Sound propagation in saturated gas-vapor-droplet suspensions with droplet evaporation and nonlinear relaxation Open Access

The propagation of sound in a gas containing suspended droplets is of considerable interest in many important technical applications such as combustion chamber instability, and jet noise mitigation by water injection. Theories of sound propagation in saturated gas-vapor-droplet mixtures have been presented by Marble,¹ Marble and Wooten,² Cole and Dobbins,³ and Davidson,⁴ among others.

In all the foregoing theories the particulate relaxation models for the sound attenuation are based on linear laws: Stokes drag (linear drag law), pure conduction limit (linear heat transfer), and pure mass diffusion limit (linear mass transfer). It is well known that for sufficiently large particle relaxation times, the linear laws break down.⁵ The linear theories do not account for or explain the attenuation behavior characterized by a spectral peak experimentally observed in the spatial absorption coefficient in supersonic jets.^6,7

The present work addresses the role of nonlinear relaxation on the attenuation and dispersion of sound in evaporating gas-droplet mixtures initially under thermodynamic equilibrium. The approach is based on an extension of the theory of Davidson⁴ to accommodate nonlinear particle relaxation processes.

Davidson⁴ considered plane wave solutions to the conservation equations for saturated gas-vapor-liquid systems by considering a solution to each dependent variable f in the form

where

and the complex wave number K is a function of several dimensionless variables

Here $ω$ is the circular frequency of sound, x is the coordinate distance, $c0$ is the sound speed in undisturbed (particle-free) gaseous mixture (=$γR0T0$⁠), $γ$ is the mixture specific heat ratio (isentropic exponent), and $R0$ and $T0$ denote, respectively, the gas constant and temperature of the undisturbed mixture, $Cm$ is the droplet mass loading, $Cv$ is the vapor mass loading, $γv$ is the specific heat ratio for vapor, $Rv$ and $Rg$ are the gas constants for the vapor and the gas, respectively, $Hg$ is the ratio of gas to mixture specific heats, $Hv$ is the ratio of vapor to mixture specific heats, $Hp$ is the ratio of particle to mixture specific heats, $L$ the dimensionless latent heat parameter (⁠$hL0/RvT0$⁠), $hL0$ is the latent heat of vaporization of undisturbed mixture. The time constants $τd,τt,τc$ are defined by

The subscripts $d,t,c$⁠, respectively, denote momentum, heat, and mass transfer, respectively. The Prandtl number $Pr$⁠, the Lewis number $Le$⁠, and the Schimdt number $Sc$ of the gas are

Physically the quantities $τd,τt,τc$ represent, respectively, the response of the particle to momentum, heat and mass transfer.

The dimensionless energy attenuation coefficient $α$ and the propagation speed $c$ are related to the complex wave number $K$ by the relation

The final results of Davidson⁴ for the complex wave number contained a few typographical errors.⁸ 

Stokes law for the drag force is valid for $ωτd≈1$⁠, provided that $ρ0/ρp≪1 and (ωdp2/8ν0)1/2≪1$⁠.⁵ Similar considerations apply for the linear laws for heat and mass transfer. Thus for a given frequency, the linear theories hold only for small droplet relaxation times $τd$ or droplet Reynolds numbers $Rep$⁠. Higher droplet Reynolds numbers are generally associated with higher droplet concentration. At high droplet Reynolds number and droplet relaxation time, the droplet viscous drag, heat and mass transfer are governed by nonlinear laws. It is evident that for $ωτd$ exceeding about unity (post-peak viscous region) and at high droplet concentration, significant non-linear influence on sound attenuation is envisaged.

The present analysis extends the work of Davidson⁴ to accommodate the effect of nonlinear relaxation on the mass, momentum and heat transfer between the droplet and the surrounding gas. Without any loss of generality the propagation of sound accounting for particle nonlinearity may be expressed by relations similar to Eqs. (2) as follows:

where the quantities $τd1,τt1,τc1$ correspond to the relaxation times in the presence of nonlinear relaxation. From physical considerations nonlinear relaxation is expected to alter the drag, heat and mass transfer which enter into the droplet relaxation times. The effect of nonlinearity on other physical parameters is believed to be of secondary importance to the present problem. The quantities $τd1,τt1,τc1$ are related to the relaxation times $τd,τt,τc$ by the relations

where

with $CD1,Nu1,Sh1$ stand, respectively, for the drag coefficient, Nusselt number and Sherwood number corresponding to circumstances of nonlinear particle relaxation. In the linear limit, we have

corresponding to the Stokesian drag law, pure heat conduction limit, and pure mass diffusion limit, respectively.

