NONLINEAR DYNAMICS MODEL OF A PULSATING COMBUSTION SYSTEM
Frederick H. Reardon
Department of Mechanical Engineering
California State University, Sacramento
Sacramento, California 95819-6031
ABSTRACT
A nonlinear dynamic model of a liquid-fuel, air breathing combustion system is described. The system
consists of an air inlet duct, a fuel supply system, a combustion chamber, and a choked exhaust nozzle. The
dynamics of the air inlet duct are represented by a pair of equations involving the pressures and mass flow rates at
inlet and outlet and the sonic wave travel time. Quasi-steady representations of the fuel supply system and exhaust
nozzle are used. The dynamics of the combustion process are represented by a burning rate expression involving a
constant combustion time lag. Calculations are presented showing the air inlet response to a sinusoidally-varying
combustion chamber pressure. Some results of the complete model are also presented, showing the effects of
combustion time lag, air inlet length, and fuel injection location on the oscillatory behavior of the system.
INTRODUCTION
Many combustion devices, such as those in jet propulsion engines and electric power plants, are designed to
operate in a steady-state mode, i.e., with constant flow rates of reactants and products, and with constant pressures
and temperatures throughout the device. Oscillatory operation, in which pressures, temperatures, and flow rates vary
periodically, has been regarded as a problem, since the pressure and temperature oscillations can lead to destructive
vibrations and heat transfer rates (Ref. 1). Recently, attention has been turned to the beneficial aspects of
oscillatory operation of combustion devices, termed "pulsating combustors." It has been shown that, with proper
design, improved efficiency of the combustion process can be achieved, the increased heat transfer rates can result in
more effective heat exchanger operation, and the fluctuating flow rates may be used in certain process operations to
improve mixing (Ref. 2).
The problem facing the designer, whether the objective is to prevent or promote oscillatory operation, is
gaining a quantitative understanding of the effects of design parameters on the system operation so that proper
choices can be made of the design parameter values. At present, no adequate design-oriented combustion oscillation
model is available. A number of years ago, such a model was developed based on a linearized mathematical analysis
(Ref. 3). It was quite successful in predicting whether or not a simple combustor would oscillate under the specified
conditions. However, a model based on linearized analysis cannot make any predictions about the amplitude of
oscillation, which is a particularly important parameter for the designer of a pulsating combustor. Only a nonlinear
dynamics model can provide the necessary information to the designer.
Oscillation frequencies in air-breathing, liquid-fuel combustors have been observed to vary widely,
depending on the size of the engine. One of the most common types of oscillation is termed "very-low-frequency" or
"bulk-mode" oscillation. In this type of oscillation, the pressure variations measured throughout the combustion
chamber are very nearly in phase. For example, for a typical combustor with L/D = 3, there may be a 10-20 degree
phase difference between the inlet-end and nozzle-end pressure oscillations. In a propulsion system, very-lowfrequency oscillations can disrupt the compression process that occurs in the engine inlet, thus preventing the engine
from producing adequate thrust. On the other hand, where oscillatory operation is desired, bulk-mode oscillations
are easier to contain, since higher-frequency modes are usually accompanied by severe wall heating in the
combustion chamber. The very- low-frequency type of oscillatory behavior is the subject of the analysis described in
this paper.
THE NONLINEAR DYNAMICS MODEL
GENERAL CONSIDERATIONS
The combustion system configuration being considered is shown in Figure 1, which also illustrates most of
the nomenclature. In this analysis, both the air and the combustion products are assumed to be ideal gases. In fact,
the combustion products are assumed to be just hot air. Some additional simplifications are the following:
The Mach numbers of the flow throughout are low, so that the static temperature is very nearly the same as
the stagnation temperature.
A mean value of the specific heat of the gaseous components is used.
The air and fuel are supplied at constant temperatures.
The pressure is assumed to be uniform in the combustor.
