(c)2001 american institute of aeronautics & … of a small combustor attached to a long pipe....

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A01-16109AiAA

Al A A 2001-0194LES of Combustion Dynamics in aPulse CombustorK. Tajirl and S. MenonSchool of Aerospace EngineeringGeorgia Institute of TechnologyAtlanta, Georgia 30332-0150

39th AIAA Aerospace Sciences Meeting &Exhibit

January 8-11, 2001 / Reno, NVFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191


LES of Combustion Dynamics in a Pulse

Combustor

K. Tajiri* and S. Menon*School of Aerospace EngineeringGeorgia Institute of TechnologyAtlanta, Georgia 30332-0150

Simulation of combustion process in a pulse combustor is carried out to understandthe dynamics of flameholding, flow-chemistry interaction and the resulting pulsation. Tosimulate natural pulsation, a characteristic based boundary condition was modified tosimulate the behavior of the flapper valve. Large-eddy simulations (LES) in an axisym-metric configuration are carried out for a range of combustor dimensions. The inflowboundary conditions automatically synchronizes with the acoustic mode and results insteady-state pulsation in this device. Results show that strain-induced extinction up-stream of the flameholder is critical in achieving oscillation at the expected frequency. Ascaling analysis is also carried out whereby the pulse combustor dimensions are reducedfrom typical systems of the order of 1 meter to as small as 15 cm to determine if thedynamics can be reproduced even in such small devices. Potential application of thesesmall pulse combustors for micro-scale heat transfer is also discussed.

1 IntroductionThe pulse combustor is a simple device consist-

ing of a small combustor attached to a long pipe.Fuel-air mixture enters the combustor from the op-posite end of the tail pipe location through an inletvalve. Periodic (pulse) combustion in the combustoris driven/maintained by the acoustic mode in the tailpipe. The concept of the pulse combustor is not newand many such devices are in operation. A key ap-plication of this device is in heat transfer since thecombustion efficiency of this device is very high.

Two types of pulse combustors have been developedand they differ primarily in the design of their inletvalve. The aerodynamical valve type is designed sothat the pressure loss in the direction of the reverseflow in the inlet is higher than the pressure loss inthe forward direction. The mechanical-valve type useseither a flapper valve or a rotary valve. The flappervalve, which is very popular, are installed at the in-let to prevent combustion flashback into the inlet andpermits a simple control of air and fuel flow rates. Onthe other hand, this type of combustor has movingparts which can lead to fatigue problem. The rotaryvalve type requires a feedback system to synchronizethe rotation of the valve and the optimum operatingfrequency of the combustor, and therefore, has notbeen very popular.

The operation of the pulse combustor occurs in threephases. In the first phase, ignition of the combustiblemixture occurs by the remaining hot burned gas fromthe previous cycle. The burned gas expands and moves

* Graduate Research Assistant, AIAA Student Membert Professor, Associate Fellow

Copyright © 2001 by K. Tajiri and S. Menon. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc. with permission.

outward towards the inlet and the exit of the combus-tion chamber. In the second phase the burned gasmoves out through the tail pipe and the pressure inthe combustion chamber falls. The pressure eventu-ally falls below the atmospheric pressure (or upstreampressure) due to the momentum effect of the outwardmoving gases and eventually the outflow slows down.In the third phase the combustible fluid-air mixtureenter the combustor. The momentum effects cause thepressure to rise above the ambient value. Due to hightemperature in the combustor this fresh mixture self-ignites and thus, the entire oscillatory cycle repeatsitself.

The primary advantage of the pulse combustor isthe enhancement of the momentum, heat, and masstransfer process due to pulsating combustion.1 Fornon-premixed combustion, the improved mixing of fueland air within the combustor reduces the combus-tor size and the amount of the excess air required toachieve complete combustion. The reduced amount ofthe excess air decreases the exhaust flow losses, andtherefore, increases the thermal efficiency.

Another benefit is the reduction of NOX formation.High heat transfer rates to the combustor walls andthe rapid and effective mixing of the cooler combustionchamber gases with the hot combustion products re-sult in a short residence time for combustion productsat high temperature. This decreases the production ofthermal NOX.

The physical mechanism by which this systemachieves self-sustained oscillation is also well knownand is related to the Rayleigh criterion. In orderto achieve pulsating combustion, heat release spon-taneously excites pressure (acoustic) oscillation andwhen the periodic heat release is in-phase with the nat-

1American Institute of Aeronautics and Astronautics Paper 2001-0194


ural acoustic mode of the tail pipe then energy is addedto the acoustic mode and the system begins to resonateat the natural mode of the tail pipe. Typically, the pul-sation occurs at the tail pipe's quarter-wave acousticmode frequency.

