Bulk Viscosity Model for Near-Equilibrium Acoustic Wave Attenuation

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Bulk viscosity model for near-equilibrium acoustic wave attenuation Jeffrey Lin,1, a) Carlo Scalo,2 and Lambertus Hesselink1 1)Department of Electrical Engineering, Stanford University, Stanford, CA, 94305 USA 2)School of Mechanical Engineering, Purdue University, West Lafayette, IN, 47907 USA (Dated: July 20, 2017) Acoustic wave attenuation due to vibrational and rotational molecular relaxation, under simplifying assumptions of near-thermodynamic equilibrium and absence of molecular dissociations, can be accounted for by specifying a bulk viscosity coefficient µB. In this paper, we propose a simple frequency-dependent bulk viscosity model which, under such assumptions, accurately captures wave attenuation rates from infrasonic to ultrasonic frequencies in Navier–Stokes and lattice Boltzmann simulations. The proposed model can be extended to any gas mixture for which molecular relaxation timescales and attenuation measurements are available. The performance of the model is assessed for air by varying the base temperature, pressure, relative humidity hr, and acoustic frequency. Since the vibrational relaxation timescales of oxygen and nitrogen are a function of humidity, for certain frequencies an intermediate value of hr can be found which maximizes µB. The contribution to bulk viscosity due to rotational relaxation is verified to be a function of temperature, confirming recent findings in the literature. While µB decreases with higher frequencies, its effects on wave attenuation become more significant, as shown via a dimensionless analysis. The proposed bulk viscosity model is designed for frequency-domain linear acoustic formulations but is also extensible to time-domain simulations of narrow-band frequency content flows.

PACS numbers: PACS: 43.28.Bj, 43.28.Js, 43.35.Ae, 43.35.Fj

I. INTRODUCTION so as to affect acoustic energy but not so intense asto require a full description of molecular-level energy ex- 2–7 The viscous stress tensor of an isotropic Newtonian change dynamics. Bulk viscosity can significantly af- fluid reads fect the attenuation rate of freely propagating waves in mixtures composed of multi-atomic species—for exam- 1,8 ′ ∂uk ple, µB ≃ 2000µ for CO2 or for dry air at 110 Hz— σij = −pδij + τij = −p + µ δij + 2µSij , (1) ∂xk and its effect accumulates over each cycle of acoustic wave propagation, making it very important at higher (e.g., ul- where p is the thermodynamic pressure, ui and xi are the trasonic) acoustic frequencies. The accurate knowledge velocity component and spatial coordinate along the i-th of bulk viscosity values is desired in high-fidelity simu- direction, Sij = 1/2 (∂ui/∂xj + ∂uj/∂xi) is the strain- lations of aeroacoustics problems, with applications in- rate tensor, and µ′ is the second coefficient of viscosity cluding thermoacoustic energy conversion and imaging, acoustic energy transfer (AET)9, ultrasonic air-coupled ′ 2 10 µ = µB − µ , (2) non-destructive examination, and supersonic flow. In 3 other applications, such as for acoustic trapping, the accurate prediction of heat generation due to both shear where µB and µ are the bulk (or volume) and shear (or 11 dynamic) viscosity, respectively. While shear viscosity and bulk viscosity specifically is critical. Bulk viscosity is a well-characterized fluid property, bulk viscosity is effects are also very important in the presence of strong neglected in most fluid problems due to its often un- spatial density gradients, such as in shock waves, which known value. The commonly adopted Stokes’ hypothe- yield high (negative) values of the velocity divergence sis, which holds for dilute, monatomic gases, in fact reads term in Eq. (1). The distinction between losses due to µ′ = − 2 µ, implying µ = 0. However, for many fluids shear or bulk viscosity effects is also relevant to acous- 3 B tic problems in which the phasing of dilatation can be and flows of interest, µ cannot be neglected.1 Further- B decoupled from that of pressure. more, in the context of wave attenuation modeling, bulk

arXiv:1707.05876v1 [physics.flu-dyn] 18 Jul 2017 Numerical simulations of wave attenuation due to bulk viscosity is not interpreted as a pure fluid property, as it viscosity effects and estimates of bulk viscosity coefficient depends on flow conditions, e.g., acoustic frequency. values have been attempted at various scales of fidelity Experimental and semi-empirical characterizations of by previous authors. Eu and Ohr12 have shown that wave attenuation in air confirm that bulk viscosity ef- Navier–Stokes simulations without bulk viscosity fail to fects are due to molecular relaxation, a non-equilibrium fully capture absorption and dispersion for polyatomic thermodynamic process which is sufficiently important gases, and Claycomb et al.13,14 have shown that under rarefied flow conditions and Mach numbers up to 15, the introduction of bulk viscosity can improve predictions of shock thickness and separation in a double cone geom- a)Electronic mail: [email protected]; Corresponding author. etry. Wagnild et al.15 evaluated vibrational relaxation 2

∗ effects in hypersonic boundary layers using direct numeri- where nk/n is the species mole fraction, and Tvib,k and 16 cal simulations, and Valentini et al. used direct molecu- Tvib,k are, respectively, the characteristic molecular vi- lar simulation to predict rotational and vibrational relax- bration temperature and species vibrational tempera- ation effects in high-temperature nitrogen. Salomons et ture. At thermodynamic equilibrium, the fluid tempera- al.17 performed lattice Boltzmann method (LBM) simu- ture T is also equal to the apparent vibrational tempera- lations without accounting for bulk viscosity effects, find- tures Tvib,k, and there is zero net energy exchange among ing that the accuracy of simulated acoustic wave attenu- the degrees of freedom. ation is limited by LBM stability and dissipation to short However, under thermodynamic non-equilibrium con- propagation distances, low frequencies, and high viscos- ditions, both vibrational and rotational energies diverge 18 ity. Viggen developed a LBM approach which captures from their equilibrium values, and the process of relax- absorption and dispersion due to both shear and bulk ation to equilibrium conditions is represented by the bulk viscosity for a hypothetical single-species fluid, with re- viscosity coefficient. The equivalent relaxation process sults in agreement with theoretical predictions. Ern and for translational degrees of freedom is embedded within 19–21 Giovangigli developed multicomponent transport al- the shear viscosity coefficient. Such attenuative pro- gorithms to accurately calculate various transport coeffi- cesses are of particular interest in sound propagation, cients, including bulk viscosity associated with rotational where pressure and velocity fluctuations (i.e. fluctua- molecular relaxation. tions in translational molecular energy) establish a cycle To the authors’ knowledge, established methods pro- of compressions and expansions accompanied by temper- viding the value of µB for the prediction of acoustic wave ature fluctuations. However, unlike shear viscosity, ab- attenuation rates are currently missing in the literature. sorption of sound is the primary effect of bulk viscos- In this manuscript, we present a model for µB valid in ity, and the experimental characterization of bulk vis- near-equilibrium acoustic energy absorption conditions, cosity in various materials has predominantly been com- which can be adopted in frequency-domain formulations pleted through various forms of acoustic wave attenua- as well as narrow-band time-domain simulations. tion measurements.22–25

