A synergistic hierarchy of analytical and computational fluid dynamic techniques is used to analyze three-dimensional combustion instabilities in liquid rocket engines. A mixed finite difference/spectral procedure is employed to study the effects of a distributed vaporization zone on standing and spinning instability modes within the chamber. Droplet atomization and vaporization are treated by a variety of classical models found in the literature. A multi-zone, linearized analytical solution is used to validate the accuracy of the numerical simulations at small amplitudes for a distributed vaporization region. This comparison indicates excellent amplitude and phase agreement under both stable and unstable operating conditions when amplitudes are small and proper grid resolution is used. As amplitudes get larger, expected nonlinearities are observed. The effect of liquid droplet temperature fluctuations was found to be of critical importance in driving the instabilities of the combustion chamber

Grenda, Jeffrey

Merkle, Charles L.

Venkateswaran, Sankaran

English

NASA Technical Reports Server

DEVELOPMENT OF A COMPUTATIONAL TESTBED FOR NUMERICAL
SIMULATION OF COMBUSTION INSTABILITY
Jeffrey Grenda, Sankaran Venkateswaran, 1 Charles L. Merkle:
Propulsion Engineering Research Center
and
Department of Mechanical Engineering
The Pennsylvania State University
University Park, PA 16802
SUMMARY:
A synergistic hierarchy of analytical and computational fluid dynamic techniques is used to analyze three-dimensional
combustion instabilities in liquid rocket engines. A mixed finite difference/spectral procedure is employed to study
the effects of a distributed vaporization zone on standing and spinning instability modes within the chamber. Droplet
atomization and vaporization are treated by a variety of classical models found in the literature. A multi-zone,
linearized analytical solution is used to validate the accuracy of the numerical simulations at small amplitudes for a
distributed vaporization region. This comparison indicates excellent amplitude and phase agreement under both stable
and unstable operating conditions when amplitudes are small and proper grid resolution is used. As amplitudes get
larger, expected nonlinearities are observed. The effect of liquid droplet temperature fluctuations was found to be of
critical importance in driving the instabilities of the combustion chamber.
TECHNICAL DISCUSSION:
Current understanding of liquid rocket combustion instability has been obtained through the implementation of two
primary tools: experimental investigations and analytical models. Computational capabilities have recently advanced
to the point where they provide a third potential investigative tool to complement these existing approaches. The
dramatic progress in the field of computational fluid dynamics (CFD) over the past decade has demonstrated the
ability to model highly complicated flow phenomena typical of combustion chambers. CFD methods offer a promising
methodology to model not only the important subprocesses such as atomization and vaporization, but also to directly
simulate and accurately capture the acoustical physics of the combustion chamber.
Research directed towards developing computational instability models for liquid propellant engines has been
undertaken in several research groups. Early work using computational fluid dynamics in studying combustion
instability was performed by Habiballah et al. [1] and Liang and Ungewitter [2], and followed later by Bhatia and
Sirignano [3], Jeng and Litchford [4], Kim et al. [5], Wang et al.[6], as well as the current authors [7]. Typical results
using these time-marching approaches have found that droplet size, mixture ratio, and mean chamber conditions are
important physical parameters. Due to computational restrictions, most analyses have considered axisymmetric or
*Graduate Research Assistant
! Research Associate
Distinguishcd Alumni Profcssor
88
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annulargeometries,althoughsomepreliminarythree-dimensionalresultsforathinor"collapsed"combustionzone
havepreviouslybeenreportedbythecurrentauthors[7].
The current work develops a computational testbed for modelling a distributed vaporization zone by coupling CFD