Although many experimental studies have been car-ried out for the pulse combustor only a few numericalstudies have been reported in the literature. Further-more, all reported studies employed simplifications orad hoc boundary conditions in order to create andsustain the pulsations. For example, Tsujimoto eta/.2 analyzed the Helmholtz-type pulse combustor nu-merically using the method of characteristics. First,they examined a simple acoustic model to show that aphase lag in the combustion plays an important role inpulse combustion. Then they assumed that combus-tion started at a certain time after the introductionof fresh air and fuel and showed that a constant com-bustion delay time model could simulate most of theimportant features observed in the pulse combustor.

Barr et a/.3 developed a one-dimensional numeri-cal model of a pulse combustor. The unsteady, one-dimensional equations of continuity, momentum, andenergy were solved using MacCormack method withvariable-area geometry, including losses due to fric-tion and heat transfer. The periodic injection of massinto the combustion chamber was handled by switch-ing between solid and open boundary conditions. Tosimulate the periodic influx of reactants and to excitethe oscillation, the inflow was explicitly forced at theexpected natural mode of the inlet pipe. Since themass flow was forced, a fixed mass influx was forcedinto the combustion chamber when the valve was open.For the modeling of the combustion and heat release,they distributed the heat uniformly within the com-bustion chamber.

In later studies, Barr et a/.4'5 investigated the ef-fect of fuel-air injection, mixing, chemical kinetics andflame extinction. They observed that the operatingfrequency was not very sensitive to changes in theheat transfer coefficient, while the magnitude of thepulsations could be more sensitive to the heat trans-fer. Their study of flame extinction was based on thevortex dynamics method combined with a flame sheetalgorithm. They showed that flame extinction playsa dominant role in delaying reaction until later in thecycle.

Benelli et a/.6 used a commercial CFD softwareto model the Helmholtz type pulse combustor withself-sustained acoustic vibrations. Numerical modelused an axisymmetric k — c turbulence model withthe Magnussen-Hjertager turbulent reaction rate clo-sure. The inlet valves, when opened, were consideredas orifices with a given cross-section and a characteris-tic pressure drop curve was used to define a constraintbetween the velocity and the pressure change acrossthe valves.

Moller et a/.7 examined the effects of inlet geom-etry change using Large Eddy Simulation (LES) in acommercial CFD code. The geometry was changedby altering the position of the stagnation plate nearthe inlet. The operating characteristics in terms oftime-resolved pressure and heat release variations, andtime-averaged profiles of the temperature and concen-trations of 02 and NOX in the tail pipe were investi-gated. In this study, the flapper valve was modeled bya prescribed mass inflow of air and fuel using a sinu-soidal curve for the first half of the combustor cycleand zero mass flow for the second half of the cycle.

As noted above, in all the previous studies, inflowwas explicitly specified. In fact, in all the reportedcases, the inlet was forced at the expected pulsationfrequency or a condition was enforced on the mass flowto achieve the expected pulsation. Therefore, none ofthe previous studies were able to properly capture thebehavior of the inlet valve and did not excite naturalpulse combustion in the device.

The present study addresses this issue and demon-strates a simulation model that incorporates properphysical inflow condition in order to capture the pul-sation without requiring ad hoc forcing of the in-flow. These simulations employ a LES approachand are used to investigate the dynamics of ignition,flame holding, gas expansion and inflow. A gas-fired,mechanically-valved, Helmholtz-type pulse combustorof a scale that has been experimentally studied is firstsimulated in order to validate the results and to in-vestigate the dynamics of flow-chemistry interaction.Subsequently, a series of simulations are carried outfor different size pulse combustors with tail pipe lengthranging from 100 cm to 15 cm. The goal of this para-metric study was to determine if the dynamics of pulsecombustion remains the same over a wide range of de-vice length and if scaling laws can be devised. It isfeasible that very small pulse combustors could findnew applications as energy transfer devices for smallsystems such a micro air vehicles, etc.

2 Governing EquationsThe governing conservation equations of motion for

mass, momentum, energy, and species in a compress-ible, reacting fluid are:

I pUj = Q

"r= 0

(1)

Here, p is the mass density, p is the pressure, E is thetotal energy per unit mass, Ui is the velocity vector, qiis the heat flux vector, r^ is the viscous stress tensor,and N is the total number of chemical species. The in-dividual species mass fraction, diffusion velocities, andmass reaction rate per unit volume are, respectively,

American Institute of Aeronautics and Astronautics Paper 2001-0194


,m, and wm. The viscous stress tensor is r^ =dxj + duj/dxi) — %p>(duk/dxk)8ij where p, is

the molecular viscosity coefficient approximated usingSutherland's law. The diffusion velocities are approx-imated by Pick's law: VijTn = (-Dm/Ym)(dYm/dxi)Here, Dm is the ra-th species mixture averaged molec-ular diffusion coefficient. The pressure is determinedfrom the equation of state for a perfect gas mixture

(2)P = pT YmRu/Wnm=l

where T is the temperature, Ru is the universal gasconstant, and Wm the species molecular weight. Thetotal energy per unit volume is determined from pE =p(e + ̂ u\) where e is the internal energy per unit massgiven by e = ]Cm=i Ymhm - P/p and hm is the speciesenthalpy. Finally, the caloric equation of state is givenby

/JT°cp,m(T)dT (3)

where AA0jm is the standard heat of formation at tem-

perature T° and Cp)Tn is the m-th species specific heatat constant pressure.

