TRANSIENT PIPE PLOW 1)EHIVED BY PEHlODlC HEAT RELEASE

l I-Zun kanp and Lsmall ('elili

Department of Llechanical and Aerospace Engineerinc best \ irglnia C~nlversity

?loreantown, k b Z 6 5 O t i - i j l U 1

The heat release resulting from chemical react~ons in a combustor/tail pipe system usually induces an instability in the gas flow. This instability may lead to a stable periodic motion under certain combinations of combustion heat release and combustor geometry. This paper reports a n~merical study of the unsteady (periodic) gas flow which is driven by a periodic heat release prescribed empirically. The one-dimensional transient equations of motion and energy are derived by integration from the more general two-dimensional equations. The combustion heat release is added to the energy equation a s a source term. These equations are solved using the explicit, predictor - corrector method of MacCormack. Some predictions a re compared with measurements. The effects of the wall friction, heat transfer, and the amplitude and frequency of combustion heat release on the velocity and pressure waves are investigated. The results indicate that pulsation amplitude is a stronq function of the heat release rate and i t shows a maximum near an equivalence ratio value of one, where the heat release is near i ts maximum; this is in conformity with the experimental data. A method for calculating the natural operation frequency of pulse combustor is suggested.

Nomenclature -

cross-sectional area speed of sound f r i c t ion coefficient heat t ransfer coefficient specif ic heat a t constant volume hydraulic diameter t o t a l energy per un i t mass frequency, fo = fundamental frequency combustion heat release r a t e reactor length mass flow ra te pressure wall heat t ransfer r a t e gas constant time temperature adiabatic flame temperature ambient temperature flow velocity axia l coordinate density equivalence r a t i o wall shear s t r e s s period (znf/,,)i/2, frequency parameter kinematic viscosity time averaged quantity over a cycle

The combustion heat release in a combustor- tail pipe assembly can result, under certain conditions, in acoustic oscillations in the system. In some cases, these oscillations cause high level noise and mechanical vibrations which are undesirable and should be suppressed1. On the other hand, the combusiton induced oscillations may enhance combusiton efficiency by speeding up the diffusion limited reaction rates. The convective heat transfer to the pipe walls also increase drastically2 a s a result of these oscillations. Another advantage of pulse combustors is that i t pumps fresh air by aspiration through a hydrodynamically operating valve. In this sense i t i s self sustaining. These favorable features has led to design of very efficient ( u p to 96%) pulse-combustion heaters (e.g. Lennox's pulsed combustion air heater, hydrotherm Incorporation's pulse combusiton hydronic unit). The design of such equipment relies on mostly empirical information, e.nd trial and error procedures3. Though there is a great deal of experimental effort reported in the literature in this area, for obvious reasons, the attention is focused on quite simple geometries (e.g. a simple dump combustor and a tail pipe attached to it). A literature survey conducted by Celik and wang4 shows a limited number of numerical investigations of the wave characteristics induced by the combustion heat release. There is a great need for improved numerical models with which the fluid dynamics and heat transfer phenomena in the combustor-tail pipe assembly can be studied with relative ease compared to the experiments in complex geometries and under extreme reactor conditions such a s high temperature and high pressure.

This paper reports on some new findings of an ongoing research on numerical modeling of combustion induced wave patterns in the combustion chamber and in the tail pipe of pulse-combustors. A s in experimental investi- gations, many simplifications had to be introduced a t the intial stages to develop and validate a mathematical model. The mathematical model and the numerical scheme a re briefly summarized and tested. A simple Helmholz type combustor model with premixed injection is considered inspired by the experimental work of Dec and ~ e l l e r ~ . The influence of wall friction and heat transfer coefficients, the equivalence ratio (and hence the frequency of forced oscillations and the rate of combustion heat release) on the wave forms are investigated. Some results obtained from the calcualtions a re compared with measurements. A method of calculating the natural operation frequency, which i s otherwise prescribed empirically, i s suggested.

Copyright@ 1988 by the American Ins t i tu te of Aeronautics and Astronautics, Inc. , A l l r igh t s reserved.

