Full text of DOI:10.2514/2.3965

J. SPACECRAFT, VOL. 40, NO. 3: ENGINEERING NOTES
437
3Bishop, R. H., and Azimov, D. M., “Analytical Trajectories of an Ex-
tremal Motion with Low-Thrust Exhaust-Modulated Propulsion,” Journal
of Guidance, Control, and Dynamics, Vol. 38, No. 6, 2001, pp. 897–903.
4Azizov, A. G., and Korshunova,N. A., “On an Analytical Solutionof the
Optimum Trajectory Problem in a Gravitation Field,” Celestial Mechanics,
Vol. 38, No. 4, 1986, pp. 297–306.
couplestwo differentcomputercodespreviouslydevelopedby these
authors:a computational uid dynamics(CFD) code to simulate the
 uid  ow inside a rocket nozzle2 and a material thermal response
code to study the in-depth response of TPS materials exposed to
the hot gas  owing through the rocketnozzle.3 These two codes are
explicitly coupled through an energy balance at the common wall
boundary of the nozzle. This Note presents in detail the coupling
technique of the two codesand comparesthe computedresultswith
the test results generated in house as well as those available in the
literature.
5Azimov, D. M., and Bishop, R. H., “Extremal Rocket Motion with Max-
imum Thrust in a Linear Central Field,” Journal of Spacecraft and Rockets,
Vol. 38, No. 5, 2001, pp. 765–776.
4
6Tapley, B. D., “Regularization and the Computation of Optimal Trajec-
tories,” Celestial Mechanics, Vol. 2, No. 3, 1970, pp. 319–333.
7McCue, G. A., “Quazilinearization Determination of Optimum Finite-
Thrust Orbital Transfers,” AIAA Journal, Vol. 5, No. 4, 1967, pp. 755–763.
8Enright, P. J., and Conway, B. A., “Optimal Finite-Thrust Spacecraft
Trajectories Using Collocation and Nonlinear Programming,” Journal of
Guidance, Control, and Dynamics, Vol. 14, No. 5, 1991, pp. 981–985.
9Kornhauser, A. L., Lion, P. M., and Hazelrigg, G. A., “An Analytic
Solution for Constant-Thrust, Optimal Cost, Minimum-Propellant Space
Description of the Problem
In a solid rocket nozzle, normally the inner wall is formed with
an ablativeliner material such as carbonphenolicor silica phenolic,
followed by its structural backup. While the rocket motor is in op-
eration, the liner is subjectedto the  ow of hot gaseous combustion
products, as a result heat  ows from the hot gas into the liner mate-
rial. The ablative liner absorbs the heat and protects the underlying
structureby keeping its temperaturewithin tolerablelimits. Thus, it
is essential for a nozzle thermal designer to compute accurately the
heat  ow rate from the hot gas into the liner material and to predict
accurately the response of charring ablators exposed to the hot gas
 ow of combustion products.
Trajectories,” AIAA Journal, Vol. 9, No. 7, 1971, pp. 1234–1239.
10
z
Novoselov, V. S., Analytical Theory of Trajectory Optimi ation in the
Gravitational Fields, Leningrad State Univ. Press, Leningrad, U.S.S.R.,
1972, p. 317.
11Azizov, A. G., and Korshunova, N. A., “On Optimal Trajectories in
Gravitational Fields, Admitting Approximation by Central Linear,” Space
Research, Vol. 29, No. 4, 1991, pp. 525–531.
12Jezewski, D. E., “Optimal Analytic Multi-Burn Trajectories,” AIAA
Journal, Vol. 10, No. 5, 1972, pp. 680–685.
During the days when the high-speed computing systems were
13Szebehely, V. G., Adventures in Celestial Mechanics, Univ. of Texas
Press, Austin, TX, 1993, pp. 56–81.