Following Erlebacher et a/.,8 the flow variables canbe decomposed into the supergrid (i.e., resolved) andsubgrid (i.e., unresolved) components by a spatial fil-tering operation such that / = / + /" where ~ and" denote resolved supergrid and unresolved fluctuat-ing subgrid quantities, respectively. The LES-resolvedsupergrid quantities are determined by Favre filtering:

(4)

where the overbar represents spatial filtering defined__f ( x i t t ) = (5)

Here, G/ is the filter kernel and the integral is overthe entire domain. Applying the filtering operation(a low-pass filter of grid size A) to the Navier-Stokesequations, the following LES equations are obtained:

— n~" u

= 0

(6)Here, Tij and q{ are approximated in terms of thefiltered velocity. The unclosed subgrid terms repre-senting respectively, the subgrid stress tensor, subgridheat flux, unresolved viscous work, species mass flux,diffusive mass flux, and filtered reaction rate are:

9891 = p\Sm- (7)

The subgrid stress tensor, r'f9, and subgrid heat fluxH*9*, have been extensively modeled in the past byemploying the subgrid kinetic energy equation, k898,and are, therefore, only briefly discussed. The unre-solved viscous work, a*9*r, and the diffusive mass flux,Oifm, are neglected in this study. Closure of the speciesmass flux, $*}̂ , is carried out in this study using agradient-diffusion model. Closure of the filtered reac-tion rate term, wm, is also discussed below.

2.1 Subgrid ClosureIn the present approach, the subgrid stress tensor,

rij9, is determined by using the local grid size, A, asthe characteristic length scale and the subgrid kineticenergy, k898, as the characteristic velocity scale. Thesubgrid kinetic energy, k898 = |[u| — S|] is obtainedby solving the following transport equation:9

dk'<>°\Prt dxi

(8)where Prt is the turbulent Prandtl number (taken asconstant and equal to 0.90 for this study), P898 andD898 are the production and dissipation of subgridkinetic energy, respectively. The production term isdefined as, P898 = -T9f9(dSi/dxi)J where r?f9 is themodeled subgrid stress tensor, r??8 is modeled as

(9)

with the eddy viscosity, vt = C7l/(fe*p*)1/2A where A isthe characteristicJjES grid size and the resolved rate-of-strain tensor, Sy = \(dui/dxj + duj/dxi).

Finally, the_dissipation term is modeled as D898 =C€p(k'")2/3/!&. The coefficients, Cv and C€ can bedynamically determined10 but will be taken as con-stants and equal to 0.2 and 0.916, respectively. Itshould be noted that 3-D isotropic turbulent scalinglaws were used to derive the preceding relations, andtherefore, may not be completely applicable in thisaxi-symmetric study.

The closure of the subgrid heat flux, Hi",is achieved using a conventional gradient-diffusionmodel:

- "* ®h /m\= -P^r-i£r (10)

8?8 = p[u&j-where h is the resolved scale total mixture enthalpyper unit mass, h = A/£m -h /^, cpim(T)dT + Ju|.



A conventional closure for the subgrid scalar velocityfluctuation term, $*^, is a gradient-diffusion modelanalogous to that used for H*9*:

_Sct (H)

where Set is the turbulent Schmidt number found fromthe product, Set = PrtLem (Lem is the species Lewisnumber). It should be noted that this form of closureeffectively treats small scale fluid dynamic effects asmolecular process.2.2 Reaction Rate Closure

A single step, global chemical mechanism formethane/air combustion is currently employed witha molar reaction rate in Arrhenius form.