Bulk viscosity, µB, as defined in the present paper by Eqs. (1) and (2), is sometimes referred to in the literature as volume viscosity26, dilatational viscosity27, expansion A. Bulk viscosity origins and measurements viscosity28,29, or the second coefficient of viscosity30. The second coefficient of viscosity µ′, as defined in the present While bulk viscosity is negligible for dilute monatomic paper by Eq. (2),31 has also been referred to by some gases, as confirmed theoretically and experimentally,1 authors28,29 as the bulk viscosity; this definition results it is generally non-zero for polyatomic gases. For ex- in a (misleadingly) negative value for bulk viscosity in ample, a diatomic molecule has six degrees of freedom: dilute, monatomic gases. According to the convention three translational, two rotational, and one vibrational. adopted in the present paper, bulk viscosity is µB = 0 The translational degrees of freedom relax to equilib- for dilute monatomic gases, consistent with literature in rium conditions relatively quickly, while the rotational acoustics and molecular kinetics.25 and vibrational degrees of freedom have longer relaxation The physical complexity of the relaxation processes times, hence delaying thermodynamic equilibrium when modeled by bulk viscosity contribute further to the con- excited. At thermodynamic equilibrium, the average en- fusion in the literature. Some authors4 attribute bulk ergy for each translational and rotational degree of free- viscosity to rotational molecular relaxation only, while dom is kBT/2, where kB is Boltzmann’s constant and others8 also include vibrational effects. In the present the fluid temperature T is equal to the apparent transla- manuscript, both the effects of vibrational and rotational tional temperature Ttr and rotational temperature Trot. molecular relaxation are incorporated within µB by def- The corresponding translational and rotational energies inition. are given by In acoustics, bulk viscosity effects due to rotational re- 3 laxation are of secondary importance; for air µ ≈ E = R T (3) B,rot tr 2 gas tr 0.6µ for approximately all frequencies, with some depen- 1 5 − 3γ dency on temperature,4 as discussed later. On the other E = R T , (4) rot 2 γ − 1 gas rot hand, the vibrational relaxation contribution can be relatively large, e.g., µB,vib ≈ 25µ at 1000 Hz in dry air where Rgas is the specific gas constant and γ is the ratio under standard atmospheric conditions. of specific heats. The vibrational energy associated with The combined effect of rotational and vibrational re- the k-th molecular species is derived from the Boltzmann laxation can be inferred from the absorption coefficient, distribution and is given by α—a measure of the relative wave amplitude attenuation per propagation distance—which is of particular interest n T ∗ k ∗ vib,k in atmospheric acoustics. For a purely traveling tonal Evib,k = RgasTvib,k exp − , (5) n Tvib,k disturbance, spatial attenuation follows the exponential 3 law complete attenuation relation

αλ[−(x−x0)/λ] " 2 # Pamp = Pamp,0 e (6a) ω2 4 (γ − 1) κ α = µ + + µB , (10) αλ[−(t−t0)ω/2π] 3 = Pamp,0 e , (6b) 2ρ0a0 3 γRgas where Pamp,0 is the amplitude of the traveling wave at which is at the core of the proposed method to estimate position x = x0 or, equivalently, at time t = t0, and ω is the value of µB. the angular frequency and λ is the wavelength. The classical acoustic absorption coefficient, or the Stokes-Kirchoff attenuation coefficient, only accounts for B. Paper Outline thermoviscous attenuation,32 " # ω2 4 (γ − 1)2 κ The remainder of the paper is organized as follows: αsk = 3 µ + , (7) Section II presents the derivation of the proposed bulk 2ρ0a 3 γRgas 0 viscosity model; Section III presents results from Navier– Stokes and LBM computations reproducing acoustic at- where ρ0 and a0 are the base density and speed of sound, and κ is the thermal conductivity. The first term on tenuation rates using the proposed bulk viscosity model; the right-hand side of Eq. (7) represents the effect of Section IV presents a discussion of the bulk viscosity coef- viscous dissipation and the second term represents the ficient as a function of pressure, temperature, frequency, effect of heat conduction; both processes have character- and humidity and its non-dimensionalization; and finally, istic timescales in the gigahertz range for gases such as Section V discusses the limitations of the current ap- argon, helium, and neon.32 As a result, the rate of lossy, proach and presents concluding thoughts. Appendix A 1 thermoviscous momentum transfer is small, compared to and Appendix A 2, respectively, provide details on the momentum transfer due to ideal wave propagation. The Navier-Stokes and LBM implementations used. classical absorption coefficient is derived from simplifying the more general form of the complex thermoviscous wave number of a damped monochromatic traveling wave, II. PROPOSED BULK VISCOSITY MODEL " s # ω (1 + iγτκω) (1 + iτν ω) αsk = Im (8a) The attenuation contributions from rotational and vi- a0 (1 + iτκω) brational molecular relaxation are collapsed into one bulk τ ω2 (γ − 1) τ ω2 viscosity coefficient, such that ≃ ν + κ (8b) 2a0 2a0 µB = µB(p0,T0, f, hr) (11a) 2 X (k) where τν = 4µ/3ρ0a0 is the viscous relaxation time, = µB,rot + µ , (11b) 2 B,vib τκ = κ/cp0ρ0a0 is the conduction relaxation time where k cp0 is the isobaric heat capacity, and the approximation 4 is taken by neglecting higher-order terms in the quan- where µB,rot is a function of only temperature T0, and tities τν ω and τκω. We note that τκ here should not µB,vib is a function of temperature, base pressure p0, fre- be confused with τk, which is the vibrational molecular quency f, and relative humidity hr, hence yielding the relaxation time explained in section II. general functional form of Eq. (11a). This form of µB 33 Bulk viscosity effects are attributed to the difference is sometimes referred to in the literature as frequency- between the observed attenuation rate and the rate pre- dependent bulk viscosity, as absorption of acoustic en- dicted by Eq. (7). The total attenuation rate therefore ergy associated with a given frequency is the primary includes contributions from rotational and vibrational technique for experimentally measuring the bulk viscos- molecular relaxation effects: ity in a gas. Such experimental measurements are in conditions of near-equilibrium, consistent with the as- X (k) sumptions made in the proposed model. In the following α = αsk + αrot + αvib , (9) k sections, the vibrational and rotational relaxation contributions are discussed separately. where the superscript k indicates the contribution from the k-th species. For air, the species in Eq. (9) are diatomic nitrogen (k = N2) and oxygen (k = O2). Other species such as water vapor do not contribute directly to A. Vibrational bulk viscosity Eq. (9), but instead act to adjust the vibrational relaxational frequencies. Based on the dispersion relation for plane traveling Combining the rotational and vibrational relaxation waves4 and statistical thermodynamics for the vibra- effects into one bulk viscosity coefficient µB yields the tional energy states, the attenuation contribution due to 4

