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Thermal Barrier Effects on Nonlinear Pressure WavePropagation in Gases

NicholasDiZinno,GeorgeVradisand VolkanÖtügenPolytechnic University, Brooklyn, NY 11201

A numerical study of wave propagation through gases with non-uniform temperaturedistributions is presented. The concept of using regions of hot gas inside an ambientenvironment has potential in aeroacoustic applications, such as jet screech mitigation. Themain objective of this study is to determine the impact of temperature gradients on high-intensity, initially sinusoidal pressure waves. Particular emphasis is paid to wave reflection,transmission, and any influence a high temperature region may have on nonlinear behavior.Ultimately, the performance of thermal barriers in attenuating nonlinear waves is evaluated.This analysis considers the one-dimensional compressible unsteady Euler’s equations withan ideal gas state equation. The shape and extent of the high temperature zone is varied tostudy the effect of this region on wave propagation. Wave reflection and transmission isstudied for a range of wave and thermal field parameters. Results for non-linear pressurewaves are compared to linear acoustic waves.

Nomenclature

CR = coefficientof reflectionF = flux vector in Euler’s equationsL = Lengthof thehigh-temperaturezonep = gaspressurep’ = pressure amplitudein waveT = gastemperaturet = time coordinateU = solution vector in Euler’sequationsu = gasbulk velocityx = spatial coordinateAT = transmissioncoefficientγ = gasspecific heatratioλ = wavelength of initial sinusoidρ = gasdensityw = frequencyof initial sinusoid

Subscripts

0 = ambientconditions1 = upstreamof thepressure wave2 = within thepressurewavei = insidethehigh-temperaturezoneo = outside thehigh-temperature zone

45th AIAA Aerospace Sciences Meeting and Exhibit8 - 11 January 2007, Reno, Nevada

AIAA 2007-832

Copyright © 2007 by Nicholas DiZinno. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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I. Introduction

Noisegeneratedby aircraft androcketsinvolves high intensity acoustic disturbancesandnonlinear waves.In thesehigh amplitude waves,nonlinear propagation effectscould substantially influence the wave’s form aswellas its energy content. An understandingof the fundamentalphysics of nonlinear wave behavior is necessary todevelop effective noisecontrol strategies. It is thepurpose of this paper to quantitativelyassessthesignificanceofthesenonlinear effectsin certainnoisecontrol scenarios.

Thehigh-intensity wavesmentionedabove aregenerally undesiredphenomena. Therefore,it is warrantedto developa noisecontrol strategy. Commonnoisereduction techniquesgenerally fall into oneof two categories: (i)control of noise at the source; and (ii) absorptionof generatedsound.Since it is frequently found that jet noisereduction measureshavesignificant adverseeffects on performance, it is desirableto developnon-intrusive noisecontrol strategies. Several methodshavebeen utilized to mitigate noise,such asmodification of engine operatingparametersand tailoring of nozzle geometry,andthe useof mechanical devices,suchasreflectorsandmicro-jets1.However,thesetechniqueshavetheir limi ts. In thequestfor noisereduction, a new strategywassought.

In the early 1980’s, experiments indicated that shock waves passing througha zone of weakly ionizedplasmamight undergosubstantialmodification2. In theseexperiments,the shockwave was observedto becomeweakeranddiffuse.This andotherrecentactivity hasinspirednew lines of researchthat explore the possibility ofusing plasmain applications such as air vehicle drag reduction3 and flow control4. At Polytechnic University,research hasfocusedon the potential useof plasma in aeroacoustic applications,particularly the containment ofaircraft noise.

Whenglow dischargeplasmais generatedin a gas,signif icant gastemperaturegradients formonbothsidesof the plasma. A pressurewave propagating through this region encounters a sharply changingspeedof sound(indexof refraction) along thepropagationdirection. A portionof thewave is reflectedin thesetemperaturegradientzones while the forward moving wavepassingthrough the plasma is attenuated.This is the so-called “boundary”mechanism of wave modifi cation. While some effort has been made to explain the previously mentionedexperimentalobservations usinginherent plasma mechanisms5, it is generally believedthat thesethermal gradients,generated in the gas by the plasma(i.e. the boundary mechanism),are the primary cause for the observedwavemodifications.