Mathematical Model

The general equations of motion representing conservation of mass, momentum and energy are integrated over the variable cross-sectional area of a duct to obtain the governing equations for one-dimensional, unsteady compressible flow including the effects of wall friction, wall-heat transfer and the combustion heat release. The set of equations consists of:

Conservation of mass

Conservation of momentum

Conservation of energy

Equation of state

P = p R T

where Es = (cvT+U2/2) is the total energy per unit mass, p is the density, A is the cross-sectional area, t and x a re the time and space coordinates, respectively, U is the cross-sectional average velocity, P is the pressure, Dh is the hydraulic diameter, T~ is the wall shear s t ress , R is the gas constant, AH is the combustion heat release rate per unit mass, and Gw i s the rate of heat transfer through the wall per unit area per unit degree Kelvin. The wall shear s t ress and the heat transfer through the walls a re calculated from

where Cf and Ch are the friction and heat transfer coefficients, respectively, and To is the ambient temperature surrounding the combustor and the tail pipe. In the above equations the variation of pressure, density and velocity over the cross-sectional area a re neglected, except for the wall shear s t ress and the heat transfer a t the walls, The equations a re for the mixture of fuel and air. A well st irred (fully mixed) reactor is assumed so that the mixing of fuel and air is not a controlling mechanism. Boundary and initial conditions a re needed for the solution of the non-linear coupled, hyperbolic differential equation set (Eqs. 1-4). The details of boundary and initial conditions a re elaborated for each application separately in the following sections.

Numerical Method

MacCormack's explicit predictor/corrector numerical method (see for example Anderson, e t a16) is utilized. The method i s a step-by-step numerical scheme which sweeps over a single time

step a t a time. At each step, the predictor and corrector use simple forward and backward finite-differences of first-order accuracy, respectively. The combined process gives second-order accuracy.

The equation set (1-3) can be written in the generalized form

where

The corresponding difference equations are:

Predictor

Corrector

The features of MacCormack's method are described in detail by Maccormack7, Anderson e t a16. The method is capable of capturing basic features of nonlinear wave propagation. In particular i t can predict the propagation of both the compression and expansion waves a t the correct speed and amplitude provided that sufficiently small grid size is used for the stability of the method. The numerical stability for this method requires6 that the Courant number be less than unity. The instabilities caused by relatively large source terms8 (G > 0) can be eliminated by refining the grid size which is discussed later in the text.

Combustor and Flow Conditions

The pulse combustor-tail pipe assembly (Fig. 1) adapted for this study represents the one studied experimentally by Dec and ~ e l l e r ~ . I t consists of a short combustion chamber and a long tail pipe that a re joined by a transition section having a linearly varying side lengths. The system has a square cross-section except a t the combustor inlet. The air and fuel (methane) a re premixed in an axisymnetric mixing chamber upstream of the combustion chamber. In the experiments the fuel and air mass flow rates were metered and controlled with sonic nozzles. In the simulation, following Barr e t the perfectly mixed fuel and air is forced into the combustor a t the experimentally measured rate with the time variation shown in Fig. 2. In this figure the dimensionless mass flow rate, ni* is defined a s

wherem(t) i s the instantaneous mass flow rate and the symbol < > denotes a time averaged quantity over the period, e , e.g.

Another important parameter is the average mass

flux, < ~ f > a t a given cross-section. Thls is given by

The combustion heat release rate per unit mass of mixture, AH, can be calculated from the stoichiometry of fuel and oxidant once the equivalence ratio, $, is specified. Assuming the complete combustion of fuel injected per cycle, < A H > is estimated from the stoichiometry of methane and air; air is assumed to contain 21 percent oxygen and 79 percent nitrogen a t an estimated flame temperature. The flame temperature is a primary unknown in these calculations. The combustion heat release is assumed to be uniformly distributed in the combustion chamber and zero elsewhere; the time variation of it is prescribed according to the normalized distribution shown in Fig. 2; where A F I ' ( ~ ) = AHIt) / <AH>. I t is through these two forced periodic functions, namely & ( t ) and AH*(^), the wave pattern is obtained in the system. The frequency, f, of the oscillations is also prescribed according to the experiments; f is a strong function of the equivalence ratio (hence of < A H > ) a s observed experimentally.

