Standing Waves in a Rectangular Resonator Containing Acoustically Active Gases

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1 ARCHIVES OF ACOUSTICS Vol.41No.1 pp ) Copyright c 2016byPAN IPPT DOI: /aoa Standing Waves in a Rectangular Resonator Containing Acoustically Active Gases Anna PERELOMOVA Faculty of Applied Physics and Mathematics Gdansk University of Technology Narutowicza 11/ Gdańsk Poland; anpe@mif.pg.gda.pl received May ; accepted October ) The distribution of perturbations of pressure and velocity in a rectangular resonator is considered. A resonator contains a gas where thermodynamic processes take place such as exothermic chemical reaction or excitation of vibrational degrees of a molecule s freedom. These processes make the gas acoustically active under some conditions. We conclude that the incident and reflected compounds of asoundbeamdonotinteractintheleadingorderinthecaseoftheperiodicsoundwithzeromeanpressure including waveforms with discontinuities. The acoustic field before and after forming of discontinuities is described. The acoustic heating or cooling in a resonator is discussed. Keywords: standing waves; acoustically active gas; resonator; nonlinear propagation of sound. PACS no Fj. 1. Introduction The boundaries of a vessel over which sound spreads have a significant impact on the distribution of acoustic perturbations in the volume of the vessel. In contrast to a free space the presence of boundaries determines the discrete spectrum of wave numbers of perturbations and may result in the standing waves in the volume of the resonator. The physically valid conditions on the boundaries of the resonator determine the magnitude and phase of the reflected waves. In general intersecting nonlinear waves do interact. Many studies are devoted to wave fields in resonators ofdifferentshapesfirstofallthosefilledwiththe Newtonian fluidschester 1964; Mortell et al. 2009; Biwa Yazaki 2010; Keller 1977). Kaner Rudenko Khokhlov1977) introduced an analytical method which applies different scales in description of the temporal evolution of standing waves in one-dimensional resonators. This method was successfully applied in studies of quasi-planar weakly nonlinear and weakly damping sound in non-dispersive flows Rudenko Soluyan 2005). It imposes a slow dependenceoftheshapeofprogressivewavesandtheirfast dependence on the retarded time. In connection to this waveforms before and after forming of discontinuities may be considered individually. In this way it becomes possible to simplify wave equations in many cases. As for the perturbations in a one-dimensional resonator the method makes it possible to subdivide the wave field into a sum of non-interacting planar waves which travel in opposite directions and to consider them individually. The total field represents a superposition of these planar waves. This is conditioned by their periodicity and zero mean pressureochmann 1985). Shock waves in the resonators filled with Newtonian fluids have been considered in Biwa Yazaki2010) and Keller1977). In the last decades the interest to the non- Newtonian fluids and media with irreversible thermodynamic processes constantly growschu 1970; Parker 1972; Makaryan Molevich 2007). The non-equilibrium molecular physics began to develop quickly due to the laser revolution in physics and chemistryosipov Uvarov 1992; Clarke McChesney 1976). Since that the attention is concentrated on media which may reveal anomalous dispersion and attenuationbauer Bass 1973; Molevich 2001). Under some conditions the bulk viscosity of a fluid takes negative values and sound increases its magnitudeinthecourseofpropagation.asfarastheauthor knows a weakly nonlinear propagation of sound inaresonatorwhichisfilledwitharelaxinggaswhere relaxation may be irreversible is a new subject of stud-