These observedwave modifications suggested that glow discharge plasma might be applied to themanagementof jet noise.Theuseof plasmaasa meansof jet noisecontrol is attractive for severalreasons. First, byremotely introducing energy into the gasmedium, it canchangelocal gas parameters. Second,the plasmacanbegeneratedby electrical or optical meansin a target region of gas, therebyfulfilling the non-intrusive requirementmentionedabove. Finally, the presenceof chargedparticles in the plasmacanallow for the efficient depositionofexternally acquired energyinto a highly localized region of the gas. Therefore, it appearsthat plasmamay be aneffective method for thecontrolof jet noise.

Sincethe effect of the boundary mechanismis due to the presenceof thermalgradients generatedin thegas,the plasma canbe modeled asa regionof high-temperature gas separated by regions of ambient-temperaturegas. This approximation ignores any constituent plasmadynamics. The boundarymodel is validatedthroughthework of Tarau6,7 and Stepaniuket al8-11. Thus,theproblem of pressurewavepropagation througha glow dischargecan bereduced to thatof determining theimpact of temperaturegradient regionson thewaves’propagation.

Stepaniuk et al8-11 reportedon experimentsconducted in an anechoic chamber where the attenuationofsingle-tonesoundpropagating througha sheet of plasma wasmeasured.In thesesystematic experiments, a rangeofair pressure,plasmaparametersand sound frequencieswascovered.Normal incidencewastheonly relative positionbetweenthesoundbeamandtheplasma sheetconsidered.Tarau6,7 contributednumerical calculationsusingtheone-dimensional Euler equationsfor thepropagation of planar sound wavesthrough regionsof hot gas representingtheplasma. The shapeandextentof the high-temperature region wasvaried in orderto investigatethe thermaleffectsbroughton by the plasmaheating of thegas. Both the experimental and thecomputational resultsshow that thereisconsiderableattenuation of the soundpressureamplitudeby the plasma.In addition to the normal incidencecase,experimentswerecarried out for a 45 degrees angle of incidence wherea single tone(12.5 kHz) soundwave was

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sent through the thermal field generated by glow discharge plasma.While at normal incidencethe measuredattenuationwas 7-9 dB, at oblique incidence (45̊ ), the highestattenuation was about23 dB. In this case, soundreflection from the glow discharge region wasalsoobserved. Basedon theseresults, Tarau6,7 and Stepaniuk et al8-11

suggest that the dominantmechanism responsiblefor the sound wave attenuationis the change in the index ofrefraction dueto gastemperature gradient at the plasmaboundaries,and that the high-temperaturezonemodelis areasonableone.

The above-mentioned studies confined the scopeof their researchto acoustic (linear)waves. This wasappropriateto begin assessing theplausibility of using glow dischargeplasmaasa jet noisemanagement tool. In thepresentwork, we extendthis analysis to the consideration of nonlinear waves. This is directedtowardassessing thevalueof using glow dischargeplasmain the containmentof high-intensity rocketnoise,jet noiseandin contributingtoward the understanding of the screech phenomena. In the next section,the formulation of the problem and thesolution technique are described. In Section 3, numerical results are presented, followed by conclusionsof thisstudy’s results in Section4 . Finally, suggestedtopicsof future studiesareidentified in Section5.

II. Problem Formulation and Numerical Implementation

The propagation of a pressure wave through a non-uniform media is studied by directly solving the unsteady one-dimensionalmass, momentum andenergy conservationequations along with the equationof statefor a perfect gasusinga second-orderaccurate numerical scheme.The mass,momentum andenergyequations in differential formareas follows:

0=∂∂+

∂∂

x

F

t

U (1)

where U andF arethesolutionvector andflux vector, respectively. Theyaredefinedas

=

e

uU ρρ

++=

upe

pu

u

F

)(

2ρρ

(2)

where ρ is the gasdensity, u is the fluid velocity, e is the internal energy per unit massof fluid, andp is the gaspressure. Theaboveequationsrepresentconservation of mass,momentum, andenergy. To close thesystem,we usethe idealgasequationof state,

TRp ρ= (3)

The effectsof viscosity andthermal conductivity areneglectedin this study.