I t is difficult to estimate the exact values of the wall friction coefficient and the wall heat transfer coefficient. Using the criteria suggested by sarpkayal0, it is determined that the oscillatory flow cases considered here a re in the turbulent flow regime. The Reynolds number based on the steady mean velocity is about 4000; based on the amplitude, it ranges from 2000-16000 and the dimensionless frequency parameter n = ( 2 = f / ~ . ) ~ ~ ~ >50. There is experimental evidencell for the cycle - averaged friction coefficient to be large for oscillatory turbulent flow than the corresponding steady mean flow. A typical value for the steady flow conditions (i.e. excluding the oscillatory part of the velocity) is calculated12 to be Cf-0.0025 at the middle of the tail-pipe. The friction coefficient for the oscillatory flow is taken somewhat arbitrarily to be 0.01. Two other values, 0.005 and 0.015 are also used to study the influence of Cf on the wave structure. The heat transfer coefficient, Ch, is taken, a s suggested by Barr e t a ~ . ~ , to be 13 W/m2-Ii and 20 W/mZ-K for the combustion chamber and the tail pipe, respectively, with a linear variation in the transition region. These values are crude estimates; sensitivity to variations is investigated.

Combustion Chamber Transition Tail Pipe

T i n:o

C

30

Fig. 1 Geometry of the combustor-tail pipe assembly; all dimensions a re in mm.

Dimensionless Time t k = t/8

Fig. 2 Normalized mass injection and heat release rate functions.

Initial and Boundary Conditions

Initially the pulse combustor is assumed to be filled with cold gas. The simulation is initiated by repeatedly imposing the reactant mass injection and heat release profiles (Fig. 2 ) a t the prespecified frenquency until a repeatable wave pattern develops (see Fig. 3 and 4 ) . At the inlet of the combustion chamber, the reactant injection is forced into the combustor in pulses; when the valve is open the mixture of cold air and fuel flows into the combustor, when the valve is closed, the mass injection stops. At the outlet of the tail pipe, the pressure is specified a s atmospheric pressure. The conditions a t the inlet a re varied depending on the direction of the flow. The boundary conditions a re listed in Table 1. Once the pressure and the temperature are calculated using the conditions of Table 1, the density a t the boundaries is calculated from the equation of state (Eq. 4). In case of unknown boundary conditions zero derivative (zeroth order extrapolation) conditions were imposed somewhat arbitrarily. Sensitivity to higher order extrapolations needs to be investigated.

Table 1. Boundary Conditions

C a s t i o n Chamber Entrance Tail Pipe-Exit

During IValve closed injection *

Results and Discussion

dP/dx = 0

T = 300 K

u = u ( T ) = ~ ( ~ ) / P A

Computational Details

The results from the preliminary calculations4 for the combustor system show that the calculated pressure distribution was approximately sinusoidal in shape and it was in phase with the combustion heat release. This is in accordance with the Rayleigh criterion13 which states that the energy release must be in phase with the maximum resonant pressure in order to derive flow pulsations. The calculated velocity profiles were also approximately sinusoidal shape which were

dP/dx = o a ~ / a x = o

U = 0

P = P atm

aT/ax=O

au/ax=o

-.a J,, , . . , , . . , , , , , . , , . , , , .T~-, . , , , , . , . 0 5 I0 16 20 25

C Y C L E NUMBER

KPa 120 4

C Y C L E NUMBER

Fig. 3 Profile development of pressure and velocity a t x = 400mm over many cycles from the initiation of the simulation. + = 0.62, f =48Hz and mass flux ~ 0 . 9 kg/s-m2.

about 90' out of phase with the pressure wave, in accordance with the linearized acoustic theory1$.

The computational grid consists of 1000 time steps per cycle, and a uniform grid distribution in axial direction with Ax = 20mm. Other combinations of At and Ax were also used to check grid dependency of the results. For instance, the calculations from a 500 time steps per cycle and Ax = 40mm were also stable and reproduced approximately the same wave pattern a s 1000 time steps and Ax = 20mm. A s long a s a periodic solution is obtained the results were practically the same for the range of Courant numbers from 0.4 to 0.7 for different frequencies imposed. I t should be noted that there is no unique Courant number for this flow. Here the cycle averaged value of the gas velocity and speed of sound are used a t a given axial location. The variation along the reactor is given a s a range. The time for convergence in the sence of obtaining periodic solutions is of course different for each Courant number; convergence is faster for Courant numbers close to unity.