105 k = O2 N2 nk/n 0.21 0.78 104 T ∗ 2239.1 K 3352.0 K (Hz) vib,k

2 3 10 −3 −2 O τk (hr = 0) 6.632 · 10 s 1.789 · 10 s f 2 −6 −4 10 τk (hr = 20.0%) 8.559 · 10 s 7.644 · 10 s 101 Table I. Input parameters for air and example τk for Eqs. (12) 104 and (13). τk are calculated for p0 = 1 atm and T0 = 300 K. 103 (Hz)

2 102 N f 101 100 10 1 100 101 102 provided in table I. The relaxation frequencies (in Hz) of − 6 hr (%) species in air are as derived by Bass et al. : −1 Figure 1. Vibrational molecular relaxation frequencies for fO2 = (2πτO2 ) oxygen f = (2πτ )−1 and for nitrogen f = (2πτ )−1 at O2 O2 N2 N2 p0 4 0.02 + h T0 = 300 K, plotted against relative humidity hr, for pressures = 24 + 4.04 · 10 h (14a) patm 0.391 + h p0 = 0.1 atm ( ), 1.0 atm ( ), 10.0 atm ( ), and 100.0 −1 atm ( ). fN2 = (2πτN2 )

p T 1/2 3 0 atm 10− = 9+ 5 × patm T0 ( " 1/3 #) ! 4 Tatm

s) 280h · exp −4.17 − 1 · T 3 0 (Pa (14b)

vib 1.261 ) 2

k log (psat/patm) = −6.8346 (T3p/T0) + 4.6151 ,

( B, 10

µ (14c) 1 where p = 101325 Pa is the reference atmospheric 0 atm 200 300 400 500 600 700 pressure; Tatm = 293.15 K, the reference atmospheric T0 (K) temperature; psat, the calculated saturation vapor pressure; and T3p = 273.16 K, the triple-point isotherm tem- (k) Figure 2. Modeled vibrational bulk viscosity µB,vib for di- perature. The contribution of water vapor to the relax- atomic nitrogen k = N2 ( ), and oxygen k = O2 ( ), ation frequency of each species in air is determined via plotted against temperature T0 (Eq. (16)). Results presented the relative humidity hr, which is related to the absolute are for ω/2π = 1 kHz in air at hr = 20.0% and p0 = 1 atm. humidity h as

psat h = hr . (15) vibrational molecular relaxation for the k-th species is: p0

(k) (γ − 1) cv,k ωτk Numerical constants in Eqs. (14a) and (14b) are dimen- αvib = 2 (12) sional. 2a0/ω cp 1 + (ωτk) ∗ 2 nk Tvib,k ∗ cv,k = Rgas exp(−Tvib,k/Tvib,k) , (13) n Tvib,k Sample curves for fO2 and fN2 plotted against relative humidity are shown in Fig. 1. While the above re- where τk is the vibrational molecular relaxation time, lations provided above are valid for air, the vibrational assigned according to the relaxation frequencies fk = bulk viscosity for a different mixture can be calculated −1 (2πτk) and given for air by the semi-empirical relation- provided species-specific characteristic molecular vibra- ∗ ships Eqs. (14a) and (14b), and cp is the isobaric specific tion temperatures Tvib,k and relaxation frequencies fk heat capacity. Assuming quasi-equilibrium, Tvib,k is set are known. equal to the base temperature T0. For air, the species involved are that of oxygen (k = O2) and nitrogen (k = N2), with relevant parameters Combining Eqs. (9), (11b), (12) and (14), the total 5 vibrational bulk viscosity coefficient can be written as 1.4

X (k) 1.2 µB,vib = µB,vib k 1.0 3 /µ (k) 2ρ0a0 (k)

rot 0.8 µB,vib = 2 αvib ω B, µ 0.6 ∗ 2 p0 2 nk Tvib,k = (γ − 1) 0.4 2π n T0 f 0.2 exp(−T ∗ /T ) k . (16) 200 300 400 500 600 700 800 900 1000 vib,k 0 f 2 + f 2 k T0 (K) For dry air at standard ambient temperature and pres- Figure 3. Ratio of the modeled rotational bulk viscos- sure at 1000 Hz, µB,vib ≃ 22µ, and an example of the ity and shear (or dynamic) viscosity, µB,rot/µ ( ), plotted dependence of Eq. (16) on temperature is shown in Fig. 2 against temperature (Eqs. (19) and (20)). Values for the ra- 19 for air with hr = 20% under standard ambient pressure. tio µB,rot/µ by Ern and Giovangigli ( ). Results presented are for air.

