A procedure which can accurately characterize injector designs for large thrust (0.5 to 1.5 million pounds), high pressure (500 to 3000 psia) LOX/hydrocarbon engines is currently under development. In this procedure, a rectangular cross-sectional combustion chamber is to be used to simulate the lower traverse frequency modes of the large scale chamber. The chamber will be sized so that the first width mode of the rectangular chamber corresponds to the first tangential mode of the full-scale chamber. Test data to be obtained from the rectangular chamber will be used to assess the full scale engine stability. This requires the development of combustion stability models for rectangular chambers. As part of the combustion stability model development, a computer code, NOAD based on existing theory was developed to calculate the nozzle admittances for both rectangular and axisymmetric nozzles. This code is detailed

Nguyen, Thong V.

English

NASA Technical Reports Server

N90-28630
COMPUTER CODE FOR THE PREDICTION OF NOZZLE ADMITTANCE
Thong V. Nguyen
Aerojet TechSystems Company
Sacramento, California 95813
ACKNOWLEDGEMENTS
This work was supported by the Air Force Astronautics
Laboratory, under contract F04611-85-C-0100,
Oxygen/Hydrocarbon Injector Characterization. The AFAL
program manager Is Capt. M. Walker, the ATC program
manager is Dr. M. Young, and the ATC project engineer is
Mr. J. Pieper.
I. INTRODUCTION
Aerojet TechSystems Company is currently conducting a
program (contract F04611-85-C-0100) to formulate a procedure
(Ref. i) which can accurately characterize injector designs
for large thrust (0.5 to 1.5 million pounds), high pressure
(500 to 3000 psia) LOX/hydrocarbon engines. In this
procedure, a rectangular cross-sectional (hereafter will be
refered to simply as rectangular) combustion chamber is to
be used to simulate the lower tranverse frequency modes of
the large scale chamber. The chamber will be sized so that
the first width mode of the rectangular chamber corresponds
to the first tangential mode of the full-scale chamber.
Test data to be obtained from the rectangular chamber will
be used to assess the full scale engine stability. This
requires the development of combustion stability models for
rectangular chambers.
As part of the combustion stability model development,
a computer code, NOAD based on existing theory (Ref. 4) has
been developed to calculate the nozzle admittances for both
rectangular and axisymmetric nozzles.
Distribution authorized to U.S. goverment agencies and
their contractors; Critical Technology; April 1988. Other
requests for this document shall be referred to the Air
Force Astronautics Laboratory, Edwards AFB, California,
93523-5000.
302
https://ntrs.nasa.gov/search.jsp?R=19900019314 2020-03-19T21:40:38+00:00Z
i. Nozzle Admittance
Combustion instability, characterized by organized
pressure oscillations in rocket combustion chamber, can
cause severe vibrations on various engine system components
and payloads. In addition, combustion instabilities may
cause excessive mechanical stresses and heat loads on the
injector and combustion chamber walls. These may result in
extensive damage to both the engine and to the payloads.
Combustion instabilities have been generally classified
according to their frequency range: low, intermediate, and
high frequency. Significant efforts have been devoted to
the understanding of high frequency instability because it
is the most common in new engine developments and is the
most destructive. Analytical models capable of
characterizing this undesirable phenomenon are obviously
useful and valuable to engine designers during the
development stage. High frequency instability results from
the coupling between the combustion process and the acoustic
waves in the chamber, which, in general consists of a
straight section and a converging section. The converging
section is also considered as part of the nozzle, hence the
beginning of the converging section is usually called the
nozzle entrance. The analysis can be greatly simplified if
the acoustic wave equations are solved separately for each
of the sections and the solutions are to be matched at the
interface of the two sections (at the nozzle entrance). For
instance, wave equations are solved for acoustic properties
in the converging section; the properties are then evaluated
at the nozzle entrance and used as the boundary condition to
solve the wave equations in the straight section. The
nozzle admittance which is defined as the ratio of the
acoustic axial velocity to the local acoustic pressure at
the nozzle entrance provides a boundary condition for
solving the wave equations in the straight section of the
chamber.
References 5 and 7 discussed another valuable aspect of
the nozzle admittance, that the relative stabilizing effects
of different nozzle designs can be determined from the
computed nozzle admittance.
2. Previous works
2.1 Experimental study
A literature search indicates that existing
experimental data on nozzle admittances are only for
axisymmetric nozzles.
In reference 10, Crocco et al. performed a series of
303
experiments to determine nozzle admittances by directly
measuring the velocity and pressure oscillations at the
nozzle entrance. A hotwire anemometer and a diaphragm
pressure transducer were installed at the nozzle entrance to
measure the velocity and the pressure. A technique was
devised to make corrections to the real part of the measured
nozzle admittance because this component results from the
part of the velocity oscillations that are in phase with
pressure and temperature oscillations, both of which
directly affect the measurements of the velocities. The use
of a hotwire anemometer to measure velocity oscillations
also limits the accuracy of the data due to the high
turbulent noise to signal ratio.