Starting with the gas a t res t and room temperature (=300K),the calculations converged to a periodic solution in about 20 - 30 cycles. To assure full convergence, the calculations were repeated for more cycles. The number of cycles required for convergence depends on the input conditions. For example, for a given mass injection rate per cycle, convergence with higher pulsation frequency requires more cycles than that with lower frequency. The development of the repeatable periodic solution for the velocity and pressure a t x = 400mm and a t the middle of the tail pipe (=1200mm) are displayed in Fig. 3 and Fig. 4 respectively. These results are for Case 1 of Table 2. The calculation s t a r t s from zero gas velocity and atmospheric pressure. After enough reactant is injected and burned during the initial process, the repeatable periodic wave pattern is

CYCLE NIJHEER

CYCLE NUHEER

Fig. 4 Profile development of pressure and velocity a t x = 1200mm over many cycles from the initiation of the simulation. + = 0.62, f= 48Hz and mass flux = 0.9 kg/s-m2.

KPa

Fig. 5 Variations of pressure, velocity, temperature and mixture density in spatial coordinate, 4 = 0.62, f = 48 Hz and mass flux = O.Skg/s-mz. - , the valve is fully open; - - - , the valve is fully closed.

established. The velocity amplitude increases and the pressure amplitude decreases a t the middle of the tail pipe in comparison with those a t x:400mm. (Fig. 3 and Fig. 4 ) .

The spatial distributions of pressure, velocity, temperature and density are shown in Fig. 5 for Case 1 after 25 cycles. The profiles are shown a t two temporal points, namely, when the inlet valve is closed and when it is fully open. The pressure node is a t the inlet and antinode is a t the outlet a s it is the case for a Helmholtz type resonator. The velocity is zero a t the inlet when the valve is closed and it has a finite small value when the valve is open; the corresponding pressure is low. The gas velocity increases in the combustion chamber significantly a s a result of combustion -heat release, and i t keeps increasing gradually till the end of the tail pipe. The opposite trend i s seen when the valve is fully closed; reverse flow occurs over most of the system. The changes in temperature and density compensate the change in the pressure. The sharp gradients in the temperature and density immediately after the inlet is noteworthy. Such sharp gradients require utilization of shock capturing numerical methods for accuracy such a s the one used in this paper.

Comparison of humerical and Experimental Results

The numerical method has been applied to some experimental situations reported by Dec and Iieller5. The measurements show that the frequency of the oscillations changes drastically with the equivalence ratio, 0, and hence with the heat release. In the calculations, we changed f with 0 according to the experiments. Furthermore, the calculation of combustion heat release per cycle, <AH>, requires that the adiabatic flame temperature Tf be known. This also changes significantly with vairiations in O. The wave amplitude a s well as the shape is a strong function of <AH>. TO eliminate the uncertainty in the following calculations, <AH> is kept a s a free parameter and adjusted until the calculated wave amplitude approximately matched the measure- ments. The important parameters pertaining to simulated cases a re listed in Table 2, where < A H > n is the normalized heat release rate w.r.t <AH> = 70864 kJ/kg-s for 0 = 1.0, <fo> is the time averaged fundamental frequency calculated from <c>/4L, L is the length of the combustor system and <c> is the time averaged speed of sound. The mass flux per cycle is fixed a t 0.9 kg/s-m2 for all three cases a s in the experiments.

Fig. 6 Influence of equivalence ratio on velocity and pressure variations a t various axial locations,

( a ) x = 400 mm, (b) x = 1200mm, (c ) x = 60mm; - 0 = 1.0, - - - o = 0.62, - - - = 1.27;

experiment ( D e c and Keller 5 ) : r~ 0 ~ 0 . 6 2 , A 0 ~ 1 . 2 7 , + O=1.0.