B. Rotational bulk viscosity mates, similarly ranging from 0.433 to 1.21 between 300 K to 1000 K, as shown in Fig. 3. The shear viscosity µ The rotational molecular relaxation contribution to the used here is given by overall attenuation rate, as per Eqs. (9), (10) and (11b), is nν µ = µref (T/Tref) (20) ω2 αrot = 3 µB,rot (17) where nν = 0.76 is the viscosity power-law exponent and 2ρ0a −5 −1 −1 0 µref = 1.98 × 10 kg m s and Tref = 300 K are the reference viscosity and temperature. Alternative formu- where µ is the rotational bulk viscosity. B,rot lations such as Sutherland’s law35 can also be used. Under the ideal gas assumption, the contribution to The overall absorption predicted using the com- bulk viscosity from rotational relaxation is expected to bined effective bulk viscosity developed in the preceding only be a function of the equilibrium temperature T .4,8 0 Eqs. (16) and (19) is shown in Fig. 5; these curves accu- In early literature,4,5,34 the rotational bulk viscosity for rately replicate experimental measurements of absorption air is described as being between µ /µ = 0.60 − 0.62, B,rot in air. where the ratio is independent of temperature. However, 33 more recent studies suggest that the ratio µB,rot/µ should increase with temperature. Ern and Giovangigli19 calculated the rotational bulk viscosity via a linear sys- III. TIME-DOMAIN ACOUSTIC WAVE ATTENUATION SIMULATIONS tems approach by using expansion functions of the energy levels of each species and approximations for collision in- tegrals, finding that the above ratio does in fact increase The bulk viscosity model as developed in section II is with temperature, as shown in Fig. 3. coupled with both a Navier–Stokes and a lattice Boltz- It follows from Bass et al.6,7 that the semi-empirical mann method (LBM) solver for verification purposes. expression for the combined attenuation from viscous, Implementation details can be found in appendix A 1 heat conduction, and rotational bulk viscosity effects (in for the Navier–Stokes solver and in appendix A 2 for the air) is LBM solver. The computational setup is identical for both solvers. r T p Monochromatic planar traveling wave simulations with α + α = 0 atm c f 2 , (18) sk rot T p 1 4096 points per wavelength have been performed in a atm 0 periodic domain. Frequencies in the range f = 101 − 105 −11 2 Hz have been tested with relative humidity levels of dry where c1 = 1.84 · 10 s /m. Combining Eqs. (9), (10), (17) and (18) yields the final air, 20%, and saturated air. Results for relative humidity expression for rotational bulk viscosity used in this paper: level of 20% only are shown in Fig. 5.

r 2 γpatm γRgas 4 (γ − 1) κ µB,rot = 2 c1T0 − µ − . (19) A. Fully compressible Navier–Stokes simulations 2π Tatm 3 γRgas

The resulting rotational bulk viscosity yields µB,rot/µ In the fully compressible Navier–Stokes simulations, ratios in good agreement with Ern and Giovangigli’s esti- the governing equations for mass, momentum, and energy 6

tω/2π 0 100 200 300 400 500 section II. 10 The wave attenuation rate reproduced by the LBM simulations matches the semi-empirical curves with an (Pa) 7 error of under 15% for all tested combinations. Tun-

amp ing of the MRT model, in particular that of the non- P 5 dimensionless viscosity and the chosen diagonalization, 0 100 200 x/λ 300 400 500 is specific for different fluid problems, and can reduce the observed discrepancy. 10 5

(Pa) 0 IV. DISCUSSION 0

p 5 − 10 A. Parametric Study −500.0 500.2 500.4 500.6 500.8 501.0 x/λ or tω/2π In this section, we explore the functional dependency Figure 4. Time history of pressure amplitudes (top) of a of Eq. (11a) for air only. The dependence of µB,vib and freely-traveling wave for ω/2π = 1 kHz with zero bulk viscos- µB,rot on temperature, pressure, and frequency is shown ity ( ) and with effective bulk viscosity ( ), for ω/2π = 10 in Fig. 7 for air with a relative humidity of hr = 80%, kHz with zero bulk viscosity ( ) and with effective bulk vis- Fig. 8 for air with a relative humidity of hr = 1%, and cosity ( ). Pressure fluctuation versus propagation distance Fig. 9 for dry air. The bulk viscosity is plotted as a sum in acoustic cycles for x/λ = tω/2π > 500 (bottom). Results of both nitrogen and oxygen species contributions. presented are from the Navier–Stokes solver with conditions Several trends are evident. First, the value of bulk of x0 = t0 = 0, hr = 20%, p0 = 101325 Pa, T0 =300 K, and viscosity is larger at low frequencies; while this does not P = 10 Pa. amp,0 necessarily mean that overall attenuation rate is larger at low frequencies, it does suggest that bulk viscosity effects cannot be completely neglected at low frequencies are solved in conservative form (see appendix A 1). (e.g., infrasonic atmospheric wave propagation). Second, Due to the monochromatic nature of the wave prop- the bulk viscosity value increases with pressure at lower agation, a fixed value of the effective bulk viscosity has frequencies, noting the plot normalization with respect been used for any given frequency. Initial conditions were to the base pressure. Third, peaks, which at low pres- set as a pure isentropic traveling wave with initial pressures can be identified as being due to either nitrogen or sure amplitude Pamp,0 = 10 Pa. Numerical experiments oxygen vibrational frequencies, tend to merge at higher yielded α as annotated in Fig. 5, which were extracted pressures, as illustrated by figures 7a and 7b. from time-series decaying pressure amplitudes, as shown ′ When plotted against frequency, the vibrational bulk in Fig. 4, where instantaneous pressure p and amplitude viscosity decreases exponentially beyond the vibrational Pamp are plotted against non-dimensional timescales as relaxation frequency of each species, as seen in subplots introduced in Eq. (6). Attenuation matched to within (d-f) of Figs. 7 to 9. At higher frequencies, the bulk 2% for all tested combinations. viscosity contribution from rotational relaxation defines the minimum value of µB. The presence of water vapor drastically affects the B. Lattice Boltzmann simulations magnitude of the vibrational bulk viscosity, as seen in Figs. 7 to 9. Reducing humidity from hr = 80% to 1% The lattice Boltzmann method is derived from Boltz- increases the maximum bulk viscosity value across the mann’s kinetic theory of gases and is analogous to a finite considered temperature and pressure combinations, and difference method for solving the Boltzmann equation. also shifts the peak towards higher temperatures. This While often used to simulate incompressible flow prob- shift demonstrates the need to evaluate bulk viscosity as lems, LBM has been successfully applied to compressible a function of humidity, temperature, as well as pressure. flow and acoustic problems as well, and the derivation of As a numerical example, at T0 = 273.15 K, reducing the compressible Navier–Stokes equations from the lat- humidity from 80% to 1% increases bulk viscosity for tice Boltzmann equations is well established.36 frequencies approximately below 500 Hz, but decreases The implementation of the proposed bulk viscosity bulk viscosity at mid frequency ranges (up through 20– model was carried out with a simple 2D LBM solver, 30 kHz) and at higher frequency ranges (above 30 kHz). using the common D2Q9 (two-dimensional, 9 velocities) Increasing humidity increases the relaxation frequency of scheme, with results shown in Fig. 5. The LBM solver is both nitrogen and oxygen species, and as seen in Eq. (16), extended with a multiple relaxation time (MRT) model, bulk viscosity will, for each species, reach a maximum at allowing the introduction of a separate bulk viscosity pa- a different combination of pressure and temperature. Be- rameter (see appendix A 2). The MRT model was cou- cause oxygen has a higher relaxation frequency than that pled with the bulk viscosity coefficient as calculated in of nitrogen, there is a region between the two frequencies 7

