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Attenuation of high-intensity sound in a droplet laden gas
Journal of Sound and Vibration (1977)51(2), 219-235
ATTENUATION OF HIGH-INTENSITY SOUND IN A DROPLET LADEN GAS F. A. LYMAN
Department of Mechanical and Aerospace Engineering, Syracuse Uniuersity, Syracuse, New York 13210, U.S.A. (Received 29 June 1976, trodin revisedform 2 November 1976) The effect of acoustic streaming on the attenuation of a high-intensity plane sound wave propagating in a gas containing a small concentration of monodisperse suspended droplets is considered. This work was undertaken to explain why some recent experiments on droplet-laden flows in pipes and nozzles indicated considerably larger attenuation than could be explained by existing theories. Acoustic streaming set up in the vicinity of a droplet by the oscillating flow due to the sound wave is analyzed by the method of matched asymptotic expansions. The resulting flow field is used to calculate the enhancement of heat and mass transfer from the droplet by convection. The effect on sound attenuation is determined by modifying existing theories of aerosol acoustics to account for the enhanced heat and mass transfer. The results indicate that the increased sound attenuation due to the proposed mechanism is significant only at droplet Reynolds numbers greater than I, which for droplets of 1 pm radius implies sound pressure levels of at least 160 dB. Also, owing to the tendency of small droplets to follow the oscillating flow, the effect is considerably reduced at frequencies below the characteristic frequency for velocity equilibration of a droplet, which is roughly 16 kHz for a 1 pm radius water droplet. It is concluded that other phenomena must be responsible for the anomalously high attenuation observed in the experiments.
1. INTRODUCTION It has long been recognized that the attenuation of sound in a gas containing suspended particles or droplets may be considerably larger than that in a clean gas. The additional attenuation is due to the fact that momentum, heat, and in the case of volatile droplets, mass, are transferred at finite rates, so that the velocity, temperature and vapor pressure of droplets lag behind the fluctuations in these quantities set up by the sound wave in the surrounding gas-vapor mixture. As a result, there is net dissipation of energy over each cycle. Several investigators [1-5] have contributed to the theory of the attenuation of plane sound waves of infinitesimal amplitude in particle- or droplet-laden gases, and their results have been well verified by experiments [6, 7]. Recently, however, experiments [8-10] on air-water mixtures containing droplets formed by condensation in duct and nozzle flows seem to indicate that the attenuation of high-intensity sound may be considerably larger than that predicted by the existing theories. Although these experiments do not provide conclusive evidence of anomalously high attenuation, because the droplet sizes were not measured within the flow, they do indicate that the introduction of a small concentration of micronsized droplets into a gas flow can produce sufficient attenuation to make it of interest as practical technique for reducing noise in aircraft turbofan engines, air conditioning ducts, and gas pipelines. 9The idea of injecting droplets into the inlets of turbofan engines to attenuate fan noise was 219
220
r.A. LYMAN
recently proposed by Marble and Candel [11]. Their estimates, based on small-amplitude theory, indicate attenuation of only about 3 dB per meter for a 1 percent mass concentration of uniformly distributed monodisperse droplets of 1 Fm radius, and such attenuation is rather marginal for a practical noise reduction technique. If, on the other hand, the actual attenuation due to droplets were much larger, as the experiments seem to indicate, droplet injection would warrant further consideration. Accordingly, it was deemed important to investigate why the actual attenuation might be larger than theory predicts. One phenomenon leading to greater attenuation at high sound levels is acoustic streaming, wherein a stead), secondary flow is set up in the vicinity of a droplet by the Reynolds stresses associated with the oscillating flow induced by the sound wave. It is well known that acoustic streaming can increase the rates of heat and mass transfer and chemical reactions at solidfluid or fluid-fluid interfaces. The effect can be significant, even though the characteristic velocity of acoustic streaming is generally much smaller than the instantaneous velocity of the oscillating flow. Since the magnitude of the acoustic streaming velocity is proportional to the intensity of the sound wave, this effect would be expected to be important at high sound intensities. Although the effect of acoustic streaming on sound attenuation in a clean gas has been studied [12], the corresponding situation in a droplet-laden gas apparently has not been previously investigated. The present analysis was carried out to determine quantitatively the increase in sound attenuation due to the enhancement of heat and mass transfer from the droplets by acoustic streaming. The main steps in the analysis are as follows: (a) determination of the flow field near a single droplet and calculation of the drag and acoustic streaming, (b) solution of the convection equation for heat and mass transfer due to acoustic streaming, and (c) calculation of the effect of increased heat and mass transfer on sound attenuation. These steps are described in greater detail in the following sections.
2. ACOUSTIC STREAMING Consider a spherical droplet of radius a suspended in a gas-vapor mixture through which a plane sound wave propagates. The velocity of the gas-vapor mixture due to the sound wave is U= = Uoe Ic'~'-k~. (la) Since ka < I i n all cases of interest here, the flow about the droplet can be considered incompressible, with an oscillating free-stream velocity
Vow(t) = Vo elo' .
(lb)
For the purposes of this paper, the velocity amplitude Uo will be related to the sound pressure level by the linear acoustics relations
Vo = "v"-~p.,,~lpc = (v'~p,./pc)
lO ",'~'o,
(2):
where p is the density of the gas-vapor mixture, c is the speed of sound in the mixture, Lp is the sound pressure level (SPL) in dB, prms is the r.m.s, pressure fluctuation, andp,ef = 2 x 10-5 N/m 2. Equation (2) will be used only to express the results in terms of SPL, inasmuch as no simple relation between pressure and velocity amplitudes exists in non-linear acoustics. It is convenient to adopt a system of spherica ! polar co-ordinates (r,0) with origin at the center of the moving droplet and polar axis along the axis of oscillation, so that far from the droplet the velocity of the flow relative to it is
V(t) = U=(t)- Va(t)= Vo e''~' ,
(3)
ATTENUATION 1N A DROPLET-LADEN GAS
221
where Ua(t) is the droplet velocity, and Vo is in general complex. Owing to the tendency of the droplet to follow the oscillating flow, ] Vo] will in general be smaller than Uo. Due to the symmetry of the flow about the axis of oscillation, the velocity compoents may be derived from a stream function tp by means of the relations 1 ar 1 ar ltr = r2 sin 0 00 ' tto = r sin 0 Or (4) When written in terms of dimensionless variables t = tot ', r = r7a, u = u'/[ Vol, ~k = ~'/I Vola z (primes denoting dimensional quantities), equations (4) are unchanged whereas the equation for the stream function [13] becomes ~ J )SP-) D2 "~ M 2 - ~0( D 2 ~k) + R _[I I r 2 a(~,
+= 2 D2~tpL~b9/
D4 ~k,
(5)
It ' 0 1 0 1 -- pZ Or " b - -att
(6)
where # - cos0,
0z 1 -- 112 0 z D2 -~ Or2 q" r 2 0112'
L-
The dimensionless parameters in equation (5) are M = a(to]v) 1/2,
(7)
which is the ratio of droplet radius to the thickness of the oscillatory boundary layer, and R = IV ol a/v, (8) the droplet Reynolds number. Note that M 2 = R S , where S = t o a / I Vol (9) is the Strouhal number, or the ratio of the droplet radius to the amplitude of oscillation. Equation (5) is to be solved subject to the boundary conditions of no slip or normal velocity at the surface of the sphere, tp = O~O/Or= 0 on r = I, (lOa) 9and a uniform oscillating flow at infinity, ~ kr2(l --p2) el' as r --~ co. (10b)