Using the impedance tube method, Zinn et al. (Ref. 3)
experimentally determined nozzle admittances for a number of
nozzle geometries. This method has the advantage over the
direct measurement approach because velocity oscillations do
not have to be measured. In this method, the nozzle
admittance is shown analytically to be related to a standing
wave pattern in the impedance tube and the wave pattern is
determined from the pressure amplitudes measured at several
axial locations along the tube.
2.2 Theoretical background
The first analytical study of small amplitude wave
propagation in axisymmetric nozzles was made by Tsien (Ref.
6). In this work, the perturbations are assumed to be
one-dimensional, isothermal; and the mean axial velocity is
assumed to vary linearly with the axial distance from the
nozzle entrance. Tsien then derives expressions for the
transfer function of the rocket nozzle for either very low
or very high frequency range. The transfer function of the
rocket nozzle is defined as the sum of the ratios of
perturbation density and perturbation axial velocity to
perturbation pressure at the nozzle entrance. The
perturbations are non-dimensionalized appropriately by their
local means.
The original study by Tsien was extended by Crocco
(Ref. 8) to include non-isothermal case, non-linear
velocity distribution and all frequency ranges. The
one-dimensional perturbations assumption is however still
retained in this study.
In reference 4, Culick developed a more generalized
model in which the perturbations no longer have to be
assumed one-dimensional. The mean axial velocity is assumed
to vary only in the axial direction and is assumed to follow
the equation applicable to one-dimensional, compressible,
isentropic flows. Although only the axisymmetric case is
304
discussed in detail in the reference, the theory can be
simply extended to rectangular nozzles.
Following Culick's theory, Waugh developed a computer
model at Aerojet TechSystems Company to calculate the nozzle
admittances of rectangular nozzles.
In reference 5, Crccco and Sirigano derived equations
to describe the oscillations in a nozzle. Both axisymmetric
and rectangular nozzles are considered. Calculations of
nozzle admittance of a specific axisymmetric nozzle geometry
are made over a wide range of frequency and entrance Mach
numbers, and the results are tabulated. A scaling technique
was developed to calculate, using the tabulatedresults, the
nozzle admittances of nozzles having different sizes but
similar shapes.
Bell and Zinn (Ref. 2) extended the above theory to
include wave-amplitude attenuations. A computer code was
developed to determine the nozzle admittances of
axisymmetric nozzles. Calculations are made and compared
with experimental data. Comparisons show good agreement
between the prediction and the measurement.
3. Objective of the present work
While the theories developed in all of the previous
works readily apply or can be simply extended to apply to
rectangular nozzles, only one computer code has been
developed by Waugh to calculate nozzle admittances of
rectangular nozzles. Unfortunately, the listing of the
computer code is incompleted.
The objective of the present work is to formulate a
computer code to calculate the nozzle admittances of not
only rectangular but also axisymmetric nozzles. The code
was to be formulated such that:
i. It predictes accurately the nozzle admittances over
a wide range of frequency and nozzle contours and for any
resonance modes.
2. It accounts for the variations of wave amplitudes
with time.
3. It generates a file which interfaces directly with
the combustion stability computer codes, or it can be easily
converted to be used as a subroutine in other combustion
stability computer codes.
4. It requires minimum computer time and storage.
3O5
II. THEORY
The theory used in the present work follows closely
reference 4. First, the continuity and momentum equations
are written for an ideal gas. Then, all dependent
variables, e.g. pressure, density, velocity, etc., are
decomposed into their mean and fluctuating components. The
mean components do not vary with time and are assumed to
vary only in the axial direction. The fluctuating
components, however, vary in all directions and are
functions of time. These fluctuating components are assumed
to be small so that the products of any two components can
be neglected. As a result, equations for the fluctuating
components are linear in time, thus their oscillations can
be assumed to be sinusoidal. The flow is assumed to be
irrotational and the fluctuating velocity components are
defined to be the gradients of a velocity potential
function. The continuity and momentum equations and an
isentropic relation are combined to yield a governing
equation for the velocity potential function. The resulting
equation is then written for a curvelinear coordinate system
which follows the streamline of the mean flow. Using
separation of variables technique, the partial differential
equation governing the evolution of the potential function
is separated into three second-degree ordinary differential
equations. For rectangular nozzles, using boundary
conditions at the nozzle walls, two of the equations in the
tranverse and lateral directions are solved explicitly to
give eigenvalues that correspond to tranverse and lateral
resonance modes. Similarly for axisymmetric nozzles,
boundary conditions at the chamber wall and at the axis of
symmetry are used to calculate the eigenvalues that
correspond to radial resonance modes. The eigenvalues that
correspond to tangential resonance modes are determined by
requiring the solutions to the differential equation to be
single value functions.