2 68 Archives of Acoustics Volume 41 Number ies.aswellasinthecaseofresonatorscontaining a Newtonian fluid the planar waves both incident and reflecteddonotinteractintheleadingorderinthe volumeofaresonatoriftheyareperiodicandzero on averagesec. 3). The standing acoustic waves in a rectangular resonator are considered before and after formation of discontinuities for two examples of thermodynamic relaxation in gases. The generic parameter which describes thermodynamic processes in a gas B is positive in an acoustically active gas and negative otherwise. The Newtonian attenuation that is mechanical viscosity of a gas and its thermal conduction will be ignored. Shock waves with discontinuities in thewaveprofilealwaysappearinaplanarwavewhich propagates in an unbounded volume of an acoustically activemediumaswellasinaresonator.thesound s magnitude increases in the course of sound propagation in the unbounded medium but the nonlinear attenuationonthefrontofthesaw-toothwaveleadsto stabilization of the peak magnitude of the shock wave which tends to some positive value depending on B as the time increasesperelomova 2012). The total fieldinaresonatorrepresentsasumofincidentand reflected waves which at first stage do not contain discontinuities. However they necessarily appear as the time increases; the time of formation is conditioned by B the Mach number and parameter of nonlinearityofagas.intheequilibriumregimewithnormal dispersion and attenuation the magnitude of acoustic field in a resonator gets smaller and discontinuity may not form at allperelomova 2012). The non-linear generation of non-acoustic modes such as the entropy and vorticity modes is anomalous in the acoustically active gases. In a resonator these effects accumulate over time. In particular cooling of a medium takes place instead of heatingsec. 4). 2. Examples: gases with excited vibrational degrees of molecule s freedom and gases with chemical reactions Thefirstexamplerelatestoagaswhosesteady state is maintained by pumping energy into the vibrational degrees of molecule s freedom Osipov Uvarov 1992; Clarke McChesney 1976). The quantity B = γ 1)2 T 0 2c 3 Cv τ R + ε ε eq τ 2 R ) dτ dt 1) is positive in the acoustically active gas and negative otherwise. It is evaluated at unperturbed pressureandtemperatureofagas p 0 T 0 where γisthe specificheatsratioforanidealgas cistheequilibriumspeedofthesoundofaninfinitelysmallmagnitude C v =dε eq /dtistheequilibriumspecificheat ataconstantvolumeε eq istheequilibriumvalueof thevibrationalenergy ε)and τ R denotesthevibrational relaxation time. The relaxation time in the most importantcasesmaybethoughtofasafunctionexclusively of the temperatureosipov Uvarov 1992; Zeldovich Raizer 1966). The non-equilibrium excitation is possible in principle due to the negative dτ R /dt. B ineq.1)isthedecrementorincrement if positive) of acoustic magnitudes in the highfrequencyoscillationswhen ωτ R 1where ωdenotes the characteristic frequency of the sound. In the low-frequency regime the attenuationor enhancement) of the sound is insignificant Perelomova 2010). Inthecaseofgasesinwhichexothermicchemical reaction occursbauer Bass 1973; Molevich 2001) B = Q 0γ 1)Q ρ +γ 1)Q T ) 2c 2 m 2) isthequantityevaluatedatunperturbed p 0 T 0 Y 0 where Y denotesthemassfractionofareagent A in A B exothermicreactions;qistheheatproduced inamediumperonemoleculeduetoachemicalreaction Q 0 = QT 0 ρ 0 Y 0 )and misthemolecule smass. Thedimensionlessquantities Q T Q ρ aredetermined by means of partial derivatives of the heat produced duetoachemicalreactionwithrespecttothetemperature and density of the mixture