The factors $ψd,ψt,ψc$ corresponding to nonlinear relaxation can be expressed by

Equation (9a) is based on a recommended correlation.⁹ Equations (9b) and (9c) are the well-known Ranz-Marshall correlations,¹⁰ and constitute an analogy between convective heat and mass transfer.¹¹ 

The determination of particle Reynolds number required in the evaluation of the functions $ψ1,ψ2,ψ3$ in Eq. (9a) is exceedingly complex. On the basis of a comparison between the theory (in the absence of mass transfer) and data on supersonic jets with water droplets, the author (Kandula⁶) proposed a satisfactory correlation of the particle Reynolds number as follows:

For a given set of parameters, Eq. (2) is solved for $k1,k2$ as a function of $ωτt$⁠. The attenuation and dispersion coefficients are then determined from Eq. (5). All calculations are performed with the aid of Mathcad.

Figure 1 shows a comparison of linear and nonlinear theories with the data of Cole and Dobbins¹² which corresponds to low droplet concentration (⁠$Cm$ of the order of 0.01). The data are obtained in a Wilson cloud chamber in which a cloud of nearly uniform size droplets is generated. Thus polydispersion effects are absent. For the present comparison, the test conditions considered correspond to those indicated by Davidson⁴ in their comparisons corresponding to an average gas temperature of $T0=276$ K.

These parameters describe mean fog conditions in the experiment. Only data on attenuation were measured, and no data for sound dispersion were obtained.

The results suggest that the nonlinear theory shifts the curve of the linear theory to the left, but the differences at this concentration level are not appreciable. The shift of the nonlinear prediction to lower frequencies can be interpreted phenomenologically on account of a slightly larger relaxation time due to nonlinearity, as evident from Eqs. (7a) and (9a). The attenuation peak for the nonlinear theory slightly exceeds that for the linear theory. Both the linear and nonlinear theories show the same level of agreement with the data at low droplet concentration.

Calculations are performed for sound propagation in air-water droplet mixtures at relatively high droplet mass concentration (⁠$Cm$ ranging from 0.1 to 3) to reflect a range of circumstances in typical practical applications involving air-water system. Although the droplet mass loadings $Cm$ (0.1 to 0.3) considered here appear to be very high compared to anything that occurs in nature or the laboratory, they are characteristic of applications such as rocket exhaust noise mitigation with water injection, where water to exhaust gas mass flow rates of the order of 1 to 4 are usually considered. These high mass loadings generally lead to average drop diameters of the order of 100 $μm$ (compared with about 2–10 $μm$ in atmospheric fog), and thus involve high droplet relaxation times. For example, average droplet mass loadings of 0.1 to 10 were discussed by Gumerov and Nigmatulin.¹³ Since the density of the droplet (water) is much greater than that of the surrounding gaseous mixture, the volume fraction of the liquid may be neglected compared with that of the gaseous species.³ Thus interaction between neighboring particles may be neglected even for relatively large values of the droplet mass fraction.

A vapor mass loading of $Cv=0.1$ corresponding to $T0$= 327 K (at a pressure of 0.1 MPa) is considered here. The relevant parameters are as follows:

Figure 2 presents the effect of nonlinear particle relaxation on the variation of the dimensionless absorption coefficient with the particle relaxation time for a wide range of droplet concentration $Cm$ (0.01, 0.1, 0.3, 1.0, and 3.0). Two distinct peaks in attenuation are noticed in the absorption spectrum for $Cm=$ 0.01 and 0.1: one corresponds to phase transition, and the other due to viscous drag and heat conduction effects, the relative magnitudes of the two peaks depending on the value of $Cm$⁠. At these low droplet mass concentrations, the phase transition effect dominates that due to viscous drag and heat conduction effects, with the peak frequency occurring at $ωτt/Cm≈1$⁠. On account of the low values of $Cm$⁠, the two peaks are well separated. In general, the peak due to phase change is observed at $ωτc≈cm$⁠, and that due to viscous and heat conduction effects at $ωτd≈1$ and $ωτt≅1$⁠, respectively.² This fact differentiates the phase transition peaks from the thermovisous peaks. The major effects of mass transfer at low mass loadings appear at low frequencies. For $τd≈τt$⁠, which is usually the case, the two peaks nearly coincide. The results also indicate that for droplet mass concentrations $Cm$ less than about 0.1, the peak attenuation is relatively independent of $Cm$⁠.² For $Cm=0.3$ and above, only one distinct attenuation peak is observed, suggesting the existence of interaction between mass, momentum and heat transfer. The attenuation peaks increase with an increase in $Cm$⁠, as is to be expected. The peak frequencies associated with the viscous drag and heat conduction are stationed at $ωτt≈1$⁠, with the peak frequency slightly increasing with an increase in $Cm$⁠.