5
yet to be invented, a closed-form equation, which could be hand
computed, was used for estimating nozzle wall heat  ux. With the
availabilityof high-speedcomputing systems, a more sophisticated
solution1 could be adopted for heat- ux computation.However, for
high-altitude rocket nozzles it is always desirable to carry out a
detailed study by solving Navier–Stokes (N-S) equations for com-
puting nozzle wall heat  ux and validate it with realistic test results
so as to design an optimum TPS. Such a detailed analysis has now
been possible, and the same has been dealt with in the following
sections.
D. L. Edwards
Associate Editor
Performance Analysis of Thermal
Protection System of a Solid
Rocket Nozzle
CFD in Rocket Nozzles
Jones and Shukla2 described an analysis of the  ow in a rocket
nozzle and the development of the associated computer code.
The steady viscous turbulent compressible  ow in a converging–
diverging axisymmetric nozzle is simulated through a computer
code using a time-marching explicit scheme. The N-S equations
governingan axisymmetric ow for the physicaldomain of a rocket
nozzle are transformedto a rectangularcomputationaldomain with
a boundary-Žtted coordinate system. The effect of turbulence is in-
corporated in the code by using the Baldwin–Lomax model. The
equationsare cast into Ž nite differenceform in a variablemesh net-
work.The functionalvaluesatthe interiormesh pointsare computed
using MacCormack’s explicit predictor–corrector Ž nite difference
scheme; a two-step characteristic scheme is adopted for applying
boundary conditions using two independentvariables.
V. Jones¤ and K. N. Shukla†
Vikram Sarabhai Space Center,
Trivandrum 695 022, India
Introduction
BLATIVE materials are commonly used to protect the solid
rocketnozzlewallsexposedto high-temperaturegaseouscom-
A
bustion products.An accurate predictionof the thermal responseof
these materialsis essentialfor a nozzlethermal designerto success-
fully carry out the design of an optimum thermal protection system
(TPS). Numerous test results on heat transfer in sea-level nozzles
are available. Therefore, it is not too difŽ cult for a nozzle thermal
designer to design an optimum TPS in the case of sea-level noz-
zles. However, test results for high-altitudenozzles are very scarce.
Consequently, a considerable degree of uncertainty exists in the
estimation of wall heat  ux, as well as the design of a TPS for
a high-altitude rocket nozzle. The most widely used approach for
computing wall heat  ux in a rocket nozzle is the boundary-layer
method developedby Elliott et al.1 It is always desirableto perform
a detailed study in order to supplement the existing design code
and validate it with realistic test results so as to design an optimum
TPS for the high-altitude rocket nozzles. The present investigation
Inadditiontothecomputationoftheusual owŽ eldvariablessuch
as density, pressure, velocity, temperature,etc., the momentum and
energythicknessesare also computedusing the respectiveintegrals,
for the estimation of heat  ux to the wall. The convective heat  ux
q
to the wall
is computed as follows:
w
qw D Ch Uc
T
ad
¡ Tw
½
.
/
(1)
p
Theexpressionsforcomputingotherparameterssuchasskinfriction
f , Stanton number h , etc., are given by Bartz.6 Boundary-layer
C
C
n D
C
for Ž lm property conditions
f
interaction exponent
are assumed.
0:1 and
Received 13 June 2002; revision received 12 January 2003; accepted for
°c
publication30 January 2003.Copyright 2003by theAmerican Instituteof
In-Depth Response of Wall Materials
Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper
may be made for personal or internal use, on condition that the copier pay
the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose-
wood Drive, Danvers, MA 01923; include the code 0022-4650/03 $10.00 in
correspondence with the CCC.
The nozzle wall is assumed to consist of a charring ablator fol-
lowedbya noncharringstructuralbackup.An analysisandthe devel-
opment of a computercode to predictthe in-depthresponseof char-
ringablatorsexposedto high-temperatureenvironmentsis available
elsewhere.3 The governingmathematical equations are derived in a
Ž xed coordinatesystem tied to the originalsurface.The adoptionof
¤Scientist.
