Band Structure and Effective Properties of One-Dimensional Thermoacoustic Bloch Waves
BBand Structure and Effective Properties ofOne-Dimensional Thermoacoustic Bloch Waves
Haitian Hao, ∗ Carlo Scalo, † and Fabio Semperlotti ‡ School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA
We investigate the dispersion characteristics and the effective properties of acoustic waves prop-agating in a one-dimensional duct equipped with periodic thermoacoustic coupling elements. Eachcoupling element consists in a classical thermoacoustic regenerator subject to a spatial temperaturegradient. When acoustic waves pass through the regenerator, thermal-to-acoustic energy conversiontakes place and can either amplify or attenuate the wave, depending on the direction of propa-gation of the wave. The presence of the spatial gradient naturally induces a loss of reciprocity.This study provides a comprehensive theoretical model as well as an in-depth numerical analysis ofthe band structure and of the propagation properties of this thermoacoustically-coupled, tunable,one-dimensional metamaterial. Among the most significant findings, it is shown that the acousticmetamaterial is capable of supporting non-reciprocal thermoacoustic Bloch waves that are asso-ciated with a particular form of unidirectional energy transport. Remarkably, the thermoacousticcoupling also allows achieving effective zero compressibility and zero refractive index that ultimatelylead to the phase invariance of the propagating sound waves. This single zero effective property isalso shown to have very interesting implications in the attainment of acoustic cloaking.
I. INTRODUCTION
In recent years, the study of acoustic metamaterialshas focused on the possibility to break reciprocity andon the resulting effects on the dispersion and propaga-tion of sound [1–6]. In conventional acoustic waveguides(e.g. a hollow duct), sound waves are reciprocally trans-mitted between two points of the domain. Exciting thedomain at a source A and measuring its response at apoint B would yield the exact same response if the sourceand the observation point were inverted. However, thisreciprocal wave transmission mechanism might not al-ways be a desired feature. There are certain applica-tions such as, medical imaging [7] or telecommunicationsdevices [8] whose performance can be significantly im-proved in presence of unidirectional sound transmission.For completeness, it is worth mentioning that unidirec-tional propagation has been observed and studied eitherin non-reciprocal or topological systems. While, undercertain conditions, the effect of the two systems on thewave propagation characteristics might be conflated, thetwo systems lead to unidirectional propagation by meansof very different mechanisms. Indeed, many topologicallynon-trivial systems are still reciprocal in nature [9–11].Focusing on non-reciprocity, non-reciprocal waves havebeen achieved in a variety of systems leveraging, as anexample, rotating fluids [1], active materials with spa-tiotemporal modulation [2, 3], near zero refractive indexmaterials [12] and materials with strong nonlinearities[4–6]. Non-reciprocal propagation was also observed inthermoacoustically coupled systems consisting in torus-shaped thermoacoustic (TA) engines [13–16]. It was this ∗ [email protected] † [email protected] ‡ Corresponding Author: [email protected] class of systems that later inspired the design of TAdiodes [17] and TA amplifiers [18]. Despite the long anddistinguished history of thermoacoustics and the morerecent analysis of diodes and amplifiers, the systematicanalysis and in-depth understanding of the dispersionand propagation properties in periodic TA waveguideshave never been undertaken.The current study specifically addresses this latterpoint by presenting a comprehensive theoretical andnumerical analysis of the dispersion and propagationproperties of thermoacoustically coupled waves in one-dimensional ducts embedded with a periodic distribu-tion of regenerators. In the following, we will refer tothe acoustic waves supported by this type of waveguidesas thermoacoustic Bloch waves . This specific type of 1Dwaveguides can be seen as a form of semi-active acous-tic metamaterial where thermo-acoustic energy conver-sion occurs periodically when the fluid passes throughthe evenly-spaced regenerators. The semi-active natureof the system is due to the fact that energy is eitherprovided or extracted from the acoustic wave as a con-sequence of the imposed static thermal gradient on theregenerators. The regenerator (REG) (also known, inmore traditional thermoacoustic studies, as a stack) con-sists of a porous material specifically designed to facil-itate thermo-acoustic energy conversion. Indeed, fromthermoacoustic principles, it is well-known that the en-ergy conversion is particularly significant when in pres-ence of a spatial temperature gradient imposed on theregenerator; a common setup for thermoacoustic engineapplications [19–22]. In the following we will show that,other than powering TA engines, the TA coupling canalso be leveraged to manipulate the propagation of TABloch waves and to shape its dispersion.One of the most immediate consequences of the pe-riodic exchange of thermoacoustic energy and of theisothermal condition imposed to the regenerators, the pe-riodic TA waveguide is non-conservative and can either a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b result in an effective lossy or amplifying medium (de-pending on the direction of propagation of the wave withrespect to the thermal gradient). While every naturalmaterial includes some degree of non-conservative effects(the most immediate being the mechanical energy dissi-pation associated with structural damping), these effectswere often deemed negligible so that the early literatureon periodic media and metamaterials had predominantlyfocused on the analysis of lossless conservative media.In this context, the classical dispersion analysis led toa real-valued band structure (RBS) for all propagatingmodes. In recent years, however, the intentional use ofnon-conservative effects has started drawing considerableattention and has established itself as a possible way tofurther manipulate the wave propagation characteristicsof the host medium . Typical examples consist in meta-material systems exploiting viscoelastic inserts as one ofthe constitutive phases [23–25], or in dissipative periodicacoustic waveguides [26, 27]. For this class of lossy ma-terials, the RBS approach was not applicable anymore,and the analysis of the dispersion behavior required acomplex-valued band structure (CBS) approach [23]. InCBS, the real part of the solution characterizes the prop-agating waves, while the imaginary part captures eitherthe dissipation or the spatial attenuation of the waves.As previously mentioned, in TA systems a significantenergy exchange occurs between the mechanical compo-nent (carried by the acoustic wave) and the thermal com-ponent (produced by the heat source), hence giving riseto non-conservative behavior. It follows that, as for theexample of viscoelastic metamaterials [23], the analysisof the band structure in TA periodic systems will also re-quire a CBS approach. In addition, and differently fromviscoleastic metamaterials, TA periodic systems are alsointrinsically non-reciprocal (due to the presence of thespatial thermal gradient on the REG). We will show thatCBS is still well equipped to capture the response of TAperiodic system and that its application allows uncover-ing the existence of an anomalous unidirectional energytransport phenomenon. The dispersion calculation willbe followed by an analysis, in the long-wavelength limit,of the effective properties of the TA unit cell. Resultswill reveal the ability of the TA metamaterial to act asa single-zero effective medium which, in turns, leads toa zero refraction index material. Interestingly, these ef-fective properties and the bandwidth of effective zero re-fractive index can be tuned via controlling the intensityof the thermal gradient, and can potentially open a routeto a tunable single (and, possibly, double) zero mediumto achieve, among others, energy squeezing and cloakingeffects.Finally, but not less significant, the model approachand analysis proposed in this study furthers the under-standing of the amplification and attenuation character-istics of TA waves in the small-channel limit, which mayhave significant implications for the optimal design ofthermoacoustic amplifiers [18] and diodes [17] and, morein general, for TA engines in traveling wave configuration. Modelled Region:
REG ( a )( b ) Isothermal,No-slip Adiabatic, Slip FloquetFloquet
FIG. 1. (a) (top) Schematic of the unit cell of the periodicthermoacoustic waveguide, and (bottom) the section of thewaveguide being modelled. (b) The mean temperature distri-bution along the unit cell. A spatial temperature gradient isimposed on the REG. An ambient heat exchanger is locatedat x/L = x . II. PROBLEM STATEMENT
The system under investigation consists of a one-dimensional infinite periodic waveguide whose fundamen-tal unit cell is made of a straight duct and a regenerator(see Fig. 1(a)). The regenerator can be thought of as astack of short parallel plates separated by thin pores sothat, at low frequency, viscous and thermal conductionlosses cannot be neglected. A temperature spatial gradi-ent is imposed on the regenerator (Fig. 1(b)) so to elevatethe temperature from ambient temperature T c at one endto the hot temperature T h at the other end. The hot endof the regenerator is followed by a thermal buffer tube(TBT), terminated by an ambient heat exchanger thatallows recovering the reference ambient temperature T c .The resulting temperature distribution in the unit cell isplotted in Fig. 1(b). Note that the TBT enables a conti-nuity of temperature at the cell ends, while also acting asa local scatterer due to the temperature variation from T c .In order to analyze the system, and without losing gen-erality, we adopt a plane wave assumption for the wavepropagating in the 1D duct outside the regenerator. Thissame assumption is not valid for the waves inside thesmall pores of the regenerator due to the thermo-viscouseffects. Therefore, inside the regenerator channels, thesolution is developed according to Rott’s thermoacousticlinear theory [19, 28]:d p d x = − ρ − f v (i ω ) u (1)d u d x = − γ − f k γP (i ω ) p + gu (2)where g = f k − f v (1 − f v )(1 − Pr) 1 T d T d x (3) u and p are first-order cross-sectionally averaged parti-cle velocity and pressure, respectively. ρ , P and T arezeroth-order (mean-state) density, pressure and temper-ature, respectively. γ and Pr are specific heat ratio andPrandtl number, respectively. f k and f v are complexfunctions expressed as: f (cid:3) = tanh[(1 + i)( h/ /δ (cid:3) ][(1 + i)( h/ /δ (cid:3) ] (4)where (cid:3) can take either the subscript of v or k . h is thewidth of the straight section. The viscous and thermalpenetration depths are expressed as: δ v = (cid:112) ν/ω, δ k = (cid:112) κ/ω (5)where ν and κ are the dynamic viscosity and the thermaldiffusivity, respectively. Note that the thermo-viscouscoupling is particularly strong when the characteristicratio h/ δ ∗ is small. This latter condition can occur ei-ther when in presence of thin channels (i.e. small h ) orof low frequency waves (i.e. large δ ∗ , Eqn 5). Whenthe thermo-viscous effects are negligible, that is when f v = f k = 0, the Helmholtz equation is recovered fromEqns. 1 and 2. Considering the plane wave assumptionfor the wide sections (outside the regenerator), as well asthe fact that the pores in the regenerator are identicalto each other, we simplify the modeling by only calcu-lating the acoustic field in a minimal unit (including asingle pore) [15], outlined by the dashed lines in Fig. 1.The Floquet boundary conditions are applied to the cell(minimal unit) ends: u ( L ) = exp( − i kL ) u (0) (6) p ( L ) = exp( − i kL ) p (0) (7)Recall that the wavenumber k is complex-valued for theCBS analysis.Mathematically, the CBS could be formulated in threedifferent ways: (1) complex ω (frequency) versus real k (wavenumber), (2) real ω versus complex k , and (3) com-plex ω versus complex k . However, only the former tworepresentations are physically significant. The first rep-resentation, appropriate for free wave propagation, con-siders a complex frequency ω = Re[ ω ] + iIm[ ω ] undera given real wavenumber k , where the imaginary partIm[ ω ] denotes the temporal growth (Im[ ω ] <
0) or de-cay (Im[ ω ] >
0) of the transient wave. The complexfrequency is especially relevant to TA engines in orderto describe the transient exponentially growing motiondue to TA instability [29–33]. The imaginary part of ω is also widely used to represent the decay rate of thefree vibration of a lossy material. The second represen-tation is better suited for a time-harmonic wave propa-gation. The imaginary part of the complex wavenumber k = Re[ k ] + iIm[ k ] allows capturing either the spatial at-tenuation or amplification of the time-harmonic wave atsteady state. In this study, we adopt this latter (time-harmonic) description of CBS in which a forcing fre-quency ω is taken as the real independent variable forthe solution of a complex wavenumber k = Re[ k ] + iIm[ k ] [23, 34], in order to describe either the spatial amplifica-tion or attenuation of thermoacoustic Bloch waves.The discretization of Eqns 1 and 2 (for the REG andfor the duct, respectively) combined with the Floquetboundary conditions (Eqns 6 and 7) yields a generalizedeigenvalue problem:[ A − exp( − i kL ) B ] (cid:20) pu (cid:21) = 0 (8)where A and B are coefficient matrices in which cer-tain elements are frequency-related. The eigenfunction [p, u] T consists of the discrete distribution of pressureand velocity. Given the