Thecalculation domainis a uniform gaswith uniform propertiesseparatedby a high temperaturezone.Thepressurewave is introduced as part of the initial conditions; a certain distance upstreamof the high-temperaturezone, a sinusoidal variation in pressureis introduced.It is definedasfollows:

( )wtkxppp +−+= sin'12 (4)

The corresponding variation in bulk (mass)velocity and density are thendevelopedfrom characteristics theory,

−

−=

−

11

2 2

1

γγ

γγ

pu1

2

2

11

−

−+=

γγρ u (5)

4

With thesedefinitions, the wave was superimposedon the uniform gas. It is important to note that the initialconditionsthusdefined are associatedwith a pure tone,or single-frequency wave.

Theabovedefinedequationsand associatedinitial conditionsweresolved using themethod of Space-Time- Conservation ElementandSolution Element (CE/SE)12-14. Thisschemeis second-order accuratein spaceandtime,featuresa usercontrolled diffusion parameter to control oscillations typically encounteredin problemsas the oneaddressedin this work, and is well suitedto aeroacoustic applications. We will summarize the major tenets of theschemein a qualitativesensehere. For mathematical details, thereaderis referred to theliterature.

The CE/SE schemeis based on the conceptof flux conservation. In a one-dimensional problem, theconservation laws are re-formulated into a staggeredtwo-dimensional space-time mesh. Within this coordinatesystem,both localand global flux conservation is enforced.Theflux evaluation at an interfacebetweenneighboringcomputational cells is generated asa part of the solution procedure,and requires no interpolationor extrapolation.The schemeuseslocal variablessuchthat the setof variablesin any oneof the numericalequationsto besolved isassociatedwith a set of immediately neighboring cells. Finally, mesh valuesof dependent variables and theirderivativesareconsideredasindependentvariables,and aresolvedsimultaneously. Stability is controlled usingthefamiliar CFL criteria.

Sincethis is a one-dimensional analysis,it is convenient to prescribea domainlarge enoughto preventanydisturbance from reaching the boundaries. However, for numerical reasons, we will prescribe non-reflectingboundaryconditions to prevent any numericalnoisefrom being reflectedback into the domainandcontaminatingthe solution. Thus, the boundary conditionsareeffectively kept constantat the valuessetby the initial conditions.However,thecode will permitoutgoingfluctuations to exit the domain.

III. Numerical Results

A computational codewas written to implementthenumericalschemeoutlined above.In orderto validatethe code,a numberof fundamental problems were solved. Results are shown herefor the case of a single-cyclesinusoidalwave propagating througha uniform background environment. Caseswererun at varioussoundpressurelevels (SPL), which are determined accordingto thefollowingdefinition:

=

refp

pSPL

'

10log20 (6)

where p’ is the actual pressure amplitudeof the wave and pref = 2 x 10-5 Pa is the reference pressureamplituderepresenting the faintestdetectablesoundat 1,000Hz. Figure 1 below showsthe evolution of this wave,which asexpectedretaineditsoriginal shape,speed andwavelength.

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10x / λ0

( p-

p0)

/(p

m-

p0)

1000DT

3000DT

5000DT

7000DT

``

Figure 1: Evolution of a linear sinusoidal wave of dB = 133

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Next, thepropagation of non-linear (high dB) wavesin a uniform temperaturemedium wasstudied. Figure2 shows theevolution of a sinusoidalwaveof dB = 180. As the wavepropagatesthroughthe uniform backgroundmedium,it evolvesinto a sawtooth form. Thedistanceover which thesinusoidal waveevolves into a saw tooth onedependson thedB level, this distancedecreasingwith increasingdB levels.

