Finite-amplitude Acoustic Wave Propagation through Gas-droplet Mixtures : a Burgers' Equation Model

An extended Burgers' enuation, suitable for media which exhibit both thermoviscous acoustic absorption as well as various mechanisms °f relaxational absorption, is presented. The equation is applied to the specific problem or piane wave propagation through an air-water fog, a medium which contains relaxation sources arising both from gas molecular motions and from mass, momentum, and energy transfer interactions between the gas and the suspended water droplets. The Burgers' equation approach provides information on infinitesimal wave attenuation and dispersion, and on finite-amplitude wave effects in air-water fogs. 1. List of symbols. A 1+C VL(L-1)(Y V-1)H V B RY-lCL-lJ-llKYv-lJCL-ilHy-Ry/RgirYAj3 3 C (d rop le t mass/cm ) / ( fog mass/cm ) 3 3 Cv (water vapour mass/cm ) / ( f og mass/cm ) c speed of sound i n fog c spec i f i c heat o f d rop le t 1/2 c speed of sound in a i r = (yRgT-% C l-c/c0 D diffusivity of water vapour in air f sound frequency g dimensionless retarded time =io(t-x/c) H droplet specific heat/air specific heat at p constant volume Hv ratio of water vapour to air specific heats at constant volume h droplet heat of vaporization 1 (-D K. relaxation strength of relaxation process j k velocity attenuation per wavelength L h/RvTQ a dimensionless distance = ux/c n droplet number density P Pra.ndtl number R„ v ideal gas constants of air and water vapour r water droplet radius T equilibrium temperature of fog t time u velocity perturbation due to acoustic field/cQ x distance a sound attenuation coefficient Y specific heat ratio of air Y v specific heat ratio of water vapour n (u/2p0cQ ) |4/3+(Y-l)/Pr| K thermal conductivity of air u absolute viscosity of air p equilibrium density of air p density of droplet material _ 2 T C mass transfer relaxation time =pr /3Dp _ 2 ir, momentum relaxation time = 2 pr /9y 2 ij heat transfer relaxation time = per / 3 K T. relaxation time of relaxation process j co sound frequency (radians/sec.) Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979806 JOURNAL DE PHYSIQUE C830 2. In t roduc t ion . Wave propagation processes have been successtul l y s tud ied using s ing le , r e l a t i v e l y simple d i f f e r e n t i a l equations t o model the f u l l s e t o f f l u i d mechanical conservat ion equation. Whitham ! 1 1 , f o r example, discusses the d e r i v a t i o n o f the wave equation, Korteweg Devries equation, and Burgers'equation from more general conservat ion laws. These equations can then be used i n the t h e o r e t i c a l study o f var ious types of wave problems. For f i n i t e amplitude, plane wave propagation through gaseous media, such authors as L i g h t h i l l (21, Blackstock 13) , and Lesser and Cr ighton 141 have demonstrated tne u t i l i t y o f the Burgers' equation approach. I n previous papers, the Burgers' equation technique has been extended t o describe wave propaga t ion through aerosol media. I n i t i a l l y , aerosols cons is t ing of monodisperse, s o l i d p a r t i c l e s suspended i n an i d e a l gas were considered 15 1 . Subsequentl y , mass t r a n s f e r e f f e c t s were inc luded f o r aerosols made up o f l i q u i d d rop le ts suspended i n an i d e a l gas m ix tu re o f 1 i q u i d vapour and an i n e r t gas 16 ( . F i n a l l y , the technique was extended t o inc lude d rop le t p o l y d i s p e r s i t y and gas molecular re laxa t ion , two e f f e c t s which should be inc luded i n a r e a l i s i c model f o r most fogs171. This l a t t e r equation has so f a r on ly been app l ied t o i n f i n i t e s i m a l , plane wave propagation through a i rzwate r fogs 171. The aim o f t h i s paper i s t o use the Burgers' equation approach t o study f i n i te-ampli tude e f f e c t s i n such media. 3. e o b l e m formulat ion. A dimensionless Burgers' equation f o r a i r -wa te r fogs may be shown t o be 16,71