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The sound of a pulsating sphere in a rarefied gas: continuum breakdown at short length and time scales

Published online by Cambridge University Press:  24 May 2019

Y. Ben-Ami  and
A. Manela Open the ORCID record for A. Manela [Opens in a new window]
Y. Ben-Ami
Affiliation:
Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel
A. Manela*
Affiliation:
Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel
*
†Email address for correspondence: [email protected]
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Abstract

The pressure field of a pulsating sphere is a canonical problem in classical acoustics, used to illustrate the acoustic efficiency of a monopole source at continuum conditions. We consider the counterpart vibroacoustic and thermoacoustic problems in a rarefied gas, to investigate the effect of continuum breakdown on monopole radiation. Focusing on small-amplitude normal-to-boundary mechanical and heat-flux excitations, the perturbation field is analysed in the entire range of gas rarefaction and input frequencies. Numerical calculations are carried out via the direct simulation Monte Carlo method, and are used to validate analytical predictions in the free-molecular and near-continuum regimes. In the latter, the regularized thirteen moments model (R13) is applied, to capture the system response at states where the Navier–Stokes–Fourier (NSF) description breaks down. Comparing with the continuum inviscid solution, the results quantitate the dampening effect of gas rarefaction on source point-wise strength and acoustic power. At near-continuum conditions, the acoustic field is composed of exponentially decaying ‘compression’, ‘thermal’ and ‘Knudsen-layer’ modes, reflecting thermoviscous and higher-order rarefaction effects. With reducing rarefaction, the contributions of the latter two modes vanish, and the former degenerates into the ideal-flow inverse-to-distance decaying wave. Stronger attenuation is obtained with increasing rarefaction, where boundary sphericity results in a ‘geometric reduction’ of the molecular layer affected by the source. Notably, while the R13 model at low frequencies appears valid up to moderate gas rarefaction rates, both the NSF and R13 descriptions break down at common low Knudsen numbers at higher frequencies. Further study should therefore be carried out to extend the applicability of moment models to unsteady flows with short time scales.

JFM classification

Acoustics: Acoustics Rarefied Gas Flow: Rarefied Gas Flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 871 , 25 July 2019 , pp. 668 - 693
Copyright
© 2019 Cambridge University Press 

