The steady-state Navier-Stokes equations are solved for hypersonic flow about blunt axisymmetric bodies. The equations of motion are solved by successive approximations using an implicit finite-difference scheme. The results are compared with viscous shock-layer theory, experimental data, and time-dependent solutions of the Navier-Stokes equations. It is demonstrated that viscous shock-layer theory is sufficiently accurate for the range of flight conditions normally encountered by entry vehicles

Anderson, E. C.

Moss, J. N.

English

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NASA TECHNICAL NASA TM X- 71977
MEMORANDUM COPY NO. , w .
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Langley Research Center .d
Hampton, Virginia _
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tnvised, or may be incorporated in another publication
NUMERICAL SOLUTIONATIONAL AERONAUTICS AND SPAC OF THE STE ADMINISTRATION
NAVIER-STOKES EQUATIONS FOR HYP ENTERSONIC, HAMPTON, VIRGINIA 23665
FLOW ABOUT BLUNT AXISYMMETRIC BODIES
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
LANGLEY RESEARCH CENTER, HAMPTON, VIRGINIA 23665
https://ntrs.nasa.gov/search.jsp?R=19740019638 2020-03-23T07:08:44+00:00Z
1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.
TM X-71977
4. Title and Subtitle 5. Report Date
Numerical Solution of the Steady-State Navier-Stokes June 1974
Equations for Hypersonic Flow About Blunt Axisymmetric 6. Performing Organizatioo Code
Bodies
7. Author(s) 8. Performing Organization Report No.
E. C. Anderson and James N. Moss
10. Work Unit No.
9. Performing Organization Name and Address
Langley Research Center 11. Contract or Grant No.
Hampton, VA 23665'
13. Type of Report and Period Covered
12. Sponsoring Agency Name and Address Technical Memorandum
NASA Langley 14. Sponsoring Agency Code
15. Supplementary Notes
Article proposed for publication in the Proceedings of the 10th Anniversary
Meeting of the Society of Engineering Science, 1974.
16. Abstract
The steady-state Navier-Stokes equations are solved for hypersonic flow
about blunt axisymmetric bodies. The equations of motion are solved by
successive approximations using an implicit finite-difference scheme.
The results are compared with viscous shock-layer theory, experimental data,
and time-dependent solutions of the Navier-Stokes equations. It is demonstrated
that viscous shock-layer theory is sufficiently accurate for the range of
flight conditions normally encountered by entry vehicles.
17. Key Words (Suggested by Author(s)) (STAR category underlined) 18. Distribution Statement
Navier-Stokes
Shock layer Viscous
Hypersonic flow Unclassified - unlimited
Blunt body
Perfect gas
19. Security Claif. (of this report) 20. i3ecurity Classif. (of this page) 21. No. of Pages 22. Price*
Unclassified Unclassified 6 $5.00
The National Technical Information Service, Springfield, Virginia 22151
*Available from
STIF/NASA Scientific and Technical Information Facility, P.O. Box 33, College Park. MD 20740
NUMERICAL SOLUTION OF THE STEADY-STATE NAVIER-STOKES EQUATIONS
FOR HYPERSONIC FLOW ABOUT BLUNT AXISYMMETRIC BODIES
E. C. Anderson
O.D.U. Research Foundation, Norfolk, Va.
and
James N. Moss
NASA Langley Research Center, Hampton, Va.
ABSTRACT
The steady-state Navier-Stokes equations are solved for
hypersonic flow about blunt axisymmetric bodies. The equations
of motion are solved by successive approximations using an
implicit finite-difference scheme. The results are compared
with viscous shock-layer theory, experimental data, and time-
dependent solutions of the Navier-Stokes equations. It is demon-
strated that viscous shock-layer theory is sufficiently accurate
for the range of flight conditions normally encountered by entry
vehicles.
INTRODUCTION
The solution for the flow field about blunt axisymmetric
bodies at zero angle of attack has been determined for moderate
to high Reynolds number conditions using the combination of
inviscid and boundary-layer theories or viscous shock-layer
theory. However, in a recent publication, Li [1] questions the