CH4 + 202 + 7.527V2 -> CO2 + 2H2O + 7.527V2

A pre-exponential term A = 3.0 x 108(sec.~1) and anactivation energy Ea = 29,38Q(cal/gmol) are used.These values are chosen from Smith et a/.11

For LES application, the filtered reaction rate hasto be obtained from a closure model. Closure of thefluctuating reaction rate, wm, has been accomplishedusing a modified Eddy-Break-Up (EBU) model basedon a model developed earlier. In the EBU model,the time scale required for complete molecular mix-ing is modeled as the time for one subgrid eddy to becompletely dissipated (mixed on the molecular level).Since a subgrid eddy can be viewed as having a uni-form composition, the time needed to molecularly dif-fuse the species is the same that needed for completevelocity dissipation. The subgrid fluid dynamic / mix-ing time, Tmix, is proportional to the subgrid turbulentkinetic energy, k*9*, and its dissipation rate, e*9*, bythe relation:

*••• (12)

Here the scaling constant, CEBU, is set to unity follow-ing Pureby et a/.12 The mixing time-scale reaction ratefor the methane-air premixed lean reaction is, then

1 •[CH4 (13)

The effective chemical reaction rate, wet,u, is found bytaking the minimum of the mixing reaction rate (fluidtime scale) and the kinetic rate (chemical time), Wkin,i.e.;

webu = mm(wmix, wkin). (14)

3 Numerical MethodAxisymmetric, compressible Navier-Stokes equa-

tions are solved in the present study using a finite-volume scheme that is second-order-accurate in space

and time. Nearly all practical pulse combustors areaxisymmetric in configuration and since the pulsationis primarily driven by the low-frequency longitudinalacoustic mode in the tail pipe, axisymmetric formula-tion is sufficient to capture all the expected dynamicsin the pulse combustor. Some finer details such asthe breakdown of the vortices into small-scale eddies(which requires full three-dimensional simulations) arenot captured in this approach but they are not con-sidered an important feature for the pulse combustor.Furthermore, from a computational point of view, 3Dsimulations of the full pulse combustor will requiresignificantly long simulation time (to resolve the longtail pipe and to simulate multiple cyclic pulsations)and therefore, are considered very expensive. Furtherdetails of the numerical model and the boundary con-ditions are given below.

3.1 Combustor GeometryThe initial geometry used for validation is nearly

identical to the one used in earlier experiments andsimulations by Moller et al.7 This pulse combus-tor is a Helmholtz-type device with a flapper valve.Fuel and air are premixed before they enter the com-bustion chamber. Although the experimental facilityhas a rectangular combustion chamber, a cylindrical(axisymmetric) one with the same cross section areais used in this study. In any case, most practicalpulse combustors are axisymmetric in cross-section. Aschematic of the simulated device is shown in Fig. 1.

The combustion chamber has 45.2mm radius and is150mm long. It is connected to a tail pipe with aninner radius of 18mm and is 1430mm long. A flameholder (12.5mm radius) is mounted on to a 2.5mm ra-dius rod and is placed on the symmetric axis of thecombustor. The inner radius of the inlet tube to thecombustion chamber is 10.25mm. Other similar de-vices were also simulated. Table 1 summarizes thevarious combustors simulated in the present study.

Relatively high grid resolution has been employedhere. The grid resolution is 190 x 80 in the combus-tion chamber and 353 x 48 in the tail pipe with gridclustering near the walls to resolve the boundary lay-ers. The effect of grid resolution was studied earlier todetermine an appropriate resolution in the combustorand in the tail pipe. The grid resolution in the combus-tor was chosen to resolved the flame structure duringthe entire period of pulsation while the tail pipe resolu-tion was chosen to resolve the outflow motion and theacoustic mode without requiring significant axial gridstretching. In fact, grid stretching was kept around 5percent to maintain spatial accuracy of the simulatedflow field.3.2 Boundary Conditions

To simulate the behavior of the flapper valve as itis driven by the acoustic oscillation in the tail pipe,"natural" inflow-outflow boundary conditions were de-



combustorcombustion chamber

length (m)combustion chamber

radius (m)inlet area (m2)

stagnation plateradius (m)

stagnation platelocation (m) (from inlet)

tail pipe length (m)tail pipe radius (m)

A

0.15

0.0452

3.14 x ID"4

0.0125

0.013

1.430.018

B

0.045

0.0136

2.82 x 10~5

0.00375

0.0039

0.430.0054

C

0.015

0.0045

5.9 x 10~7

0.00125

0.0013

0.1430.0018

Table 1 Geometries of various combustors.

veloped. No-slip conditions are prescribed on all thewalls, while the symmetric (slip) conditions are pre-scribed on the symmetric axis. For the thermal wallcondition, the combustion chamber walls are treatedas adiabatic while a isothermal condition is employedin the tail pipe wall using a linear temperature profile(Fig. 3).13