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References

Abramowitz, M. 1953 Evaluation of the integral ∫0 ∞ e-u 2-x/u  du . J. Math. Phys. 32, 188–192.Google Scholar
Arnold, H. D. & Crandall, I. B. 1917 The thermophone as a precision source of sound. Phys. Rev. 10, 22–38.Google Scholar
Bargatin, I., Kozinsky, I. & Roukes, M. L. 2007 Efficient electrothermal actuation of multiple modes of high-frequency nanoelectromechanical resonators. Appl. Phys. Lett. 90, 093116.Google Scholar
Beckmann, A. F., Rana, A. S., Torrilhon, M. & Struchtrup, H. 2018 Evaporation boundary conditions for the linear R13 equations based on the Onsager theory. Entropy 20, 680–708.Google Scholar
Ben Ami, Y. & Manela, A. 2017 Acoustic field of a pulsating cylinder in a rarefied gas: thermoviscous and curvature effects. Phys. Rev. Fluids 2, 093401.Google Scholar
Bird, G. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon.Google Scholar
Claydon, R., Shrestha, A., Rana, A. S., Sprittles, J. E. & Lockerby, D. A. 2017 Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects. J. Fluid Mech. 833, R4.Google Scholar
Crut, A., Maioli, P., Fatti, N. D. & Vallée, F. 2015 Acoustic vibrations of metal nano-objects: time-domain investigations. Phys. Rep. 549, 1–43.Google Scholar
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 2, 331–407.Google Scholar
Greenspan, M. 1956 Propagation of sound in five monatomic gases. J. Acoust. Soc. Am. 28, 644–648.Google Scholar
Gu, X. & Emerson, D. 2007 A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions. J. Comput. Phys. 225, 263–283.Google Scholar
Gu, X. & Emerson, D. 2009 A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. J. Fluid Mech. 636, 177–216.Google Scholar
Hadjiconstantinou, N. G. 2002 Sound wave propagation in transition-regime micro- and nanochannels. Phys. Fluids 14, 802–809.Google Scholar
Hadjiconstantinou, N. G. & Garcia, A. L. 2001 Molecular simulations of sound wave propagation in simple gases. Phys. Fluids 13, 1040–1046.Google Scholar
Hu, H., Wang, Z., Wu, H. & Wang, Y. 2012 Analysis of spherical thermo-acoustic radiation in gas. AIP Adv. 2, 032106.Google Scholar
Iannacci, J., Huhn, M., Tschoban, C. & Potter, H. 2016 RF-MEMS technology for future mobile and high-frequency applications: reconfigurable 8-bit power attenuator tested up to 110 GHz. IEEE Electron. Device Lett. 37, 1646–1649.Google Scholar
Julius, S., Gold, R., Kleiman, A., Leizeronok, B. & Cukurel, B. 2018 Modeling and experimental demonstration of heat flux driven noise cancellation on source boundary. J. Sound Vib. 434, 442–455.Google Scholar
Juvé, V., Crut, A., Maioli, P., Pellarin, M., Broyer, M., Fatti, N. D. & Vallée, F. 2010 Probing elasticity at the nanoscale: terahertz acoustic vibration of small metal nanoparticles. Nano Lett. 10, 1853–1858.Google Scholar
Kalempa, D. & Sharipov, F. 2014 Numerical modelling of thermoacoustic waves in a rarefied gas confined between coaxial cylinders. Vacuum 109, 326–332.Google Scholar
Kogan, M. N. 1969 Rarefied Gas Dynamics. Plenum.Google Scholar
Landau, L. & Lifshitz, E. 1959 Fluid Mechanics. Pergamon.Google Scholar
Loyalka, S. K. & Cheng, T. C. 1979 Sound-wave propagation in a rarefied gas. Phys. Fluids 22, 830–836.Google Scholar
Manela, A. & Hadjiconstantinou, N. G. 2007 On the motion induced in a gas confined in a small-scale gap due to instantaneous boundary heating. J. Fluid Mech. 593, 453–462.Google Scholar
Manela, A. & Hadjiconstantinou, N. G. 2010 Gas-flow animation by unsteady heating in a microchannel. Phys. Fluids 22, 062001.Google Scholar
Manela, A. & Pogorelyuk, L. 2014 Cloaking via heating: approach to acoustic cloaking of an actuated boundary in a rarefied gas. Phys. Fluids 26, 062003.Google Scholar
Manela, A. & Pogorelyuk, L. 2015 Active noise control of a vibrating surface: continuum and non-continuum investigations on vibroacoustic sound reduction by a secondary heat-flux source. J. Sound Vib. 358, 20–34.Google Scholar
Manela, A., Radtke, G. A. & Pogorelyuk, L. 2014 On the damping effect of gas rarefaction on propagation of acoustic waves in a microchannel. Phys. Fluids 26, 032001.Google Scholar
Marty, R., Arbouet, A., Girard, C., Mlayah, A., Paillard, V., Lin, V. K., Teo, S. L. & Tripathy, S. 2011 Damping of the acoustic vibrations of individual gold nanoparticles. Nano Lett. 11, 3301–3306.Google Scholar
McDonald, F. A. & Wetsel, G. C. 1978 Generalized theory of the photoacoustic effect. J. Appl. Phys. 49, 2313–2322.Google Scholar
Morse, P. M. 1948 Vibration and Sound. McGraw-Hill.Google Scholar
Nassios, J., Yap, Y. W. & Sader, J. E. 2016 Flow generated by oscillatory uniform heating of a rarefied gas in a channel. J. Fluid Mech. 800, 433–483.Google Scholar
Pelton, M., Sader, J. E., Burgin, J., Liu, M., Guyot-Sionnest, P. & Gosztola, D. 2009 Damping of acoustic vibrations in gold nanoparticles. Nat. Nanotechnol. 4, 492–495.Google Scholar
Rana, A. S., Lockerby, D. A. & Sprittles, J. E. 2018 Evaporation-driven vapour microflows: analytical solutions from moment methods. J. Fluid Mech. 841, 962–988.Google Scholar
Shinoda, H., Nakajima, T., Ueno, K. & Koshida, N. 1999 Thermally induced ultrasonic emission from porous silicon. Nature 400, 853–855.Google Scholar
Sirovich, L. & Thurber, J. K. 1965 Propagation of forced sound waves in rarefied gasdynamics. J. Acoust. Soc. Am. 37, 329–339.Google Scholar
Sone, Y. 1965 Effect of sudden change of wall temperature in rarefied gas. J. Phys. Soc. Japan 20, 222–229.Google Scholar
Struchtrup, H. 2005 Macroscopic Transport Equations for Rarefied Gas Flows. Springer.Google Scholar
Struchtrup, H. 2012 Resonance in rarefied gases. Contin. Mech. Thermodyn. 24, 361–376.Google Scholar
Struchtrup, H. & Torrilhon, M. 2003 Regularization of Grad’s 13 moment equations: derivation and linear analysis. Phys. Fluids 15, 2668–2680.Google Scholar
Tamayo, J. 2005 Study of the noise of micromechanical oscillators under quality factor enhancement via driving force control. J. Appl. Phys. 97, 044903.Google Scholar
Torrilhon, M. 2016 Modeling nonequilibrium gas flow based on moment equations. Annu. Rev. Fluid Mech. 48, 429–458.Google Scholar
Torrilhon, M. & Struchtrup, H. 2008 Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227, 1982–2011.Google Scholar
Wadsworth, D. C., Erwin, D. A. & Muntz, E. P. 1993 Transient motion of a confined rarefied gas due to wall heating or cooling. J. Fluid Mech. 20, 219–235.Google Scholar
Wente, E. 1922 The thermophone. Phys. Rev. 19, 333–345.Google Scholar
Yap, Y. W. & Sader, J. E. 2016 Sphere oscillating in a rarefied gas. J. Fluid Mech. 794, 109–153.Google Scholar
Yariv, E. & Brenner, H. 2004 Flow animation by unsteady temperature fields. Phys. Fluids 16, L95–L98.Google Scholar