validity of existing theories in the range of Reynolds numbers
encountered by entry vehicles in the altitude range 200,000 to
300,000 feet. Li presents a time-dependent finite-difference
solution of the Navier-Stokes equations and applies the method
to typical shuttle trajectory points in this altitude range.
This method of solution is severely restricted by storage
requirements and excessive computing cost and, as a result,
limited in application to the nose region of the body.
In a previous publication,' Davis [2] presents a solution of
the viscous shock-layer equations and shows good agreement with
experimental data for Reynolds numbers as low as 22. The viscous
shock-la er equations presented by Davis [2] contain all te rns up
to 0 (E2), where e is the Van Dyke 13] expansion parametez
defined as
= E[(Tref)/p U RN 1/2
and
L-9200
Tref = U/Cp
RN = body nose radius
U = free-stream velocity
i = coefficient of viscosity
Consideration of the order of magnitude analysis of the
Navier-Stokes equations given by Van Dyke [3] and the viscous
shock-layer equations presented by Davis [2], shows that the
solution of the two sets of equations will have significant dif-
ferences only when E ~ O(1). This suggests that a method of
successive approximations in which the starting solution is given
by the viscous shock-layer equations can be used to determine the
solution of the steady-state Navier-Stokes equations for most
flight conditions of practical interest.
The solutions of the Navier-Stokes equations presented herein
are restricted to a perfect gas having a constant ratio of spe-
cific heats and the bow shock is assumed distinct. However,
appropriate slip and jump boundary conditions are included. The
solutions of the a-equations are compared wit viscous
shock-layer theory, experimental data, and the transient solution
presented by Li [1].
E -LOD OF SOLUTAOlU
The Navier-Stokes equations are expressed in the shock-body
oriented coordinate system defined by Davis [2]. After deter-
mining the solution of the viscous shock-layer equations using
the implicit finite-difference scheme described in Reference [2],
the terms of higher order are evaluated using these flow-field
data. The higher order terms are held constant during the solu-
tion for the first approximation to the Navier-Stokes equation.
After the flow-field data for the first approximation are deter-
mined, the higher order terms are reevaluated and held constant
during the solution for the second approximation. This procedure
is repeated until the flow-field data corresponding to successive
approximations converge within a specified limit. For details of
the numerical method, the reader is referred to the paper pre-
sented by Davis [2].
RESULTS AND DISCUSSION
A comparison of the temperature profiles along the stagna-
tion streamline and at a distance of 0.7 nose radii downstream
of the stagnation point is shown in Figure 1 for solutions corre-
sponding to the steady-state Navier-Stokes equations, the viscous
shock-layer equations, and Li's transient solution of the Navier-
Stokes equations. The flight conditions for this case correspond
2
160 - E sRN 07
TIT 80 
/N
40( - PRESENT STEADY STATE RESULTS
---- Li's TRANSIENT RESULTS. [11
0 .0 .08 .12 .16 .20
n/RN
Figure 1. Comparisons of temperature profiles
for transient and steady-state solutions.
to a Mach number of 24 at an altitude of 250,000 feet, a free-
stream Reynolds number of 14,400 based on a 2-foot nose radius,
and an E of 0.04. The body considered is a 200 h perboloid.
For these conditions, the present solution of the Navier-Stokes
equations and the solution of the the viscous shock-layer equations
differed by less than 0.5% and these results are shown as a
single curve. The solutions given by Li are seen to exhibit
large oscillations at both body locations. Since the maximum
number of node points across the shock layer in Li's solution is
less than 35, these oscillations cm probably be removed by
increasing the number of node points. For the present analysis,
a minimum of 50 grid points was necessary for adequate resolu-