†Scientist. Associate Fellow AIAA.

438
J. SPACECRAFT, VOL. 40, NO. 3: ENGINEERING NOTES
a Ž xed coordinatesystemeliminatesthe use of a convectioncompo-
nent, which is inherent in a moving coordinatesystem while giving
the same results as that of the latter. In-depth pyrolysis kinetics is
assumed to obey an Arrhenius law in a reaction zone, and the ther-
mophysical properties are treated as temperature as well as state
(virgin, partially charred, or fully charred) dependent. These equa-
tions are transformed into Ž nite difference form implicit in time,
and a computercode is developedfor solution.The parameterssuch
as erosion, char depth, charring ablator/structural backup interface
temperature, in-depth density, and temperature Ž elds, etc., can be
estimated by this code.
Coupling of CFD and Material Response Codes
The governing equations describing the conservation of mass,
momentum, and energy for  uid  ow in the rocket nozzle and the
equations of mass and energy to characterize the thermal response
of the charring wall material and the development of respective
computer codes were discussed earlier. The two codes are coupled
along the active surface of the nozzle wall by the following energy
balance:
qcond D qw C qrad ¡ mP H ¡ mP
H
c
g1
c1
(2)
g
q
q
w
where
is the energy conductedinto the material, is the con-
cond
Fig. 1 Comparison of heat  ux for sample problem 1.
q
vective heat  ux to the wall obtained from the CFD code, and
rad
is the radiative heat exchange. The third and fourth terms on the
right-hand side represent the energy convected away by the pyrol-
ysis gases passing through the surface and the energy consumed
for surface char ablation, respectively.The surface char ablation is
assumed to take place once the temperature of the charred surface
£
is dividedinto 45 21 mesh points.The solutionconvergesin 2644
iterations and takes 14.2 s, CPU time. As shown in Fig. 1, nozzle
wall heat  ux computed by the CFD code compares well with the
test results.
T
reaches a particular temperature ab, known as the char ablation
temperature,whose value varies from material to material.
The couplingis done explicitlysuch that the convectiveheat  ux
qw
is computedfrom the CFD code, using Eq. (1), at each boundary
Sample Problem 2
node of the nozzle wall for the initial chamber pressure and the
ambient surface temperatureof the wall. The computed heat  ux is
fed into the thermal responsecode, and this code is time marched at
each nodal point for a Ž xed time step. The new surface temperature
of the nodal points and the chamber pressure for the corresponding
time, taken from the pressure vs time arrays of the predicted motor
performance,aretheninputto theCFD code.The processisrepeated
for each time step until the end of motor operation. CPU time is
considerably reduced by assuming the CFD solution as the initial
condition for the subsequent time step of the duration of motor
operation.
The static test of a solid rocket motor tested under high-altitude
conditionsis describedelsewhere.7 The motor has a contour nozzle
D
with throat diameter of 8.0 cm and expansion ratio of " 50:0.
D
The throat insert extending to " 2:7 is made of graphite. Carbon
D
phenolic is provided as liner for the expansion ratio " 2:7–13.9,
and thereafter it is silica phenolic. Both the composites are tape
wound, parallel to the nozzle axis. The motor operated for 15.5 s
with an average pressure of 4.78 MPa. The pressure vs time trace
recorded from the static test is reported in Ref. 7. Properties of the
D
mN D
c D
combustionproducts are as follows:° 1:19;
29:4;
2198
p
¡ K T D
c¤ D
J/(kg
/;
3414 K, and
1469 m/s.
