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Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

Numerical Analysis Model forRadial Combustion Instability of Liquid Propellant Engine

Liu weidong, Wang zhenguo, Zhou jin, Zhuang fengchenDept. of Aerospace Technology

National University of Defense TechnologyChangsha, 410073, P. R. China

ABSTRACT

Numerical analysis of radial combustion instability in liquid propellant rocket engine wasperformed with PISO algorithm. The PISO algorithm employed an implicit prediction step and twostep of explicit correction steps to yield a time-accurate result at even,- time level when it was usedto calculate unsteady flow. Physical models such as stochastic spray model. ZKS dropletevaporation model. Low-ReynoIds-Number K-K. model and E.B.U. turbulent combustion model wereused to describe the combustion sub-processes. The response of the acoustic resonator to pressurefluctuation was taken into account through the classic model described by Helmholtz. A pressurebomb distributed in radial direction as sinusoid wave was imposed on the steady flow field, and thepropagation of pressure wave produced by the bomb was simulated to analyze the damping effect ofthe acoustic resonator. The numerical calculations show that the resonator has no obvious dampingeffect to the pressure waves.

Introduction

Combustion instability in liquid propellant rocket engine usually was characterized by thefrequency of chamber pressure oscillation as chug(low frequency instability, coupling betweenpressure oscillation and propellant feed system). buzz( intermediate frequency instability, its drivemechanism is still uncertain), scream (high frequency instability, coupling between gas flow andcombustion)[l]. Of main interest is the high frequency instability because they are known to be themost destructive by grossly enhancing heat transfer[2]. There are three basic types of acoustic modeof high frequency combustion instability, namely, longitudinal mode, tangential mode and radialmode. The tangential mode is the most dangerous one of the three modes because the pressureamplitudes of the tangential oscillations reach their maximum at the chamber wall. The radial modealso results in pressure oscillation but its largest amplitude locates at the center. If a chamber issusceptible to combustion instability, it is usually sensitive to these two modes because the.cylindrical chamber has less damping effects to them[3]. To eliminate the combustion instability,baffles and resonators were used as the main damping devices in the past time, so an attempt try tostudy the damping effects of the resonator was present in the paper.

In spite of the endeavor devoted to the combustion instability in the past decades, the problemis still be a challenge difficulty to the rocket designer. Because of the limitation of the

Copyrigth (C),1997. by the Americal Institute of Aeronauticsand Astronautics. Inc. All right reserved.


methodologies, theoretical analyses were performed mostly with small perturbation technique forlinearized or weakly nonlinear oscillations. The results of linear analysis become unreliable whenthe amplitudes of the oscillation grow to large values. Another reason for this is that the combustioninstability itself is a nonlinear phenomenon, linear analysis may leads to false stability. However., itis too difficult to get really useful results by nonlinear theoretical analysis because the spraycombustion processes in a rocket engine are too complex. In recent years, as the development ofcomputer technology and numerical methods used in fluid How calculation, it is possible to usenumerical simulation to analyze the combustion instability^.5,6]. Some plausible results havealready shown that numerical methods will be the main analyzing tool in engine designing[7|.

The combustion instability usually is observed as pressure wave in the chamber, in order tosimulate the propagation and reflection of pressure wave, numerical methods must possess at leastsecond order accuracy of time and space. The extended PISO algorithm and a second order up-wind/central scheme developed by the authors were used to calculate the unsteady combustionflow[8]. In fact, more accuracy scheme should be developed for capturing the wave characteristics,but high order accuracy scheme usually has dispersion and results in numerical oscillation, so ahigh order accuracy scheme must ensure that numerical oscillation should not overlap the physicaloscillation..

Physical models employed in numerical calculation include spray model which is applied todetermine the droplet initial -conditions, evaporation model which is used to describe the dropletevaporate behavior in high pressure environment and turbulent combustion model, etc. Thecombustion instability is general understood as the coupling of combustion and gas flow, therefore,the coupling relations should be embodied in these physical models of combustion sub-processes.However, most of the available physical models were developed in quasi-steady or steady casesand no dynamic response to the pressure oscillation was taken into account. So more attention.should be paid to the study of" combustion sub-processes such as propellant injection, atomization.evaporation and chemical reaction etc. to improve the existing models. In the present study, the two-parameter combustion instability model proposed by Priem[9] was used to set up the couplingrelations.