tion of this problem. As indicated by Li, this is apparently a
prohibitive requirement in his solution. It is noted that the
temperature behind the shock at the downstream station computed
using the transient method of solution is approximately 15% lower
than predicted by the present method. The temperature predicted
by Li closely corresponds with the value obtained assuming that
the local shock angle is the same as that at the body. The com-
parison of the present solution of the steady-state Navier-Stokes
equations and the viscous shock-layer equations with the tran-
sient solution of Reference [1] indicates that the transient
solution is an unacceptable alternate method for solving the
viscous blunt-body problem.
The present solution of the Navier-Stokes equations is com-
pared with the solution of the viscous shock-layer equations\and
experimental data for Reynolds numbers in the range 90 to 31,160
in Figure 2. The experimental data are given by Little [4]. The
comparisons of the total drag coefficient, CD, shown in Figure 2
are representative of the differences noted in heat transfer,
3
2.0
N Re, Tw Tw , R E M
1.8 1 31,160 .08 474 .024 19.10
0 2 1.035 .79 83 .140 9.92
0 3 90 .65 174 .366 9.90
1.6 -
- NAVIER-STOKES
\ VISCOUS SHOCK-LAYER
1.4\ 0, 0 EXPERIMENT (4]1.4 \
3
CD  1.2
1.0 2
.8 o
10
.4 I  ,
0 1 2 4 5 6
rIRN
Figure 2. Total drag coefficient for a 450 hyperboloid.
skin friction, and surface pressure distribution using the two
sets of governing equations. For the higher Reynolds number
case, the two solutions differed by less than 0.1% and are shownas a single curve. At a Reynolds number of 1035, the maximumdifference in the two solutions is less than 5%, and at aReynolds number of 90, the two solutions differ by approximately15% in the stagnation region and less than 2% in the downstreamregion. The solution of both the viscous shock-layer equationsand the Navier-Stokes equations is in good agreement with the
experimental data. Figure 3 shows the skin friction and heat-transfer distributions for the case of a Reynolds number of 90.
The solutions of the Navier-Stokes equations presentedherein are seen to reduce to the solution of the viscous shock-layer equations for high Reynolds number flow and significantdifferences occur only at very low Reynolds numbers. These
results are in agreement with the requirements of Van Dyke's [3]order of magnitude analysis. The cases presented were solved toa distance of 8 nose radii downstream of the stagnation point
4
NAVIER-STOKES SOLUTION
--- VISCOUS SHOCK-LAYER SOLUTION
12
q. Btulft2 -s 8
4-
I I I I
0 1 2 3 4 5 6
siR
N
HEAT TRANSFER DISTRIBUTION
NAVIER-STOKES SOLUTION
----- VISCOUS SHOCK-LAYER SOLUTION
0 1 2 3 4 5 6
slRN
SKIN FRICTION DISTRIBUTION
Figure 3. Comparisons of results for a 450 hyperboloid.
using a CDC-6600 digital computer. The maximum computing time
was less than 3 minutes and the storage requirement is less
than 50,0008.
The results of this investigation reveal that there is little
need for a solution of the Navier-Stokes equations for viscous
blunt-body flows in the range of flow conditions of practical
interest. However, a method which is accurate and computatilonally
efficient has been developed for their solution. The numerical
method with proper modifications can be extended to merged layer
problems and reacting gas mixtures.
REFERENCES
[1]C. P. Li, "Numerical Solution of Viscous Reacting Blunt Body
Flows of a Multicomponent Mixture," AIAA Paper No. 73-202,
1973.
[2]R. T. Davis, AIAA J. 8, 843-851 (1970).
[3]M. Van Dyke, "A Review and Extension of Second-Order Hyper-
sonic Boundary-Layer Theory," Rarefied Gas Dynamics, Fluid
Symposium Supplement 2, Vol. II, Academic Press, New York,
1963.
[4]H. R. Little, "An Experimental Investigation of Surface Con-
ditions on Hyperboloids and Paraboloids at a Mach Number of
10," M. S. Thesis, University of Tennessee, 1969.
6


Numerical solution of the steady-state Navier-Stokes equations for hypersonic flow about blunt axisymmetric bodies

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