Usually the air inlet is long enough that the wave processes are important. The equations given below are
derived from the one-dimensional wave equation, assuming the flow velocity to be small compared to the sound
velocity.
The combustion products are exhausted through a choked nozzle, which is also assumed to be short, so that
wave effects are negligible.
In this analysis, all of the equations are given in nondimensional form. The quantities used for
nondimensionalizing are the combustion chamber diameter, the mean combustion chamber pressure, and the
temperature, density, specific heat, and sound velocity of the incoming air at the dump plane.
COMBUSTION CHAMBER DYNAMICS
The combustor is taken as the control volume for this analysis. The air inlet, fuel supply, and exhaust
nozzle are external systems that are connected to the combustor.
Conservation of mass is expressed by
dM(t)/dt = ma(t) + mf(t) - me(t)
(1)
where M(t) is the total mass in the combustor at time t, ma is the rate of air flow into the combustor, mf is the fuel
injection rate, and me is the flow rate through the exhaust nozzle.
Conservation of energy in the combustor can be written as
dE(t)/dt = Ein(t) + Ecomb(t) - Eout(t)
(2)
where E(t) is the energy in the combustor at any time, Ein is the rate at which energy is convected in, primarily by the
air inlet (since the fuel flow rate is only about 5% of the air flow rate), Ecomb is the rate at which energy is released by
the combustion process, and Eout is the rate at which energy is convected out through the exhaust nozzle. The
combustor walls are assumed to be adiabatic.
The rate of energy influx is given by
Ein(t) = ma(t) + mf(t) Cf Tf(t)
where Cf is the specific heat of the liquid fuel, and T f is the fuel temperature.
(3)
Unless the oscillations are of very large amplitude, the energy in the combustor can be found by assuming
that the mean temperature is approximately constant, so that
E(t) = Lc Ac Pc(t)
(4)
where Lc is the combustor length and Ac is the cross-sectional area. A more complicated formulation can be worked
out to include the variation of the combustor exit temperature, which would be needed for large amplitude
oscillations. However, actual oscillation amplitudes in ramjets are quite small, on the order of 10-20% of the mean
combustion pressure.
AIR SUPPLY SYSTEM DYNAMICS
The wave equation governing the air inlet line can be solved to yield the following equations, in which P i
and mi are the pressure and mass flow rate at the upstream end of the inlet line and P a and ma are the pressure and
mass flow rate at the plane where the air is discharged into the combustor (the so-called "dump" plane)
ma(t) = Z Pi(t-ta) + mi(t-ta) - Z Pa(t) - R [mi(t-ta) + ma(t)]^2
(5a)
mi(t) = Z Pi(t) + ma(t-ta) - Z Pa(t-ta) - R [mi(t) + ma(t-ta)]^2
(5b)
where Z = Aa/k, Aa is the cross-sectional area of the inlet, k is the air specific heat ratio, and ta is the air inlet time
lag, the acoustic wave travel time in the air inlet duct (i.e., ta = La when nondimensionalized). These equations were
derived from the analysis presented by Nestlerode, et al (Ref. 1, Chapter 3, Section 3.2) but with added frictional
damping terms proportional to the square of the mean flow velocity.
At the dump plane, the following semi-empirical equation is used to relate the pressure in the inlet line and
the combustion chamber pressure, Pc,
Pa = Pc + Rdump ma2
(6)
where Rdump = (Pamean - Pcmean)/mamean2 is an empirical pressure drop coefficient.
At the upstream end of the air inlet, there are a number of possible boundary conditions, depending on the
actual experimental setup. In some cases, Pi is constant because the air is supplied from a large tank. In other cases,
the flow rate Mi is constant because there is a sonic choke in the air supply line. In actual flight operation of a
supersonic ramjet, there is a normal shock at the upstream end. At the present time, the computer program based on
this nonlinear model includes on the case of Pi = constant. Most of the tests for which data are available (Ref. 4)
were made with air supplied from a relatively large air heater, for which the constant P i condition seems to be
reasonable. This is the condition used for the linear model and the one used in the present one.