respectively: Q T = T 0 Q 0 Q ρ = ρ 0 Q 0 ) Q T T 0ρ 0Y 0 ) Q. ρ T 0ρ 0Y 0 The characteristic time of chemical reaction is 3) τ c = HmY 0 Q 0 Q Y 4) where H isthereactionenthalpyperunitmassof ) reagenta Q Y = Y0 Q Q 0 Y. BgivenbyEq.2) T 0ρ 0Y 0 is the decrementor increment if positive) of acoustic magnitudesif ωτ c 1.Inthelow-frequencyregime aswellasinthecaseofvibrationallyexcitedgasesthe attenuationor enhancement) of the sound is insignificant.atlowfrequenciesagashassufficienttimeto relaxoverthesoundperiodandbehaveslikeanewtonian fluid. This applies to both considered cases of relaxation. 3. Wave perturbations in a waveguide An equation which describes dynamics of the potentialofvelocityϕ: ϕ =v)ingaseswithtype

3 A. Perelomova Standing Waves in a Rectangular Resonator Containing Acoustically Active Gases 69 of relaxation described in the previous section in the leading order is 2 ϕ t 2 c2 ϕ 2cB ϕ t = 2 ϕ ϕ ) t γ 1) ϕ ϕ t. 5) This wave equation includes the term responsible for dispersionit is proportional to B) and differs from the well-known equation describing lossless flows. Even in the case of a lossless perfect gas no analytical solutions are available for the unsteady flow apart from those for planar waveshamilton Morfey 1998). We will consider the velocity potential in a resonator consisting of four parts specifying periodic planar waves: ϕ 1 τ 1 ϑ = µt) ϕ 3 τ 3 ϑ) ϕ 2 τ 2 ϑ) ϕ 4 τ 4 ϑ) 6) where µ is a generic small parameter that characterizes slow variations in the waveforms due to nonlinearityandrelaxation k x k y arecomponentsofthewave vectorand ω = c kx 2 +k2 y and τ 1 = t k x ω x+ k y ω y τ 2 = t k x ω x k y ω y τ 3 = t+ k x ω x+ k y ω y τ 4 = t+ k x ω x k y ω y. Our primary objective is to derive model equations validatorder µ 2 forsoundperturbationsinaresonator.equation5)with accountfor6) maybe rewritten in the leading order as follows: 2 ϕ i ) 2µ τ i ϑ 2cB ϕ i = γ 1 2 ϕ i τ i c 2 τ 2 i ) ϕ i ϕ i 2 ϕ i 2 τi 2 + ij=1i j α ij k 2 x +β ijk 2 y ω 2 ϕ i 2 ϕ j 7) τ 2 j where α 12 = β 12 = 1 α 13 = β 13 = 1 α 14 = β 14 = 1 α 23 = β 23 = 1 α 24 = β 24 = 1 α 34 = β 34 = 1.Averagingalltermsoverperiods in τ 2 τ 3 and τ 4 thefirstequationinthesetwhich follows)andsoonandreturningtothevariable twe readily subdivide Eq.7) into four equations governing ϕ 1 τ 1 t) ϕ 2 τ 2 t) ϕ 3 τ 3 t)and ϕ 4 τ 4 t): 2 ϕ i t cb ϕ i = γ +1 ϕ i 2 ϕ i 2c 2 i = 1...4). τ 2 i 8) Equations8) are readily integrated: ϕ i t cbϕ i = γ +1 ) 2 ϕi 4c 2 i = 1...4). 9) Equations8)9)arevalidif ϕ i areperiodicfunctionsof τ i.inthiscasethefourwaveconstituents do not interact in the volume of the resonator in theleadingorder.since u x = 4 u xi = 4 k x ω ϕ1 τ 1 ϕ2 τ 2 + ϕ3 τ 3 + ϕ4 ϕ i y = ky ω τ 4 ) ϕ i x = and u y = 4 u yi = ϕ1 τ 1 ϕ2 τ 2 + ϕ3 τ 3 ϕ4 τ 4 )theaveraged over periods components of velocity are equal to zero foreachofthefourcomponentsofthefield.thecomponents of the individual velocities are described by the equations which follow from Eqs.8): u xi t u yi t cbu xi = χ i γ +1 2c 2 ω k x u xi u xi cbu yi = δ i γ +1 2c 2 ω k y u yi u yi 10) χ 1 = χ 2 = 1 = χ 3 = χ 4 = 1 δ 1 = δ 3 = δ 2 = δ 4 = 1.Theequationswhichdescribeacousticpressures belonging to each wave are p i t cbp i = γ +1 2c 2 ρ 0 p i p i i = 1...4) 11) sincethetotalacousticpressure p ρ 0 ϕi ). Equations10)11) relate to the waveforms without discontinuities and to the waveforms with discontinuities. The mean values of all components of velocity perturbations and acoustic pressure are zero in symmetric acoustic pulses. In the new dimensionless variables τ = ωt η i = ωτ i i = Y = ωy c X = ωx c K x = ck x ω K y = ck y ω b = cb ω U = uexp bτ) P = p exp bτ) Mc Mc 2 ρ 0 θ = expbτ) 1 12) where MistheMachnumberEqs.10)11)takethe formsi = 1...4) U xi θ U yi θ χ ig K x U xi U xi η i = 0 δ ig K y U yi U yi η i = 0 P i θ GP P i i = 0 η i 13)