Damping characteristic of the acoustic resonator to ideal pressure disturb was analyzed first bya numerical model. The resonator was analogized to a spring-mass-dashpot system[10], and therelations between mass flow in the orifice of resonator and pressure disturb were obtained. Theacoustic resonator has maximum damping effect to the pressure disturb which frequency is near toits intrinsic frequency. But in reality, the intrinsic frequency of a resonator is difficult to measurebecause the temperature in resonator cavity changes with time, hi numerical calculation it is alsodifficult to determine the frequency of the pressure disturb in examining the effects of the acousticresonator.

Numerical pressure bombs were imposed on the steady flow field to investigate the combustionstability of a rocket engine which use liquid hydrogen and oxygen propellants. The numericalcalculation shows that acoustic resonator in die studied engine has no obvious damping effects tothe pressure disturbs.

Numerical Method


Governing EquationsThe governing equations of the combustion flow in liquid propellant rocket engine are transfered

to the general curvilinear coordinate system to treat the irregular boundary of combustion chamber.All equations can be written in the same form for convienent numerical completion.

ct rJ cq rJ c'rf rJ dq J cq rJ erf J dr] * '''** c *where (f> = p,u,vji,k,£,mf stands for different variables. U,V are components of velocity

vector in computational plane.S^,Sd^,S c < t > are source terms due to coordinator transform, two

phase flow and chemical reaction, respectively.

PISO AlgorithmThe governing equations were discretized by finite-volume approach on collocated grid and

solved by extended PISO[11,12,13] algorithm with a second order accuracy up-wind/centralscheme developed by the authors[8]. The coefficients of the algebra equations were calculated as:

a (j) = ci[;(f)E + aw(j)w +aN<f>N +ov^v

where: ap = aE + aw + aN + av + aEE + anw + am

aE = De+[\-2Fe,Q] • aEE=-[\-Fe/2,(aw = Dw+[\2Fw,Q\\ • aww =

fls=JD1+[|2F,,0|]

" StAll dependent and independent variables were stored at the same grid points when collocated

grid was used, this may produced pressure oscillation[14]. To eliminate the pressure oscillation, theso-called momentum interpolation numerical techniques proposed by Demirdzic[15]were studiedand successfully applied in the numerical simulation. The velocity component Ue at the volume facewas computed by the following formulae instead of die directly interpolation of the velocity u£ Up .

therefore, the velocity Ue was related with die pressure at the grid point E and P radier than dievelocity // // . This has die same influences as die staged grid.

"" E. P

+b

ap a (Pe-Pp) (3)p

where die subscript "e" denoted the east face of the control volume.

Physical Models

Spray ModelIn the numerical simulation of the spray combustion in liquid rocket engine, droplets initial

conditions(such as velocity components Ud ,Vd ) must be given. In the present study, droplets in thespray were divided into groups by their size according to the Rosin-Rammler distribution. The


velocities and angles in which they were injected into the combustion chamber were 'determinedby stochastic functions, this random spray model was based on the atomization mechanism of swirlcoaxial injector[ 15].

/(.,) = -^^expt-^^l] ' ( 4 )V2;rcr

where x represents the random variables, namely, droplet velocity and enter angle u, Q,respectively. The mass How which computational droplets stand for was calculated by:

mlt,=mqiIKl (5)where m was total mass flux of the liquid propellant. q was the mass fraction of the / th droplet

group. K is the computational droplet number in the group. The trajectories of droplets weredescribed by lagrangian method and the liquid source terms were incorporated into the gas phasecalculation by PSIC model [16]. In the computation of unsteady two-phase flow, this method justis a quasi-steady approximation to the real flow.

Droplet Evaporation ModelThe droplets travel in the combustion chamber with high pressure and temperatiire. their

evaporation rate, diameter and temperature were calculated by ZKS model [17].. t . 2vpDr,Nu°ml r^-BYvxevaporation rate: m =—-——-———Inl————j (6)

AB l~BYVJ

diameter- s - \ ^ \ *"* ^' ' 1 (T\Uttl lil^-l^l , „— l --——-———-—-- - -f- —————. • \ /

dt 47a-;pt 3p; dT dtLl J. - l. —'SVJL T f&'r V>- r -i „ JL „ i-:\L J. _ . ,temperature: —- = ————L-— —•——L — ——————————-—————:—— (8)dt PlCplr; ec-\ ACp.v+(A-" " '

mv dt pvs ASubscript s . =o denote the droplet surface and surrounding gas. Subscript /, v represent die liquid anddie vapor phase. The evaporation rate. was computed by (6).(7),(8) with iterative methodology.