101

100

1 10− 10% 2 10− 50%

(1/m) 3 100% 10− α 0% 0% 4 10−

5 10% 10− 50% 6 100% 10− 1 2 3 4 5 6 10 10 10 f (Hz) 10 10 10

Figure 5. Acoustic amplitude attenuation rate (Eq. (6)) per unit length of propagation in air versus frequency, at T0 = 300 K and p0 = 1 atm, with semi-empirical expressions ( ) (Eqs. (9), (16) and (19)) and Stokes-Kirchoff attenuation expressions with rotational bulk viscosity ( ), without rotational bulk viscosity ( ) , and without bulk viscosity nor conduction ( ) (Eq. (7)); relative humidity in percentage reported in figure. Computationally-determined absorption via the Navier–Stokes solver for a relative humidity of hr = 20% for zero bulk viscosity ( ) and for calculated effective bulk viscosity ( ); absorption in the lattice Boltzmann solver for a relative humidity of 20% for zero bulk viscosity ( ) and for calculated effective bulk viscosity ( ).

100 B. Dimensionless scaling of bulk viscosity 1 10− 10 2 s) − 1. Effective acoustic pressure · 3

(Pa 10−

B 4 µ 10− Effects related to bulk viscosity are largest near vi- 5 brational energy “peaks,” where the acoustic frequency 10− approaches the natural frequency of a vibrational mode. 10 6 −101 102 103 104 105 106 107 108 In order to evaluate the effect of bulk viscosity under dif- f (Hz) ferent conditions, we consider its effect on the acoustic effective (or mechanical) pressure. Bulk viscosity effects Figure 6. Bulk viscosity relative to frequency as calculated within the momentum and energy equations (Eq. (A1)) for air at T0 = 273.15 K and p0 = 1.0 atm for relative humid- result in an adjustment to the thermodynamic pressure, ity levels of h = 1% ( ), 40% ( ), and 80% ( ). r ′ peff = p − µB∇ · ⃗u , (21)

an expression which absorbs µB into the effective pressure gradient term. Per linear acoustics and assuming idealized oscillations, the continuity equation is for which µB achieves a maximum relative to relative hu- 1 ∂p′ midity; at lower frequencies, bulk viscosity decreases as + ∇ · u⃗′ = 0 . (22) ρ a2 ∂t humidity increases, and at higher frequencies, bulk vis- 0 0 cosity increases as humidity increases, as seen in Fig. 6. Adopting the harmonic convention p′ ∼ peˆ iωt and assuming either standing-wave or traveling-wave phasing Bulk viscosity in air with zero relative humidity (both corresponding to conditions where pressure oscil- ′ ◦ (Fig. 9) demonstrates no peaks with respect to temper- lations lead the velocity divergence term ∇ · ⃗u by 90 ), ature, as would be expected from Eqs. (14) and (16). the above equation simplifies to An increase in base temperature reduces vibrational re- ω laxation frequencies for the nitrogen species, but has no \⃗′ 2 |pˆ| − ∇ · u = 0 . (23) effect on the oxygen species. ρ0a0 8

(a) (d) 2 10− 1 2.0× 10− 100 Hz 2 215 K 10− 1.5 3 10−

s/atm) 4 · 10− 1.0

(Pa 5 10− 0 6 /p 10− B 0.5 µ 7 10− 0.0 10 8 200 300 400 500 600 700 800 −101 102 103 104 105 106 107 (b) (e) 3 10− 1 4.5× 10− 4.0 1 kHz 2 273 K 10− 3.5 10 3 3.0 −

s/atm) 4 · 2.5 10−

(Pa 5 2.0 10− 0 1.5 6 /p 10− B 1.0 µ 7 0.5 10− 0.0 10 8 200 300 400 500 600 700 800 −101 102 103 104 105 106 107 (c) (f) 4 10− 3 9× 10− 8 10 kHz 400 K 7 4 6 10− s/atm) · 5

(Pa 4 0 5 3 10− /p

B 2 µ 1 0 10 6 200 300 400 500 600 700 800 −101 102 103 104 105 106 107 T (K) f (Hz)

Figure 7. Pressure-normalized bulk viscosity µB /p0 for air at hr = 80%. Bulk viscosity is plotted against temperature at frequencies of 100 Hz (a), 1 kHz (b), and 10 kHz (c) with pressures p0 = 0.1 atm ( ), 1.0 atm ( ), 10.0 atm ( ), and 100.0 atm ( ). Bulk viscosity is plotted against frequency at temperatures of 215.15 K (d), 273.15 K (e), and 400 K (f) with pressures p0 = 0.1 atm ( ), 1.0 atm ( ), 10.0 atm ( ), and 100.0 atm ( ).

Finally, combining Eqs. (21) and (23) results in with gas temperature, pressure, and humidity, as shown in Fig. 10. µ ω B \⃗′ |pˆ| = µB ∇ · u = p\eff − p , (24) γp0 thus suggesting the dimensionless group 2. Dimensionless groups

µ ∇\ · u⃗′ For many fluid problems, dimensionless groups can re- B p\eff − p ∗ µBω duce the number of variables involved, helping to under- µB = = = , (25) γp0 |pˆ| |pˆ| stand the underlying phenomenon and simplifying experiments. However, the proposed bulk viscosity model ∗ where µB is a measure of the relative impact of bulk does not offer a straightforward non-dimensionalization, viscosity on pressure fluctuations. as the direct application of the Buckingham π theorem Traditional attenuation curves, such as Fig. 5, are mea- to Eqs. (11b), (16) and (19) is limited by constants in the sured relative to attenuation per meter and tend to belie relaxation frequencies (Eq. (14)) and by the saturation the effect of bulk viscosity at high frequencies. When vapor pressure. measured relative to acoustic wavelength, the bulk vis- As a result, for the general air case, neither frequency cosity contribution to attenuation can be as large as 1% f nor relative humidity hr can be part of a useful di- of pressure amplitude and has a magnitude peak varying mensionless group. Furthermore, temperature T0 is con- 9