With regard to the role of nonlinearity, the results suggest for very low droplet mass concentration of $Cm=0.01$⁠, the nonlinear particle relaxation effects are seen to be rather unimportant for frequencies characterized by $ωτt/Cm<1$⁠. For larger $Cm$⁠, the nonlinear theory departs from the linear theory in a significant manner for large $ωτt$⁠, with the nonlinear absorption coefficient considerably lower than that predicted by the linear theory. The results also suggest that with nonlinear relaxation the peak frequency is reduced relative to that indicated by the linear theory. The frequency shift due to nonlinearity seems to persist over the entire range of the droplet mass concentration considered here. In general, the spectrum width of dimensionless attenuation coefficient is reduced relative to that predicted by the linear theory. This suggests the importance of nonlinear relaxation.

Figure 3 presents the spectral distribution of the scaled attenuation coefficient $α/αmax$ as a function of $ω/ωmax$ for various values of $Cm$ ranging from 0.1 to 3.0. Here the quantities $αmax,ωmax$ correspond to the peak attenuation. It is noticed that beyond the peak frequency ratio, the scaled attenuation coefficient becomes independent of the droplet mass concentration. However, for frequencies below the peak value, the shape of the spectrum of the scaled attenuation broadens with a decrease in the value of $Cm$⁠. For large values of $Cm$ = 0.3 and above, the shape of the scaled attenuation curve is characterized by a universal spectrum independent of $Cm$⁠. This finding marks an important contribution of the present work. The spectral broadening effect for small values of $Cm$ arises as a consequence of the importance of phase transition effect.

It may be remarked that the double peaks and/or shoulders manifested in the spectral absorption are primarily related to the superposition effects among the relaxation processes associated with the phase transition and thermoviscous effects, as illustrated by Marble.¹ There appears to be little evidence as to the existence of secondary relaxation processes or damping mechanisms.

Figure 4 displays the variation of the dispersion coefficient β = (c₀/c)²−1 with particle relaxation time $ωτt$⁠. It is evident that the nonlinear effects become important for $ωτt$ exceeding about 0.2 to 0.4, where viscous and heat conduction effects become important. In this range of $ωτt$⁠, the dispersion coefficient according to the nonlinear theory is smaller than that provided by the linear relaxation. Further measurements with moderate to high droplet concentration and large droplet size will be valuable in our understanding of the role of nonlinear relaxation on sound attenuation and dispersion.

In view of the relatively high droplet sizes encountered in technical applications (of the order of 100 $μm$⁠), the improvement of the predictions of sound attenuation due to the nonlinear theory is indeed physically pertinent for conditions of $ωτ$ exceeding about unity (beyond thermoviscous peaks).

The results suggest that at sufficiently high droplet concentration, significant interaction between the transport processes result in only one distinct attenuation peak. It is shown that with nonlinear relaxation the peak frequency is reduced relative to that indicated by the linear theory. The frequency shift due to nonlinearity persists over the whole range of the droplet mass concentration considered here. The spectrum width of dimensionless attenuation is reduced relative to that predicted by the linear theory. The nonlinear effects appear to become important for $ωτt$ exceeding about 0.2 to 0.4, where viscous and heat conduction effects become predominant. In this range of $ωτt$⁠, the dispersion coefficient according to the nonlinear theory is smaller than that provided by the linear relaxation. It is found that at large values of the droplet mass concentration, the scaled attenuation spectrum exhibits a universal shape independent of the droplet mass concentration. The results also point to the existence of spectral broadening for low droplet mass concentration.

This work was primarily derived from the author’s recent work.¹⁴ The author thanks the reviewers for detailed comments and suggestions. Thanks are also due to Stanley Starr (Chief, Applied Physics Branch) of NASA Kennedy Space Center for review and helpful suggestions. Papers by the author from 1973 to 1982 were published with the name K. Mastanaiah.