Numerical Calculation of Effective Density and Compressibility Tensors in Periodic Porous Media: A Multi-Scale Asymptotic Method

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1 Ecerpt rom the Proceedings o the COMSOL Conerence 8 Boston Numerical Calculation o Eective Density and Compressibility Tensors in Periodic Porous Media: A Multi-Scale Asymptotic Method Chang-Yong Lee, Michael J. Leamy *,, and Jason H. Nadler School o Mechanical Engineering, Georgia Tech Research Institute (GTRI), Georgia Institute o Technology, Atlanta, Georgia 333 * Michael J. Leamy: GWW School o Mechanical Engineering, 77 Ferst Drive N.W., Atlanta, GA 333 michael.leamy@me.gatech.edu Abstract: A maor issue in predicting and controlling (via design) absorption properties o rigid porous media is the determination o the requency-dependent eective density and compressibility tensors. Unlike previous research eorts which employ in-house and, otentimes, multiple numerical procedures or determining these two essential tensors, we ormulate their solution in terms o a set o micro-scale governing equations (and associated boundary conditions) resulting rom a multi-scale asymptotic analysis. The orm o these equations is ideally suited or incorporation into the inite element analysis package, COMSOL Multiphysics. Incorporating the equations directly into COMSOL allows or arbitrary three-dimensional model generation, unit cell meshing, and ultimate analysis using a single sotware package. We demonstrate the validity o this approach by comparing our numerical results with those published in the literature speciically, we analyze porous media composed o rigid, ace centered cubic (FCC) packed spheres. Keywords: porous media, eective density, eective compressibility, acoustic absorption.. Introduction Noise reduction is currently a maor issue in the automobile, aeronautical, and building industries. One promising way to reduce noise is through the design and optimization o porous structures, which demonstrate avorable acoustic absorption properties due to luid losses associated with large wetted areas. The important measurement parameters characterizing and predicting acoustic absorption perormance o these structures are their eective density and compressibility tensors. For this reason, research attention devoted to the theoretical ormulation o these two tensors has received considerable attention in the past three decades -3. Although the ormulations in the cited works are given in an eplicit analytical orm, they can only be evaluated in cases o very simple pore geometries assuming isotropic material properties, such as low in a uniorm cylinder. More recently, several numerical methods have been suggested in the literature 4-, which oer the possibility o etending the evaluation to general geometries and materials using desktopbased workstations. However, most o these analysis techniques require the use o (multiple) in-house codes and complicated numerical procedures. Furthermore, many o the cited numerical treatments are still not suitable or both arbitrary pore geometry and materials. Thereore, a need eists or a general computational approach or predicting the acoustic properties o periodic, porous materials which, ideally, can be cast in a orm suitable or analysis using a commercial package, such as COMSOL Multiphysics. In this paper, we present a multi-scale numerical approach or determining the eective density and compressibility tensors or periodic porous media with arbitrary 3-D unit cells. To the authors knowledge, the approach presented is the irst posed in a consistent and general manner such that the resulting micro-scale equations can be implemented in a single commercial o-the-shel (COTS) simulation code. We begin by reviewing a mathematical procedure suitable or analyzing acoustic porous medium: the multi-scale asymptotic method (MAM) 3,4,,3. MAM enables one to determine the macroscopic material description rom knowledge o the physics and geometry at the microscopic level. We use this method to derive a set o requency-domain partial dierential equations, and boundary conditions, or coupled