The variations of the velocity and pressure waves with the equivalence ratio are deplcted in Fig. 6 along with the variation of these quantities with the axial distance. The calculated velocity profiles are compared with measurements a t x - 400mm. A suprisinply good agreement is observed between calculations and experiments for O 1.0 and 0 = 0.62 (Fig. 6a). For the case with O ~ 1 . 2 7 (fuel rich mixture) the calculations seem to follow the same trend a s the measurements, but with a more irregular wave pattern a s compared to the measurements a t x = 400mm. I t is noteworthy, though, to observe that a simple one-dimensional model does show the trend that the velocity amplitude f i rs t decreases and then increases as the equivalence ratio increases from a fuel lean to a fuel rich mixture. In plotting Fig. 6, the zero- time was determined from the downward zero point, the zero line being a t P = <P>. This method was also used in the experiments. In this respect the phase angle of the velocity profiles (i.e. angle a t minimum velocity) a s compared to the experiments is also predicted quite well by the model (Fig. 6a). The pressure and velocity waves shown a t different locations along the reactor show drastically different t rends with the equivalence ratio. In the combustion chamber O 1

1.0 produces maximum pressure oscillation, but in the tail pipe maximum oscillations a re seen for O = 0.62, whereas O = 1.0 produces the minimum oscillations among the three cases investigated. The velocity amplitude Ua follows a similar trend with increasing o a s the pressure amplitude, Pa, in the tail pipe (x-400mm). But in the combustion chamber the velocity wave shows a totally irregular behavior.

Figure 7 compares the predicted and experimentally determined amplitudes for tail pipe velocity and combustor pressure. In this figure, the average fundamental frequency, <fo> = <c>/4L, determined from calculations is also compared with the experimentally determined pulsation frequency, f, which was imposed in the calculaions. The good agreement between the calculated and the measured velocity amplitudes (Table 2) is not suprising since this was achieved by adjusting the heat release rate. However, the good agreement between the predicted and measured pressure amplitudes for cases 2 and 3 is noteworthy (Fig. 7). For case 1 the calculated Pa i s significantly larger than the measured one (see Table 2). This case has the highest velocity oscillations in magnitude. The discrepancy is most probably due to under estimation of the friction coefficient ( C f = 0.01 for the whole system) a s discussed in the next subsection.

E x p e r i m e n t s o COtiEUSTION CHAMBER PkE5SURE Rf lS

EQUIVALENCE R A T I O

Fig. 7 Comparison of predictions with experiments (Dec and Keller 5): Variation of frequency and pressure amplitude with equivalence ratio; predictions: +pressure , 6 frequency, <fo>.

Influence of Friction Coeffient

The influence of the friction coefficient on the velocity and pressure variations is depicted in Fig. 8. These results are calculated for Case 1 with 0~0.62 and f = 48 Hz by varying the friction coefficient Cf. The increased Cf results in damping in the wave a s well a s changing the phase angle. The wave with a larger Cf value lags the one with a smaller value. The peak values of the amplitudes of the velocity and the pressure for Cf=O are about twice a s those for Cf-0.01. In this case, i t has been found interestingly that when Cf = 0 (i.e., there is no friction in the system), the velocity and pressure wave motion exibit a "beating phenomena" a s depicted in Fig. 9 where the results for Cf = 0.01 is also shown for comparison. I t is known that the "beating phenomena" can occur under the condition when the forcing frequency is close to the fundamental frequency of the system without damping15. When the forcing frequency is equal to the fundamental frenquenc y, resonance occurs and the system is unstable. The fundamental frequency defined a s fox c/4L i s a function of the tempearature T. Since the speed of sound is related to T by c-constJT, fo changes both in spatial and

Table 2. Important parameters for the simulated cases

Case f [Hz1 <fo> [Hz] <AH>, Pa = Pr.m.s Ua = U r . m . s

a t x=60 mm a t x=400 mm No. O ( imposed ) a t x=60 nun

Expt . Calc . 1 0.62 48 62 0.30 5.2 11.0 20.50 --

2 1.00 82 80 1.00 3.4 3.2 2.5

Fig. 8 lnfluence of wall friction coefficient, C f , on wave pattern a t x= 120Umm with 4~0.62, f = 48 Hz, < A H > = 21473 kJ/kg-s;

c'f=O.Ol, - - - Cf=0.005, - - - Cf=O.015.