This problem has been attacked by several authors, most recently Wang [14], Riley [15] and Mednikov [16]. Wang and Riley treat the case S >> I : i.e., small-amplitude oscillations. For micron-sized droplets subjected to sound pressure levels in excess of 100 dB, however, the amplitude of oscillation is at least of the same order as the droplet radius, and may be much larger. Figure 1 shows "maps" relating the dimensionless parameters R,'~, S and M z to the droplet radius a, frequency f = to/2z and sound pressure level (SPL). It is apparent that for micron-sized droplets and frequencies in the audible range, M2,~ 1, whereas R~ ranges from 10-4 to almost 10 as the SPL increases from 100 to 160 dB. The fact that M 2 ,~ 1 suggests treating the flow as quasi-steady, i.e., neglecting the first term on the left of equation (5), which implies that the oscillations of the fluid are everywhere in phase. This was the approach of Mednikov, who adapted the well-known solution of Proudman and Pearson [17] for steady flow about a sphere at low Reynolds numbers to the case of an oscillating free stream by simply replacing the Reynolds number by a periodic function of time. By integrating over a cycle and using certain symmetry arguments, he obtained the steady component of the stream function. Mednikov overlooked the fact, to be shown subsequently, that at sufficiently large distances from the sphere unsteady effects cannot be neglected. Therefore his far-field solution is of questionable validity,, and because i" Since the amplitude [Vol of the relative velocity is not known a priori, Figure l(a)shows the acoustic Reynolds number R~= Uoa/vrather than R. In general, R ~
222
F. A. LYbfAN l
I
I
I
la) SPL (dB) ,,/160 140 "
2 0 o
E
-2 -4 (c) -6
I
-2
,
-i
I
I
I
o
I
2
I
I
I
I0
IOg~o 0 (/din) =
l
i
I
I
I
I i
I
SPL {dB}-
/ioo
2
//~zo
-
o
f[kHz)'" . IO0
0-1 0.0
~:~ - 2
o
-4
S/,,//,,~o 0
-6
-2
-8
-4
-2
o
2 IOglo
4
6
-10
-2
-I
of (p.m/sl
o
I
log ~o o ( p m )
Figure 1. Dimensionless parameters for droplet of radius a in the oscillating flow set up by a plane sound wave of frequency f. (a) Acoustic Reynolds number, R, = Uoalv; (b) Strouhal number, S = 2naflUo; (c) M 2 = R o S = 2nfa'lv. Acoustic impedance pc and kinematic viscosity v are those of dry air at 1 arm and 15~ (pc = 414 N s/m3; v = 1-46 x 10-5 m2/s).
the stream function near the sphere is obtained by matching with the outer solution, this casts doubt on the inner solution. In the region near the sphere one may neglect the unsteady term in equation (5) and solve the resulting equation by means of the regular perturbation expansion
~k= ~o-t- R~t + ....
(I1)
The zero-order solution satisfying boundary conditions (10) is 4o = 882r2 - 3r + I/r) (1 - pz) cos t,
(12)
which represents quasi-steady oscillating Stokes flow. The periodic behaviour in time of equation (12) implies that to this level of approximation there is no acoustic streaming, and one must proceed to the next term in the expansion (11) to obtain it. The next term in the inner expansion is obtained by substituting equation (12) for ~kin the convective term in equation (5), which results in D 4 ~kl = - ~ (2/r 2 -- 3/r 3 + l/r 5) p(l -- p2)cos 2 t. (13) The solution of this equation which satisfies the inner boundary condition (10a) and is symmetric with respect to the plane 0 = n/2 is ~, = - - ~ 2 ( 2 r 2 - 3 r
+ 1 - I + L ' ~ Pr '(/ l r
--'/L2)COS21'q" (I//1)c'
(14)
where (~1)c is the complementary solution of equation (13) which also satisfies boundary conditions (10a) and has the requisite symmetry. It is to be determined by matching with the outer solution. It is well known [17, 18] that these inner solutions break down when Rr = O(1). This can
223
ATTENUATION IN A DROPLET-LADEN GAS
be shown by using equation (12) to obtain the following estimate for the ratio of the neglected convective term to the viscous term: convective/viscous = O(Rr) as r --~ oo. 05) Besides this, there is an additional restriction in unsteady flow that appears to be less well known. An order-of-magnitude analysis using equation (12) to estimate the ratio of the unsteady and convective terms in equation (5) discloses that unsteady/convective = O(Sr) as r -+ 0% (16) which, together with equation (l 5), implies that unsteady/viscous = O ( M 2 r2). (l 7) Therefore the unsteady term must be retained in the region beyond the point where M r = 0(1). In the most general case, in which M 2 and R are unequal and both less than l, the flow should be divided into three separate regions, in each of which different sets of terms in equation (5) would be important, and a separate perturbation expansion would be required. If, however, M 2 = O(R2), then the two outermost regions merge into a single one, in which both unsteady and convective terms are important. In the region Rr > 1 the new variables ~ = R2 ~, p =Rr (18a, b) are introduced, following Proudman and Pearson [17]. This makes the coefficients of the viscous and convective terms the same and transforms equation (5) into ~_) ~ ( ~ 2 ~ u ) + ~ _
a(p,~,)
+2~2~L'e~
=~4~,
(19)
where ~,,2 and .~. are the differential operators obtained by replacing r by p in the expressions of equation (6) for D 2 and L, respectively. To solve equation (19), the expansion ~t' = !Po(p, i~, t) + RtPI(p, la, t) + . . . (20) used by Proudman and Pearson is attempted. Obviously this cannot work for arbitrary M, because the coefficient of the unsteady term would become infinite as R ~ 0 for fixed p. Therefore, it is assumed that M 2 = KR 2, (21) where ~c is a constant of order unity. This is the simplest and most convenient assumption which corresponds reasonably well to the present situation.t The zero-order term in the expansion (20) is simply an oscillating uniform flow, q~o = ~P2( 1 -- p2) cos t. (22) Substitution of this and equation (20) into equation (l 9) leads to the following equation for ~/Jl: K _ ~ / ( ~ z ~ , ) q _ C O S I ( 1 --/'t2 0
P
~+/a0-~)~2~,=~4~
,.
(23)
This is equivalent to Oseen's equation for an oscillatory free stream. The time-dependent coefficient makes it intractable to the usual techniques of solution, however. Fortunately, Ockendon [19] has recently managed by an ingenious technique to obtain a solution to the unsteady Oseen equation for the case of a sphere accelerating in an arbitrary manner. He considered the effect of the inner boundary condition on the outer solution to be correctly simulated by the introduction of a delta-function term to the Navier-Stokes equation. The integral of this singular term corresponds to the Stokes drag on the sphere. t Although Figure 1 indicates that for droplet radii of 1-10/~m, R, is larger than M 2 when SPL > 140 dB andfis in the audible frequencyrange, the fact that R < Roin general and is much less than R~for these droplets at audible frequenciescompensates for this.
224
F.A. LYMAN
The modified equation with the singularity at the origin was solved by Fourier transforms in the three spatial co-ordinates over the entire region, including the origin. The resulting expression for the stream function is expanded in powers of R to obtain the following for the inner limit o f the outer solution: lim (~l)outer = - - t ~ r 2/a(l -/~2) cos 2 t + 89 - l/2) n(t). (24) R---~ 0
The second term represents an unsteady but uniform stream with velocity H(t) along the axis of oscillation of the sphere. The function H(t) was obtained by Ockendon in terms of a convolution integral involving the velocity ofthe sphere relative to the fluid. The general form of this function and its behavior in the case of oscillating motion are discussed in the Appendix. Choosing the complementary function in equation (14) to match the second term in equation (24), one obtains for the inner solution ~'1='~
2r2-3r+l-
+
p(l-lt2)cos2t+ 88
2r2-3r+
(1-/a2).