Turbulent Combustion ModelTurbulent viscosity was obtained by the low-Reynolds-Number K - n model proposed by

C.K.Lam and K.Bremhost[18,19]. Compared with the standard « - t- model, this turbulent modelneeds dense grid in the near wall region, and the constants Cr C, of the standard model wererevised to improve the calculation in the near wall region.

dps d . . d , . d r , . <%•..—— +— (pue) + — (pvs ) = —[(//,+ jut )— ]a ex ay ex exds £ £~ (9)dy k k

turbulence viscosity was computed by:tt^cJrftfls (10)

where /^ = [1 -exp(-4 Rer)]2(l + 4 /Re,) ;/ ,= l + (4//y;) ; /,=l-«cp(-Re r)

The chemical reaction rate was calculated by the E. B. U. model and Arrehnius model.


(11)

(12,)

The chemical reaction rate finally was determined by the smaller one of the two rates.

, .„.,, ,„....., (1 3>

Resonator ModelThe acoustic resonator was used widely in the liquid rocket engines as the main damping device

to the combustion instability. In order to simulate the resonator effect by numerical methods, theclassical Helmholtz model showed in Fig.2 was employed in the calculation. The mass flow in theresonator orifice was calculated by the following partial equations which were derived from theanalogy of a mechanical mass-spring-dashpot system[l].

dmdt D p vcav

m = ~pAf>— (15)dt

where:lef. ,A,yV is efficient length, area of the orifice and volume of the cavity, respectively.

R D — ITplT is the resistant coefficient. The intrinsic frequency of the Helmlioltz resonator wascomputed by the following equation.

(16)

The mass flow in orifices of the acoustic resonators were considered as the mass source terms tothe gas phase at the near wall control volumes. Hence, the effects of resonators to the pressurewave in combustion chamber were embodied bv this wav.

Unstable Combustion ModelBecause most of the physical models available were developed as quasi-steady or steady

model to the combustion environment, they are unable to reflect the responses in nonlinear 'dynamic relations when combustion instability takes place. To compensate the dynamic response,die Priercfs two-parameter model[9] was used to relate combustion perturbation and pressuredisturb. However, die relations given by Priem's model are still in linear form and have nostrong support from the "experiments. Obviously, models describe the sub-processes ofcombustion must be improved in the future to satisfy die needs of combustion analysis.

. , fh . n ,mL, dp' .._.nf=av(—)/?'+#,(——)-:r (1?)p pa a

p^aA'+A.^f ' <«>p pa aw here av, as , J3V, fls are constants which can be adjusted to change the response intensity. Thesuperscripts overbar and prime denote the steady and fluctuation parameters, receptively. But inthis numerical study, the second terms at die right of the equations were neglected for thesimplicity. Hie pressure disturbs imposed on the steady flow field was given in the followingform [7]:


t<t0

where: ~p,0x -. is steady chamber pressure, a is adjustable coefficient used to examine the

instability margin. f3r describes the space distribution in radial direction. This finite amplitudebomb test may irritate nonlinear combustion instability, it is very important in stability assessmentbecause the engine may be linear stable but nonlinear unstable. That means if a engine is linearstable, all small disturb can be suppressed, but large disturb may results in combustion instability.In the case of linear unstable combustion, whatever small disturb may be amplified and finally leadto finite amplitude oscillation.

Results and Discussion

The numerical and physical models described in the paper were used to simulate the unsteadyflow in a LOX/LH2 rocket engine to assess the damping effects of the acoustic resonators topressure bomb wave. This is part of the plan to develop a numerical analyzing tool for diecombustion instability problem. Physical models for the sub-processes as propellant atomization,droplet evaporation will be studied for real nonlinear models in another plan.

Fig. 1 shows the schematic of the model chamber and resonators at the chamber wall nearinjection face. Although existing resonators, the continuous wall boundary was postulated yetbecause the flow between the resonators and chamber were considered as source terms to thechamber gas phase. Fig.2 is the resonator model which was analogized as mechanical vibrationsystem. Its intrinsic frequency was calculated by theoretical formula and an ideal sinusoidalpressure fluctuation P(t)=Asin(o)t) was imposed to it to analyze the influence of pressure frequency.

Fig.3 is the pressure bomb distributes in radial direction and declins with time. Fig.4 shows theresponse of the resonator to pressure disturbs at different frequency. Pressure disturbs at low andhigh frequency (compared with the resonator intrinsic frequency) results in small oscillation of themass flow in the resonator orifice. This indicates that the resonator has obvious damping effect onlyto these pressure wave which frequency near its natural frequency.