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10x / λ0

( p- p

0)/(

pm

- p0)

1000DT

3000DT

5000DT

7000DT

Figure 2: Evolution of a nonlinear wave of dB = 180

Next, the domain was divided into three different zones. Zones1 and 3 at the two ends had a uniformtemperatureT0, while zone2 (in-betweenzones1 and2) had a uniform temperatureT2 higherthan that of zones1and3. Thetemperatureratio between thesezoneswas variedbetween2.0 and 4.0, andtransition from onezonetothe next takes placein a stepfunction fashion. The domaindescribedin this mannerwill be referredto asthe deltabarrier. Figures 3 and4 show thepropagation of a nonlinear (dB = 180) wave througha deltabarrier of T2/T0 = 2.0.The length of the high temperature zoneis two λ0.

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5 6 7 8 9 10 11 12

x / λ0

p/ ρ

0a 0

2

0.9

1.1

1.3

1.5

1.7

1.9

2.1

T/T

0

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5 6 7 8 9 10 11 12

x / λ0

p/ ρ

0a 0

2

0.9

1.1

1.3

1.5

1.7

1.9

2.1

T/T

0

Figure 3: Propagation of a 180 dB wave through adelta barrier, T2/T0 = 2.0, L/λ0 = 2.

Figure 4: Propagation of a 180 dB wave through adelta barrier, T2/T0 = 2.0, L/λ0 = 2.

Theinitial sinusoidal wavepropagates in the uniform temperaturezone(zone1) andquickly evolvesinto asaw-tooth shapedwave. This wave crossesthe barrier between zones1 and 2 with some lossof energy,whichmanifestsitself in the form of a reflected wave. The samepattern is realizedat theright boundary of zone2, asthewave leavesthe high temperature zone2 andentersthe lower temperature zone3. Eventually a saw tooth shapedwave propagatesthroughzone3 from left to right, while a morecomplex pattern of reflectedwaves emerges in zone1 moving from right to left. The wave reflected from the cold-to-hot interface encounters a decreasedcompressibili ty of the gas(resulting from the increasedacoustic speed)and experiencesa π phaseshift. The wavereflected from the hot-to-cold interface, encountering an increasedcompressibilit y of the gas, experiencesno phaseshift. A small residual waveis trappedin thehigh temperature zone.

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To evaluatethe amount of transmissionor reflection through (or from) the high-temperaturebarrier, it isnecessary to measure the energy of the wave. For linear sinusoidalwaves, it is appropriateto use the squareofpressureasa measureof waveintensity(energy flux). However,for nonlinearwaves this doesnot give anaccuratepictureof the energyof the wave15. To measurethe intensity(energy flux) associated with a nonlinear wave, wemustusethe generaldefinition

( ) ( )∫ ∆⋅∆=T

dttutpT

I0

1(7)

where∆p is theacoustic pressure (overpressure),∆u is the massvelocity of the wave, and the integration is carriedout over the time associated with one period T. For nonlinearwaves, we define an intensity-basedtransmissioncoefficientas

inc

trans

I

I=TA (8)

where Itrans andIinc are theintensitiesof thetransmitted andreflectedwaves,computedusingequation(7) above.

Similarly, the reflectioncoeffic ient is definedas

inc

ref

I

I=RC (9)

where Iref is theintensity of thereflected waveandIinc is theintensityof theoriginal wave.

The effect of the lengthof the high temperature zone is shownin Figures5 and 6. In thesefigures, thetransmission andreflectioncoefficientsare shownagainstthelength of thethermal zone,L, for 180dB andT/T0 = 2.In these,andall thefigures that follow, interrogation points areidentified in each zoneof thedomain. Thevaluesofpressureandvelocity are recordedasthe wave passesthesefixed points. The intensity of the incident, transmittedand reflected wavesare found by numerically integrating equation (7). The transmission coefficient (AT) andreflection coefficient(CR) arethenfound from equations(8) and (9), respectively.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5 6 7 8 9

L / λ0

AT

0

0.01

0.02

0.03

0.04

0.05

0.06

0 1 2 3 4 5 6 7 8 9

L / λ0

CR

Figure 5: Transmission coefficient vs. thermal zonelength for a 180 dB wave traveling through a delta

barrier, T/T0 = 2

Figure 6: Reflection coefficient vs. thermal zone lengthfor a 180 dB wave traveling through a delta barrier,

T/T0 = 2.