At the inflow, a modified version of the characteris-tic boundary conditions is imposed. Since the inflow issubsonic, three conditions is prescribed and one con-dition is computed from inside the computational do-main. Here, the stagnation pressure pt, the stagnationtemperature Ti, and the angle 6 of the inflow directionare specified as 101325Pa, 300K and 0°, respectively.Using these quantities other properties such as, pres-sure, temperature, and radial velocity are obtained asa function of axial flow velocity as follows

where 7, Cv are specific heat ratio and constant-volume specific heat, respectively. To obtain the axialinflow velocity, a characteristic relation for the up-stream moving acoustic wave of the form

dpat

du du

is solved at the inflow. With these boundary condi-tions, the axial inflow velocity adjusts naturally to thedifference between the pressure upstream of the inletboundary (as determined by the above equations) andthe pressure in the combustor just downstream of theinlet. This behavior allows us to simulate the openingand closing of the flapper valve without explicitly con-trolling the mass inflow. When the interior pressureapproaches the upstream (inlet) stagnation pressure,the flow velocity approaches zero and this is used asan indication that the flapper valve is closed. Then,

the boundary conditions of the inlet are changed toa no-slip condition such that the upstream stagnationpressure and the pressure downstream of the inlet arematched and no inflow or reverse flow occurs at theinlet.

At a later stage, when the outflow through the tailpipe occurs and the pressure in the combustor dropsbelow the inlet stagnation pressure, the inflow beginsand the cycle repeats itself. Thus, the present inflowcondition does not require an explicit specification ofthe oscillating mass flow and allows the inflow bound-ary conditions to respond to the pressure oscillation inthe combustor. The implications and some surprisingconsequences of this boundary conditions are discussedlater.

For the outflow conditions only the exit pressure isprescribed at the atmospheric condition and charac-teristic outflow boundary conditions are solved for thethree outgoing waves. The four characteristic relationsat the exit are

However, the last condition is replaced the pre-specified pressure condition in the numerical imple-mentation.

3.3 Starting ProcedureAlthough the starting procedure is primarily a nu-

merical artifact, proper initialization and a numericalignition procedure is required initiate the combustionprocess and the subsequent pulsations. Here, we triedto follow the procedure used in the experiments todemonstrate the ability of LES method to mimic real-istic starting conditions. In the actual pulse combustorit is common to provide a small blower to force a suf-ficient flow of starting gas. After the pulsation hasstarted the blower and the igniter are turned off.



This procedure is also modeled in the simulations.At the beginning, the combustor is filled with an in-ert (e.g., burned products) gas. When the simulationstarts, the pressure downstream of the tail pipe is re-duced to a value slightly below atmospheric (98,000Pa) and the temperature in the upper combustionchamber wall is set to 2000K to ignite the mixture(Fig. 1). This ignition process is similar to the ap-proach used in the experiments.14 Due to the pressuredifference between the inlet and the exit of the com-bustor, the fuel-air mixture flows into the combustorand this flow pushes out the initial inert gas. After thecombustion chamber is filled up with the fuel-air mix-ture and the gas starts reacting, the igniter is turnedoff. Although the first cycle takes longer time to com-plete compared to the steady operation, the steadystate is achieved in 2-3 cycles. Once the combustorstarts pulsating, the exit pressure which was reducedfor the starting process is raised and reset to the at-mospheric value.

For the long tail pipe combustor (combustor A),to achieve sustained oscillation at the expected lowfrequency of around 100 Hz requires a very long sim-ulation. However, the code is highly optimized forparallel processing and has high scalability. Thus,although a large number of time steps have to be sim-ulated, a typical simulation (of around 5 flow throughtimes after the initial transients) requires around 360single processor hours on the Cray T3E. Of course, forthe small devices the computational cost drops consid-erably.

4 Results and DiscussionIn this section, we summarize the results obtained

in this study. First, the baseline configuration is sim-ulated to validate the simulation model and then theother sub-scale devices are studied.

Here cycle time is defined as follows: the onset ofthe inflow is 0, the end of the cycle (and the beginningof the next cycle) is 1.0. The fuel-air mixture entersthe combustor between cycle time 0 and 0.5, and theflapper valve closes between 0.5 and 1.0.4.1 Self-Sustained Pulsation

To validate the present model, simulations were car-ried for the test configuration described earlier7 forwhich data is available for validation. Fig. 1 shows theschematic of this device and Table 1 summarizes thepertinent dimensions.

The first series of simulations employed the LESmodel using the EBU closure model for the reactionrate. For these studies no effort was made to takeinto account the effect of strain-induced extinction ob-served in the experiments.15 Fig. 6a-f shows a time se-quence of the oscillating filtered reaction rate contoursand Fig. 7a-f shows the corresponding temperaturecontours. As can be seen, during the inflow (Fig. 6b-

d), the hot remnant gases from the previous cycleignites the fuel-air mixture as it enters the combustorand gets deflected to the upper wall. The flame zoneis a narrow band surrounding the reactant mixture.The flame is anchored at two locations: the stagnationplate and the wall along the inlet port. Most of thecombustion is completed before the inlet port closesand the flame structure collapses to a region near theinlet. It can be seen that there is still some unburnedmixture near the inlet that appears to maintain theflame before the next cycle begins anew. Althoughthis simulation achieved self-sustained steady pulsa-tion the predicted frequency and the mode shape didnot agree with experimental observation. This is dis-cussed below.