0
q
T
w; "/ is generatedby the CFD code
Alternativelya matrix of
.
w
The computationaldomainof the precedingnozzleis dividedinto
for the average chamber pressure, where " is the area ratio of the
section of the nozzle, which varies from the nozzle inlet to nozzle
£
54 24 mesh points.For the computationof cold wall heat  ux, the
CFD solution converges in 2348 iterations and takes 28.1 s, CPU
T
w
exit, and
varies from ambient temperature to the char ablation
q
T
q
time. For the generation of
.
w; "/ matrix, is computed for
w
w
temperatureof the material. Once this matrix is generated,the ther-
T
T D
21 values of
with
305; 400; 500; : : : ; 2100; 2173; 2273 K,
w
w
q
q
mal response code need not invoke CFD code for obtaining
.
.
w
w
where 305 K is the ambient temperature and 2173 and 2273 K are
assumed to be the ablation temperatures of carbon phenolic and
silica phenolic, respectively.
q
T
w; "/ matrix and obtainthe required
Instead,it can scan the
.
w
This procedure has further reduced the CPU time drastically.
The coupled code is to be operated for the performance anal-
ysis of the TPS. The liner thickness of the nozzle varies from
Validation of the Code
The modules developed and presented are based on a detailed
CFD analysis of  uid  ow in rocket nozzles and heat-transferanal-
ysisof ablativewall materials.The validityof the analysisand accu-
racy of the Ž nal codecan be establishedby comparingthe computed
resultswiththetestresults.Two differentsampleproblemsforwhich
test results are available are presented. In the CFD module a con-
vergence tolerance of 0.0003% is speciŽ ed on the axial velocity in
the expansion region. The computations are performed on an IBM
RS/6000-44P-270system.
D
D
12.0 mm at " 2:7 to 7.0 mm at " 50:0. The liner material is
divided into nodes of 0.2 mm, and each node is subdivided into
eight nodelets. Behind the liner material is a structural backup of
Ž berglass material of uniform thickness 2.0 mm. The backup is
divided into nodes of 0.25 mm and is assumed as noncharring be-
cause the system is designed so as to keep the interface temper-
ature around the ambient level only. The time step assumed for
t D
this analysis is 1 0:02 s. The CPU time taken by the matrix
method is 1.12 s per run, whereas that by the direct use of CFD
code is 77.6 s. The computed results by the two different proce-
dures at two typical locations of the nozzle are given in Table 1.
The thermal properties of carbon phenolic and silica phenolic are
available in Ref. 8. Computed cold wall heat  ux for the perceding
nozzle is plotted in Fig. 2. Computed char depth values in carbon
phenolic and silica phenolic are comparedin Fig. 3 with those mea-
sured from the static test.
Sample Problem 1
Consider the measured heat  ux into the wall of a conical nozzle
D
of exit area ratio, " 20:0 and as reported in Ref. 1. The motor was
operated with a N2O4–N2H4 combination at a chamber pressure of
301 psia. Other nominal conditions are assumed to be the same as
reported in Ref. 1. The computationaldomain (nozzle region only)

J. SPACECRAFT, VOL. 40, NO. 3: ENGINEERING NOTES
439
Table 1 Comparison of char depth for sample problem 2
Nozzle
diameter, mm
Ablator
thickness, mm
Char depth qw
from matrix, mm
Char depth qw direct
from CFD code, mm
Char depth measured
average, mm
201
446
11.18
8.41
6.70
3.37
6.68
3.34
5.90
3.10
method. Thermal analysis for a working nozzle in an upper-stage
solid rocket motor is also performed. The throat diameter of the
D
nozzle is 16.6 cm and has an expansionratio at the exit of " 70:0.
The motor operates for 115.0 s. For the CFD analysis the compu-
£
tational domain is divided into 148 43 mesh points. The matrix
method takes 10.0 s for one run, whereas the coupled method takes
300.0 h, CPU time.