with available empirical and semi-empirical models of the dominant subprocesses in rocket engines. These solutions
are then compared with closed-form analytical solutions to verify their accuracy. Although the solution of the unsteady,
three-dimensional fluid dynamic equations is straightforward in principle, the required computational resources limit
the quantity of solutions that can be obtained. This is the motivation for providing companion analytical procedures,
which may be performed in a significantly more efficient manner. This approach permits important trends in relevant
physical variables to be identified rapidly while simultaneously allowing a systematic assessment of the numerical
issues involved.
The unsteady Euler equations are used to describe the fluid dynamics of the three-dimensional gas phase flowfield
within the combustion chamber. Due to the substantial computational cost associated with finite difference solutions of
the three-dimensional unsteady equations, a mixed character finite difference/spectral method is employed to decom-
pose the primary variable Q in the circumferential direction into a Fourier series. The vector Q is given as a truncated
series Q = )-'ira=0 Om,,(z, r, t)cos mO+ Q,n,, (x, r, t) sin rn0 where _),_,, = (_,, pu,, pv,, pw,, ec, O, O, O, O, O)T
and Q.m,, = (0, O, O, O, O, p,, pu, , pv ,, pw , , ._, ) v are time-dependent functions of both the axial and radial finite-
difference directions. A general equation governing the dynamics of the flowfield development in Fourier space can
then be written as
_ 1O(2,n,T OE(Qm,T) + 10._((2m,T)r rn G(Qm,r) + -fl,ot(Om,T) (1)
0----_ + Oz r Or r r
where the vectors _:, F, G, and /-/tot are conservative flux vectors written in Fourier space [7]. The source term
vector/:/_ot contains the interphase coupling terms between the gas phase and liquid phase analyses including effects
such as atomization, vaporization, and injector coupling. Representative vaporization models based upon the work
of Priem-Heidmann and Abramzon-Sirignano are summarized in Table 1. These relationships are complicated
nonlinear functions of the liquid and gas phase variables. Linearized versions of these expressions written in terms
of an appropriate set of independent variables lead directly to a combustion response function for the instability.
This combustion response function in turn allows analytical solutions to be obtained which may complement CFD
approaches.
One appropriate set of functional variables governing the droplet vaporization rate can be written as rh_p =
f(p, u, Tt) where 7] is the liquid droplet temperature. Since the pressure and velocity are fundamental flow quantities,
the evaluation of the combustion response functions is greatly simplified if the liquid temperature fluctuation can be
eliminated. By manipulating the vaporization expression and combining it with a linearized energy balance over the
droplet surface, the two-parameter expression
"' -':- + 13 (2)mva p -_ mvap or*-- *
P
provides a relationship for the rate of vaporization from a liquid droplet with a fluctuating temperature as a function
of the mean vaporization rate and the acoustic disturbances of pressure and velocity. The values of the complex
linearization constants a* and t* are determined from the conditions within the combustion chamber. It is sufficient
to note here that it is expected that the values of a* and _* will directly determine the stability of the combustor in the
linearized case.
For a constant mean flow, a small amplitude version of Eqn. 1 can be solved analytically. When the vaporization
zone is treated with physical models in the form of Eqn. 2, the problem can be reduced to an eigenvalue solution in
89
termsof thevaporizationconstantsandmeanflowparameters.Thedispersionrelationshipfortheeigenvaluesare
obtained by manipulating the equations of motion to provide a single wave equation in the chamber,
v'. + + + = -T +/3. ) + + -- (3)fi 1
where the subscripts denote differentiation with respect to the particular variable. This expression also holds in regions
where the mean volumetric vaporization term N_ vanishes. Substitution of the classical acoustic form of solutions for
pressure and velocity produces a dispersion relationship of the form