Fig. 4 shows the pressure oscillation at three loca-tions in the tail pipe. It can be seen that near thecombustor exit the pressure oscillation is quite highand reaches around 8 percent peak-to-peak but closerto the exit the pressure amplitude is much lower, lessthan 2 percent peak-to-peak. Although all locationsshow a similar frequency, the pressure signature showsa traveling wave.

Analysis of the above pressure signature in termsof mode shape and phase is given in Fig. 5. It canbe seen that a three-quarter acoustic mode shape isexcited in the tail pipe. The associate frequency is 266Hz. In contrast, the experimental results reported aquarter-wave mode at a frequency of 103 Hz. Clearly,the simulation discussed above is missing some keyphysics.

A feature noted in the earlier experiments was thatstrain-induced extinction occurred in the combustor.However, the above noted simulation ignored this fea-ture. Furthermore, the EBU model described abovedid not have this feature. The global chemical mecha-nism does not have any extinction/re-ignition capa-bility and it was considered too expensive to makethe chemical kinetic model any more complex for thepresent study. Furthermore, replacing the EBU modelwith a more accurate but perhaps more expensive sub-grid combustion model was not considered acceptablefor this study.

Therefore, an attempt was made to determine if theEBU model could be modified in a very simplisticmanner to mimic local extinction. First, the resultsof the above noted simulation was analyzed to de-termine which part of the EBU model, Eqn. 14, wasdominating and where. It was determined that duringinjection and the formation of the initial shear regions(see Fig. 6c-e and Fig. 7c-e) the turbulence mixingtime is very small and thus, the EBU model was ignit-ing the mixture using the Arrhenius kinetics. In fact,in most of the near field of the injector combustionwas kinetically controlled. If one considers the inverseof the mixing time as a measure of the local strain, itbecomes obvious that there are regions in the reacting



zone where strain exceeds 1000 sec"1. Clearly, in areal system such a high value of strain should causelocal extinction16 but this feature was not modeled inthe above simulation.

Based on this information, another simulation wascarried out where an extinction model modification tothe EBU was implemented in a very simplistic manner.All regions where the mixing time scale exceeded 1000sec"1 was considered extinct and the reaction rate wasset to zero. This modification made a drastic change tothe oscillation mode and frequency in the combustor.

Figs. 8 and 9 show respectively, the filtered reactionrate and the temperature contours in the combustorwith this extinction modification. Significant changesin the combustion process can be observed by com-paring these results with the earlier results withoutextinction (Figs. 6 and 7). Fig. 8 (when comparedto Fig. 6) shows that with extinction the reactionzone is no longer a narrow region, rather it is spreadout further downstream of the stagnation plate. Theflame is still anchored at the plate but since the highstrain regions are no longer burning, reactions occurfurther downstream of the stagnation plate. In theearlier experiments,15 combustion was seen to occurdownstream of the stagnation plate and the presentsimulation appears to capture this as well.

The impact of the change in the location of burn-ing on the pressure mode is immediately apparent.Figs. 10 and 11 show respectively, the pressure fluc-tuation and the mode shape for this case. It can beclearly seen that the pressure oscillation is in-phaseat all locations in the tail pipe. The peak-to-peakvalue are not significantly affected by the inclusion ofthe extinction criteria. The frequency and the modeshape, however, drastically changed from the earliercase. The frequency has reduced to 109 Hz and themode shape changed from the three-quarter wave tothe quarter-wave shape. It is, therefore, clear that theprocess of combustion and the location of heat releasein the combustor can play an important role in deter-mining the oscillation properties.

4.2 Scaling StudiesTo determine if the same simulation model is capa-

ble of predicting pulsation in much smaller devices,two other combustors (dimensions are given in Ta-ble 1) were also simulated using the same model withno changes to the extinction criteria. The dimensionsof the devices were scaled down from the referencecase to mimic smaller pulse combustors. Both thesecombustors B and C showed similar steady-state pul-sation but at higher frequencies (see Table 2), as wasexpected. Therefore, only representative results arediscussed here.

Fig. 12a shows the mode shape and phase for Com-bustor B and 12b shows the corresponding results forthe combustor C. Clearly, the extinction model and

the inflow/outflow boundary conditions employed inthe present study allows simulation of pulse combus-tion over a wide range of device scales.