frequency ω , the eigenvalueexp( − i kL ) can be obtained by applying any availableeigenvalue solver. The complex wavenumber k can thenbe easily extracted. Repeating the process for differentvalues of the frequency ω spanning a given range leadsto the CBS of the system.Recall that the thermoacoustic coupling results fromthe combined effect of the thermo-viscous behavior ( f v (cid:54) =0 and f k (cid:54) = 0) and of the temperature gradient along theregenerator ( T h (cid:54) = T c ). For a better understanding ofthe TA Bloch waves, we perform a CBS analysis of thewaveguide under three configurations employing differentassumptions: (1) pure acoustics, or equivalently, losslessacoustics ( f v = f k = 0, T h = T c ), (2) thermo-viscousacoustics ( f v (cid:54) = 0 , f k (cid:54) = 0, T h = T c ), and (3) thermoa-coustics ( f v (cid:54) = 0 , f k (cid:54) = 0, T h = 1 . T c ). In all threecases, both the geometrical and the material propertiesare maintained the same, as listed in Table I. Cases (1)and (2) will serve as a reference to better understand thebehavior observed in Case (3) that represents the actualTA Bloch waves. Note that cases (1) and (2) are expectedto be reciprocal due to the lack of the thermal gradient.In addition, in Case (1) the imaginary part shall be non-zero only in the band gaps (it will be zero outside a bandgap due to the lossless assumption). L [m] h s [mm] h s /h x x x P [Pa]0.5 0.25 0.75 0.1 0.12 0.45 101325 T c = T ref [K] ρ ref [kg / m ] Pr γ µ ( T )[Pa · s]300 1.2 0.72 1.4 1 . × − ( T /T ref ) . TABLE I. Geometrical and material parameters of the TAunit cell.
III. RESULTS AND DISCUSSION
Figure 2(a) and (b) show the CBS of the periodic sys-tem under pure acoustics and thermo-viscous acousticsassumptions. Band gaps appear in Fig. 2 (a.1) at theband crossings. These band gaps are the result of Braggscattering occurring at the abrupt cross-sectional areachanges at the REG ends. The scattering is particularlystrong when the length of the unit cell is approximatelya multiple of the half wavelength. The Im[ k ] is non-zeroin the band gaps, indicating the presence of evanescentwaves. The symmetry of the CBS also suggests that reci-procity is preserved. This behavior is clearly not surpris-ing and consistent with the well-known response of classi-cal non-resonant periodic media with periodic mechanicalimpedance mismatch. In the CBS plots, modes are col-ored by the sign (or, equivalently, the direction of propa-gation) of the cycle-averaged acoustic intensity. The in-tensity is expressed as I = 0 . pu ], where the over-bardenotes complex conjugate quantities. Remember thatthe sign of the acoustic intensity indicates the directionof acoustic energy transport. The colors yellow and bluerepresent positive and negative intensity, while the greencolor denotes a zero intensity (which only appears forevanescent modes). The band gaps in Fig. 2 (a.1) disap-pear in Fig. 2 (b.1) due to the significant thermo-viscouslosses in the REG pores. The symmetrical distributionof k is still preserved in these configurations (Fig. 2 (b.1)and (b.2)) because the temperature gradient is not acti-vated. It is worth mentioning that the viscous and heat-conduction losses break time-reversal symmetry, but donot affect reciprocity. It is also notable that in Fig. 2(b.2) all non-zero Im[ k ] correspond to spatial attenua-tion. Considering the exp( − i kL ) notation for Floquetconditions, a forward propagating wave (with positiveintensity, yellow) attenuates along x if Im[ k ] <
0, whilea backward propagating wave (with negative intensity,blue) attenuates along − x if Im[ k ] > f v (cid:54) = 0 , f k (cid:54) = 0, T h = 1 . T c ). In the Re[CBS], theband crossing that in the (reciprocal) thermo-viscous case(Fig. 2(b.1)) occurred at Re[ kL ] = ± π is now shifted toRe[ kL ] = − .
07; as a consequence, an opening appearsaround π in the symmetric half of the first Brillouin zone(BZ) (See Fig. 3(a.2)). The shift of the band crossingis a direct result of space-inversion symmetry breaking,leading to non-reciprocal wave propagation.Recall that, under the thermo-viscous assumptions, ei-ther a forward or a backward propagating wave (markedin yellow or blue in Fig. 2(b), respectively) is always as-sociated with either a negative or a positive Im[ k ], whichindicates spatial attenuation along the direction of prop-agation of the wave. Unlike the thermo-viscous case, inthe TA case the forward propagating waves can be spa-tially amplified if T h (cid:54) = T c . This latter condition is indi-cated by yellow dots with positive Im[ k ] in Fig. 3(b). Atthe same time, the backward propagating wave still at-tenuates along − x , according to the positive Im[ k ]. Suchasymmetric non-conservative behavior (characterized byeither attenuation or amplification) is a distinctive fea-ture of the non-reciprocal response of TA Bloch waves.We highlight that the non-reciprocal propagation can beeffectively exploited to achieve an effective one-way soundtransmission in the 1D TA waveguide. Figure 4 shows thesimulated results of a pressure pulse applied at the center of a finite length waveguides consisting of 30 unit cells.The propagation of the pulse is studied for the three con-figurations discussed above, that is Pure Acoustics (PA),Thermo-Viscous Acoustics (TVA), and TA assumptionsdescribed in Section II. Clearly, in the PA and TVA cases,the pulse splits in two equal parts that propagate in op-posite directions maintaining a symmetric behavior. Thesame situation occurs for the TVA configuration, howeverthe two pulses experience a spatial attenuation due tothermoviscous losses. The small fluctuations near p = 0are due to the inter-cell reflections. In the TA case, af-ter the initial pulse separates into two fronts, the for-ward propagating front is spatially amplified while thebackward moving one is attenuated. The spatial ampli-fication of acoustic waves relies on the heat provided tothe system at the hot side of REG. The amplitude ofthe amplified pulse will eventually reach a steady statevalue, balanced by nonlinear saturation [15, 35]. It fol-lows that, in the far field, only the forward propagatingpulse survives while the backward one disappears. Thisone-directional, non-decaying propagation may find in-teresting applications in long range acoustic communi-cation [36]. The spatial amplification of time-harmonicwaves in a waveguide formed by a cascade of several unitcells had been experimentally demonstrated in Refs. 17and 18, although without providing the underlying the-oretical framework. We merely note that the ability ofthe TA propagating Block wave to reach a consistentlystable amplitude could effectively result, in the far field,in a cloaking behavior. Indeed, any reduction in the waveamplitude due to scattering effects would be recovered,upon propagation, due to the TA coupling; hence notleaving any detectable trace of scattering in the far field.