FUEL SUPPLY SYSTEM
In this analysis it is assumed that the fuel supply lines are short, much shorter than the air inlet, and the fuel
supply pressure is constant. In most ramjet engines, the fuel is injected into the air inlet line. In other systems, the
fuel may be injected directly into the combustion chamber.
When the fuel is injected into the air stream close to the dump plane, the fuel flow rate is given by the
orifice equation
mf2(t) = 2 Cd Af (Pf - Pa(t))
(7)
where Cd is the discharge coefficient, Af is the total fuel orifice area, and Pf is the fuel supply pressure. Inertance
effects are not usually important in these feed systems.
An alternative approach that has been used is to inject the fuel directly into the combustion chamber just
downstream of the dump plane. In that case, Eq (7) becomes
mf2(t) = 2 Cd Af (Pf - Pc(t))
(7a)
Some results for both of these schemes are presented below.
EXHAUST NOZZLE
The energy convected out through the nozzle is simply
Eout = me(t) Cp Tb(t)
(8)
where me is the mass flow rate of hot air leaving the combustor and T b is the temperature after combustion, which
can be represented by the adiabatic flame temperature multiplied by a combustion efficiency. The mass flow rate is
related to the combustor pressure and temperature by the isentropic equation
me(t) = At Zy Pc(t) / (Tb(t))1/2
(9)
where Zy = [(2/(k+1)((k+1)/(k-1))]1/2, k is the ratio of specific heats of the air, and At is the throat area of the nozzle.
In some low-pressure pulsating combustion systems, the choked exhaust nozzle is replaced by a long tube.
Such an exhaust configuration is not included in the present model, but can be treated by a scheme similar to that
used for the air inlet duct.
COMBUSTION PROCESS
The dynamics of the combustion process are represented in this analysis by means of a combustion time lag,
an approach that has been successfully applied to low frequency oscillations in liquid propellant rocket engines (Ref.
1, Chapter 5). This approach was also used in the linearized analysis.
The rate of energy release can be represented as
Ecomb = mb(t)Qc(t)
(10)
where the mass burning rate mb is given by
mb(t) = ma(t-tc) + mf(t-tc)
(11)
where tc is the combustion time lag. The energy release per unit mass, Qc, is a function primarily of the fuel/air ratio,
r, i.e.,
Qc(t) = F(r(t-tc))
(12)
r(t) = mf(t)/ma(t)
(13)
where
The temperature at the nozzle entrance, T b, is
Tb(t) = 1 + Qc(t)
(14)
The function F(r) is essentially empirical since it represents the product of the ideal adiabatic temperature rise due to
chemical reaction and a combustion efficiency. The combustion efficiency ranges from 60-90% and depends on the
combustor L/D, the air supply temperature, the mean combustor pressure level, and how the fuel is injected. For this
analysis it can be taken to be constant. An empirical relationship derived from Air Force ramjet testing (Ref. 4) is
given by the equation
F(r) = combeff (68.212 r - 101.87 r2 - 2654.6 r3)
(15)
COMBUSTION TIME LAG
The simplest physical model of the time lag is that it is the transport time of the unburned reactant mixture
from the dump plane to the location at which it reacts. Since the combustion zone is spread out in space, the time lag
must represent a mean value of such a transport time. One approach to estimating the time lag that seems to work
well is to divide the distance from the dump plane to the axial location at which 50% of the temperature rise has been
achieved, sometimes called L50, by the mean velocity of the unburned air-fuel mixture (Ref. 1, Chapter 5, and Ref.
5). Both experimental combustion efficiency data and numerical simulations have been used to estimate the
combustion time lag (Ref. 6); the two approaches agree pretty well. For very-low-frequency oscillations, assuming
that the combustion time lag is constant seems to be satisfactory.