(25)
Consider first the steady component of the stream function, obtained by averaging equation (25) over a period. Obviously ~ko has no steady component. Furthermore, one would expect that the time-average of H(t) should also be zero because of the antisymmetry of the second term in equation (25) about the equatorial plane (0 = 7r/2 o r / t = 0), which is inappropriate for acoustic streaming. In the Appendix it is shown that for an oscillating sphere the behavior of H ( t ) as t ~ co is indeed periodic, with zero mean value. Therefore the steady component of the stream function near the droplet is i~ = - ~ R (2r 2 _ 3r + 1 - l/r + 1/r2) sin2 0 cos 0, (26) from which the relations (4) yield the velocity components
3R/2_L+ L l+l 3cos20-1 r2-7
7/
i ( 3 l uo=~R 2 - ~rr -I-
r
l)
2r 3
2
r-4. sin0cos0.
(27) '
(28)
Note that these are proportional to R, which increases rapidly with the sound pressure level. The streamlines in one quadrant of the meridional plane are plotted in Figure 2 for the particular case R = I. The streamlines in the other quadrants are of course symmetric with respect to 0 = 0 and n/2.
Figure 2. Streamlines of steady acoustic streaming flow for R = 1. The dircction of the oscillating free stream is indicated by ~-~.
ATTENUATION IN A DROPLET-LADEN GAS
225
The drag on the droplet can be calculated to order R from equations (l l), (12) and (25). The first term in equation (25) contributes no net force, because of the symmetry of its velocity components about 0 = 0 and n/2. The drag is due to the Stokes stream function @o and the second term in @1, and since the latter has the same dependence on r and 0 as the former, the result is Fx = 6rratll 1Iol [cos t +RHCt)], (29) where '1 is the viscosity of the gas-vapor mixture. From equation (A19) of the appendix, as t--* co, RH(t)... Mcost. Thus the first-order correction to the Stokes drag formula for oscillatory flow about a sphere is O(M), instead of O(R) as in the Oseen formula for steady flow [17]. The physical reason is that the additional drag in steady flow is due to the wake formed by the slow convection and diffusion of vorticity far from the sphere [20]. In an oscillating flow such a wake cannot form, owing to the continual reversals of the free stream. For the small droplets considered in this paper, .~it 2 ,~ 1, and the correction to the Stokes drag can be neglected. This result suggests that Mednikov's treatment [21] of the absorption of large-amplitude sound in an aerosol is incorrect, because he assumes not only that the steady-state Oseen drag correction is valid for oscillatory motion, but also that it may be approximated by taking its maximum value. 3. ANALYSIS OF CONVECTIVE HEAT AND MASS TRANSFER BY ACOUSTIC STREAMING 3. I. FORMULATION OF THE PROBLEM AND METHOD OF SOLUTION
The steady flow obtained in the preceding section is used to determine the convective transport of heat and mass between the droplet and the fluid. The energy equation can be written in the dimensionless form V2 T = . R ( u,0T+-- u~ aT.I or 7 b-a-J '
(30)
where V2
1 a/ 2a'~ l ;(sin0a__~) -=~'-~r~ r ~r} q" r2sin----~
In equation (30) a = v/ct, is the Prandtl number, at is the thermal diffusivity of the gas-vapor mixture, and T represents the normalized temperature difference, obtained by subtracting the temperature T~0 at large distances from the droplet from the temperature, and dividing by the difference Ts - T~, where T~ is the surface temperature of the droplet. The boundary conditions are therefore T= 1 on r = l, (31) T --~ 0 as r -+ co. (32) The mass transfer problem is formulated in the same way, except that a is replaced by the Schmidt number v/D, where D is the coefficient of diffusion of vapor through the gas, and T by the normalized concentration difference. Although equation (30) is linear, the fact that it is non-separable for the flow described by equations (27) and (28) makes it unlikely that exact solutions can be obtained by analytical means. If R and a are small, a perturbation solution in these quantities is possible. Acrivos and Taylor [22] solved the corresponding heat and mass transfer problem for a sphere in Stokes flow by the method of matched asymptotic expansions, and Rimmer [23] extended the solution to higher Reynolds number by using the corrections to Stokes flow obtained by Proudman and Pearson [17]. In the present problem, however, the flow far from the sphere is not a uniform stream, which makes it difficult to obtain an outer solution valid for all angles.
226
r.A. LY.MAN
Therefore it was necessary to resort to numerical techniques. The series truncation method developed by Dennis et aL [24] was found to be particularly well suited to this problem. Substituting the expressions (27) and (28) for the velocity components into equation (30) and transforming to the variables ~ = lnr and tt = cos0, one obtains a 2T OT a f 0 T ] e -r
ae
(33)
e
where g = - ~ a R z,
d? = 89
2 - l ) f ( O e ~,
f ( ~ ) = 2e ~ - 3 + e -~ - e -2~ + e -a'.
The temperature is expanded in a series of Legendre polynomials T ( ~ , p ) = ~ t.(~) P.(p).
(34)
n=O
In the present case the symmetry of the acoustic streaming flow about the midplane it = 0 requires that T ( ~ , p ) also be symmetric about this plane, so that n is restricted to even integers. The functions t.(~) satisfy the boundary conditions t.(0) =
1, 0,
n = 0, n # 0,
(35)
t. ~ 0 as ~ ~ m. (36) By substituting the expansion (34) into equation (33) and using the properties of the Legendre polynomials, one obtains an infinite set o f ordinary differential equations t~, + (1 - A.) t,; - [n(n + 1) + B.] t. = S., (37) where in the present problem A.(~) = - - ~ e ~ . . f ( O ,
B . ( O = --lzefl.. g ( Q ,
o0
S.(~) = - 5 e ~o{~.~f({) t;,({) + fl.kg({) t~(r k•n
d g(O = e - e ~ - [eef(O] = 4e~ - 3 + e -2r - 2e -3r . The summation in S.(~) is over ~ e n integer values of k, excluding k = n, for which the corresponding terms have been included on the left side of equation (37). The constants ~.k and/~.k are defined by the following definite integrals: +1
ffnk
~--"
4
!. fj --1
+1
P.(p) Pk(pJ(3p a -- 1) dp,
flnk
=.,2n 4+
1
fj
--1
These are evaluated by using the recursion relations for Legendre polynomials, with the results n(n + 1) 3 (n + 1)(n + 2) 3 n(n - 1) Ct"k= (2n -- 1) (2n + 3) 6.k + -~ (2n + 3) (2n + 5) 6.+=, k + ~ (2n -- 3) (2n -- 1) &.-2.k, n(n - I)(n - 2) a ] ' 1 [ n(n + 1) (n + l)(n + 2)(n + 3)6.+= ~ ('~n'--'l)--~'--'~-"-2'~] /~"~ = ~- I.(2n --- l]-(2n + 35 a"~+ (2n + 3)(2n + 5) '
where 6.k equals 1 when k = n and is zero otherwise. Thus the differential equation (37) for t.({) contains on the right side only terms involving t._2({) and t.+=({) and their first derivatives. In practice, the series (34) is truncated at a finite number, n = n o . The finite system of equations is solved numerically, using central differences to approximate the derivatives. The corresponding finite-difference equations are of the t'orm al t.(i - 1) + bi t.(i) + ct t.(i + 1) = h z S.(i), (38)
A T T E N U A T I O N I N g D R O P L E T - L A D E N GAS
227
where tn(i ) -- tn(~,) denotes the value of the function at the ith mesh point ~, = ih, and a, = 1 - 89 [1 - A,(i)], b, = -{2 + h2[n(n + 1) + R~(i)]}, c, -- 2 - at. For each n and i = I t o / , the tridiagonal system of equations (38) is solved by standard techniques [25]. Inasmuch as Sn contains the function tn+2(~) and its derivative, and this function is unknown at the time tn(~) is being generated by solving the finite-difference equations, an iterative procedure is necessary. At each step in the iteration procedure Sn is calculated by using the function tn+2(~) generated during the previous iteration, but the latest values of the function tn-2(O. This iterative, procedure was followed to solve equations (38) on a digital computer for = 0.1, 0.5, 1.0, 1-5 and 2.0. For a Prandtl number tr = 0-72, this corresponds to Reynolds numbers R = 0-86, 1-92, 2.72, 3.33 and 3.85. For most ofthe calculations a step size h = 0.025 was used, and the calculations were carried out to ~l = 5.0, corresponding to a radial distance of 148.4 times the radius of the sphere. In order to ensure convergence, it was found expedient to use the "interpolation" procedure suggested by Dennis et al. [24]. At the completion of t h e j t h iteration, the functions tt~J)(~) have been calculated. On the next iteration, as each function, which will be denoted fn(~), is calculated, the new ( j + l) approximation to tn is obtained by interpolating between in and t~J), as follows: t~+~'(~) = E~(~) + (1 - E)ttJ'(~). The parameter E was selected by trial to prevent divergence of the iterative procedure. The value E---- 0.05 was used for most of the calculations. For e = 0.1, the convergence was obtained at values of E up to 0.2, however. The iterative procedure was considered to have converged when It,'~+-(O- t'.J'(r < 10-4 for all ~ and n ~
to(~j) = -t;(O) e -~ + ]-6
f ( O t2(O d~,
(39)
r
where co
t6(0 ) =--1 +~-0 ~ f ( ~ ) t2(~)d~.