(a) (d) 2 10− 1 8× 10− 100 Hz 2 215 K 7 10− 6 3 10− 5 s/atm) 4 · 10− 4

(Pa 5 10− 0 3 6 /p 10− B 2 µ 7 1 10− 0 10 8 200 300 400 500 600 700 800 −101 102 103 104 105 106 107 (b) (e) 2 10− 0 2.5× 10 1 1 kHz 10− 273 K 2.0 2 10− 10 3 s/atm) 1.5 − · 4 10− (Pa 5 0 1.0 10−

/p 6 B 10−

µ 0.5 7 10− 0.0 10 8 200 300 400 500 600 700 800 −101 102 103 104 105 106 107 (c) (f) 3 10− 1 7× 10− 10 kHz 2 400 K 6 10− 5 3 10− s/atm) · 4 4 10− (Pa 3 0 5 10− /p 2 B

µ 6 1 10− 0 10 7 200 300 400 500 600 700 800 −101 102 103 104 105 106 107 T (K) f (Hz)

Figure 8. Pressure-normalized bulk viscosity µB /p0 for air at hr = 1%. Bulk viscosity is plotted against temperature at frequencies of 100 Hz (a), 1 kHz (b), and 10 kHz (c) with pressures p0 = 0.1 atm ( ), 1.0 atm ( ), 10.0 atm ( ), and 100.0 atm ( ). Bulk viscosity is plotted against frequency at temperatures of 215.15 K (d), 273.15 K (e), and 400 K (f) with pressures p0 = 0.1 atm ( ), 1.0 atm ( ), 10.0 atm ( ), and 100.0 atm ( ).

strained by the triple-point isotherm temperature T3p sociated with rotational bulk viscosity may be neglected and pressure p0 is similarly constrained due to the satu- at lower frequencies. Considering instead only the vi- ration pressure, psat, which depends on T0. brational bulk viscosity, a dimensionless vibrational bulk To find useful scaling parameters, we consider two spe- viscosity cial cases. In the first case, relative humidity hr is allowed P µ(k) ω to vary without bounds. Combining Eqs. (16) and (19) ∗ k B,vib suggests that the dimensionless group µ∗ remains con- µB,vib = (27) B γp0 stant so long as scaling is applied simultaneously to frequency, pressure, and relative humidity, not only follows the scaling law Eq. (26) but also remains constant if the heat capacity ratio γ is scaled as ∗ µBω µB = = fn (f0, p0, hr,T0) γp0 ∗ µB,vib = fn (f0, p0, hr,T0, γ) = fn (b1f0, b1p0, b1hr,T0) , (26) ! (γ − 1)2 = b2 2 fn (b1f0, b1p0, b1hr,T0, b2γ) , for positive scaling parameter b1. (b2γ − 1) The second case considers splitting the dimensionless (28) bulk viscosity into rotational and vibrational relaxational components. The rotational bulk viscosity, as noted be- where both b1 and b2 are arbitrary scaling parameters. fore, solely depends on temperature, and the effect as- 10

(a) (d) 1 1 10 10− 100 Hz 10 2 215 K 100 − 3 10− 1 s/atm) 10 4 · − 10−

(Pa 5 2 10− 0 10− 6 /p 10− B 3 µ 10− 7 10− 10 4 10 8 −200 300 400 500 600 700 800 −101 102 103 104 105 106 107 (b) (e) 100 100 1 kHz 10 1 273 K 1 − 10− 2 10− 2 10 3 s/atm) 10 − · − 4 10− (Pa 3 5 0 10− 10−

/p 6 B 4 10−

µ 10 − 7 10− 10 5 10 8 −200 300 400 500 600 700 800 −101 102 103 104 105 106 107 (c) (f) 2 1 10− 10 10 kHz 100 400 K 3 1 10− 10− 10 2 s/atm) − · 4 3 10− 10− (Pa 4 0 10−

/p 5 5 B 10− 10− µ 6 10− 10 6 10 7 −200 300 400 500 600 700 800 −101 102 103 104 105 106 107 T (K) f (Hz)

Figure 9. Pressure-normalized bulk viscosity µB /p0 for dry air. Bulk viscosity is plotted against temperature at frequencies of 100 Hz (a), 1 kHz (b), and 10 kHz (c) with pressures p0 = 0.1 atm ( ), 1.0 atm ( ), 10.0 atm ( ), and 100.0 atm ( ). Bulk viscosity is plotted against frequency at temperatures of 215.15 K (d), 273.15 K (e), and 400 K (f) with pressures p0 = 0.1 atm ( ), 1.0 atm ( ), 10.0 atm ( ), and 100.0 atm ( ).

∗ ∗ For Eq. (28), µB,vib ≈ µB for small f, as shown in Fig. 11. ABC Because γ is related to the degrees of freedom of the fluid f (kHz) 0.5 1 2 in question, it is unlikely that scaling with b2 can be used without disrupting the general model. p0 (atm) 0.5 1 2 An example of both forms of scaling is shown in Fig. 11. hr (ref.) 10% 10% 10% Because the vibrational bulk viscosity has a dominant ∗ effect in the low-frequency regime, the scaling of µB ac- hr (similitude) 5% 10% 20% cording to Eq. (28) differs by less than 1% for frequencies under 10 kHz. Table II. Input parameters for self-similar simulations, as used in Fig. 12.