What is important to notefrom thesegraphsis the relative trend for transmission and reflectioncoefficients.For 180dB and T/T0 = 2, L/λ0 was variedbetween0 and9. In this case,thetransmissioncoefficientexhibitsseveralmild peaksand valleys. For values of L/λ0 up to approximately the length of the wave train, the reflectioncoefficient also exhibits several peaks and valleys. However, beyondthis critical length, there is no effect ofthermal zone length on reflection. This behavior is consistent with the manner in which the incident wave is

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modifiedby the thermal zone. Theoriginal waveexperiences a reflection from both the upstreamanddownstreamboundariesof the thermal zone. For small thermalzonelengths,the two reflectedwavesinterferewith eachother,resulting in a single wave of a unique form. This happensbecausethe reflected wave from the downstreamboundarytravelsthroughthe thermal zone,which hasa higher acoustic speed. Thus, for part of its’ propagation,this wavetravels fasterthanthewavereflected from theupstream boundary,which travels entirely in zone1 with alower acoustic speed. Ultimately, as the lengthof the high temperature zonebecomeslong enough, even with thisincreasedspeed, the wave reflected from the downstream boundary cannot catch the wave reflected from theupstreamboundary. At this point, two distinct reflected waves are observed. Below this critical length, thereflection coefficientvaries substantially reflectingthe interferenceof these two waves. Beyondthis critical length,we measure thefirst reflected waveonly, and seeno further changein reflection coefficient.

Next, theeffectof thenumber of cyclesin thewave train on transmissionand reflection is studied. ShowninFigures 7 and 8 are the transmissionand reflection coeff icients versusthermalzonelength for a two-cycle wavetrain andalso for a three-cyclewavetrain, bothof 180 dB and T/T0 = 2. The transmissioncoefficientsfor thetwo-and three-cycle wave trains are very similar. The reflection coefficientsare consistentwith the behaviorseeninFigure 6. A wavetrain with fewer cycles requires lesslengthof thermal zoneto reachthe critical length to avoidinterference. This is clearly seen asthe point wherethe graphbecomesconstant shifts to the right with increasingcyclesin Figure 8.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5 3 3.5 4

L / λ0

AT 2 Cycles

3 Cycles

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5 2 2.5 3 3.5 4

L / λ0

CR 2 Cycles

3 Cycles

Figure 7: Transmission coefficient vs. length ofthermal zone for a 180 dB wave traveling through a

delta barrier, 2 Cycles and 3 Cycles, T/T0 = 2

Figure 8: Reflection coefficient vs. length of thermalzone for a 180 dB wave traveling through a delta

barrier, 2 Cycles and 3 Cycles, T/T0 = 2

Figures 9 and 10 show the variation of transmissionandreflection coefficients for a two-cycle wavetrainof variousdB levels propagatingthrougha delta barrier of L/λ0 = 2 andT/T0 = 2.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

130 135 140 145 150 155 160 165 170 175 180

dB

AT

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

130 135 140 145 150 155 160 165 170 175 180

dB

CR

Figure 9: Transmission coefficient vs. dB for a deltabarrier, T/T0 = 2, L/λ0 = 2.

Figure 10: Reflection coefficient vs. dB for a deltabarrier, T/T0 = 2, L/λ0 = 2.

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As seen,as the strength of the wave increasesbeyond160 dB and the disturbancebecomesprogressivelynonlinear, the transmissioncoefficient (defined according to Equation 8) and the reflection coefficient (definedaccording to equation 9) both decrease. It is suspected that the formation of side-band frequency components innonlinearwavesis responsiblefor thisbehavior.