Conclusions
An attempt is made to provide a detailed analysis technique for
the performance of thermal protection systems of solid rocket noz-
zles. The present algorithm and the associated computer code are
validatedby comparing the computed results with test results. This
work is aimed at providinga detailed method of analysis to supple-
mentthe existinganalysistechniquesin viewofthelackofsufŽ cient
test results for high-altituderocket nozzles, so as to design an opti-
mum TPS having a minimum weight conŽ guration. The algorithm
isapplicableto sea-levelnozzlesalso. In high-altituderocketmotors
the inert weight of the system has a direct or near-directbearing on
the payload of the launch vehicle.
Fig. 2 Computed cold wall heat  ux for sample problem 2.
Acknowledgments
The authors express their sincere gratitude to A. K. Verma, E. V.
Zoby, T. C. Lin, and the anonymousrefereesfor criticallyreviewing
the manuscript and providingvaluablesuggestions.The authors are
also thankful to S. V. Subba Rao, M. S. Padmanabhan, B. C. Pillai,
and M. C. Uttam for their keen interestin the successfulcompletion
of this work.
References
1Elliott, D. G., Bartz, D. R., and Silver, S., “Calculation of Turbulent
Boundary Layer Growth and Heat Transfer in Axisymmetric Nozzles,” Jet
Propulsion Lab., California Inst. of Technology, TR 32-387, Pasadena, CA,
Feb. 1963.
2Jones, V., and Shukla, K. N., “Fluid Flow in Rocket Nozzles,” AIAA
Paper 98-3840, July 1998.
3Jones, V., and Shukla, K. N., “In-Depth Response of Charring Ablators
to High Temperature Environments,” AIAA Paper 99-3462, June 1999.
4Jones, V., and Shukla, K. N., “Performance Analysis of Thermal Pro-
tection System of a Solid Rocket Nozzle,” AIAA Paper 2002-4196, July
2002.
Fig. 3 Comparison of char depth for sample problem 2.
Discussion
5Bartz, D. R., “A Simple Equationfor Rapid Estimation of Rocket Nozzle
ConvectiveHeat Transfer CoefŽ cient,” Jet Propulsion, Vol. 27, No. 1, 1957,
pp. 49–51.
Generally,an elementof uncertaintyexistsin the modelingof sur-
face erosion in ablative materials. In the case of sample problem 2,
posttest evaluation showed no visible erosion in the nozzle. This is
because the motor operated for the duration of 15.5 s only. How-
ever, the computations predicted a small erosion of 0.50 mm at the
6Bartz, D. R., “Turbulent Boundary-Layer Heat Transfer from Rapidly
Accelerating Flow of Rocket Combustion Gases and Heated Air,”Advances
in Heat Transfer, Vol. 2, 1965, pp. 1–108.
D
D
fore end (" 2:7) of the nozzle and diminished to zero at " 5:0,
7Krishnamoorthy, P. A., Pillai, L. A., Padmanabhan, M. S., and Pillai,
B. C., “Subscale Performance Simulation Testing for High Altitude Solid
Motors,” 21st International Symposium on Space Technology and Science,
Japan Society for Aeronautical and Space Sciences, Tokyo, May 1998,
ISTS Paper 98-a-1-13.
but still the computed char depth matches the experimental results
D
D
well. From " 5:0 to " 50:0, there is absolutely no erosion ei-
ther experimentally or theoretically. Again as seen from Fig. 3 the
D
measured char depth at " 49:0 is much less than that computed.
8“Testing of Ablative Specimens at Elevated Temperatures,” Energy Ma-
terials Testing Lab., ME-04005,Final Rept. EMTL-1059,FMI-EMTL-W/A
No. 5751, Biddeford, ME, Jan. 1986.
There was some quantity of alumina deposit along the exit region
of the nozzle. This alumina shielded the nozzle liner from the expo-
sure to the hot gas and hence reduced the char depth. Thus, this test
provides an excellent set of results for comparison with theory, in
that it is free from many uncertainties inherent in surface erosion.
As seen from Table 1, the computed results obtained by the cou-
pled method are closer to the test results than those of the matrix
T. C. Lin
Associate Editor