be utilized to account for oxygen freezing and
not recombining as in the equilibrium flight situation.
III. DATA CORRELATION
The heat transfer at locations downstream of the stagnation point of
blunt nosed slender bodies, which still exhibit the nonequilibrium relax-
ation process originating from the stagnation point, will now be considered.
The stagnation flow freezes in the rapid expansion around the blunt nose
and its flow characteristics near the sonic line are indicative of the
amount of heat transfer further downstream. When freezing occurs, the
chemical enthalpy becomes unavailable for conversion to kinetic energy
until much further downstream when equilibrium is achieved. The frozen
situation is indicative of blunt nosed slender bodies while the equilibrium
case corresponds to sharp cone flow for the same cone half angle.
An estimate is made of the reduction in heat transfer at a point
immediately downstream of the stagnation point due to the nose bluntness-
induced freezing of chemical energy. The estimate for the blunt nosed,
conical body heat transfer coefficient in terms of the sharp cone value
for the same conditions can be related to the enthalpy by
CH blunt/CH sharp U hblunt/hsharp = hnot frozen /hequilibrium (7)
The expressions for enthalpy are easily written for a vibrationally excited
gas made up of N2, 02, N and 0 assuming that there is neither any ionization
nor any electronic excitation. Two cases are examined: (1) the case where
4
oxygen is completely dissociated, and (2) the case where oxygen and
nitrogen are completely dissociated. Case (1) corresponds to rarefied
flow with temperatures behind the shock from about 4,000 OK to 5,000 OK
and Equation (7) gives values from 0.59 to 0.64. Case (2) corresponds
to rarefied flow with temperatures from about 6,000 OK to 12,000 OK and
Equation (7) gives values from 0.23 to 0.37.
The spread of CH blunt/CH sharp from 0.23 to 0.64 for case (1) and
(2) predicts the lowest value before ionization. As the flow reaches
equilibrium far downstream of the nose the ratio approaches 1.0. Since
more sharp cone data exists than blunt cone data for low Reynolds numbers,
this range of 0.23 to 1.0 gives a rough estimate of the value for CH blunt
and an idea of the degree of dissociation.
Figures 1 and 2 show the heat transfer coefficient in terms of the
rarefaction parameter for sharp and blunt cones, respectively, Ref. 9 -
11. For design purposes least squares correlations of the data are
obtained in the form
3
LOG CH (1-H /H )/SIN 9c] =s as (LOG10  )s (8)
where H is the total enthalpy,
X = Remj/(1 Y C* COS 9) = (PO L) Tr/ (U, rCOS c' (9)
Re, = p, U, L/ , (10)
C* = r Tj.pT r ' (11)
T T + (T -T )/2 - T COS2 9 /3 (12)
r w 0 w o c
and for sharp cones ao = -0.34407400, al = -0.34912983, a2 = -0.10445498,
and a3 = +0.022766463 while for blunt cones ao = -0.66125881, al = -0.33391435,
a2 = -0.035024979, and a3 = +0.0072725000. Note that the rarefaction param-
eter in Equation (9) can be written in terms of the similarity parameters
discussed earlier.
The estimates of the heat transfer coefficient from Equation (7) are
shown in Figure 3 along with the curve fit for sharp and blunt cones. For
large Reynolds numbers the nonequilibrium effects are not to prevalent;
however, for values of L 10 the gas appears to be completely dissociated
in 02 and partially dissociated in N2. The band of data for blunt nosed
cones is all above the value CH blunt = .37 CH sharp and since there is 5
very little dependence upon the amount of nose bluntness, all blunt cone
values for CH can be described by Equation (8) or the inequality .37 L
CH blunt/CH sharp 1.
IV. CONCLUSIONS
In summary, the duplication of hypersonic, rarefied blunt nose and
sharp nose heat transfer requires the use of air over a geometrically
similar model with the same free stream velocity, wall temperature, and
product of free stream density and characteristic dimension. This is
based on the binary scaling assumption.
Estimates of the heat transfer coefficient on blunt nosed cones are
found to be in the range from .37 to 1.0 (as the flow relaxes) times the
value on sharp nosed cones for the same half angle and distance downstream.
Physically, this reduction is due to the nose bluntness-induced freezing
of chemical energy. Least squares correlations are also presented for
sharp and blunt cones. These correlations may be utilized for preliminary
estimates of hypersonic aeroheating in the transitional flow regime.
REFERENCES
1. Hayes, W. D., and Probstein, R. F., Hypersonic Flow Theory, Academic
Press, New York, 1959, p. 30.