i/3"_2N_ ic_*_2-_" + w 2 - e2A + - 0 (4)
k2(_2 + _2) + k -2_w + P_ P P
Here, ,X is the radial wavenumber, and k and o: are the complex wavenumber and frequency, respectively. Once these
have been determined, the oscillatory flow quantities which determine the stability behavior can be evaluated. In this
way, the analytical solutions provide a valuable means of validating and complementing the more general nonlinear
numerical approaches described earlier. This capability of plays an important role in establishing a validated testbed
upon which more comprehensive physical modelling can be added.
RESULTS:
First, we consider the analytical dispersion relationship presented in Eqn. 4 for a uniformly distributed vaporization
process. For simplicity, we neglect droplet production processes and instead specify an initial droplet size of 140# and
a total mass flow rate of 5 kg/sec (H2 and LOX) entering the combustion chamber. This corresponds to a vaporization
zone 95% of the length of the combustion chamber.
In the absence of droplet temperature fluctuations, _" and/_* are real and depend on droplet Reynolds number and
gas static pressure. A typical stability map for the constant liquid temperature case is presented in Fig. 1. The stability
plane for the distributed vaporization model includes both stable and unstable regions depending on the values of the
vaporization coefficients c_* and 8*. Positive contours indicate unstable growth of disturbances, and negative values
indicate stable decay. The neutral stability curve is indicated by the dashed contour and separates the unstable and
stable regions which are labelled U and S, respectively. For this typical example, unstable regions are centralized in
the regions of positive c_* and negative/3*. This indicates that both pressure and velocity are important in determining
the stability although pressure effects may be more important under certain conditions.
The estimated ranges of the two vaporization models are indicated by the shaded regions on the stability plane.
These values are approximated by allowing possible variations in the properties and other constant terms in the
vaporization expressions. The locations of the Priem-Heidmann and Abramzon-Sirignano models indicate that both
lie in the stable region of the stability domain. These predicted results indicate that the coupling of the vaporization
process alone for constant temperature liquid droplets is insufficient to sustain pressure oscillations characteristic
of combustion instability. Similar stable behavior has also been found for a wide range of operating conditions.
These generalized results confirm our previous numerical findings [7] that droplet vaporization without temperature
fluctuations cannot account for unstable growth of pressure oscillations.
When liquid droplet temperature is permitted to fluctuate, the linearization coefficients in Eqn. 2 become complex
numbers since they are functions of the complex frequency eigenvalue. Since both the real and imaginary components
of c_* and B* may be important, the stability diagram becomes a more complicated four-dimensional space. The overall
result is that the presence of a temperature f uctuation drives both the Priem-Heidmann and Abramzon-Sirignano models
more towards instability. Both stable and unstable modes are possible, depending on the physical operating conditions
within the combustion chamber.
90
Althoughtheeffectsofthegasphaseandliquidphaseflowvariablesarestronglycoupledandnoteasilyseparated,
experiencehasdemonstratedhatdropletdiameteranddroplettemperaturearetwocriticalparametersthatinfluence
stabilitybehavior.Fig.2presentsa tabilitycontourwhich asbeendeterminedcomputationallybyaparametricstudy
ofthesevariables.Here,wespecifyatotalmassflowrateof3kg/secwithameanchambertemperatureof3000Kand
pressureof40atm.Thetriangularupperboundofthestabilityisdeterminedbytheconstraintthatthedropletsmust
totallyvaporizewithinthechamber.Theneutralstabilitycurveisagainindicatedbyadashedcontourandseparates
theunstableandstableregionswhicharelabelledUandS,respectively.Theresultsindicatethatsmallerinitialdroplet
sizesandcolderinitialdroplettemperaturest ndtoresultindestabilizedcombustorbehavior.
Inordertodemonstrateth applicabilityofthetwo-zoneanalysiswithfluctuatingtemperaturedroplets,thepredicted