There are some qualitative differences in the reac-tion rate and temperature field for these cases whencompared to combustor A. Fig. 13 and 14 shows re-spectively, the time sequence of the filtered reactionrate and temperature for the combustor C.

In the combustor A there exists high temperatureregion upstream of the stagnation plate while thecombustor C doesn't have such a region. Also highreaction rate and low temperature zone advances fardownstream of the plate in the combustor A, whereasthe reaction completes near the stagnation plate inthe combustor C. One reason of this may be thatalthough the inlet size was decreased the stagnationplate was not changed. This modification can signifi-cantly change the combustion dynamics since the tur-bulent mixing and flame holding can be changed due togeometrical modifications. In fact, it was reported ear-lier that even a slight change of the inlet geometry canhave strong effect on the performance.7 Nevertheless,the ability of the present simulation model to captureeven subtle changes in flow and geometrical parame-ters provides some confidence that this simulation toolmay be useful to study design of pulse combustors evenat the small scale.

5 ConclusionsSimulation of combustion process in a pulse combus-

tor is carried out to understand the dynamics of flame-holding, flow-chemistry interaction and the resultingpulsation. To simulate natural pulsation, a character-istic based inflow condition was modified to simulatethe behavior of the flapper valve. Large-eddy simu-lations in an axisymmetric configuration are carriedout for a range of combustor dimensions. The inflowboundary conditions automatically synchronizes withthe acoustic mode and results in steady-state pulsa-tion in this device. Results show that strain-inducedextinction upstream of the flameholder is critical inachieving oscillation at the expected frequency. Ascaling analysis is also carried out whereby the pulsecombustor dimensions are reduced from typical sys-tems of the order of 1 meter to as small as 15 cm todetermine if the dynamics can be reproduced even insuch small devices. Results suggest that the presentmodel is capable of simulating self-excited pulsationeven in very small devices. Although no data is avail-able for such small devices, the predicted frequenciesand mode shapes are as expected.

Additional studies are planned to further addressthe strain-induced extinction effect in more detail. Thepresent study employed a fixed location of the stag-nation plate relative to the inlet. Furthermore, thescaling of the size of the stagnation plate was notaddressed. By changing the location and the size of



combustor B

experimentno

extinctionmodel

extinctionmodel

extinctionmodel

extinctionmodel

frequency(Hz) 103 266 109 217 404

5.05 x 10~3 6.79 x 10~3 7.55 x 10~3 4.65 x HT4 7.93 x 10~6mass flowrate (kg/s)

Table 2 Frequencies and mass flow rates for various configuration.

the stagnation plate, the effective mixing process andhence, the mixing time could be modified which in turnwould change the local strain rate. Furthermore, sinceflameholding is a critical issue for self-sustained oscil-lation, the location and size of the stagnation plate canplay a major role. These issues will be addressed inthe future. Finally, the ability of the simulation modelto capture pulsation in very small devices suggests anopportunity to study even smaller pulse combustors ofthe order of 1 cm or smaller to determine if this type ofdevice could be used for efficient heat transfer at thesescales. Potential applications could be as propulsionor heat transfer devices in micro-air vehicles and othersimilar small-scale devices.

References1 Zinn, B. T., "Pulsating Combustion," Mechanical

Engineering, Vol. 107, No. 8, 1985, pp. 36-41.

2 Tsujimoto, Y. and Machii, N., "Numerical Anal-ysis of a Pulse Combustion Burner," Twenty-firstSymposium (International) on Combustion, 1986,pp. 539-546.

3 Barr, P. K., Dwyer, H. A., and Bramlette, T. T.,"A One-Dimensional Model of a Pulse Combustor,"Combustion Science and Technology, Vol. 58, 1988,pp. 315-336.

4 Barr, P. K., Keller, J. O., Bramlette, T. T., West-brook, C. K., and Dec, J. E., "Pulse Combus-tor Modeling Demonstration of the Importanceof Characteristic Times," Combustion and Flame,Vol. 82, 1990, pp. 252-269.

5 Barr, P. K. and Keller, J. O., "Premixed Combus-tion in a Periodic Flow Field. Part II: The Im-portance of Flame Extinction by Fluid DynamicStrain," Combustion and Flame, Vol. 99, 1994,pp. 43-52.

6 Benelli, G., De Michele, G., Cossalter, V., Da Lio,M., and Rossi, G., "Simulation of Large Non-Linear Thermo-Acoustic Vibrations in a PulsatingCombustion," Twenty-fourth Symposium (Interna-tional) on Combustion, 1992, pp. 1307-1313.

7 Moller, S. I. and Lindholm, A., "Theoretical andExperimental Investigation of the Operating Char-acteristics of a Helmholtz Type Pulse Combustordue to Changes in the Inlet Geometry," CombustionScience and Technology, Vol. 149, 1999, pp. 389-406.