Cycle Time (a) C y c l e Time

Cycle Time ( b )

Cvcle Tlme

Fig. 9 Profiles with different friction coefficients for the case of f = 48 Hz, 4 = 0.62. ( a ) Dynamic respons with Cf = 0.01. ( b ) "Beating phenomena" with Cf = 0.

temporal coordinates a s T changes. A cycle averaged frequency, <fo>, can then be calculated a t every section of the combustor - tial pipe system. The frequency calculated In this way at the inlet of the combustion chamber is found to be approximately 44 Hz for Case 1 when Cf = 0 (Fig.Sb). I t i s believed that the "beating phenomena" occurs because <fo> a t the inlet is close to the imposed frquency of oscillations which is 48 Hz. For the other two cases listed in Table 2, this does not occur when Cf=O because <fo> at the inlet is significantly different than f imposed for these cases; <fo> is 47 Hz and 46 Hz a s compared to f being 82 and 63 Hz, for cases 2 and 3, repectively. I t is possible that the presently observed "beating phenomena" include a non-ph y sical variable such a s the Courant number. This needs further investigation. How- ever, these results show clearly that the wall friction can be an important factor in the design of an efficient pulsating combustor and in eliminating undesirable high level noise and mechanical vibrations.

Fig.10 Influence of heat transfer coefficient, Ch, on wave pattern a t x = 1200mm, + = 0.62, f = 48Hz, < A H > = 21473kJ/kg-s; base case Ch=13W/mZ-K for the combustion chamber, and Ch= 20 W/m2-K for the tail- pipe;

- - - - Ch = 0 every where; - - - Ch = 0 in the combustion chamber.

The time averaged fundamental frequency calculated a s <fo> = <c>/4L in the combustion chamber is compared with the imposed frequency in Table 2 (see also Fig.7) which shows that <fo> follows the same trend a s f, a s well a s having magnitudes fairly close to it. This suggests that in the mathematical model, <fo> can be used to force the oscillations whenever empirical information is not available. Further, the natural operation frequency of the system can be determined by an iterative guess and correct method; i.e., s tar t with an estimated f , (say f = fo a t the adiabatic flame temperature), perform calculations, calculate <fo> and use this a s the new opertion frequency, iterate until f does not change, and apply under relaxation if needed. This procedure would eliminate the need for prescribing f empirically.

Influence of &at_Transfer Coeffied

The influence of the heat transfer coefficient is illustrated in Fig. 10, where, again, a large influence is seen. Insulating the whole system (Ch-0) increases the effective heat per unit volume significantly. This leads to a d a m ~ i n g in the pressure and velocity amplitudes a t x=1200mm. This is in accordance with experiments if increasing equivalence ratio in the range O < 1 is taken a s a measure for increasing heat release. Setting Ch = 0 in the combustion chamber only, does not show any significant changes probably due to the relatively short length of this section compared to the tail pipe length. The phase of the oscillations also change significantly with changing Ch.

Fig. 8 and Fig. 10 show profiles a t x = 1200mm (the middle of the tail pipe). Similar trends were observed a t other locations along the pipe when Cf and Ch were changed4.

Conclusions

A numerical model has been constructed for calculating the characteristics of wave propagation in pipes with variable cross-sectional area. The model is one dimensional and i t includes the

effects of combustion heat release, the wall friction and heat transfer through the walls. I t uses MacCormack's explicit predictor corrector- finite difference method, whlch 1s capable of capturing the sharp gradient variations in flow variables such as that in a shocktube. The numerical model is applied to a model combustor- tail pipe assembly to study the influence of various parameters on the wave structure. The acoustic waves are generated by periodically imposing mass and heat release rates prescribed empirically. The following conclusions can be drawn from the present study:

The most important parameters that govern the wave structure in this problem are: ( a ) amount of the combustion heat release per cycle which is a function of the mass injection rate and the equivalence ratio, ( b ) the wall heat transfer coefficient and ( c ) the wall friction coefficient. The frequency of oscillations, which is a dependent variable in a self sustaining pulse combustor, can not be prescribed independent of the equivalence ratio (hence of combustion heat releas; doing so map lead to erroneous conclusions about the trends of wave properties. The wave properties in the tailpipe are a complex function of the above parameters and they show drastic changes with distance. Therefore conclusions concerning trends must refer to specific sections in the pip?; this may not be applicable to other sections. The one-dimensional model does predict the experimentally measured trends with regard to the amplitude of pressure and velocity waves with changing equivalence ration provided that the correct amount of combustion heat release is prescribed. At a section in the tail pipe close to the combustor, these amplitudes decrease with increasing $ in the fuel lean range $ 6 1.0, the opposite trend is observed in the fuel rich range $ -' 1.0. Larger wall friction coefficient causes a damping in the acoustic waves. Using a fixed friction coefficient may not be appropriate in all ranges of wave amplitudes. The cycle averaged fundamental frequency in the combustion chamber follows the same trend a s the operation frequency. This suggests that the cycle averaged frequency can be used in a mathematical model with the help of an iterative procedure in stead of the empirically prescribed operation frequecy. If the cycle averaged fundametal frequency a t the inlet is close to the externally imposed frequency, a "beating phenomena'' may occur, this has been observed by setting Cf = 0 for one of the case considered in this paper.

Acknowledgment

This work was sponsored by the West Virginia University Energy and Water Research Center (EWRC) under project No.: CMB-32-88. The calculations were performed on the State of West Virginia Computer Network (WVNET).

SIVASEGARAPI, S., and WHITELAh, J.H., (1987), 'Oscillations in Axisymmetric Dump Combustors," Combust. Sci. and Tech., Val. 52, pp. 413-426.

HARGRAVE, G.K., BILHAM, J.K. and WILLIAMS,A. (1986), "Operating Characteristics and Convective Heat Transfer of a Natural-Gas- Fired Pulsating Combustor," .I. of t,he Inst. of Energy, June issue, pp. 53-67.

BRENCHLEY, D. L,. and BOMELBURG, H.J. (1985) "Opportunities in Pulse Combustion," Report No. PNL-5563, Pacific Northwest Laboratory Richland, Washington 99352, U.S.DOE Contract No. DE-AC06-76RLO-1830

CEIJK, I., Y.-2. WANG, (1988), "An Analytical Study of Pulsating Coal Combustion Phenomena: A One - Dimensional Transient Model," Final Report, No. MAE/88/IC3, prepared for the Energy and Water Research Center IEWRC), West Virginia University, Morgantown, WV,under Project No. CMB-32-88.

DEC, J.E. and KELLER, J.O. (1986) "The Effect of Fuel Burn Rate on Pulse Combustor Tail Pipe Velocities," Sandia National Laboratories, Livermore, CA, Rept. No. SAND86-875.

ANDERSON, D.A., TANNEHILL, J.C. and PLETCHER, R.H., (1984), Computational l?M Mechanics and Heat Transfer, Hemisphere Publishing Corp., New York.

MacCORMACK, R.N., (1969), "The Effect of Viscosity on Hypervelocity Impact Cratering," AIAA Paper 69-354, Cincinnati, Ohio.

ROACHE, P.J., (19821, Computational Fluid Dynamics, 5th edition, Hermosa Publishers, Albuquerque, NM.

BARR, P.K., DWYER, H.A., and BRAMLETTE, T.T., (1987), "A One-Dimensional Model of a Pulse Combustor," Sandia Report No. SAND 86-8864, Sandia National Laboratories, Livermore, CA 94550.

SARPKAYA,T.,(1966),"Experimental determination of the Critical Reynolds Number for Pulsating Poiseille Flow", J. of Basic Engineering, Sept., pp. 589-598.

RAMAPRIAN, B.R., (1984), "A Review of Experi- ments in Periodic Turbulent Pipe Flow", Symp. on Unsteady Turbulent Boundary Layers and Friction, New Orleans, LA, Feb. 12-15, ASME publication No. FED. Vol. 12, pp. 1-16.

WHITE, F.M., (1974), Viscous Fluid Flow, McGraw Hill Inc., New York.

RAYLEIGH, J.W.S. (1945), The Theory of Sound, Vol. 2, Dover Publications, New York.

TEMKIN, S, (1981), Elements of Acoustic Theory, John Wiley & Sons, New York.

HUTTON, D. V. (1981) Applied Mechanical Vibrations, McGraw Hill, New York.