MODEL IMPLEMENTATION
The dynamics model has been implemented in a Fortran computer program that allows the user to simulate
the unsteady operation of a wide variety of combustion systems. The combustion chamber, air inlet, and fuel supply
dimensions, the mean operating pressure and flow rates, and the fuel and air property values are entered in the form
of an input data file. Parameters that may change from run to run are entered from the keyboard. The calculations
are carried out in the time domain. After the steady-state calculation has stabilized, an initial perturbation, consisting
of a half-cycle of a square wave, is introduced into either the fuel supply pressure or the nozzle throat area. The
magnitude and duration of the perturbation are input by the user. The output file name and the number of calculation
steps complete the input. Currently, the output quantities are time, combustion chamber pressure, air flow rate, fuel
flow rate, and air inlet pressure (Pa).
MODEL VERIFICATION AND VALIDATION
AIR INLET DYNAMICS
Since the air inlet dynamics play an important role in the operation of a pulsating combustion system,
special consideration was given to this part of the model. A test apparatus (Figure 2) has been built so that
experimental validation of the model can be carried out. To date, however, only limited testing has been carried out.
Therefore, the nonlinear model results have also been compared to the results of a linearized model using the
Nestlerode equations (Ref. 1, Chapter 3).
Since the linearized model works in the frequency domain, the present computer program was modified,
replacing the combustion chamber and exhaust nozzle dynamics equations with a sinusoidal chamber pressure
exciting function. Calculations were made for a number of frequencies for two values of the amplitude of the forcing
function, 0.1% and 10% of the mean chamber pressure. The two "transfer functions" calculated were (1) the air inlet
fluctuation normalized by the mean pressure was divided by the normalized chamber pressure fluctuation, PA/PC,
and (2) the mass flow fluctuation, normalized by the mean flow, divided by the chamber pressure fluctuation,
MA/PC.
Figure 3 shows the magnitude of these two transfer functions and Figure 4 shows the phase, in both cases as
functions of the dimensionless frequency, for an inlet nondimensional length (time) of 6.1, which corresponds to the
experimental apparatus. It can be seen from Figure 3 that at frequencies of 0.515 and 1.03 the pressure function
PA/PC magnitude reaches a minimum value whereas the mass flow function MA/PC reaches a maximum. These
frequencies correspond to the first two resonant frequencies of a duct with two open ends. At these resonant
frequencies, the inlet pressure is in phase with the chamber pressure whereas the mass flow is 180 degrees out of
phase (Figure 4).
Figure 5 shows the same results on the complex plane, compared with the results of the (old) linearized
model. It can be seen that the new model shows a more responsive air inlet system, with an interesting variation in
the response curve when compared to the elliptical shape of the linear response function. The magnitude of the
chamber pressure oscillation did not seem to have a strong effect on the air inlet system response.
The only experimental data available at present were obtained at a dimensionless frequency of 0.97 (400
Hz). Experimentally, the magnitude of PA/PC was found to be 0.87, in good agreement with the calculated results.
The measured phase angle was -33 degrees, somewhat larger than the calculated value of -25 degrees. Additional
experiments at different frequencies are planned.
CHARACTERISTIC TIME EFFECTS
The behavior of a pulsating combustion system has been investigated by systematically varying the values
of the nondimensional combustion time lag and the nondimensional air inlet length (wave travel time). Both fuel
inlet configurations were considered, i.e., fuel injected into the air duct and fuel injected into the combustion
chamber. In each case, the mean fuel system pressure drop was the same. The parameter values were chosen to
correspond to ramjet systems tested by the Air Force (Ref. 3, 4, 6).
Figure 6 shows the results for fuel injection into the air inlet duct just upstream of the dump plane. Each of
the cases calculated has been characterized by a symbol denoting the time-varying amplitude of the chamber pressure
oscillation during the early portion of the calculation. Eventually, the oscillations that grow do reach a finiteamplitude limit cycle, as illustrated in Figure 7. The five categories of growth, slow growth, limit cycle, slow decay,
and decay are shown in Figure 8. This somewhat subjective categorization shows the general behavior of the system.