1. Unidirectional energy transport
An inspection of the acoustic intensity associated withthe TA coupled modes in the reduced frequency range( ωL ) / (2 πa ) ∈ [0 . , .
53] reveals that the intensity I of both modes has the same sign (Fig. 3(a.1)). Recallthat a is the ambient sound speed in air, i.e. a = (cid:112) γP /ρ ref = 343[m / s]. In other words, the mode la-belled with a star marker has positive group velocity, yetnegative intensity. This observation is certainly counter-intuitive because in classical wave theory the group veloc-ity is also understood as the direction of energy transport.The results suggest that, while a wave packet with thecarrier frequency around ( ωL ) / (2 πa ) = 0 .
52 propagatesforward as a whole, the energy of the carrier frequencytime-harmonic TA Bloch wave is transmitted along thedirection of the intensity, that is the − x direction. Simi-lar phenomena are also observed around the reduced fre-quency 1.55 and 2.07 (Fig. 3(a.3) and (a.4)), where bothmodes have positive intensity. The inconsistency betweenthe group velocity v g , also dubbed macroscopic energytransport velocity [37], and the microscopic energy trans-port velocity v E associated with the acoustic intensity I -3.14 -3.105 1.0080.990 0.003-0.003 0.4940.506-0.015 0.015-0.030 0.0300.9921.006 (a.1) (a.2)(b.1) (b.2) -3.14 -3.04 -0.1 0.11.0150.985 FIG. 2. Complex band structure of the unit cell under (a) pure acoustics ( f k = 0 , f v = 0 T h = T c ) assumption, and (b)thermo-viscous acoustics ( f k (cid:54) = 0 , f v (cid:54) = 0 T h = T c ) assumption. Reciprocity is preserved under both assumptions. The verticalaxis is the reduced frequency, where a is the ambient sound speed at room temperature T c , i.e. a = (cid:112) γP /ρ ref = 343[m / s]. in acoustic Bloch waves was examined by Bradley [37].Noting that v E is proportional to I . For inviscid acousticBloch waves, the sign of v E and v g can differ due to theportion of energy stored in scattering elements eventu-ally present on the wave path. In other terms, while themajority of the energy is transported by the wave in thesame direction of propagation, a small portion of energyis stored in the wave scatterers. This part of the energy,termed stagnant energy , is not accounted for in the cal-culation of the microscopic energy transport velocity v E .Indeed, the calculation of v E only considers the portion ofenergy that is effectively in transport following the time-harmonic wave. On the other side, the stagnant energy isconsidered in the calculation of v g , which is a measure ofthe propagating speed of pulses or wave packets, as wellas the energy carried by them. When the thermoacoustic effect is taken into account, the discrepancy between v g and v E is affected by the thermo-viscous effect as well asthe thermoacoustic energy production. Although it is thelatter effect which mainly contributes to the occurrenceof opposite signs for v g and v E (or equivalently I ).To further illustrate the characteristics of the energytransported by time-harmonic waves (at velocity v E ) andby a wave packet (at velocity v g ), we consider a periodicTA waveguide as shown in Fig. 5(a). A pulse initiatedat x/L = 0 (and intrinsically composed of many har-monics) will split into two fronts and travel in both di-rections, with one front being amplified and the otherbeing attenuated, as shown in Fig. 4. The energy car-ried by each front travels at a speed v g (the slope of thecurves in Fig. 3(a)) associated with these two fronts asschematically shown in Fig. 5(b). If a time-harmonic ex- -3.15 -2.900.490.55 3.143.040.550.49-3.15 -2.901.531.57 0 0.152.052.085 (a) (b) (a.1) (a.2)(a.3) (a.4) FIG. 3. Complex band structure of the unit cell under thermoacoustic ( f k (cid:54) = 0 , f v (cid:54) = 0 T h = 1 . T c ) assumption. Reciprocityis broken indicated by the asymmetric CBS. The star and the square in (a.1) denote two modes at the same frequency withopposite direction of group velocity, yet same direction of intensity. The acoustic impedance of these two modes will be plottedin Fig. 6. citation is instead generated at x/L = 0 and at a selectedfrequency (represented by either the star or the squaremarkers shown in Fig. 3(a.1)), the steady state energytransfer shows different properties. Although the ampli-tude of the time-harmonic wave increases in the directionof the rising temperature gradient, or, equivalently, de-creases in the opposite direction, the acoustic intensityalways has a negative sign. In other terms, the energyalways flows in the negative x direction. In order to pro-vide further validation and insights in this unexpectedresult, we perform both fully numerical analyses as wellas theoretical investigations. Starting with the numeri-cal analyses, we developed a finite element model (FEM)(using the commercial software Comsol Multiphysics) ofa TA coupled periodic duct consisting of ten unit cells.An impedance type boundary condition was applied atboth ends of the periodic assembly to reduce unwantedreflections. The impedance values were calculated usingour theoretical model (Eqn. 8) for the specific modes se-lected (indicated by the star and square markers in Fig.3(a.1)). A unit amplitude time-harmonic velocity exci-tation was applied at x/L = 0. Figure 5(d) shows theacoustic intensity extracted from the data produced bythe finite element model. The arrows direction and size(plotted in log scale) indicate the direction and magni-tude of the intensity, respectively, at different locationsalong the periodic waveguide. It follows that the wavefield on the right and left hand sides of the excitation correspond to the two modes indicated by the star andsquare markers in Fig. 3, respectively. Figure 5(d) showsthat, on both sides of the time-harmonic