8 Erlebacher, G., Hussaini, M. Y., Speziale, C. G.,and Zang, T. A., "Toward the Large-Eddy Simula-tion of Compressible Turbulent Flows," Journal ofFluid Mechanics, Vol. 238, 1992, pp. 155-185.

9 Menon, S., Yeung, P. K., and Kirn, W.-W., "Effectof Subgrid Models on the Computed Interscale En-ergy Transfer in Isotropic Turbulence," Computersand Fluids, Vol. 25, No. 2, 1996, pp. 165-180.

10 Kirn, W.-W., A New Dynamic Subgrid-Scale Modelfor Large-Eddy Simulation of Turbulent Flows,Ph.D. thesis, Georgia Institute of Technology, At-lanta, GA, September 1996.

11 Smith, T. M. and Menon, S., "One-DimensionalSimulations of Freely Propagating Turbulent Pre-mixed Flames," Combustion Science and Technol-ogy, Vol. 128, 1997, pp. 99-130.

12 Fureby, C. and Lofstrom, C., "Large Eddy Simu-lations of Bluff Body Stabilized Flames," Twenty-fifth Symposium (International) on Combustion,1994, pp. 1257-1264.

13 Ku, S. H., An Investigation of the Gas Fired Pul-sating Combustor, Ph.D. thesis, Georgia Instituteof Technology, Atlanta, GA, August 1988.

14 Putnam, A. A., Belles, F. E., and Kentfield, J.A. C., "Pulse Combustion," Progress in Energy andCombustion Science, Vol. 12, 1986, pp. 43-79.

15 Keller, J. O., Barr, P. K., and Gemmen, R. S.,"Premixed Combustion in a Periodic Flow Field.Part I: Experimental Investigation," Combustionand Flame, Vol. 99, 1994, pp. 29-42.

16 Kee, R. J., Miller, J. A., and Evans, G. H., "A Com-putational Model of the Structure and Extinctionof Strained, Opposed Flow, Premixed Methane-Air Flames," Twenty-second Symposium (Interna-tional) on Combustion, 1988, pp. 1479-1494.



combustionchamber0.15m

stagnation[

\

I iin0

J plate

0.013m

1

Fig. 1 Schematic of the pulse combustor.

0.10

0.05

0.00

-0.05

-0.1C0.55 0.56

time (sec)

Fig. 4 Time trace of the oscillating pressure: com-bustor A, no extinction model.

Fig. 2 Computational grid (focused on around theinlet). Combustion chamber: 190 x 80, Tail pipe:353 x 48

Fig. 3 Temperature profile for the tail pipe bound-ary condition.

0.05

0.00

180

0 Z

-180

0.0 0.5 1.0axial location (m)

1.5

Fig. 5 Pressure mode shape and phase: combustorA, no extinction model.



• 800

a) cycle time = 0.0

2000

300

a) cycle time = 0.0

b) cycle time = 0.16 b) cycle time = 0.16

c) cycle time = 0.33 c) cycle time = 0.33

d) cycle time = 0.5 d) cycle time = 0.5

e) cycle time = 0.66 e) cycle time = 0.66

f) cycle time = 0.83

Fig. 6 Time sequence of the oscillating filteredreaction rate contours (kg/m3s): combustor A, noextinction model.


Fig. 7 Time sequence of the oscillating filteredtemperature contours (K): combustor A, no extinc-tion model.



• 300

a) cycle time = 0.0

2000

I300

a) cycle time = 0.0






Fig. 8 Time sequence of the oscillating filteredreaction rate contours (kg/m3s): combustor A, withextinction model.


Fig. 9 Time sequence of the oscillating filteredtemperature contours (K): combustor A, with ex-tinction model.



0.10

0.39 0.41time (sec)

0.43

Fig. 10 Time trace of the oscillating pressure:combustor A, with extinction model.

0.02

0.000.0 0.1 0.2 0.3

axial location (m)0.4

90

0 ~

-90

0.5

a) combustor B

0.05

0.00

90

°I

-90

0.0 0.5 1.0axial location (m)

1.5

0.01

180

-180

100 0.05 0.10 0.15axial location (m)

b) combustor C

Fig. 12 Pressure mode shape and phase: combus-tor B and C, with extinction model.

Fig. 11 Pressure mode shape and phase: combus-tor A, with extinction model.



• 300

a) cycle time = 0.0

2000

300

a) cycle time = 0.0






Fig. 13 Time sequence of the oscillating filteredreaction rate contours (kg/m3s): combustor C, withextinction model.


Fig. 14 Time sequence of the oscillating filteredtemperature contours (K): combustor C, with ex-tinction model.


(c)2001 american institute of aeronautics & … of a small combustor attached to a long pipe....

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