It can be seen from Figure 6 that the tendency of the system to oscillate increases with increasing values of the two
parameters, but that increasing one parameter while holding the other constant does not necessarily lead to more
rapidly growing oscillations. Such behavior is typical of resonant systems.
Similar results were found for the case of fuel injection directly into the combustion chamber, shown in
Figure 9. In general, this type of system appears to have a greater tendency to oscillate. For any given inlet length,
there is a range of combustion time lag values that will produce oscillations. For a given value of the combustion
time, there may be one or more ranges of inlet length corresponding to pulsating behavior.
CONCLUSIONS
A nonlinear dynamic model of a liqiuid-fuel, air-breathing combustion system has been developed for the
purpose of allowing designers to simulate the oscillatory behavior of a planned system. The air inlet dynamics are
represented by a pair of equations with a time delay proportional to the length of the inlet duct. The combustion
dynamics are represented by a combustion time lag formulation. In the present model, a short, choked nozzle is
assumed, although an unchoked exhaust duct could be treated by a method similar to that used for the air inlet.
The air inlet dynamic behavior predicted by the model seems to agree fairly will with very limited
experimental measurements. The overall growth-decay characteristics predicted by the full model in terms of the
values of the combustion time lag and inlet length are at least qualitatively reasonable, although detailed comparisons
with available ramjet test results have not yet been made. Calculations made to date indicate that fuel injection
directly into the combustion chamber gives a greater tendency to oscillatory operation than fuel injection into the air
inlet duct.
Additional calculations are in progress to investigate the effect of the initial perturbation on the resulting
oscillations, both on the growth rate and the final limit-cycle amplitude. Future efforts will involve a revised burning
rate model that can include the effects of vortex structures in the flow.
REFERENCES
1. Harrje, D. T., and Reardon, F. H., eds., Liquid Propellant Rocket Combustion Instability, NASA-SP-194,
Washington, D.C., NASA Scientific and Technical Information Office, 1972.
2. Zinn, B. T., "Pulse Combustion: Recent Applications and Research Issues," Twenty-Fourth Symposium
(International) on Combustion,
3. Reardon, F. H., "Very Low Frequency Oscillations in Liquid-Fueled Ramjets," in Combustion Instabilities in
Liquid-Fueled Propulsion Systems, NATO AGARD Conference Proceedings No. 450, 1989.
4. Davis, D. L., "Coaxial Dump Ramjet Combustor Combustion Instabilities," Air Force Wright Aeronautical
Laboratories, Report No. AFWAL-TR-81-2047, 1981.
5. Reardon, F. H. and Clifford, R. J., "Calculation of the Combustion Distribution in a Liquid-Fuel Ramjet", 26th
JANNAF Combustion Meeting, 1989.
6. Reardon, F. H., and Duvvur, A., "Calculations of the Combustion Time Lag in Coaxial-Dump Ramjet
Combustors," Proceedings of the 32nd Heat Transfer and Fluid Mechanics Institute, California State University,
Sacramento, 1991.
Figure 1. Simple Pulsating Combustion System
Figure 2. Inlet Dynamics Apparatus
Figure 3. Calculated Air Inlet Response: Magnitude
Figure 4. Calculated Air Inlet Response: Phase
Figure 5. Comparison of Linear (Old) and Nonlinear (New) Air Inlet Models
Figure 6. Nonlinear Model Results for Fuel Injection into Air Inlet
Figure 7. Illustration of Typical Perturbation Development Modes
(a) Growth (b) Slow Growth (c) Limit Cycle
(d) Slow Decay (e) Decay
Figure 8. Growth of a Perturbation into a Limit-Cycle Oscillation
Figure 9. Nonlinear Model Results for Injection into Combustion Chamber