excitation, thecycle-averaged intensities are negative. On the positivehalf region of the x axis ( x/L > x/L = 0 or, equiv-alently, the mechanical excitation behaves as an effectiveenergy sink. This counter-intuitive behavior is a resultof the rectifying effect produced by the periodic distribu-tion of regenerators (REGs) in the waveguide. In fact,the REGs behave as external heat sources that provide(or extract) energy to (from) the waveguide, hence alter-ing the net transport of acoustic energy and its direction.The intensity plots in Fig. 5(d), reveal that both modesare attenuated along the energy flow. Recall that thearrows length is plotted in log scale. By inspecting thewave in the x/L > x/L > v g and I . The derivation is providedhere below.Evaluating d[ p (cid:48) u + pu (cid:48) ] / d x yields: = 0= 1= 2= 3= 4= 5 Pure Acoustics Thermo-Viscous Acoustics Thermo-Acoustics
FIG. 4. A pressure pulse is applied in the center of a 30-unit TA periodic waveguide. The pulse splits into two fronts propagatingin opposite directions. The propagation characteristics are studied for the three different cases previously defined, that is PureAcoustic (PA), Thermo-Viscous-Acoustic (TVA) and Thermo-Acoustic (TA) assumptions, respectively. In the PA and TVAcases, the two fronts propagate in opposite directions and the response remains symmetric. The TVA response shows amplitudeattenuation due to the thermo-viscous losses. In the TA case, the forward propagating front is amplified while the backwardpropagating one is attenuated due to the TA coupling. The combined effect results in an effective non-reciprocal, unidirectionalsound propagation. The small fluctuations near p = 0 are due to the inter-cell reflections. τ indicates time and τ = 0 . dd x [ p (cid:48) u + pu (cid:48) ] = E ta (9)= i ρ − f v (1 + ω f (cid:48) v − f v ) uu + i γP [1 + ( γ − f k + ω ( γ − f k (cid:48) ] pp + 2 ωρ | − f v | Im[ f v ] u (cid:48) u + 2 ωγP m ( γ − f k ] p (cid:48) p + gup (cid:48) + gu (cid:48) p + g (cid:48) up (10)where E ta is the mechanical energy distribution under TAcoupling, the over-bar denotes complex conjugate quan- tities and the prime denotes ∂/∂ω . Integrating Eqn. 10over the unit cell and incorporating the Floquet bound-ary conditions (Eqns. 6 and 7) yield: ...... = - 4 .4= - 2 .4= - 0 .4 = 0 .6= 2 .6= 4 .6 ( a )( b ) = 0 = 0 ( c ) ( d ) Pulse Excitation Time-harmonic Excitation
FIG. 5. (a) Schematic of a periodic TA waveguide including the REG and the ambient heat exchanger. The color gradient inthe REG indicates the cold (blue) and hot (red) temperature side. The vertical dashed lines represent the separation betweenconsecutive unit cells. The behavior of the waveguide is explored under either (c) pulse excitation or (d) time-harmonicexcitation. (c) A pulse is generated at x/L = 0 and splits into two fronts. The two fronts, as well as the energy they carry,travel in opposite directions. The front traveling in the direction of the temperature rise ( x/L > x/L = 0and propagates in both directions. However, at selected frequencies (e.g. the one represented by the star and square markersin Fig. 3 (a.1)) both steady-state modes can transport energy unidirectionally towards − x . (b) The cycle-averaged acousticintensity I associated with the time-harmonic wave is plotted at different locations along the periodic waveguide (highlightedby dashed boxes in (a)). The direction and length of the arrows indicate the direction and log-scale amplitude of the acousticintensity. L ∂k∂ω {
14 [ p (0) u (0) + u (0) p (0)] } exp(2Im[ k ] L ) = − [ p (cid:48) (0) u (0) + u (cid:48) (0) p (0)]exp(2Im[ k ] L −
1) + (cid:90) L E ta d x (11)Recall that: v g = ∂ω∂ Re[ k ] = 1 / Re[ ∂k∂ω ] (12) I = Re {
14 [ p (0) u (0) + u (0) p (0)] } (13)For propagating pure acoustic waves (Im[ k ] = 0, f v = f k = 0, and T h = T c ), Eqn. 11 becomes: I/v g = (cid:104) E (cid:105) = 14 L (cid:90) L ( ρ | u | + 1 γP m | p | )d x (14) where (cid:104) E (cid:105) denotes the spatially averaged mechanical en-ergy along the unit cell. Equation 14 is consistent withthe observation that the intensity of a propagating time-harmonic acoustic plane wave is always in the same di-rection as its group velocity. However, the existence ofthermo-viscous losses (nonzero f v and f k ) as well as ofthe temperature gradient (nonzero g ) gives rise to thediscrepancy between the sign of v g and I (Eqn. 11).To further substantiate the previous finding, we plotthe acoustic impedance distribution z = p/u (which isscale-independent) along the unit cell for the two modesat ( ωL ) / (2 πa ) = 0 .
52 in Fig. 6. The results are com- -Theory-Theory-COMSOL-COMSOL -400-600-2000200 -1000 -3000-5000 (a)(b)
FIG. 6. Real and imaginary parts of the impedance z = p/u of the two modes at reduce frequency ( ωL ) / (2 πa ) = 0 . z ], denoting negative intensity I . pared with a fully numerical FE solution obtained viaCOMSOL. The numerical model represents a single pore,as outlined in Fig. 1(a), with parameters given in TableI. Similarly to the other numerical FE simulations pre-sented in the earlier part of the section, the right end ofthe duct was subject to an impedance boundary condi-tion. The complex-valued impedance was extracted fromour theoretical results. The duct was excited by applyinga unit-amplitude pressure at the left end. The model wasused to calculate the steady-state response and to extractthe impedance distribution along the waveguide. The re-sults reported in Fig. 6 show a very good agreementbetween the finite element solution and our theoreticalcalculations. Recalling that Re[ z ] = I/ | u | , the nega-tive Re[ z ] confirms the occurrence of negative intensityfor both modes, hence unidirectional energy transportwithin this frequency range.
