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 assump- tions 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 0 @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 µ0 is the second coefficient of viscosity cluding thermoacoustic energy conversion and imaging, acoustic energy transfer (AET)9, ultrasonic air-coupled 0 2 10 µ = µB − µ ; (2) non-destructive examination, and supersonic flow. In 3 other applications, such as for acoustic trapping, the ac- curate 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 µ0 = − 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 µ0, 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