(40)
0
Therefore, once t2 is determined, to can be found by simple quadrature. Thus it was necessary only to calculate those tn for n ~> 2 by solving equations (38) by the iterative method, t~(~) was eliminated from the first of these equations 0l = 2) by using relation (39). To start the procedure it was assumed that tn = 0 for n >/4, and the initial t2 was obtained by the finite-difference solution of the n = 2 equation. The iterative procedure was then carried out for all equations 2 ~
228
F.A.
LYMAN
region near It = 0 (the temperature "wake"), and because the validity of the asymptotic representation is open to question, its development, described in reference [26], will not be given here. The numerical results indicated that t2 varies as e x p ( - 2 0 for large 4, regardless of the actual boundary condition applied at ~z-This behavior is also predicted by an asymptotic analysis of the differential equation for t2 when t+ is neglected [26].
3.2.
CALCULATED
RESULTS
The rates of heat and mass transfer from the droplet are the quantities of main interest. Since these rates are derived in an identical manner, only the calculation of the former will be described. The heat transfer per unit area, per unit time through a point on the surface ot the droplet located at angle 0 to the oscillating flow is
q ( O ) = - k T s - T= 0(~_) a
~
(411
=0'
where k is the thermal conductivity of the surrounding gas-vapor mixture. Thecorresponding local Nusselt number based on droplet radius is
Nu.(O) = k(Ts- T.~)= -
~=o = -
.=o ~" t;,(0) P.(cos 0).
(42)
The average value of this over the surface of the droplet is 71"
Flu. = 89J Nu.(O) sin 0 dO = -t~(0).
(43)
0
The calculation oft6(0) was carried out by quadrature, equation (40). For n ~> 2, t,~(0) was calculated by using a three-point Lagrangian formula. The local Nusselt number is plotted as a function o f 0 in Figure 3 for various values of the parameter e. As expected, the heat transfer is largest at the pole 0 = 0 and decreases monotonically towards the equator. 0"5
I
I
I
]
I
I
I
I
I
F
I
~
[
I
0-4
T
0-3
0-2
0.1 I
I0
20
30 40
50 60 70
80 90
e Figure 3. V a r i a t i o n o f local Nusselt number at droplet surface w i t h p o l a r angle 0 measured f r o m axis o f oscillation.
The average Nusselt number and Sherwood number are in this analysis a function of the parameter ~ = :-~aR z alone. For heat transfer, ais the Prandtl numberofthegas-vapor mixture, while for mass transfer it is the Schmidt number. The increase in the average Nusselt or Sherwood number due to acoustic streaming is plotted as a function ore in Figure 4. It is seen that at ~ = 2.0 the increase is about 30 percent.
ATTENUATION IN A DROPLET-LADEN o
.
3
GAS
229
~
"7
"7
0.2
0.1 z o
0
I
Convectionparameter,
2 _ 3
~r-
T~o-R2
Figure 4. Increase in Nusselt and Sherwood numbers due to acoustic streaming. The convection parameter E depends on the droplet Reynolds number R = [ Vo]a/v and a, which for heat transfer is the Prandtl number, and for mass transfer the Schmidt number.
4. EFFECT OF ACOUSTIC STREAMING ON SOUND ATTENUATION The effect of acoustic streaming on sound attenuation is due to enhanced heat and mass transfer. As pointed out at the end of section 2, acoustic streaming does not affect the drag on a droplet, which to a good approximation is given by the Stokes formula. Since convective heat and mass transfer from a droplet are governed by the linear equation (30), the corresponding transfer rates are proportional to the temperature or concentration differences (e.g., see equations (41) and (42)). Thus the heat and mass transfer from a single droplet are Q = 47rakN~vu(T~ -- To~), (44) M = 4naDFlsh(P~ -- Pv). (45) These are of the s a m e form as the expressions used by Cole and Dobbins [4] in their analysis of the attenuation of small-amplitude sound waves in an aerosol, except here the average Nusselt and Sherwood numbers are greater than unity. Since these parameters are independent of x and t, however, the analysis of sound attenuation can be carried out just as in reference [4], by incorporating N,~u and/Vsh as follows in the expressions for the relaxation times for temperature and concentration equilibration: z, = ma cp/4r~akNuu = Pl a2/3oq PNu,,,
(46)
rc = ma/ 4rcapD ~l~n = Pt a2/3p D EI~h (47) where ma = (4n/3)Pt a3 is the droplet mass, Pl is the liquid density, and cp and p are respectively the constant-pressure specific heat and density of the gas-vapor mixture. One point deserves mention: the ratio zc/rt is in the Cole and Dobbins theory equal to the Lewis number, NLe = o~t/D, whereas according to equations (46) and (47) zd
,=
(
Nt.e.