2. Cheng, H. K., "Similitude of Hypersonic Real Gas Flows Over Slender
Bodies with Blunted Noses," J. Aero/Space Sci., Vol. 26, 1959, pp. 575-585.
3. Inger, G. R., "Similitude of Hypersonic Flows Over Slender Bodies in
Nonequilibrium Dissociated Gases," AIAA Journal, Vol. 1, No. 1, Jan.
1963, pp. 46-53.
4. Cheng, H. K., "Hypersonic Shock-Layer Theory of the Stagnation Region
at Low Reynolds Number," Proceedings of the 1961 Heat Transfer and
Fluid Mechanics Institute, Stanford University Press, Stanford,
California, 1961, pp. 161-175.
5. Jain, A. C., "Hypersonic Merged Stagnation Shock Layers," AIAA 6th
Fluid and Plasma Dynamics Conference, AIAA Paper No. 73-639, Palm
Springs, California, July 1973.
6. Hendricks, W. L., "Nonequilibrium and Equilibrium Radiation to the
Space Tug from the Shock Layer," NASA CR-61396, Sept. 1972, pp. 229-278.
7. Hall, J. G., Eschenroeder, A. Q., and Marrone, P. V., "Blunt-Nose
Inviscid Airflows with Coupled Nonequilibrium Processes," JAS, Sept.
1962, pp. 1038-1051.
8. Camm, J. C., Kivel, B., Taylor, R. L., and Teare, J. D., "Absolute
Intensity of Nonequilibrium Radiation in Air and Stagnation Heating
at High Altitudes," AVCO-Everett Res. Rept. 93, 1959.
6
9. Waldron, H. F. , "Viscous Hypersonic Flow Over Pointed Cones at Low
Reynolds Numbers," AIAA Journal, Vol. 5, No. 2, February 1967,
pp. 208-218.
10. Wilkinson, D. B. and Harrington, S. A. , "Hypersonic Force, Pressure,
and Heat Transfer Investigations of Sharp and Blunt Slender Cones,"
AEDC-TDR-63-177, August 1960.
11. Kussoy, M. I., Stewart, D. A., and Horstman, C. C. , "Hypersonic
Rarefied Flow Over Sharp Slender Cones," NASA TN D-6689, March 1972.
7
Sharp Cone Half Angle
A 5.00*
Free Molecular Flow Limit Unity Accommodation A 6.3**
Least Squares Correlat:on 0 9.0**/E 10.00*
1. 00_ G N 10. 00***
um o 20.00*
MD 0 20.00 Theoretical - Cheng
o I 0 25.0
*Waldron 1967
**Wilkinson & Harrington 1963
***Kussoy 1972 ± 20% O A
0.01
0.01 0.10 1.0 10 100 1000
Re 1
X C
Fiure 1 Shar cone heat transfer d ta v riation with rarefaction arameter.
Figure i. Sharp cone heat transfer data variation with rarefaction parameter.
Wilkinson & Harrington 1963
Blunt Cone Half Angle & Nose Radius
6 6.30 r = .125"
A 6.30 r = .25"
t0 a 9.00 r = .25"
0 9.00 r = .45"
U
Least Squares Correlation
C= 0.10-
0.10 1.00 o10.0 100.00 1000.00
RE, /N YC , c os 0
Figure 2. Blunt cone heat transfer data variation with rarefaction parameter.
1.00
.Sharp Cone Data
a0 Dissociated
Blunt Cone Data
0 & N Dissociated
0.10
0. 01
0.10 1.00 10.00 100.00 1000.00
x = Rei,/M2 ye C, COS 9
Figure 3. Comparison of blunt and sharp cone heat transfer data.
APPROVAL
SIMILITUDE REQUIREMENTS FOR HYPERSONIC,
RAREFIED, NONEQUILIBRIUM FLOW
By
William L. Hendricks
The information in this report has been reviewed for security classification.
Review of any information concerning Department of Defense or Atomic Energy
Commission Programs has been made by the MSFC Security Classification
Officer. This report, in its entirety, has been determined to be unclassified.
This document has also been reviewed and approved for technical accuracy.
Homer B. Wilson, Jr.
Chief, Thermal Environment Branch
Werner K. Dahm
Chief, Aerophysics Division
J. D. Lovin od
Director, Systems Dynamics Laboratory
NASA-MSFC
1I


Similitude requirements for hypersonic, rarefied, nonequilibrium flow

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