4 70 Archives of Acoustics Volume 41 Number where G = γ+1)m 2b is a non-dimensional quantity. The dimensionlesswavenumbers K x K y arerealandpositive their ratio determines the angle between a beam andtheboundariesofaresonatorand Kx 2 +Ky 2 = 1. The solutions of13) before formation of discontinuities are as followsrudenko Soluyan 2005): U x = U y = P = where J n nσ x ) 4 χ i sinnη i ) U xi = 2 nσ x n=1 J n nσ y ) 4 δ i sinnη i ) U yi = 2 nσ y n=1 J n nσ) 4 sinnη i ) P i = 2 nσ σ x = Gθ K x n=1 14) σ y = Gθ K y σ = Gθ. 15) The boundary conditions for an impenetrable flow at theboundariesofaresonator x = 0 x = L x y = 0 and y = L y aregivenbyequalities U x X = 0Yt) = U y X = 2n ) xπ Yt = 0 K x U y XY = 0t) = U y XY = 2n ) yπ t = 0. K y 16) The length and width of a rectangular resonator equal integer number of longitudinal or transversal wavelengths L x = n x λ x L y = n y λ y thatinfact determines the spectrum of longitudinal and vertical components of wave numbers. An acoustic pressure and vertical velocity in a rectangular resonator before formation of a discontinuity in an acoustically active gaswith B > 0areshowninFigs.1and2.Inevaluations in accordance with14) only ten first terms were taken into account. Numerical evaluations undertaken by the author reveal practically indistinguishableresultsstartingfromthefirstfivetermsinthe series. The series is quickly convergent. The dimensionlesstimeofshockformation T saw ) fortheprofileof theverticalvelocityis 1 b ln theoneforthe 1+ Ky G horizontalvelocityis 1 b ln ) 1+ Kx G andtheonefor acousticpressureequals 1 b ln 1+ G) 1. Fordimensionlesstimes σ y largerthan π/2the dimensionlessvelocities U yi i = 1...4)takethe form of the saw-tooth shapes consisting of straight-line parts δ i τ i U yi = if π τ i < π. 17) 1+Gθ/K y Individualdimensionlessacousticpressures P i i = 1...4)alsotakethesaw-toothformif σ > π/2: P i = τ i 1+Gθ if π τ i < π. 18) Thepeakmagnitude p i doesnottendtozerowhen time increases in an acoustically active gas. The nonlinearattenuationatthefrontofasaw-toothwaveis suppressed by enlargement of the sound intensity. The limit value of dimensionless peak magnitude of every partoftheverticalvelocity u yi is 2πKybc γ+1 andthatof theacousticpressureis 2πbcρ2 0 γ+1 when θ.inthe equilibrium regime normal dispersion and attenuation support nonlinear attenuation and peak values quickly tendtozero.inthiscasediscontinuityintheprofileof theverticalvelocitydoesnotformforalargeenough attenuationthatisfor Ky G 1. The peak dimensionless magnitude of the vertical velocity at the same times as in the Fig. 3 for b = 0.01and K y = 0.2equals 0.2at τ = 2T saw and 0.002at τ = 20T saw respectively. T saw isevaluated at b = 0.01.For b = 0.01and K y = 0.8disconti- nuitydoesnotformatall.figure4showsthedis- Fig.1.Dimensionlessacousticpressure p /Mc 2 ρ 0)inaresonatoratdifferentdimensionlesstimesbefore formation of discontinuity. For M = 0.01 b = 0.01 γ = 1.4 the dimensionless time of discontinuity formation T sawequals 61.

5 A. Perelomova Standing Waves in a Rectangular Resonator Containing Acoustically Active Gases 71 Fig.2.Dimensionlessverticalvelocity u y/mc)inaresonatoratdifferentdimensionlesstimesandfor differentdimensionlessvaluesof K ybeforeformationofdiscontinuity.for M = 0.01 b = 0.01 γ = 1.4 thedimensionlesstimeofdiscontinuityformation T sawequals 15if K y = 0.2and 51if K y = 0.8. Fig.3.Dimensionlessverticalvelocity u y/mc)inaresonatoratdifferentdimensionlesstimesandfor differentdimensionless K yinasaw-toothwave.for M = 0.01 b = 0.01 γ = 1.4thedimensionlesstime ofsaw-toothshapeformation T sawequals 23if K y = 0.2and 72if K y = 0.8.