2. Effective properties and zero compressibility
The characterization of TA periodic waveguides alsobenefits from an analysis of the effective properties. Inthe long-wavelength limit, the effective density ρ eff andeffective compressibility 1 /B eff are expressed as: ρ eff = i p ( L ) − p (0) ωL (cid:104) u (cid:105) (15)1 /B eff = i u ( L ) − u (0) ωL (cid:104) p (cid:105) (16)where the angle brackets denote the spatial average alongthe unit cell. By comparing Eqn 16 and Eqn 2, we an-ticipate that the effective compliance can deviate con-siderably from the reference value 1 /γP due to the ex-istence of the gu term. To enhance the thermoacoustic coupling effect, represented by the gu term, it is prefer-able to have (1) a sufficiently large temperature gradient,and (2) a locally expanded REG ( h s /h >
1) to improvethe thermo-acoustic energy conversion efficiency of theREG [18, 38, 39]. Therefore, in order to clearly illustratethe effect of the TA coupling on the effective properties,we modify the design parameters for the TA unit cellto the following values: x = 0 . , x = 0 . , x =0 . , h s = 0 . . h, and T h /T c = 3.Figure 7 shows the real and imaginary parts of theeffective density, the effective compressibility, and the ef-fective refractive index n eff in the TA waveguide. Notethat as the frequency decreases, the direction of intensityassociated with both modes becomes the same (denotedby the blue color). This phenomenon has been explainedin Section III 1. Due to the non-reciprocity, the effectiveproperties of the two modes are different. The effectivedensity is largely unaffected. The real part is slightlylower than the corresponding static value in ambient air(dashed line) but it remains almost constant for bothmodes across the selected frequency range. The devia-tion of Re[ ρ eff ] from the reference value is mainly due toviscous losses ( f v in Eqn. 1) and the dependence of thedensity on the temperature. The imaginary parts of thecomplex-valued ρ eff , /B eff and n eff , which are a resultof the non-conservative thermoacoustic coupling ( f (cid:3) (cid:54) = 0and T h (cid:54) = T c ), affect the spatial amplification or atten-uation of the TA Bloch waves. The effect of the TAcoupling is particularly visible on the effective compress-ibility. The TA coupling is induced by the term gu inEqn. 2, which is proportional to the temperature gra-dient (Eqn. 3). By comparing Eqn. 16 and Eqn. 2, itcan be seen that the term gu can significantly alter thevalue of 1 /B eff from the reference value 1 /γP . Figure7(e) shows that the real part of the effective compressibil-ity of one mode (blue) reaches zero in the low frequencyrange. This result indicates that, due to the TA coupling,the acoustic waveguide behaves as a single zero medium[40, 41]. In zero index media, acoustic waves can propa-gate without phase variation which in turns can lead toacoustic devices exhibiting intriguing properties [40, 41],such as acoustic cloaks and energy squeezing [42], andacoustic lenses [43]. Double zero properties (i.e. botheffective density and effective compressibility) are typi-cally required for an efficient preservation of both phaseand amplitude profiles of the acoustic wave front wheninteracting with scatterers or geometric inhomogeneities.However, single zero media (i.e. zero compressibility)have shown to be capable of preserving the phase profile[40, 41].The calculation of the refractive index n eff = (cid:112) ρ eff /B eff shows a near zero Re[ n eff ] value in the fre-quency range where a near zero Re[1 /B eff ] is achieved.Note that the Im[ n eff ] reflects the spatial amplificationor attenuation of the traveling wave traveling but it doesnot contribute to the phase.In order to further substantiate this observation of theoccurrence of a zero-index, we performed a numerical0 ( a )( b ) ( c )( d ) ( e )( f ) FIG. 7. Real and imaginary parts of the effective density ρ eff , the effective compressibility 1 /B eff , and the effective refractiveindex n eff . The red dashed lines indicate the reference static values of each property for ambient air at T c . simulation of the steady state harmonic response of thewaveguide via finite element modelling. The waveguideconsists of 40 TA unit cells ( x/L ∈ [0 , x/L ∈ [40 , L long and it is termi-nated at x/L = 100 with an impedance boundary con-dition that matches the impedance of air (415 [Rayls]),in order to eliminate reflections. The waveguide is ex-cited at x/L = 0 with a pressure p = exp(i θ ). Thesolid lines in Fig. 8 show the phase of the pressure field φ p = atan(tan(Im[ p ] / Re[ p ])) along the center line of thewaveguide following different input phase θ = 0 , π/ π/
3. The shaded area denotes the section of the waveg-uide taken by the 40 TA unit cells. The dashed lines showa baseline phase profile that would be obtained if the TAsection were substituted by a hollow duct filled with am-bient air (that is the entire 100 L long waveguide wouldconsists in a hollow duct filled with air). Results clearlyshow that, under near zero refraction index conditions(or, equivalently, near zero compressibility conditions),the phase remains invariant within the TA section, re-gardless of the input phase.An important feature of zero-index materials is thatthe internal defects do not alter significantly the phaseprofile [40, 42]. The concept of acoustic cloaking stronglyrelies on this feature. To further corroborate the phaseinvariant characteristic of the proposed TA waveguideand its insensitivity to internal defects or scatterers, weperformed numerical simulations on an acoustic waveg- uide composed of 40 unit cells, ten of which includeddefects. More specifically, we consider a baseline config-uration made of 40 unit cells without defects (essentiallyequivalent