(48)
230
F.A. LYMAN
Therefore the Lewis number has to be multiplied by (.K[A.Jfitsh) whenever it appears in the Cole and Dobbins results. The Nusselt and Sherwood numbers depend on the droplet Reynolds number given by equation (8), and it is necessary to determine the latter as a function of the sound pressure level and frequency. For this purpose the amplitude of the relative velocity V is required. Since the drag on the droplet can be adequately represented by the Stokes formula, the equation of motion of the droplet is d U d / d t = (l/ra)(U~ -/./4). (49) where Tn = ma/6rr~q = 2pl a2/9pv (50) is the relaxation time for velocity equilibration. The result of solving equation (49) for the oscillating free-stream velocity of equation (I) is I Vol/Uo = cord/V'I + (cora)2,
(51)
where Uo is given by equation (2). According to equation (51), at low frequencies (cora < l) the amplitude of the flow velocity relative to the droplet is significantly smaller than the peak velocity induced by the sound wave. This effect tends to limit the effect of acoustic streaming on sound attenuation to high frequencies: i.e., those for which cord > I. Another implication of equation (51) is that the Reynolds number R defined by equation (8) is frequency-dependent, which makes the Nusselt and Sherwood numbers functions of frequency as well as sound pressure level and droplet radius. The fact that R depends on a both directly and implicitly, through cord, precludes the possibility of presenting, as in reference [4], curves of attenuation per wavelength versus cord which are independent ofa. The foregoing modifications were all incorporated in the calculation of the attenuation coefficient]" a according to equation (24) of Cole and Dobbins' paper [4]. Figure 5 displays typical results of attenuation as a function of frequency for selected sound pressure levels and droplet radii. For droplets of radius I lam, the increase in sound attenuation due to acoustic streaming is perceptible only for sound pressure levels approaching 180 dB and frequencies above 16 kHz, where it amounts to about I dB/m (Figure 5(a)). This is so because corn for I lam droplets is about I at 16 kHz; much below that frequency equation (51) predicts that the droplets follow the oscillations set up in the air-vapor mixture by the sound wave. For larger droplets the increase in attenuation due to acoustic streaming appears at lower frequencies and somewhat lower sound pressure levels, but the amount is too small to be significant. This is illustrated in Figure 5(b) for 2/am radius droplets, for which the increase in attenuation is slightly less than 0.2 dB at a sound pressure level of 170 dB. Although the results in Figure 5 were calculated by using the complete expression for attenuation derived by Cole and Dobbins, it is worthwhile to give the following approximate expression, which more readily displays the effect of acoustic streaming: CWo I [A+B(flcor,) 2"1
1 [ (cora)2 .]
C. -~.t i T(~r~) z J + ~ _ i + (cora)2J'
(52)
where Co is the speed of sound in the air-vapor mixture in the absence of mass, momentum and energy transfer with the droplets, and Cm is the droplet mass fraction (mass of droplets per unit mass of the air-water vapor mixture). The coefficients~ A, B and p depend mainly on t Defined in the conventional manner by l(x) = l(0)exp(--~x), where l(x) represents the sound intensity1 after the plane wave has propagated a distance x in the medium. Here 0~has units of m-x. The attenuation in( dB/m, equal to 4.34z, is the quantity presented in the figures. In terms of the constants Ct ..... C8 definedby Cole and Dobbins [4], A = (C4 - Cs)IC2, B = (Cl - C6)[Cs ? - l + (NrsJlqLeNtu~)f, and fl= C5[C2, where ~,is the ratio of specific heats of the air-water vapor mixture, and ~" PdP is the vapor mass fraction. =
231
ATTENUATION IN A DROPLET-LADEN GAS 18
(o) i
i
1
l
i
i
i
i
1
i
i
i
16
,r
//
14 12
io 8 6 4 II)
2 L
0
~
I
I
I 4k
I Elk
I
I
I
I [ 16k 32k
I 64k
4
(b) 3
2
I
0
f
I l 16 31-.,5 621
I [ I 12.5 2.~0 , , ~
I Ik
l 2k
12Bk
FrequenCy (Hz)
Figure 5. Effect of acoustic streaming on attenuation due to droplets, for various sound pressure levels (SPL). (a) D r o p l e t radius 1 lam: , SPL < 140 dB; . . . . . , SPL = 180 dB; (b) droplet radius 2 lam: , SPL < 140 dB; . . . . , SPL = 160 dB; - - - - - , SPL = 170 dB. Temperature 23~ droplet mass fraction, 0-01.
the thermodynamic and transport properties of air and vapor, and are only weakly dependent upon the frequency and sound pressure level, the latter dependences being introduced by the modification given in equation (48). Equation (52) is valid when C,, ,~ 1 and (ogr,)2 ~ C,,, except that unsteady effects neglected in the relaxation model also impose an upper frequency Iimit such that the parameter M defined in equation (7) must be small [2, 27]. Acoustic streaming affects only the first term in equation (52), and that primarily through the Nusselt number dependence ofz,. In fact, since acoustic streaming is only important when corn and tort are of order 1 or greater, the following estimate of the maximum increase in attenuation is obtained from equation (52) by assuming tort >> I : Co(A~).,~, B v-1
cm
~
(~,)o
(~TN~ - 1) ~ 5---:-, ( ~ , , ~ - 1),
(53)
trt)o where (r~)oiS given by equation (46) with -~.u = 1. Thus the maximum increase in attenuation is proportional to the increase in the Nusselt number, which can be determined from Figure 4 at any SPL. Also, since xt is proportional to the square of the droplet radius, the amount of increase is much greater for smaller droplets.
5. CONCLUDING REMARKS The calculated results indicate that the effect of acoustic streaming is not sufficient to provide an explanation for the anomalously high sound attenuation observed in experiments with duct and nozzle flows. Although the predicted magnitude of this effect alone is perceptible
232
F . A . LYMAN
(1 dB/m) for 1 pm radius droplets at the rather high sound pressure level of 180 dB, other effects ignored in the present analysis would likely mask it under such conditions. For one thing, non-linear acoustics effects, such as the growth of higher harmonics [28, 29], are important at these amplitudes. Despite much recent research on the non-linear acoustics of aerosols [28, 29], no way has yet been found to analytically determine the total attenuation~ only the decay of the fundamental and growth of the second harmonic have been determined, although much has been learned about the evolution of the waveform. Thus it is difficult to make a meaningful comparison of streaming and amplitude distortion effects. Another phenomenon of possible importance at high amplitudes is droplet-droplet collisions, leading to shattering and/or coalesence. This would most likely affect attenuation by altering the droplet size distribution although there does not appear to be a significant direct energy loss in collisions at the low droplet number densities considered here. The results obtained here for the enhancement of heat and mass transfer from droplets by acoustic streaming may also prove to be useful in other applications, such as the evaporation and combustion of liquid droplets in the intense sound fields set up in combustion chambers. The analysis presented in sections 2 and 3 is valid for droplet sizes and frequencies such that 1)12 ,~ 1, which according to Figure l is true for droplets of radius less than l0 llm at all audible frequencies. For larger droplets or higher frequencies, such that M 2 > I, the flow near the droplet is quite different from that shown in Figure 2, the most conspicuous feature being a shear layer near the droplet of thickness O(M -t) in which the vorticity generated at the surface is confined [15]. This would lead to a much larger increase in heat and mass transfer than in the caseconsidered here. Finally, it should be noted that the continuum approach of the present work excludes from consideration very small droplets (radii less than 0-1 lam). Such droplets have been shown [8, 10] to cause extraordinarily large attenuation when they are formed by sudden condensation from a supersaturated gas-vapor mixture undergoing rapid expansion through a nozzle. Since the radii of these droplets are smaller than the mean free paths of the gas and vapor molecules, the mass, momentum and energy transfer rates must be appropriately modified, as in reference [I0]. But if the flow about the droplets were in the free-molecular or transition regimes, it is doubtful whether acoustic streaming would even occur.-]" Also, the very small droplets are extremely short-lived. They rapidly grow or evaporate (depending on whether the mixture is super-saturated or unsaturated); hence one would expect them to be important mainly in rapid expanding nozzle flows but not so in low-speed duct flows, especially where the presence of foreign nuclei (dust particles, etc.) would not allow significant supersaturation. ACKNOWLEDGMENT This work was supported under National Science Foundation Grant ENG 73-04257. REFERENCES 1. P. S. EPSrEINand R. R. CARHAR'r1953 Journal of the A coustical Society of America 25, 552-565. The absorption of sound in suspensions and emulsions. I. Warm air in fog. 2. S. TEr,fKIN and R. A. DOBBINS1966 Journal of the Acoustical Society of America 40, 317-324. Attenuation and dispersion of sound by particulate-relaxation processes. 3. F. E. MARBLEand D. C. WOOtEN 1970Physics of Fhdds 13, 2657-2664. Sound attenuation in a condensing vapor. t In the continuum flow regime, the acoustic streaming velocity is proportional to the Reynolds number, as in equations (27) and (28), and therefore proportional to the density.