analyticalsolutioniscomparedwithnumericalresultsusingapuremodeinitialconditionandtrackingthetemporal
evolutionof theflowfieldin time.Fig.3presentsacomparisonf theanalyticalndnumericalsolutionsforthegas
phasepressureoscillationattheupstreamcenterlineof thechamberforaninitialdropletdiameterandtemperature
of 130/_and130K, respectively.Alsoshownisacomparisonf theanalyticalndnumericalpredictionsforthe
fluctuatingliquiddroplettemperatureataaxiallocationone-fourthofthechamberlength.Theagreementisexcellent
forbothcases.
Thestabilitybehaviorfpracticalenginesi oftentestedbybombingorpulsingthecombustionchambertoinduce
pressureoscillations.Thenumericalpressuresponsetoanpulsedinitialconditioncanbevalidatedbyusingthe
long-termanalyticallypredictedgrowthrate.AsshowninFig.4,theshort-termtemporalresponsecontainsawide
varietyofmodeswithinthechamber.Inthelong-termsolution,however,weexpectthemostunstablemode(predicted
bytheanalyticalmethod)todominatewithinthecombustionchamber.Here,wehaveshownthepressurefluctuation
attheupstreamcenterlineofthecombustionchamber.A comparison with the analytically predicted solution indicates
that the growth rates correlate to within about 0.1%. It is evident that the numerical and analytical solutions demonstrate
excellent agreement, thus validating the CFD results.
REFERENCES:
.
.
.
°
.
*
.
Habiballah, M., Lourme, D. and Pit, F., "A Comprehensive Model for Combustion Stability Studies Applied
to the Ariane Viking Engine", AIAA-88-0086, AIAA 26th Aerospace Sciences Meeting, January 11-14, 1988,
Reno, NV.
Liang, R., Ungewitter, R., "Multi-Phase Simulations of Coaxial Injector Combustion", AIAA-92-0345, AIAA
30th Aerospace Sciences Meeting, January 6-9, 1992, Reno, NV.
Bhatia, R., Sirignano, W., "A One Dimensional Model of Ramjet Combustion Instability", AIAA-90-0271,
AIAA 28th Aerospace Sciences Meeting, January 8-11, 1990, Reno, NV.
Jeng, S.M. and Litchford, RJ., "Nonlinear Analysis of Longitudinal Mode Liquid Propellant Rocket Combustion
Instability", AIAA-91-1985, AIAA/SAE/ASME/ASEE 27th Joint Propulsion Conference, June 24-27, 1991,
Sacramento, CA.
Kim, Y.M., Chen, C.P., and Ziebarth, J.P. "Numerical Simulation of Combustion Instability in Liquid-Fueled
Engines", AIAA-92-0775, AIAA 30th Aerospace Sciences Meeting, January 6-9, 1992, Reno, NV.
Wang, ZJ., Yang, H.Q., Pindera, MT.., and Przekwas, AJ., "CFD Simulations of Nonlinear Acoustics in
Liquid Rocket Combustors", Penn State Propulsion Engineering Research Center and MSFC Fourth Annual
Symposium, September 9-11, 1992, MSFC, Huntsville, AL.
Grenda, J.M., Venkateswaran, S., and Merkle, C.L., "Multi-Dimensional Analysis of Combustion Instabilities
in Liquid Rocket Motors", AIAA-92-3764, AIAA/SAE/ASME/ASEE 28th Joint Propulsion Conference, July
6-8, 1992, Nashville, TN.
91
Model Vaporization HeatT_ansfer _
Priem-Heidmann
Abramzon-Sirignano
= ......Z__
rhvap ,'rd_D Sh ln [v_p,., ]
rhvap = rd_D Sh* ln[l + BM]
qv = m"°r _r'"(T:-Ta")
e=-I
Table 1: Summary of Typical Vaporization Models
1.25 I...J eli _ =_- "
13g* 0
..................... , ........... ! ........ -. Riemidaekl "'''4/10
-1.25 :"i'i.--,i::,:;_.i:i.::.i:_:.:..:.:_:'- i:":--:i:i- i:i:--:i:--"i-.
-zno ,o • 1:oo • =:oo • 3:0o • .o
-1.0
-0.5
0,0
O.S
1.0
I.S2.0
¢-
2
.9
E
E
o
0
400
300
200
100
110
Le'_ Fre,_
1 -7.-*00
12(2 130 140
DropletTem_atute (K)
Figure 1: Distributed Vaporization Stability Plane;
Constant Temperature Droplets With Initial Diameter
140/z; Most Unstable Growth Rate IT,1R
Instability Mode, Nondimensionalized Growth Rate.
Figure 2: Distributed Vaporization Stability Plane as
a Function of Droplet Diameter and Temperature;
Variable Temperature Droplets; Most Unstable
Growth Rate IT,1R Instability Mode.
150
1.5
0.75
U3
x 0
h.
-0.75
I 1
upstream P fluctuation
li(:uidT fluctuation
-1.5 I t t
0 0.00025 0.0005 0.00075
= 5 = i
numerical
analytical ....
2.5
-2.5
-5
0.001 0 0.001 0.002 0.003
TIME (see) TIME (sec)
Figure 3: Pressure and Liquid Temperature Fluctuation vs.
Time; Pure Mode 1T,1R Initial Condition,
Figure 4: Upslxeam Centerline Pressure Fluctuation vs.
Time; Pulsed (Mixed) Mode Initial Condition.


Development of a computational testbed for numerical simulation of combustion instability

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