6 72 Archives of Acoustics Volume 41 Number Fig.4.Dimensionlessacousticpressure p /Mc 2 ρ 0)inaresonatoratdifferentdimensionlesstimesin asaw-toothwave.for M = 0.01 b = 0.01 γ = 1.4thedimensionlesstimeofthesaw-toothshape formation T sawequals 84. tribution of dimensionless acoustic pressure in a volume of a resonator at different dimensionless times. The dimensionless time of discontinuity formation in apressurestandingwave T saw equals 84.Themagnitude of dimensionless acoustic pressure achieves 0.1 for b = 0.01and τ = 2T saw and for b = 0.01 and τ = 20T saw respectively. 4. Acoustic heating An excess temperature T ent which specifies the non-wave entropy mode in the field of acoustic planarwaveswhichdonotinteractisgovernedbythe leading-order equationperelomova 2012) = bm 2 exp2bτ)γ 1) in the case of vibrationally excited gases and = bm 2 γ 1)γ +2) exp2bτ) γ P 2 i 19) P 2 i 20) in the case of chemically reacting gasesperelomova Pelc-Garska 2014; 2011) where the top line denotestheaverageoverthesoundperiodand T 0 isan unperturbed temperature of a gas. Before the formation of discontinuity the rate of heat release varies as b exp 2bτ) since the averaged squared dimensionless pressure Pi 2 equals Pi 2 = Pi τ Pi i dτ i = 0.5.Itdependsonthesignof b:thetemperatureofthemedium increasesif b < 0anddecreasesotherwise.Inthecase ofthesaw-toothwave Pi 2 π = 2 31+Gexpbτ) 1)) and 2 the acoustic heating is described by equation = 4M 2 π 2 bexp2bτ) γ 1) 31+Gexpbτ) 1) 2 ) 21) in the chemically reacting gases and by equation = 4M 2 π 2γ 1)γ +2) 3γ bexp2bτ) 1+Gexpbτ) 1) 2 ) 22) in the gases with excited internal degrees of a molecule freedom. In the non-equilibrium regime enlargement inthesoundmagnitudeisfollowedbycoolingofa medium. The sound wave takes energy from the background making it cooler. 5. Concluding remarks The peculiarities of a periodic acoustic field in a rectangular resonator filled with some relaxing medium which may be acoustically active are studied. Ifthemeanvalueofacousticpressureintheincident andreflectedwavesiszerotheydonotinteractin theleadingorderinavolumeoftheresonator.that allows to consider the incident and reflected waves individually. In the equilibrium regime discontinuity maynotformatallforlargeenoughattenuationsconnectedwithrelaxationinagas.ifthesaw-toothwave forms the peak acoustic pressure rapidly decreases towards zero. Vice versa an anomalous increase in the sound amplitude along with the nonlinear attenuationresultinthesettlingofthepeakmagnitudein the shock wave in a non-equilibrium gas. The conclusions concern propagation of the planar wave in a free space in waveguidesperelomova 2012) and resonators.inthetwolastcasesthediscretesetof wave numbers is determined by the boundary conditions. There appear the additional travelling nodal pointsinthevelocityprofileofstandingwavesina flat resonatorone travelling point between two static ones). There are no additional nodal points in the profile of the acoustic pressure but their coordinates vary with time periodically. That has been discovered with respect to gases with irreversible chemical reactions

7 A. Perelomova Standing Waves in a Rectangular Resonator Containing Acoustically Active Gases 73 inperelomova Pelc-Garska 2014). The conclusions are valid also in a rectangular resonator. In contrastthenodalpointsinthestandingwavesinnewtonian fluids are static. The slowly developing in time modes which are induced in the sound field in the non-equilibrium gases also behave atypically. The anomalous cooling of a medium and streamingwith streamlines inverted as compared with direction of streamlines in a Newtonian fluid) has been recently considered in reference to aperiodic and periodic in time sound beams including