ATTENUATION IN A DROPLET-LADEN GAS
233
4. J. E. COLE, III and R. A. DOBBINS 1970 Journal of the Atmospheric Sciences 27, 426--434. Propagation of sound through atmospheric fog. 5. G. A. DAVlDSON 1975 Journal of the ,4 tmospheric Sciences 32, 2201-2205. Sound propagation in fogs. 6. S. TEMKIN and R. A. DOBBINS 1966 Journal of the Acoustical Society of America 40, 1016-1024. Measurements of attenuation and dispersion of sound by an aerosol. 7. J. E. COLE, III and R. A. DOBBINS 1971 Journal of the Atmospheric Sciences 28, 202-209. Measurements of the attenuation of sound by a warm air fog. 8. W . J . HILLER, M. J'AESCHKEand G. E. A. MEIER 1971 Journal of Sound and Vibration 17, 423--428. The influence of air humidity on pressure and density fluctuations in transonic jets. 9. K. W. BUSnELL 1974 Private communication regarding unpublished Rolls-Royce data. 10. M. JAESCHI,ZE,W. J. HlLLER and G. E. A. MHER 1975 Journal of Soundand Vibration 43, 467-481. Acoustic damping in a gas mixture with suspended submicroscopic droplets. 11. F . E . MARBLEand S. M. CANDEL 1975 American htstitute of Aeronautics and Astronautics Journal 13, 634--639. Acoustic attenuation in fans and ducts by vaporization of liquid droplets. 12. A. I. IVANOVSKY 1958 Soviet Physics--Acoustics 4, 142-152. On the connection of acoustical streaming with sound absorption. 13. S. GOLDS'rEIN(editor) 1938 l~[odern Developments bt Fhdd Dynamics. Oxford: Clarendon Press. 14. C. -Y. WANG 1965 Journal of Sound and Vibration 2, 257-269. The flow field induced by an oscillating sphere. 15. N. RILEY 1966 Quarterly Journal of Mechanics attd Applied ~,Iathematics 19, 461-472. On a sphere oscillating in a viscous fluid. 16. E. P. MEDNIKOV 1969 Soviet Physics--Acoustics 14, 487--492. Theory of acoustic streaming about very small spherical obstacles. 17. I. PROUDMANand J. R. A. PEARSON 1957 Journal of FluidAfechanics 2, 237-262. Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. 18. M. VAN DYKE 1964 Perturabtion Methods in Fhdd Mechanics. New York and London: Academic Press. 19. J. R. OCKENDON 1968 Journal of Fluid)~lechanics 34, 229-239. The unsteady motion of a small sphere in a viscous liquid. 20. G. K. BA'rCHELOR 1967 An Introduction to Fhdd Dynamics. Cambridge University Press. 21. E . P . MEDNIKOV 1970 SovietPhysics-Acoustics 15, 507-509. Absorption and dispersion of sound in aerosols at large particle velocity amplitudes. 22. A. ACRIVOSand T. D. TAYLOR 1962 Physics of Fhdds 5, 387-394. Heat and mass transfer from single spheres in Stokes flow. 23. P . L . RIMMER 1968 Journal of FhddMechanics 32, 1-7. Heat transfer from a sphere in a stream of small Reynolds number. 24. S . C . R . DENNIS, J. D. A. WALKERand J. D. HUDSON 1973 Journal of FluM Mechanics 60, 273-283. Heat transfer from a sphere at low Reynolds number. 25. F. B. HILDEBRAND 1974 bttroduction to Numerical Analysis. New York: McGraw-Hill Book Company, Inc., second edition. 26. F. A. LYMAN 1976 Syracuse University Department of Mechanical and Aerospace Engbteerhtg Report 3IAE-5192-TI. Attenuation of high-intensity sound in a droplet-laden gas. (Available from U.S. National Technical Information Service, Springfield, Virginia; Accession number PB-252985.) 27. C. L. MORrEY 1968 Jottrnal o f Sound attd Vibration 8, 156-170. Sound attenuation by small particles in a fluid. 28. G. A. DAVIDSONand D. S. SCOTT 1973 Journal of the Acoustical Society of America 53, 17171729. Finite-amplitude acoustics of aerosols. 29. G. A. DAVlOSOy 1976 Journal of Sound and Vibration 45, 473--485. A Burgers' equation for finite amplitude acoustics in fogs. 30. M. ABRAMOWITZ and I. A. STEGtrN (editors) 1964 Handbook of Afathematical Fttnctions. Washington, D.C.: U.S. Government Printing Office. APPENDIX T h e function H(t) derived b y O c k e n d o n [19] for a sphere accelerating from rest with velocity U(t) can be written as follows:
H ( t ) = Hi(t) + H2(t),
(AI)
234
F. A. LYMAN
where
Hi(t) = -(x/n) '/2 f (d U/dO (t
-
T) -1]2
dr,
(A2)
0 Z(t)
//2 -- 3(~r '/2 j G(X)T-'12 y-2 d Y,
(A3)
0
T = t - 3, r = Z ( t ) - Z(z), ~cZ'(t) = U(t),
(A4) (A5) (A6)
G(X) = (nm/2) X - ' erf(X) - exp ( - X 2) - 2X213,
(A7)
X 2 = tr Y2](4T). (A8) The constant r:, defined in equation (21), is of order one. The behavior of H(t) as t ~ oo for U(t) a periodic function of time will be deduced here. In this limit the transients associated with the start-up of the sphere will have died out, leaving only the steady-state response. In the present case U(t) = -V(t)]l Vol, (A9) because V(t) has been defined in equation (3) as the dimensional velocity of the fluid at infinity relative to the sphere, whereas Ockendon's U(t) is dimensionless and is the velocity of the sphere relative to the fluid at rest at infinity. Also, t is the dimensionless time variable (~ot'), in terms of which V(t) = Voexp(it) = I Vol exp [i(t - 6)], where 6, the argument of Vo, is related to the phase lag between the sphere and the flow. Since its precise value is immaterial to the present discussion, any convenient value may be chosen, such as 6 = n/2 so that
U(t) =
- s i n t,
(AI0)
which satisfies Ockendon's requirement that U(0) = 0. For this U(t) the integral in equation (A2) becomes
H,(t) = (2~) v" [C(V'2-~) cos t + S(V/2t/n) sin t] = (hi2) m [(cos t + sin t) - 2g( 2 v ' ~ ] , (AI 1) where C(z) and S(z) are the Fresnel cosine and sine integrals, respectively, and g(z) is an auxiliary function expressible in terms of them (see equations 7.3.1, 2, 6 of reference [30]). As t -~- oo g(2~/~t]n) = O(t-a/2); CA12) hence in this limit H,(t) is a simple sinusoid. In the present case I
Z(t) =_If U(z)dr =--I(I - cost), KJ
(Al3)
K
and 1
Y =--(cos t - cos z),
(A 14)
N
which greatly complicate evaluation of the integral (A3) for all t. To obtain the behavior for large t, however, we expand the integrand'of equation (A3) as y 2 T,12 -
2Ta/Z ,_~xn!(2n + 3) ~-'~--/"
(AI5)
Since in general [19]
T=xY/U(t)+O(Y
z)
as
Y-+0,
(AI6)
ATTENUATION IN A DROPLET-LADEN GAS
235
the integrand behaves as y-1/2 near Y = 0 and is thus integrable. The following estimate of /42(0 for large t is obtained from the leading term of equation (A15): ZCt)
3K5/2
H2(t) N - 40~ 1/2
t
y2dy 1 [ ( 1 - cost)3 5_f (COSt-- cosz)3dt1 Tst2-=4OX/-~K~( t ,'7l" 2.1 (t--z) w2 /"
o
(A17)
o
This indicates that H2(t) = O(t -5/2)
as
t --+ co.