2 dynamic and thermal response. COMSOL Multiphysics is then employed with periodic boundary conditions on a unit luid cell (UFC) to determine the eective density and compressibility tensors, which ultimately determine the acoustic absorption properties o the porous material. For a porous material composed o FCC packed spheres, comparisons o results generated in this study with those ound in the literature 5,7,8 demonstrate very good agreement.. Multi-Scale Asymptotic Method In order to describe the linear acoustics phenomenon created in rigid porous medium, one has to solve the ollowing set o harmonic viscothermal governing equations and associated boundary conditions or harmonic waves o requency ( ): In the viscothermal luid low, p ρ τ = + () P ρ T ρiωu = p+ ( λ + μ) ( u) + μδu ρ (a,b) iω = u ρ ρiωcpτ = iωp+ KΔ τ (3) On the luid-solid interace, u = and τ = (4) Eqs. (-4) represent the equation o state, momentum balance, mass balance, and energy balance. P, ρ, and T denote the pressure, density, and temperature o the air at rest, while u, p, ρ, and τ the luid velocity, pressure variation, density variation, and temperature variation. In addition, the angular requency, shear and bulk viscosities, speciic heat at constant pressure, and heat conductivity are denoted, respectively, by ω (= π ), μ, λ, C p, and K. As has been noted by several authors, any problem o linear acoustics involving porous medium can be dealt with using this ormalism. However, because the equations presented above are based on interdependent macroscopic variables, they cannot take into account eplicitly the micro-scale physics and geometry o the porous media at the microscopic level. MAM is a multi-scale approach based on an asymptotic analysis o the governing equations (-4). It is used to urther derive a set o wellposed micro-scale equations necessary or computing eective macro-scale variables, via averaging. The asymptotic approach begins by introducing two space variables: or the macro-variations and y = ε or the microvariations. Here the small parameter ε = OlL ( ) << is a scale ratio o a characteristic unit cell length l and requency-dependent wavelength L. As such, the small parameter denotes a ratio o a micro-scale characteristic length to a macro-scale characteristic length. The solution variables sought and the dierential operators are net split into their macroscopic and microscopic components via power series involving ε, u = u ( y, ) + εu ( y, ) + ε u ( y, ) + p= p( y, ) + εp( y, ) + ε p( y, ) + (5) τ = τ ( y, ) + ετ ( y, ) + ε τ ( y, ) + The gradient ( ) and Laplacian ( Δ ) operators take the orms, = + y and y y ε Δ =Δ + ε Δ + ε Δ. (6) Moreover, because one can consider the viscothermal eects to occur at the micro-scale, it is necessary to rescale the viscosity and conductivity coeicients appearing in the momentum balance (a) and energy equations (3) by ε ρ iωu = p+ ε ( λ + μ ) ( u ) + μδu (7) ρ [ ] iωcpτ = iωp+ ε KΔ τ (8) With this rescaling, MAM allows or orderly solution o the seeking variables. We start with the mass balance equation (b), combined with the state equation (). Updated using the epansions (5-6), a single multi-scaled relationship results, ( p + εp ( τ + ετ iω P T (9) = + y ( u + εu ε Separating scales, the corresponding equation at the highest order ε implies that the luid velocity can be considered as locally incompressible, u =, () y

3 while at order ε, p τ iω = P T A similar procedure carried-out on the momentum balance equation (7) yields: ρiω( u + εu = + y ( p + εp ε + ε μ Δ y y ( u + εu ε ε + ( λ + μ) + y + y ( u + εu ε ε () Separating orders, order ε yields the relationship, p = p y (, y ) = p ( ) (3) which implies that the macro-scale pressure is constant at the micro-scale. At order ε, ρiωu = yp p + μδyu. (4) For the energy equation (8), the procedure results in the multi-scale equation, ρiωcp ( τ + ετ = iω( p + εp + ε K Δ y y ( τ + ετ + ) (5) ε ε The highest order identiied is ε and thus a single relationship is obtained (macro-scale), ρ iωc τ = iωp + KΔ τ. (6) p y u. () Following the methodology introduced by Laarge et al, we assume, at a given requency, the appropriate solution orms o the macro-scale velocity ( u ) and micro-scale pressure ( p ) distributions as linear relationships with p, while the macro-scale temperature ( τ ) distribution as linearly-related to iω p, as ollows: k ( y, ω) u ( y, ) = p( ), =,,3 μ p (, y) = α ( y, ω) ( ) ˆ p + p ( ) (7) and k ( y, ω) τ (, y) = iωp ( ) (8) K Note that k and k denote the dynamic viscous and thermal permeability, and also note that the pressure ( p ) can be epressed in terms o its deviatoric part ( ˆp ) and zero mean value ( α ) on the UFC. Substituting (7-8) into (), (4), and (6) yields the ollowing decoupled set o partial dierential equations suitable or incorporation into COMSOL Multiphysics: Momentum equation with no-slip boundaries ρ iω k e α e Δk e= e μ k e = in Ω (9) k = on Γ, k & α : Ω periodic where e denotes any o three unit vectors directed along a global coordinate system. Energy equation with isothermal boundaries ρ Pr iωρk Δ k = in Ω () μ k = on Γ, k : Ω periodic, where Pr = μcp K denotes the Prandtl number. Here Ω, Ω and Γ represent the UFC volume, the luid-illed pore volume, and the luid-solid interace, respectively. Taking the volume average o Eqs. (), (7) and (8) over the UFC and using the relationship between the eective χ ) and dynamic ( β ) compressibilities, ( e ( ) β ω β ω i ω p = u, with χe = () γ P γ P,which is deined by Laarge 9, yields two macroscopic equations as ollows: ρ iω u = p, e ( ) χe iω p = u, () where μφ ρ ˆ e = k iω β( ω) ρ ˆ Pr iωk χe = = γ ( γ ) (3) γ P γp μ φ with γ as the speciic heat ratio, = dω Ω, Ω φ Ω k = φ k, and ˆk = φ k. Here, ρ e =, ˆ Ω and χ e are identiied as the eective density and compressibility tensors, which ultimately dictate the medium's acoustic absorption behavior. 3. Numerical Implementation in COMSOL