Pressure bombs distribute in radial direction as sinusoidal function are imposed on the steadyflow field. The bomb lasts certain time and then disappears, it produces pressure waves in thechamber. Fig.5 presents the first order pressure oscillation amplitude along the radial directiondeclines with time when no resonator is taken into consideration. No stable oscillation is found inthis cases. Fig.6 shows the amplitude decline of the second order radial pressure oscillation withoutresonator. Fig.7 represents the case which resonator is added to the chamber. In all cases, thepressure wave were finally damped, but with resonator, the pressure wave was damped in short time.In fact, the damping effect of the resonator was not obvious. This may be due to the erroneousdescription of boundary conditions when resonator is added to the chamber, the mass flow in theresonator orifice just treat as source terms to the gas phase is improper and cause weak couplingbetween the oscillation in chamber and in resonator cavity.

The results obtained from the calculation show that the extend PISO algorithm is suit for thenumerical simulation of unstable combustion phenomenon in liquid rocket engine. However, thephysical models used in the calculation need improving.


References

[ I ] Harrje.D.T.. Readon,F.R. Liquid Rocket Combustion Instability, NASA-SP-192,1972.[2] Vigor, Yang, William. E.A., Edited, Liquid Rocket Engine Combustion Instability. Progress in

Astronauts and Aeronautics. 1995.[3J Sun vveishcng. Combustion Instability in Solid Rocket Engine. Published in Chinese,1986.|4| Habiballah. M, Dubois.I.. PHEDER: Numerical Model for the Combustion Stability Studies

Applied lo the Ariane Viking Engine. J. Propulsion and Power.. Vol:7. No:3.1991.[51 Kini. Y.. Chen. C.P. and ZicbarthJ.p.. Numerical Simulation of Combustion Instability in

Liquid Fueled Engines, AIAA-92-OU75.[6] Grenda.J., Venkateswaran.S., and Merkle.C.L.. Liquid Rocket Combustion Instability Analysis

by CFD Methods, A1AA-91 -0285.[7] Habiballah.M.. Dubois.I.. Numerical Analysis of Engine Instability, First International

Symposium on Liquid Rocket Combustion Instability. 1993[X] Liu vveidong. Studies on Numerical Models of Combustion Instability in Liquid Propellant

Rocket Engine. Ph.D. Thesis. 1996. National University of Defense Technology.[9] Prime.R.J.. Roundrobin Calculation of Wave Characteristic in a Fixed Geometry-Operating

Condition Liquid Rocket Using Given Simplified Combustion Equations,JANNAF Workshop on Numerical Method in combustion Instability, Orlando, 1990.

[10) Demirdzic.I. and Peric.M.. Comparison of Finite -Volume Numerical Method with Stagedand Collocated Grids. Computer and Fluid, Vol:16,1988,pp389-415.

[11] lssa.R.1.. The Computation of Compressible Recirculating Flows by a Non-Iteration ImplicitScheme, J. comput. Phys.. Vol:62, pp40-82,1985

[12] Issa,R.I.. Solution of the Implicit Discretized Reacting Flow Equations by Operator-SplitingJ. Compt. Phys., Vol:62.pp40-65,1985.

[13] Chen. C.P.. Kirn. Y.M.. MAST: A Computer Code for Multiphase All-Speed TransientFlows in Complex Geometries, NASA-CR-NAG8-092,1991.

[14] Patankar.S.V., Numerical Method for Heat Transfer and Fluid Flow. 1981.[ 15] Wu. J.X.. Liu,W.D..Zhuang.F.C., Studies on the Large .Flux Coaxial Injector Spray Model

and Atomizationr AIAA-93-2160.[16] Crow. C.T.. Sharma,M.P., The Partical-Source-In-Cell Model for Gas Droplet Flows, ASME

J. of Fluid Engineering, pp325,1977.[17| Zhunag, F.C.. Theory of Spray combustion. Modeling and Appilcation in Liquid Propellant

Rocket Engine. Published in Chinese, 1995.[18] Lam, C.K., BremhostK.A.. Modified Form of the K - s Model for Predicting Wall

Turbulence. ASME J. Fluid Engineering. Vol:103,pp456-46().[19] Jones.W.P., Launder,B-E.. The Calculation of Low-Reynolds-Number Phenomena with a

Two-Equation Model of Turbulence, Liter. J. Heat and Mass Transfer, Vol:16,1973 c[20] Palel. V.C., Rodi.W.. Turbulence Models for near wall and Low-Reynolds -Number Flows

A review. AIAA Journal. Vol:23.No:9. 1981.


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Fig. 1 Schematic of the combustion chamber and resonator

Fig. 2 Schematic of the acoustic resonator

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10

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