As alreadyseenabove, a nonlinearwavechangesits shapeasit propagates. Figures 11 and12 show theinfluenceof the locationof thebeginningof the thermalzone on transmissionandreflection. Here,the locationofthe initial wave is kept constant,but the location of theupstreamedgeof thehigh temperature zone(STF) is varied.Thereappearsto bealmost no influenceof thethermal zone location on transmission coeff icient. However,there isa significant influenceof thermal zonelocation on reflection coefficient. Theseeffects aredueto the fixed locationof the interrogation pointsweretheintensityof thetransmittedandreflected waveswere measured. Thetransmittedwave was measured near the downstreamend of the domain. As the location of the thermal barrier was shifteddownstream, the transmitted wave had lessdistance to travel to reach the interrogation point. Thus, this waveexperiencedlessdiffusion. Alternatively, the reflectedwave had moredistance to travel asthe thermal barrier wasshifted,causing this waveto experience morediffusion,and hence, a decreasein intensityandreflectioncoeffi cient.The shadedregion in Figure 10 shows the rangeof valuesthe reflection coefficient can take depending on thelocation of thethermal zone.

0

0.2

0.4

0.6

0.8

1

1.2

3 4 5 6 7 8

STF

AT

133 dB

170 dB

180 dB

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

3 4 5 6 7 8 9

STF

CR

133dB

170dB

180dB

Figure 11: Transmission coefficient vs. location ofthe delta barrier for a 2 cycle wave, L/λ0 = 3,

T/T0 = 2

Figure 12: Reflection coefficient vs. location of thedelta barrier for a 2 cycle wave, L/λ0 = 3,

T/T0 = 2

Al l of the data previously presented for the delta barrier have beenfor a temperature ratio betweenthezones of T/T0=2. We would now like to investigate the influence that this temperatureratio may have on thetransmission andreflection coefficients. Figures 13 and 14 showthe transmissionand reflectioncoeffi cientsversusL/λ0 for multiple valuesof T/T0. The wavetrain is three-cycles andthedB level is 180 dB. The increaseof thermalzone temperature had a negligible effect on transmission coeffi cient, while resulting in an increased reflectioncoefficient.

9

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5 3 3.5 4

L / λ0

AT

T/T0 = 2.00

T/T0 = 2.66

T/T0 = 3.33

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.5 1 1.5 2 2.5 3 3.5 4

L / λ0

CR

T/T0 = 2.00

T/T0 = 2.66

T/T0 = 3.33

Figure 13: Transmission coefficient for 3 cycle wave(180 dB) vs. L/λ0 for multiple T/T0

Figure 14: Reflection coefficient for 3 cycle wave(180 dB) vs. L/λ0 for multiple T/T0

IV. Conclusions

A study of high-intensity nonlinear wavesthat are initially sinusoidal as they propagate throughregions of non-uniform temperaturedistribution has been initiated. TheSpace-Time-Conservation-Element and Solution Element(CE/SE)12-13 numerical schemewas employed. This is a fully second order accurate scheme(in time and space),which features a user controlled diffusionparameter to controloscillations typically encounteredin problemsastheoneaddressed in this work. Thecodewas validated by solving basic wave propagationproblems aswell asearlierwork thatfocusedon thepropagation of linearwavesthrough media with variabletemperature distributions. Resultsfor thecaseof non-linear wavesare presented, which show that as thenon-linearity (i.e. thedB level) increases,thetransmission coefficient decreases; these decrease growing rapidly for dB levels greater than about 170.Additionally, thereflection coefficient alsodecreaseswith increasing dB level. Theeffect of high temperaturezonewidth (relative to thedisturbancewavelength) is similar to thatin thecaseof linearwaves,i.e. the effect is limitedtowidths approximately that of the wave train, and is not very strong. The location of the thermal zonehasa veryweak influenceon transmission for all dB levelsconsidered. However,the locationof the thermal zone influencedthe reflection coefficient, with an increasingeffect observed for increasing dB levels. Overall, it is concludedthatwhile further researchneeds to be undertaken on this topic, it appears that energydeposition technologies suchasglow discharge plasma hold great promise in aeroacoustic applications, particularly for the attenuation andcontainment of high-intensitysound.

V. Future Studies

This studyconfined its scope to analyzing the wave in the time domain. Further studies which analyzewave transmission andreflection in the frequency domain arerequired to furtherunderstandthe physics of certaintrends observedhere. Furtherparametric studies should be performed, both in the time and frequencydomains,which explore the influenceof the shapeof the gradient region and the temperature ratio on any transmission andreflection.