3. Self-similarity of wave attenuation where

A dimensionless collapse of the spatial attenuation re- ∗ ∗ Pamp = Pamp/Pamp,0 x = (x0 − x) /λ lationships Eqs. (6) and (10) gives 2 ∗ ω 4 ∗ ω (γ − 1) κ ∗ ∗ ∗ ∗ ∗ µ = µ κ = (30) log Pamp = π [µ + κ + µB] x (29) γp0 3 γp0 γRgas 11

f/10 (Hz) (a) 1 10− 0 1 2 3 4 5 6 7 101 10 10 10 10 10 10 10 2 10− 10−

3 2 10− 10− 4 (n.d.) 10− 3 ∗ B 5 (n.d.) 10− µ 10− ∗ B µ 6 4 10− 10− 7 10− 1 2 3 4 5 6 7 10 10 10 10 10 10 10 10 5 f (Hz) −101 102 103 104 105 106 107 108 (b) 2 f (Hz) 10−

3 ∗ 10− Figure 11. Dimensionless bulk viscosity µB for air versus frequency at T0 = 300 K for conditions of unscaled f, p0 = 1 atm, hr = 10%, and γ = 1.4 ( ) and of scaled dimensionless (n.d.) 4 10− bulk viscosity (see Eq. (28)) for conditions of scaled frequency ∗ B

µ f/10 kHz, p0 = 0.1 atm, hr = 1%, and γ = 1.66 ( ). This corresponds to scaling parameters of b = 0.1 and b = 1.19, 5 1 2 10− such that b2γ = 1.66. Under conditions of b2 = 1, the plots would be identical. 4 3 2 1 0 1 2 10− 10− 10− 10− 10 10 10 hr (%) (a) 1.0

∗ 0.5 Figure 10. Dimensionless effective bulk viscosity µB plotted

(n.d.) 0.0 against frequency at T0 = 215 K ( ), 273.15 K ( ), 400 0∗

p 0.5 K( ), and 600 K ( ). Results shown for hr = 5% and − atmospheric pressure (a). Dimensionless effective bulk vis- 1.0 ∗ − cosity µB plotted against relative humidity, hr, at T0 = 215 500.0 500.2 500.4 500.6 500.8 501.0 K( ), 273.15 K ( ), 400 K ( ), and 600 K ( ). Results (b) 1.0 shown for ω/2π = 25 kHz and atmospheric pressure (b). 0.5

(n.d.) 0.0

∗ 0∗ p 0.5 are the chosen normalizations and µB is as given in − Eq. (25). 1.0 − 500.0 500.2 500.4 500.6 500.8 501.0 The similitude scaling in Eq. (26) is tested using the x/λ Navier–Stokes simulations, with both imperfect pressure- frequency scaling and pressure-frequency-humidity scal- Figure 12. Scaled dimensionless pressure fluctuations p′∗ = ing, as reported in table II. The simulation results are ′ p /Pamp,0 from Navier–Stokes simulations of a freely-traveling shown in Fig. 12; as expected, the scaling form of Eq. (26) wave for acoustic cycle x/λ > 500 (see Eqs. (29) and (30)) for produces self-similar attenuation rates if the simulations cases in table II: pressure-frequency similitude scaling (a) and account for the effect of humidity in air. pressure-frequency-humidity scaling (b).

V. CONCLUSION ity is significant at high frequencies when normalized to absolute length scales; at low frequencies, attenuation is Bulk viscosity is often neglected in fluid problems, for significant when normalized by wavelength. lack of an established model. We have presented a bulk The present model has several limitations: (1) the viscosity model valid for tonal acoustic wave propaga- frequency-dependency of µB makes the model unsuit- tion in air, with verification via companion compressible able for broadband time-domain simulations; (2) it can- Navier–Stokes simulations and lattice Boltzmann method not capture broadband wave dispersion attributed to µB; simulations. The bulk viscosity model captures acoustic and (3) the current modeling framework and the ideal gas attenuation which otherwise is severely underestimated if assumption can be invalid for dense gases—in such con- Stokes’ hypothesis is assumed, and can be a simple addi- ditions, even monatomic gases can exhibit bulk viscosity tion to both time-domain and frequency-domain solvers. effects.37 The importance of incorporating bulk viscosity depends However, the present model provides several bene- on the fluid problem: attenuation related to bulk viscos- fits, as it is: (1) algorithmically simple; (2) extensible 12 to any gaseous mixture provided attenuation measure- simulations are, respectively, ments and species-specific characteristic molecular vibra- ∗ ∂ ∂ tion temperatures T and relaxation frequencies fk are vib,k (ρ) + (ρuj) = 0 (A1a) known; and (3) applicable to frequency-domain analysis, ∂t ∂xj commonly used in acoustics, and to near-monochromatic ∂ ∂ ∂ ∂ (ρui) + (ρuiuj) = − p + τij (A1b) time-domain problems. ∂t ∂xj ∂xi ∂xj The relationship between bulk viscosity, pressure, and ∂ ∂ ∂ (ρ E) + [uj (ρ E + p)] = (uiτij − qj) (A1c) dilatation is used to construct a bulk viscosity dimen- ∂t ∂xj ∂xj ∗ sionless group µB, applicable for both standing- and traveling-wave acoustics problems. Non-dimensional where x1, x2, and x3 (equivalently, x, y, and z) are ax- analysis of bulk viscosity suggests that it is a nontrivial ial and cross-sectional coordinates, ui are the velocity function of fluid properties, with limited opportunities to components in each of those directions, and p, ρ, and E apply similitude. Nevertheless, the bulk viscosity dimen- are respectively pressure, density, and total energy per sionless group does follow pressure-frequency similitude, unit mass. The gas is assumed to be ideal, with equa- with the caveat that relative humidity must be simulta- tion of state p = ρ Rgas T and a constant ratio of specific neously varied with pressure and frequency. Because of heats, γ. The gas constant is fixed and calculated as −1 saturation humidity, such scaling cannot always be pos- Rgas = pref (Tref ρref) , based on the reference thermo- sible. dynamic density ρref, pressure pref, and temperature Tref. The viscous and conductive heat fluxes are, respectively, The difference between effective and thermodynamic pressure suggests that the effects of bulk viscosity can ′ be manipulated in situations for which the phasing of µ ∂uk τij = 2µ Sij + δij (A2a) pressure and dilatation can be decoupled. Isolating the 2µ ∂xk effects of attenuation due to bulk viscosity from that of µ Cp ∂ shear viscosity can allow for the optimization of geome- qj = − T (A2b) tries where loss is important. Pr ∂xj