beams with discontinuities in unbounded volumes of acoustically active gasesmolevich 2001; Perelomova Pelc-Garska 2011; Perelomova 2012). An increaseor decrease) in the temperature of a gas does not disturb the boundary conditions in resonators16) since the velocity associated with the entropy mode is zero in the leading order. The squared sound velocity c 2 isproportionaltothemeantemperaturewhich makes the set of longitudinal and transversal wave numbersvaryinthecourseoftimeinordertohold the sound frequency constant. Hence they enlarge in the acoustically active gasand get smaller otherwise) proportionally to the square root of the initial sound intensity and B. The effects connected with the first and second viscosities and thermal conduction are not considered in this study. They may be readily included. Newtonian attenuation alone makes the peak pressure in the saw-tooth waves in the non-equilibrium regime todecrease;atlargetimesittendstozero.account for Newtonian attenuation may prevent formation of discontinuity in the acoustically active gas. References 1. Bauer H.J. Bass H.E.1973) Sound amplification from controlled excitation reactions Phys. Fluids Biwa T. Yazaki T.2010) Observation of energy cascade creating periodic shock waves in a resonator J. Acoust. Soc. Am Chester W.1964) Resonant oscillations in closed tubes J. Fluid Mech Chu B.T.1970) Weak nonlinear waves in nonequilibrium flows[in:] Nonequilibrium flows Wegener P.P. [Ed.]vol.1part2MarcelDekkerNewYork. 5. Clarke J.F. McChesney A.1976) Dynamics of relaxing gases Butterworth UK. 6. Hamilton M. Morfey C.1998) Model equations [in:] Nonlinear Acoustics Hamilton M. Blackstock D. [Eds.] pp Academic Press New York. 7. Kaner V.A. Rudenko O.V. Khokholov R.V. 1977) Theory of nonlinear oscillations in acoustic resonators Sov. Phys. Acoust Keller J.J.1977) Nonlinear acoustic resonances in shock tubes with varying cross-sectional area J. Appl. Math. Phys Makaryan V.G. Molevich N.E.2007) Stationary shock waves in nonequilibrium media Plasma Sources Sci. Technol Molevich N.E. 2001) Sound amplification in inhomogeneous flows of nonequilibrium gas Acoustical Physics Mortell M.P. Mulchrone K.F. Seymour B.R. 2009) The evolution of macrosonic standing waves in a resonator International Journal of Engineering Sience Ochmann M.1985) Nonlinear resonant oscillations in closed tubes an application of the averaging methodj.acoust.soc.am Osipov A.I. Uvarov A.V. 1992) Kinetic and gasdynamic processes in nonequilibrium molecular physics Sov. Phys. Usp Parker D.F.1972) Propagation of damped pulses through a relaxing gas Phys. Fluids Rudenko O.V. Soluyan S.I. 2005) Theoretical foundations of nonlinear acoustics Consultants Bureau New York DOI: /jcu Perelomova A.2010) Nonlinear generation of nonacoustic modes by low-frequency sound in a vibrationally relaxing gas Canadian Journal of Physics doi: /p Perelomova A.2012) Standing acoustic waves and relative nonlinear phenomena in a vibrationally relaxing gas-filled resonator Acta Acustica Perelomova A. Pelc-Garska W.2014) Standing waves and acoustic heatingor cooling) in resonators filled with chemically reacting gas Archives of Acoustics doi: /aoa Perelomova A. Pelc-Garska W.2011) Non-wave variations in temperature caused by sound in a chemically reacting gas Acta Physica Polonica A Zeldovich Ya.B. Raizer Yu.P.1966) Physics of shock waves and high temperature hydrodynamic phenomena Academic Press New York.

Acoustic Heating Produced in the Thermoviscous Flow of a Shear-Thinning Fluid

ARCHIVES OF ACOUSTICS 36, 3, 69 64 (011 DOI: 10.478/v10168-011-0044-6 Acoustic Heating Produced in the Thermoviscous Flow of a Shear-Thinning Fluid Anna PERELOMOVA Gdansk University of Technology Faculty