4 To assess the validity o the described MAM process and numerical implementation, we calculated eective tensors or the FCC geometry 7,8 depicted in Fig. subect to mirror symmetry (y- and z-directions) and translational periodicity (-direction) boundary conditions. This igure depicts an irreducible unit cell consisting o a luid-illed interstitial space between FCC packed spheres. a. sphere contact point, where the solder radius used is 5 μm. 4. Results and Discussion For a given requency, using COMSOL Multiphysics we obtain numerical results or the macro-scale velocity and temperature distributions within an irreducible UFC. We present results where we restrict the unit vector e to the -ais. Fig. provides an eample dynamic and thermal permeability distribution at 5 Hz in a y-z plane. b. Figure (a) original FCC structure (b) corresponding meshed FCC UFC in COMSOL multiphysics. Using COMSOL Multiphysics, the numerical solutions o the dynamic viscous and heat equations are calculated rom Eqs. (9) and (). Table provides the coeicients used to arrive at the numerical solutions. Table Coeicients or a FCC stacking o beads. ρ T P μ Pr γ [kgm -3 ] [K] [Pa] [kg(ms) - ] To directly compare our numerical results with those presented in the literature 7,8, the radius o the sphere is chosen as mm. As in the cited studies, we also include a soldering neck at each Figure Magnitude o dynamic (top) and heat (bottom) permeability distributions at 5 Hz. We then employ a COMSOL script to iteratively compute the eective density and compressibility tensors (3) in a requency range o interest (here, rom to, Hz). The results are presented in Fig. 3. For validation purposes, numerical points provided in the literature 8 are also plotted. As can be seen in the subigures, the requency-dependent eective tensors obtained rom our approach are in very

5 good agreement with those provided by Re. 8 over the entire range o requencies considered. Moreover, as additional outcomes o the numerical procedure, the porosity (φ ), the static viscous permeability and tortuosity ( k, α ), the tortuosity ( α ), the static thermal permeability and tortuosity ( k, α ), and the viscous and thermal characteristic lengths ( Λ, Λ ) are automatically computed. These parameters provide an alternative, analytic-based means to calculate the eective density and compressibility tensors 9,,. The results are gathered in Table or comparison with those available in the literature 5,7,8. Real eective density [kg/m 3 ] a. COMSOL's results S.Gasser et al. (Re. 8) 3 4 Frequency [Hz] (log) presence o soldering necks. Also, the static viscous tortuosity is 4 % lower than the value obtained in this study. A possible eplanation or this disagreement is the use o a dierent deinition used to calculate the value values obtained rom Res. 7 and 8 were indirectly calculated using a very low requency 3 ( Hz), whereas our approach was directly estimated according to Re., α = k k. (3) Table Computations o the acoustic properties o a FCC sphere stacking versus those available in the literature. Re. 5 Res. 7, 8 COMSOL φ k α NA.63.4 α k NA.74.7 α NA Λ Λ NA Imaginary eective density [kg/m 3 ] (log) Eective compressibility [-5 Pa-] COMSOL's results S.Gasser et al. (Re. 8) 3 4 Frequency [Hz] (log) b. c. Real (COMSOL's results) -Imaginary (COMSOL's results) Real (S.Gasser et al. (Re. 8)) -Imaginary (S.Gasser et al. (Re. 8)) 3 4 Frequency [Hz] (log) Figure 3 (a) real and (b) imaginary parts o the eective density via requency (c) eective compressibility via requency. Table shows good agreement between published results and our numerical results. However, as concerns the viscous characteristic length, there are disagreements with three numerical values. As mentioned in Re. 7, this can be eplained by the 5. Conclusions In conclusion, we present a consistent and general approach or numerically computing requency-dependent eective density and compressibility tensors or periodic porous materials. These tensors ultimately characterize the acoustic absorption behavior o the material. The ormulation is cast in a orm suitable or incorporation into COMSOL Multiphysics, and as such, avoids the complication o working with and coupling multiple in-house codes, as done in previous investigations. Comparisons o results generated in this study with published numerical data demonstrate very good agreement over all requencies considered. 6. Reerences. G. Kirchho, Über den einluß der wärmeleitung in einem gase au die schallbewegung, Ann. Phys. Chem. 34, 77 (868).. M. A. Biot, Theory o propagation o elastic waves in a luid saturated porous solid II: Higher

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