As wasindicatedin Section 1, this analysisonly consideredthe boundaryeffect of wavemodification byglow discharge plasma. While experimentally it is very diffi cult to separate the boundaryeffect from the internaleffect, this would be quite easy to do numerically. It is suggested that a study be performed where the wavepropagatesentirely within a plasmaenvironment. This would effectively cancelout any boundaryeffects andtherelative merit of theinternaleffect couldbeassessed.

VI. References

[1] Maglieri D.J.and K.J. Plotkin, Aeroacoustics of Flight Vehicles, editedby H.H. Hubbard,NASA RP1258Vol. 1, 1991, pp. 519-561.

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[2] Klimov, A.I., Koblov, A.N., Mishin, G.I., Serov, Yu. L.,and Yavor, I.P.,“Shock Wave Propagation in aGlow Discharge”, SovietPhysicsTechnical Physics,Vol. 8, No. 4, 1982.

[3] Kremeyer,K., Sebastian,K., and Shu,C., “Lines of Pulsed Energy: Shock Mitigation and Drag Reduction”,AIAA PaperNo. 04-1131,January2004.

[4] Corke, T.C.,Jumper, E.J.,Port,M.L., and Orlov, D., “Application of Weakly Ionized Plasmas as Wing FlowControl Devices”, AIAA PaperNo. 02-0350,January 2002.

[5] Soukhomlinov, V.S.,Kolosov, V.Y., Sheverev, V.A. and Otugen,M.V. “Acoustic dispersion in glowdischarge plasma: A phenomenological analysis”, Physicsof Fluids., Vol. 14,No. 1, pp 427-429.

[6] Tarau,C., “Wave Propagation Through Regions of Non-Uniform Temperature Distribution,” Ph.D.Dissertation, Mechanical andAerospace EngineeringDept.,Polytechnic Univ., Brooklyn,NY, 2004.

[7] Tarau,C. andÖtügen,V., “Propagation of Acoustic Waves Through Regions of Non-Uniform Temperature,”InternationalJournal of Aeroacoustics,Vol. 1, No. 2, 2002, pp.165-181.

[8] Stepaniuk, V., Tarau,C., Ötügen,M.V., Sheverev, V., Soukhomlinov, V., and Raman,G., “Acoustic WaveAttenuation through Glow Discharge Plasma”, AIAA \CEASAeroacousticConferencePaperNo. 2002-2433, 17-19 June2002.

[9] Stepaniuk, V., Sheverev,V., Ötügen,V., Tarau,C., Raman, G. andSoukhomlinov, V., “Sound Attenuationby Glow Discharge Plasma”, AIAA PaperNo. 2003-0371,6-9 January, 2003.

[10] Stepaniuk, V., Sheverev,V., Ötügen,V., Soukhomlinov,V. and Raman,G. “Sound Attenuation by GlowDischarge Plasma”, APSMeeting, 23-25 November,2003.

[11] Stepaniuk, V., Sheverev,V., Ötügen,V., Tarau,C., Raman, G. andSoukhomlinov, V., “Sound Attenuationby Glow Discharge Plasma”, AIAA J.Vol. 42, pages454-550,2004.

[12] Chang,S.C.,“The Method of Space-Time Conservation Element and Solution Element – A New Approachfor Solving the Navier-Stokes and Euler Equations,” J.Computational Physics,Vol. 119,pp. 295-324,1995.

[13] Chang,S.C.,Yu, S.T., Himansu,A., Wang,X.Y., Chow,C.Y., and Loh, C.Y., “The Method of Space-TimeConservation Element and Solution Element – A New Paradigm for Numerical Solution of ConservationLaws,” Computational Fluid DynamicsReview,1996.

[14] Loh, C.Y., Hultgren, L.S. and Chang,S.C.,“Wave Computation in Compressible Flow Using Space-TimeConservation Element and Solution Element Method,” AIAA Journal, Vol. 39,No. 5, May 2001.

[15] DiZinno, N., “Thermal Barrier Effects on the Propagation of Nonlinear Pressure Waves in a Gas,” MastersThesis, Mechanical andAerospaceEngineeringDept., Polytechnic Univ., Brooklyn, NY, 2006.