where Sij is the strain-rate tensor, given by Sij = (1/2) (∂uj/∂xi + ∂ui/∂xj); Pr is the Prandtl number; and µ is the dynamic viscosity, given by µ = nν µref (T/Tref) , where nν is the viscosity power-law expo- ′ ACKNOWLEDGMENTS nent and µref is the reference viscosity. µ is the second ′ 2 viscosity defined by Eq. (2), µB ≡ µ + 3 µ, where µB is the effective bulk viscosity value capturing the combined Jeffrey Lin and Carlo Scalo acknowledge the support effects of rotational and vibrational molecular relaxation. of the Inventec Stanford Graduate Fellowship and the Simulations have been carried out with the follow- −3 Precourt Energy Efficiency Center Seed Grant at Stan- ing gas properties: γ = 1.4, ρref = 1.2 kg m , pref = −5 −1 −1 ford University. The authors also acknowledge the gen- 101 325 Pa, Tref = 300 K, µref = 1.98 × 10 kg m s , 38 erous computational allocation provided to Dr. Scalo Pr = 0.72, and nν = 0.76, valid for air. on Purdue’s latest supercomputing architecture, Rice, The governing equations are solved using CharLESX , and the technical support of Purdue’s Rosen Center for a control-volume-based, finite-volume solver for the fully Advanced Computing (RCAC). Dr. Scalo acknowledges compressible Navier–Stokes equations on unstructured the support of the Air Force Office of Scientific Research grids, developed as a joint-effort among researchers at (AFOSR) grant FA9550-16-1-0209 and the very fruitful Stanford University. CharLESX employs a three-stage, discussions with Dr. Ivett Leyva (AFOSR) on ultrasonic third-order Runge-Kutta time discretization and a grid- wave attenuation in hypersonic boundary layers. adaptive reconstruction strategy, blending a high-order polynomial interpolation with low-order upwind fluxes.39 The code is parallelized using the Message Passing Inter- face (MPI) protocol and highly scalable on a large number of processors.40 Appendix A: Bulk Viscosity Implementation in Time-Domain Simulations

2. Lattice Boltzmann Equations 1. Fully compressible Navier–Stokes

The LBM solver uses the Bhatnagar-Gross-Krook The conservation equations for mass, momentum, and (BGK) collision model. Variables given in this section energy solved in the fully compressible Navier–Stokes will be dimensionless in the LBM solver. The quadra- 13 ture points selected for each lattice node are these relaxation times introduced by MRT correspond to “ghost modes,” which have no basis in kinetic theory, but  can enhance stability and indirectly affect fluid 41flow. (0, 0) i = 0  Nevertheless, some of these modes have direct physical ⃗ei = (1, 0), (0, 1), (−1, 0), (0, −1) i = 1, 2, 3, 4 connections to decoupling bulk and shear viscosity, and  (1, 1), (−1, 1), (−1, −1), (1, −1) i = 5, 6, 7, 8 , the presented model avoids tuning the MRT parameters unphysically. LBM with MRT and the proposed bulk (A3) viscosity model (section II) can fully capture acoustic at- with corresponding weights of tenuation in air. For the chosen quadrature points and corresponding  4/9 i = 0 weights, the MRT model used is specified by the forward  and inverse matrices, wi = 1/9 i = 1, 2, 3, 4 (A4)    1/36 i = 5, 6, 7, 8 , 1 1 1 1 1 1 1 1 1   −4 −1 −1 −1 −1 2 2 2 2  and a streaming and collision update model of    4 −2 −2 −2 −2 1 1 1 1    ∗ ∗ ∗ ∗  0 1 0 −1 0 1 −1 −1 1  fi (⃗x + ⃗ei, t + 1) − fi (⃗x , t )   M =   1 ∗ ∗ eq ∗ ∗  0 −2 0 2 0 1 −1 −1 1  = − [fi (⃗x , t ) − f (⃗x , t )] , (A5)   τ ∗ i  0 0 1 0 −1 1 1 −1 −1    0 0 −2 0 2 1 1 −1 −1 where fi is the particle distribution function for each   eq   streaming direction, fi is the corresponding equilibrium  0 1 −1 1 −1 0 0 0 0  ∗ ∗ ∗ distribution, and ⃗x , t , and τ are the non-dimensional 0 0 0 0 0 1 −1 1 −1 grid, time, and relaxation time, respectively. The lattice (A9a) ∗ ∗ units are chosen to be ∆x = ∆t = 1,√ such that the ∗   non-dimensional lattice speed is a0 = 1/ 3. 4 −4 4 0 0 0 0 0 0 The relaxation time τ ∗ determines the fluid kinematic   4 −1 −2 6 −6 0 0 9 0  viscosity,   4 −1 −2 0 0 6 −6 −9 0    2τ ∗ − 1 4 −1 −2 −6 6 0 0 9 0  ∗ ∗ ∗ 2 1   ν = (τ − 1/2) (a0) = . (A6) −1   6 M = 4 −1 −2 0 0 −6 6 −9 0  36   4 2 1 6 3 6 3 0 9  For a channel of defined physical length L and nx grid     points, in order to represent a simulation of physical kine- 4 2 1 −6 −3 6 3 0 −9   matic viscosity ν0 and reference density ρ0, the conver- 4 2 1 −6 −3 −6 −3 0 9  sion from lattice units to physical units is defined as 4 2 1 6 3 −6 −3 0 −9 ∗ ∗ (A9b) x = Cxx t = Ctt ∗ ∗ ν = Cν ν ρ = Cρρ , (A7) and the diagonalization of where S = diag (0, s2, 1.4, 0, s5, 0, s7, s8, s9) , (A10) 2 2 ∗ ∗ L Cx Cxν 2τ − 1 2 Cx = Ct = = = Cx where it is noted that s1, s4, s6 can be set, without loss of nx Cν ν0 6ν0 generality, to 0, as mass and momentum are necessarily 2 Cx conserved, s5 = s7 is required and chosen to be 1.2, and Cν = Cρ = ρ0 . (A8) ∗ Ct s8 = s9 is also required and chosen to be 1/τ . The bulk viscosity is a tuned parameter and is related to s2; in the MRT model, the shear and bulk kinematic a. LBM with MRT viscosities are, respectively,

2 ∗ The lattice Boltzmann method in its simplest form as- s − 1 2τ − 1 ν∗ = 8 = (A11) sumes equal relaxation times, τ ∗, across multiple mo- 6 6 ments. Multiple relaxation time (MRT) models extend 2 − 1 ∗ s2 LBM by removing this constraint, allowing different and ν = . (A12) B 6 multiple relaxation times to be used instead. Typically, MRT models replace τ ∗ as used in the previous section For implementation, the ratio of bulk viscosity to shear with separate relaxation times for each value. Several of (dynamic) viscosity, µB/µ, is used to select s2. 14

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