(A18)
Therefore, as t --+ ~o, only the first two terms in equation (A11) remain, and after combining these terms and disregarding the phase difference, we have H(t) = ~0/2cos t. (A19)
ATTENUATION OF HIGH-INTENSITY SOUND IN A DROPLET LADEN GAS F. A. LYMAN
Department of Mechanical and Aerospace Engineering, Syracuse Uniuersity, Syracuse, New York 13210, U.S.A. (Received 29 June 1976, trodin revisedform 2 November 1976) The effect of acoustic streaming on the attenuation of a high-intensity plane sound wave propagating in a gas containing a small concentration of monodisperse suspended droplets is considered. This work was undertaken to explain why some recent experiments on droplet-laden flows in pipes and nozzles indicated considerably larger attenuation than could be explained by existing theories. Acoustic streaming set up in the vicinity of a droplet by the oscillating flow due to the sound wave is analyzed by the method of matched asymptotic expansions. The resulting flow field is used to calculate the enhancement of heat and mass transfer from the droplet by convection. The effect on sound attenuation is determined by modifying existing theories of aerosol acoustics to account for the enhanced heat and mass transfer. The results indicate that the increased sound attenuation due to the proposed mechanism is significant only at droplet Reynolds numbers greater than I, which for droplets of 1 pm radius implies sound pressure levels of at least 160 dB. Also, owing to the tendency of small droplets to follow the oscillating flow, the effect is considerably reduced at frequencies below the characteristic frequency for velocity equilibration of a droplet, which is roughly 16 kHz for a 1 pm radius water droplet. It is concluded that other phenomena must be responsible for the anomalously high attenuation observed in the experiments.
1. INTRODUCTION It has long been recognized that the attenuation of sound in a gas containing suspended particles or droplets may be considerably larger than that in a clean gas. The additional attenuation is due to the fact that momentum, heat, and in the case of volatile droplets, mass, are transferred at finite rates, so that the velocity, temperature and vapor pressure of droplets lag behind the fluctuations in these quantities set up by the sound wave in the surrounding gas-vapor mixture. As a result, there is net dissipation of energy over each cycle. Several investigators [1-5] have contributed to the theory of the attenuation of plane sound waves of infinitesimal amplitude in particle- or droplet-laden gases, and their results have been well verified by experiments [6, 7]. Recently, however, experiments [8-10] on air-water mixtures containing droplets formed by condensation in duct and nozzle flows seem to indicate that the attenuation of high-intensity sound may be considerably larger than that predicted by the existing theories. Although these experiments do not provide conclusive evidence of anomalously high attenuation, because the droplet sizes were not measured within the flow, they do indicate that the introduction of a small concentration of micronsized droplets into a gas flow can produce sufficient attenuation to make it of interest as practical technique for reducing noise in aircraft turbofan engines, air conditioning ducts, and gas pipelines. 9The idea of injecting droplets into the inlets of turbofan engines to attenuate fan noise was 219
220
r.A. LYMAN
recently proposed by Marble and Candel [11]. Their estimates, based on small-amplitude theory, indicate attenuation of only about 3 dB per meter for a 1 percent mass concentration of uniformly distributed monodisperse droplets of 1 Fm radius, and such attenuation is rather marginal for a practical noise reduction technique. If, on the other hand, the actual attenuation due to droplets were much larger, as the experiments seem to indicate, droplet injection would warrant further consideration. Accordingly, it was deemed important to investigate why the actual attenuation might be larger than theory predicts. One phenomenon leading to greater attenuation at high sound levels is acoustic streaming, wherein a stead), secondary flow is set up in the vicinity of a droplet by the Reynolds stresses associated with the oscillating flow induced by the sound wave. It is well known that acoustic streaming can increase the rates of heat and mass transfer and chemical reactions at solidfluid or fluid-fluid interfaces. The effect can be significant, even though the characteristic velocity of acoustic streaming is generally much smaller than the instantaneous velocity of the oscillating flow. Since the magnitude of the acoustic streaming velocity is proportional to the intensity of the sound wave, this effect would be expected to be important at high sound intensities. Although the effect of acoustic streaming on sound attenuation in a clean gas has been studied [12], the corresponding situation in a droplet-laden gas apparently has not been previously investigated. The present analysis was carried out to determine quantitatively the increase in sound attenuation due to the enhancement of heat and mass transfer from the droplets by acoustic streaming. The main steps in the analysis are as follows: (a) determination of the flow field near a single droplet and calculation of the drag and acoustic streaming, (b) solution of the convection equation for heat and mass transfer due to acoustic streaming, and (c) calculation of the effect of increased heat and mass transfer on sound attenuation. These steps are described in greater detail in the following sections.
2. ACOUSTIC STREAMING Consider a spherical droplet of radius a suspended in a gas-vapor mixture through which a plane sound wave propagates. The velocity of the gas-vapor mixture due to the sound wave is U= = Uoe Ic'~'-k~. (la) Since ka < I i n all cases of interest here, the flow about the droplet can be considered incompressible, with an oscillating free-stream velocity
Vow(t) = Vo elo' .