to the TA waveguide discussed above), and asecond configuration in which 10 unit cells in the middleincluded a defect. The defect consisted in a rectangularsound hard scatterer having dimensions of 0 . L long and0 . h wide, as shown in Fig. 9(a). The structure was ex-cited at x/L = 0 with a zero-phase, unit-amplitude pres-sure, that is p = exp(i0). Impedance matching conditionis applied to the other end of the structure ( x/L = 40).The remaining geometrical parameters and the appliedtemperature gradient are unchanged with respect to theresults presented in Figs. 7 and 8.Results are presented in Fig. 9 in terms of phase pro-file of the pressure field along the centerline (dashed linein Fig. 9(a)) of the 40-cell structure. Figure 9(b) showsthe pure acoustic case (i.e. without TA coupling) whichprovides a reference baseline. The blue and red curves de-note the phase along the regular and the defected waveg-uide, respectively. From these results it is seen that, inthe region downstream of the defects, a 0.26 rad ( ≈ ◦ )phase difference ∆ φ p is induced due to the presence of thedefects. Contrarily, when the TA coupling is activated,the shift in phase ∆ φ p is reduced to 0.02 rad ( ≈ ◦ ),as shown in Fig. 9(c). This drastic reduction leads toan overall phase shift that is negligible in practical ap-plications. We conclude that, as expected based on thepreviously discussed effective properties, the TA waveg-1 ( a )( b )( c ) FIG. 8. Phase of pressure along the center line of the designedwaveguide with TA section (solid), and a hollow duct filledwith ambient air as a reference (dashed). The input phase θ is (a) 0, (b) π/ π/
3. The shaded region representsthe TA section. uide is still capable of maintaining an effective invari-ant phase transition, a necessary condition for acousticcloaking. Recall that the TA waveguide is only a single-zero material, hence incapable of impedance matching,which is a key feature of double-zero materials for highenergy transmission [40, 42]. Despite this latter aspect,an interesting phenomenon can take place in this class ofwaveguides thanks to the multi-physics coupling and tothe energy provided to the system by the thermal sources(at the REG locations). Indeed, it was previously shownthat the wave traveling in the direction of the rising tem-perature gradient can be effectively amplified. We alsoknow that, in TA system, sound amplification (due tothe underlying TA instability) is balanced by nonlinearlosses, ultimately allowing to reach a steady state re-sponse [15, 19]. It is possible to envision that, under thisscenario, the amplitude decrease in the transmitted wave(due to the scattering element) can be recovered due tothe TA coupling. This means that, in the far field down-stream of the scattering section, the response of the twowaveguides (i.e. the baseline and the TA configurations)would be exactly identical (both in terms of amplitudeand phase profile), hence resulting in a perfect cloakingof the upstream scatterers. In other terms, the lack ofimpedance matching due to the presence of double-zeroeffective properties is balanced by the TA growth mech-
20 4030100210-1-200.020.080.060.04
Regular CellDefected Cell (a)(b)(c) Pure AcousticsThermoacoustics
FIG. 9. (a) Schematic of a (top) regular unit cell withoutdefect, and (bottom) a defected unit cell with a rectangularsound hard scatterer. (b) and (c) Pressure phase distribu-tion along the centerline (dashed line in (a)) of the 40-cellwaveguide under (b) pure acoustic, and (c) thermoacousticassumptions. The blue and red phase is associated with theregular and the defected waveguide, respectively. The dashedsections in the red curves denote the location of the 10 de-fected cells. anism. Note that, we did not show a numerical valida-tion of these results because our model does not integratenonlinear losses.An additional interesting aspect of this class of 1Dacoustic metamaterials consists in the ability to tune theeffective properties by adjusting the intensity of the tem-perature gradient. Figure 7(e) shows that only the lowermode is capable of effective zero refractive index in thelow frequency range. Therefore, the tuning effect of thetemperature gradient T h /T c is illustrated for this specificmode. Using the regular TA waveguide illustrated above,we performed a parametric study by varying the tempra-ture ratio T h /T c . The numerical results presented in Fig.10 show that as the temperature ratio increases, which isindicative of a more intense thermoacoustic coupling, therefractive index Re[ n eff ] decreases. More remarkably, theeffective zero range, that is the range where Re[ n eff ] ≈ T h /T c increases. When T h /T c = 3, the re-fractive index is effectively zero in the reduced frequencyrange of approximately [0 ∼ .
02] where Re[ n eff ] is lessthan 1.5% of n eff (Fig. 10 inset). n ref is the refractive in-dex of ambient air at T c , shown as the horizontal dashedline in Fig. 7(e). This range spans ± = 1 .5 = 2 = 2.5 = 3 FIG. 10. Re[ n eff ] normalized by n ref , as a function ofthe reduced frequency under different temperature gradients, T h /T c . n ref is the refractive index of ambient air at T c , shownas the horizontal dashed line in Fig. 7(e). Results show thatthe effective refractive index of the TA waveguide and thebandwidth of the zero refractive index can be tuned by con-trolling the temperature ratio T h /T c . band control. Indeed, this latter characteristic stems alsofrom the non-resonant nature of the present design.We merely suggest that the effective zero compressibil-ity of the TA waveguide could be further combined witheffective zero density, realized by resonating side branches[44], to achieve a complete impedance matching double-zero-index material [42].