A new integration variable to which nozzle admittance
can be related is defined to reduce the differential
equation in the axial direction to the first order. This
reduction is possible since we are interested in only the
ratio of the fluctuating components of the axial velocity
and the pressure but not the absolute values of the
individual quantities. The resulting first order
differential equation, having the same form for both
axisymmetric and rectangular cases, is integrated
numerically to determine the value of the integration
variable at the nozzle entrance. The nozzle admittance is
then calculated from an expression that relates the
admittance to the value of the integration variable.
No attempts are made to describe the derivations of the
306
equations here since it has been described in full details
elsewhere (Ref. 4). A computer code, NOADwas developed
using the aforementioned theory.
III. RESULTSAND DISCUSSIONS
Nozzle admittances were calculated using the computer
code NOADover a wide range of frequencies for longitudinal
mode and for mixed longitudinal and transverse modes. The
calculations were made for rectangular and axisymmetric
nozzles having different convergence angles. Calculated
results for axisymmetric nozzles are compared with the
experimental data of reference 9. In the discussion
follows, all calculated results and experimental data are
presented in non-dimensional forms. Frequencies are
non-dimensionalized by the half height (rectangular nozzles)
or by the radius (axisymmetric nozzles) of the nozzles at
the entrance, and by the stagnation sound speed. Nozzle
admittances are non-dimensionalized by the stagnation sound
speed and by the stagnation pressure.
Calculations were made for six different rectangular
and axisymmetric nozzles having convergence angles of 15, 30
and 45 degrees. The nozzle contours of the rectangular
nozzles and the axisymmetric nozzles having the same
convergence angles are similar. The radii (or the half
height) at the entrance and at the throat of the nozzles are
5.69 and 2.11 inches, respectively. Using the specific heat
ratio of 1.4 for air, the entrance Mach numbers were
calculated to be approximately 0.08 for the axisymmetric
nozzles, and they were calculated to be approximately 0.221
for the rectangular nozzles. The radii of curvature of the
nozzle contours at the entrance and at the throat are 2.5
inches.
The calculated real and imaginary parts of the
admittances for longitudinal modes are plotted against the
frequency as shown in Figures i, 2, and 3. The experimental
data obtained from reference 9 for axisymmetric nozzles are
also shown on the figures. The figures show good agreement
between the calculated results and the experimental data.
It should be noted that the data were obtained only for
axisymmetric nozzles. It can be seen that the dependence of
the nozzle admittances on the frequency for rectangular
nozzles are trendwise similar to that for axisymmetric
nozzles. The peaks and valleys appear more frequent over
the frequency range for the rectangular nozzles than for the
axisymmetric nozzles.
Calculations are then made for mixed longitudinal and
first tangential modes and the results are shown in Figures
307
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4,
3
c
_Q
Q
3.
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o
...."ii""
..._:_ ',. _ .......i_;_ _ ......
* ,., ...__>_'_
n on- d'n_,cn_ norm:_d rrcqu e¢-c F
Leger_
0 ¢lr_.. i.JUSV.
¢*.,ARP. &.KIST.
Figure 1: Nozzle admittances for axisymmetric
and rectangular nozzles having 15
convergence half angle. Longitudinal
modes.
308
u'I °°'°°'""" "°o°.
I ."" /"
_lr _ .... • • I w u • i •
II o.II OA OM 1.4 I I.t t4 !.4 LO
Legec'_4J
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¢._.._L_...
4'
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......... .,, oo
D$ , m
no_-d'rv_ns;_nc_ax_f_c_*f_'7
_ge_
0 ((fR. IdL_T.
Figure 2: Nozzle admittances for axisymmetric
and rectangular nozzles having 30
convergence half angle. Longitudinal
modes.
309
J4
o,-
o,
14 °*
0 02 0A 41-I 0_ ! I_ I_4 1.1. L8
t_gend
0 lsPm. +.qtll.
r.A t C. 116v.
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I
I
I
I
z4
Figure 3: Nozzle admittances for axisymmetric
and rectangular noz_zles having 45
convergence half angle. Longitudinal
modes.
310
4, 5 and 6. Again, the experimental data obtained from the
reference 9 for the axisymmetric nozzles are shown in the
figures for comparisons. The agreement between the
predictions and the measurements is also good.
IV. SUMMARY AND CONCLUSIONS
A computer code, NOAD was developed to calculate the
nozzle admittances of rectangular and axisymmetric nozzles.
Several statements can be made with regards to the computer
code and the calculated results:
i. The computer code is capable to calculate the
nozzle admittances of rectangular nozzles and axisymmetric
nozzles.
2. Correlations of calculated results and experimental
data with regard to the nozzle admittances of axisymmetric
nozzles are quite good. Although there is no experimental
data to verify the predictions for rectangular nozzles, it
is believed that the model also predicts accurately the
nozzle admittances for rectangular nozzles.
3. The dependencies of nozzle admittances on frequency
for rectangular nozzles and for axisymmetric nozzles having
the same nozzle contours are trendwise similar. The
variation trends of the nozzle admittances for rectangular
and axisymmetric nozzles are also similar when the acoustic
resonance goes from one mode to another.
4. A typical run of NOAD for 30 frequency values
requires approximately 16 CPU seconds on the micro-VAX at
ATC.
V. REFERENCES
i. Oxygen/Hydrocarbon Injector Characterization. Phase I
industry briefing by Aerojet TechSystems Comapny under
contract F04611-85-C-0100, Air Force Rocket Propulsion
Laboratory, February 1986.
2. Bell, W. A., and Zinn, B. T., "The Prediction of Three
Dimensional Liquid-Propellant Rocket Nozzle Admittances".
NASA CR-121129, February, 1983.
3. Zinn, B. T.. Bell, W. A., Daniel, B. R., and Smith
Jr., A. J., ,,Experimental Determination of
Three-Dimensional Liquid Rocket Nozzle Admittances". AIAA
Journal, volume ii, March, 1973.
311
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0
g I e o_
...... -,% ,°°
I
•-41 • • . i •
t.,ll I 11 2.4 t,4 1hi
n oct- o"w..en_o I_011_m_$ fneq_encT
Leger_
0 ¢gPR. *J_v.
3:t
I
"iic
15
-°_
o
• ,°°°'°"°'" "'°'°
-41 , * ! *
n o,..-d'r.e.sio _ fncqu c-c7
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/
!
L.e9er'._
0 ezra. l.Jts'S*_.
C_LC. *,_rSY.
w_.. _f_:-..
%
3.1_
Figure 4: Nozzle admittances for axisymmetric
and rectangular nozzles having 15
convergence half angle. Mixed IT and
longitudinal modes.
312
U4
,J
I
0...5.4
! "o-
........................... ...._ ........ °.°-'°
,_J (
_, ; tl ,_4 ,:, i* i
no_d_narmsd f_'_l_ e.'cl'
L_g_'_
C_.* r. aJIS'V.
!
o
"6
1
o
. o_- _,mc.13o .¢_"cmd ¢r,cqu t_'_c'O'
.°...°" .__.,._.?,:_;"
,
.°
; .
LegenO
Figure 5: Nozzle admittances for axisymmetric
and rectangular nozzles having 30
convergence half angle. Mixed IT and
longitudinal modes.
313
o41
E ooo :
_* _ O. °"
--.o ......... o ................ .o.o°° .... oo°°.o---°'°°°-
,5 n
!
e, ,
-oJ."l
°! | •
,., + ,_ :', 2% ,.o
mm-d'+_oil_zod fmqu m
Lege_
_) llPlt. +;)It+_'.
P_I.C. a,WlST.
_. +t,C...q'_'_..-
4'I
.'.I
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Z
_+ ,i
0
.E '
o.
o.
,.
,°°
oO
°,, • ,- G°° °°°'" °°
.... .++'-"-'_
-!
Le?encl
0 IIPII. _¥.
Figure 6: Nozzle admittances for axisymmetric
and rectangular nozzles having 45
convergence half angle. Mixed IT and
longitudinal modes.
314
4. Culick, F. E. C., "Stability of High Frequency
Pressure Oscillations in Gas and Liquid Rocket Combustion
Chambers". Ph.D thesis, Massachusetts Institute of
Technology, June 1961.
5. Crocco, L., and Sirignano, W. A., "Behavior of
Supercritical Nozzles under Three Dimensional Oscillatory
Conditions". Technical report number 790. Princeton
University, Department of Aerospace and Mechanical sciences,
April, 1967.
6. Tsien, H. S., "The Transfer Functions of Rocket
Nozzles". ARS journal, volume 22, May-June 1952.
7. Sigman, R. K., and Zinn, B. T., "A Finite Element
Approach for Predicting Nozzle Admittances". Journal of
Sound and Vibration, 1983.
8. Crocco, L., and Cheng, S., "Theory of Combustion
Stability in Liquid Propellant Rocket Motors". Published
for The Advisory Group for Aeronautical Research and
Development, North Atlantic Treaty Organization by
Butterworths Scientific Publications, 1956.
9. Bell, W. A., "Experimental Determination of
Three-Dimensional Liquid Rocket Nozzle Admittances", Ph.D
Thesis, Georgia Institute of Technology, July 1972.
I0. Crocco, L., Monti, R., and Grey, J., ,,Verification of
Nozzle Admittance Theory by Direct Measurement of the
Admittance Parameter". ARS Journal, June 1961.
315


Computer code for the prediction of nozzle admittance

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