3. Wave amplification and attenuation in small channels
It is widely accepted in thermoacoustics that, when atemperature gradient is established along a small channel(small-channel limit, such as those in a REG), the volu-metric velocity is proportionally amplified (attenuated)if the direction of propagation occurs along (against)the positive (i.e. rising) temperature gradient direction[19, 45]. This fact can be expressed by the relation U h /U c = T h /T c , where U and T are volumetric veloc-ity and temperature, while the subscripts denote the h otand c old ends of the small channel. We anticipate that,thanks to the modeling approach presented in the pre-vious sections, we will be able to make an importantdiscovery concerning the ideal limit behavior of the TAresponse. Indeed we will show that, unlike the conven-tional understanding of the small-channel limit behav-ior in classical thermoacoustics, the proportional relation( U h /U c = T h /T c ) only holds for the spatially attenuatingTA Bloch wave that propagates against the temperaturerise. The proportional amplification along the tempera-ture rise does not take place. This very important obser-vation may have great implications for the optimal designof thermoacoustic amplifiers [18]. Figure 3(b) shows that, as ω →
0, one branch of Im[ k ] L converges to a finite value, while the other branch con-verges to zero. This indicates that one mode propa-gates with an amplitude variation (Im[ k ] L (cid:54) = 0), whilethe other mode propagates with an unchanged amplitude(Im[ k ] L = 0). Whether the mode subject to amplitudevariation undergoes spatial amplification or attenuationdepends on the propagation direction of the mode. In thefollowing, we theoretically prove that this limit behavioris always true and that the branch of Im[ k ] L convergingto a finite value, proved to be Im[ k ] L = ln( T h /T c ), is al-ways associated with a negative intensity, indicative of aspatial attenuation.As ω → h/ δ ∗ → f v → [1 − (2 / h/ ( ω/ ν )]under second order approximation, hence, neglecting thehigher order terms, Eqns. 1 and 2 can be recast into:d p d x = − ρ ν ( h/ u (17)d u d x = − i ωP m p + d T m T m d x u (18)Considering that the effect of the temperature gradientoverpower the viscous effect, Eqn 18 becomes: u h = ( T h /T c ) u c (19)where u h and u c are the cross-sectionally averaged par-ticle velocities at the REG ends.Two solutions are possible: (1) u h = u c = 0, and (2) u h /u c = T h /T c . Solution (1) is trivial and leads to aconstant pressure p distribution according to Eqn. 17,or equivalently, Im[ k ] L = 0. Solution (2), under thelong-wavelength assumption (small Re[ k ] L ), leads to u ( L ) /u (0) = u h /u c = T h /T c = exp(Im[ k ] L ). Therefore,Im[ k ] L = ln( T h /T c ). This conclusion is also consistentwith the classical understanding of thermoacoustic wavesin the ”small-channel limit” ( h/ δ ∗ →
0) [45], which inSwift’s words is stated as: ”The volume flow rate is am-plified in proportion to the temperature rise (or attenu-ated in proportion to a temperature drop).”
It is under-stood, based on the previous analysis, that as ω → u h /u c = T h /T c .In the following, we show that in the small-channellimit ( ω →
0) only the ideal proportional attenuationalong the temperature drop (associated with a negativeintensity, or equivalently, backward propagating wave) ispossible.For the mode satisfying u h /u c = T h /T c , the velocitydistribution in the REG is proportional to T ( x ), i.e.: u ( ξ ) = C (cid:2) aξ + T c (cid:3) (20)where ξ = x − x c , a = ( T h − T c ) / ( x h − x c ), and C is anarbitrary proportional constant. According to Eqn. 17,the pressure p in the REG is expressed as: p ( ξ ) = − ρ ν ( h/ C (cid:20) aξ + T c ξ + ( T h + T c ) T c a (cid:21) (21)3 FIG. 11. Imaginary part of the CBS for the waveguide in TAconfiguration under a prescribed set of parameters (see thetext). Results show that the ideal amplification of u alongthe direction of the temperature rise (yellow-colored solutionwith the value of Im[ k ] L = ln[ T h /T c ] = 0 .
405 as ω →
0) canbe closely approached but never reached.
Therefore, the intensity at the cold end of the REG ( ξ =0) is: I = 12 Re[ pu ] = −
12 3 ρ ν ( h/ | C | ( T h + T c ) T c a < u out /u in = u h /u c = T h /T c for a single TA unit [19, 45].Based on the conventional understanding [19, 45], thistheoretical extreme can be achieved in the small-channellimit. However, the previous TA Bloch wave analysisshows that such ideal change of u proportional to thetemperature gradient is always accompanied by a nega-tive intensity I (indicating a spatial attenuation of thewave along − x ). Nevertheless, by careful design, one canstill get close to (although never reach) the ideal ampli-fication of u along the temperature rise. Fig. 11 showsthe imaginary part of the CBS of a TA Bloch wave in thelow-frequency range obtained with another set of param-eters: x = 0 . , x = 0 . , x = 0 . , h s = 0 . . h . Note that the ratio h s /h = 5 . > k ] L =ln(1.5)=0.405 has negative inten-sity (blue) as ω →
0, the intensity soon becomes positive(yellow) with increasing ω . In other terms, a forwardmoving Bloch wave, denoted by a positive intensity, isspatially amplified with a rate that reaches the maxi- mum theoretical amplification rate Im[ k ] L ≈ ln( T h /T c ).However, as ω →
0, the intensity associated with bothmodes become negative (denoted by blue curves in Fig.11), which is a sign of unidirectional energy transport.This aspect has been discussed in detail in Section III 1.This underlying behavior in the small-channel limitwas not observed before and it might have substantialimplications to guide the optimal design of thermoacous-tic diodes and amplifiers.
IV. CONCLUSIONS
This study presented an in-depth theoretical and nu-merical investigation of the dispersion and propagationcharacteristics of Bloch waves occurring in an acousticperiodic waveguide in presence of thermoacoustic cou-pling. This class of waves was dubbed thermoacousticBlock waves. The work highlighted several noteworthyfindings concerning the basic physical behavior of thiswave type as well as their potential impact on futureapplications. While the static temperature gradient im-posed on each regenerator led, as expected, to breakingthe intrinsic reciprocity of the waveguide, it also high-lighted very intriguing and unexpected propagation phe-nomena. Indeed, by leveraging a complex band structureapproach, we uncovered an anomalous unidirectional en-ergy transport unique to this type of waveguides. Theenergy transport was also found to be significantly dif-ferent depending on the nature of the acoustic wave, thatis a wave packet or a harmonic wave, potentially resultingin contrasting directions for the transfer of macroscopicand microscopic energy.Also remarkable was the ability of the TA waveguide toact as a broadband, tunable, effective zero refractive in-dex material. In selected frequency ranges, the waveguidewas shown to achieve zero effective compressibility, hencezero effective refractive index. In addition, both the ef-fective properties and refractive index could be tuned bysimply controlling the intensity of the temperature gra-dient. The single-zero nature (i.e. zero compressibility)of the TA waveguide was shown to enable phase invari-ance of the pressure field within the TA waveguide, henceshowing potential for application of efficient energy trans-mission and acoustic cloaking devices. 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