(lb)
For the purposes of this paper, the velocity amplitude Uo will be related to the sound pressure level by the linear acoustics relations
Vo = "v"-~p.,,~lpc = (v'~p,./pc)
lO ",'~'o,
(2):
where p is the density of the gas-vapor mixture, c is the speed of sound in the mixture, Lp is the sound pressure level (SPL) in dB, prms is the r.m.s, pressure fluctuation, andp,ef = 2 x 10-5 N/m 2. Equation (2) will be used only to express the results in terms of SPL, inasmuch as no simple relation between pressure and velocity amplitudes exists in non-linear acoustics. It is convenient to adopt a system of spherica ! polar co-ordinates (r,0) with origin at the center of the moving droplet and polar axis along the axis of oscillation, so that far from the droplet the velocity of the flow relative to it is
V(t) = U=(t)- Va(t)= Vo e''~' ,
(3)
ATTENUATION 1N A DROPLET-LADEN GAS
221
where Ua(t) is the droplet velocity, and Vo is in general complex. Owing to the tendency of the droplet to follow the oscillating flow, ] Vo] will in general be smaller than Uo. Due to the symmetry of the flow about the axis of oscillation, the velocity compoents may be derived from a stream function tp by means of the relations 1 ar 1 ar ltr = r2 sin 0 00 ' tto = r sin 0 Or (4) When written in terms of dimensionless variables t = tot ', r = r7a, u = u'/[ Vol, ~k = ~'/I Vola z (primes denoting dimensional quantities), equations (4) are unchanged whereas the equation for the stream function [13] becomes ~ J )SP-) D2 "~ M 2 - ~0( D 2 ~k) + R _[I I r 2 a(~,
+= 2 D2~tpL~b9/
D4 ~k,
(5)
It ' 0 1 0 1 -- pZ Or " b - -att
(6)
where # - cos0,
0z 1 -- 112 0 z D2 -~ Or2 q" r 2 0112'
L-
The dimensionless parameters in equation (5) are M = a(to]v) 1/2,
(7)
which is the ratio of droplet radius to the thickness of the oscillatory boundary layer, and R = IV ol a/v, (8) the droplet Reynolds number. Note that M 2 = R S , where S = t o a / I Vol (9) is the Strouhal number, or the ratio of the droplet radius to the amplitude of oscillation. Equation (5) is to be solved subject to the boundary conditions of no slip or normal velocity at the surface of the sphere, tp = O~O/Or= 0 on r = I, (lOa) 9and a uniform oscillating flow at infinity, ~ kr2(l --p2) el' as r --~ co. (10b)
This problem has been attacked by several authors, most recently Wang [14], Riley [15] and Mednikov [16]. Wang and Riley treat the case S >> I : i.e., small-amplitude oscillations. For micron-sized droplets subjected to sound pressure levels in excess of 100 dB, however, the amplitude of oscillation is at least of the same order as the droplet radius, and may be much larger. Figure 1 shows "maps" relating the dimensionless parameters R,'~, S and M z to the droplet radius a, frequency f = to/2z and sound pressure level (SPL). It is apparent that for micron-sized droplets and frequencies in the audible range, M2,~ 1, whereas R~ ranges from 10-4 to almost 10 as the SPL increases from 100 to 160 dB. The fact that M 2 ,~ 1 suggests treating the flow as quasi-steady, i.e., neglecting the first term on the left of equation (5), which implies that the oscillations of the fluid are everywhere in phase. This was the approach of Mednikov, who adapted the well-known solution of Proudman and Pearson [17] for steady flow about a sphere at low Reynolds numbers to the case of an oscillating free stream by simply replacing the Reynolds number by a periodic function of time. By integrating over a cycle and using certain symmetry arguments, he obtained the steady component of the stream function. Mednikov overlooked the fact, to be shown subsequently, that at sufficiently large distances from the sphere unsteady effects cannot be neglected. Therefore his far-field solution is of questionable validity,, and because i" Since the amplitude [Vol of the relative velocity is not known a priori, Figure l(a)shows the acoustic Reynolds number R~= Uoa/vrather than R. In general, R ~
222
F. A. LYbfAN l
I
I
I
la) SPL (dB) ,,/160 140 "
2 0 o
E
-2 -4 (c) -6
I
-2
,
-i
I
I
I
o
I
2
I
I
I
I0
IOg~o 0 (/din) =
l
i
I
I
I
I i
I
SPL {dB}-
/ioo
2
//~zo
-
o
f[kHz)'" . IO0
0-1 0.0
~:~ - 2
o
-4
S/,,//,,~o 0
-6
-2
-8
-4
-2
o
2 IOglo
4
6
-10
-2
-I
of (p.m/sl
o
I
log ~o o ( p m )
Figure 1. Dimensionless parameters for droplet of radius a in the oscillating flow set up by a plane sound wave of frequency f. (a) Acoustic Reynolds number, R, = Uoalv; (b) Strouhal number, S = 2naflUo; (c) M 2 = R o S = 2nfa'lv. Acoustic impedance pc and kinematic viscosity v are those of dry air at 1 arm and 15~ (pc = 414 N s/m3; v = 1-46 x 10-5 m2/s).
the stream function near the sphere is obtained by matching with the outer solution, this casts doubt on the inner solution. In the region near the sphere one may neglect the unsteady term in equation (5) and solve the resulting equation by means of the regular perturbation expansion
~k= ~o-t- R~t + ....
(I1)
The zero-order solution satisfying boundary conditions (10) is 4o = 882r2 - 3r + I/r) (1 - pz) cos t,
(12)
which represents quasi-steady oscillating Stokes flow. The periodic behaviour in time of equation (12) implies that to this level of approximation there is no acoustic streaming, and one must proceed to the next term in the expansion (11) to obtain it. The next term in the inner expansion is obtained by substituting equation (12) for ~kin the convective term in equation (5), which results in D 4 ~kl = - ~ (2/r 2 -- 3/r 3 + l/r 5) p(l -- p2)cos 2 t. (13) The solution of this equation which satisfies the inner boundary condition (10a) and is symmetric with respect to the plane 0 = n/2 is ~, = - - ~ 2 ( 2 r 2 - 3 r
+ 1 - I + L ' ~ Pr '(/ l r
--'/L2)COS21'q" (I//1)c'
(14)
where (~1)c is the complementary solution of equation (13) which also satisfies boundary conditions (10a) and has the requisite symmetry. It is to be determined by matching with the outer solution. It is well known [17, 18] that these inner solutions break down when Rr = O(1). This can
223
ATTENUATION IN A DROPLET-LADEN GAS
be shown by using equation (12) to obtain the following estimate for the ratio of the neglected convective term to the viscous term: convective/viscous = O(Rr) as r --~ oo. 05) Besides this, there is an additional restriction in unsteady flow that appears to be less well known. An order-of-magnitude analysis using equation (12) to estimate the ratio of the unsteady and convective terms in equation (5) discloses that unsteady/convective = O(Sr) as r -+ 0% (16) which, together with equation (l 5), implies that unsteady/viscous = O ( M 2 r2). (l 7) Therefore the unsteady term must be retained in the region beyond the point where M r = 0(1). In the most general case, in which M 2 and R are unequal and both less than l, the flow should be divided into three separate regions, in each of which different sets of terms in equation (5) would be important, and a separate perturbation expansion would be required. If, however, M 2 = O(R2), then the two outermost regions merge into a single one, in which both unsteady and convective terms are important. In the region Rr > 1 the new variables ~ = R2 ~, p =Rr (18a, b) are introduced, following Proudman and Pearson [17]. This makes the coefficients of the viscous and convective terms the same and transforms equation (5) into ~_) ~ ( ~ 2 ~ u ) + ~ _
a(p,~,)
+2~2~L'e~
=~4~,
(19)
where ~,,2 and .~. are the differential operators obtained by replacing r by p in the expressions of equation (6) for D 2 and L, respectively. To solve equation (19), the expansion ~t' = !Po(p, i~, t) + RtPI(p, la, t) + . . . (20) used by Proudman and Pearson is attempted. Obviously this cannot work for arbitrary M, because the coefficient of the unsteady term would become infinite as R ~ 0 for fixed p. Therefore, it is assumed that M 2 = KR 2, (21) where ~c is a constant of order unity. This is the simplest and most convenient assumption which corresponds reasonably well to the present situation.t The zero-order term in the expansion (20) is simply an oscillating uniform flow, q~o = ~P2( 1 -- p2) cos t. (22) Substitution of this and equation (20) into equation (l 9) leads to the following equation for ~/Jl: K _ ~ / ( ~ z ~ , ) q _ C O S I ( 1 --/'t2 0
P
~+/a0-~)~2~,=~4~
,.
(23)
This is equivalent to Oseen's equation for an oscillatory free stream. The time-dependent coefficient makes it intractable to the usual techniques of solution, however. Fortunately, Ockendon [19] has recently managed by an ingenious technique to obtain a solution to the unsteady Oseen equation for the case of a sphere accelerating in an arbitrary manner. He considered the effect of the inner boundary condition on the outer solution to be correctly simulated by the introduction of a delta-function term to the Navier-Stokes equation. The integral of this singular term corresponds to the Stokes drag on the sphere. t Although Figure 1 indicates that for droplet radii of 1-10/~m, R, is larger than M 2 when SPL > 140 dB andfis in the audible frequencyrange, the fact that R < Roin general and is much less than R~for these droplets at audible frequenciescompensates for this.
224
F.A. LYMAN
The modified equation with the singularity at the origin was solved by Fourier transforms in the three spatial co-ordinates over the entire region, including the origin. The resulting expression for the stream function is expanded in powers of R to obtain the following for the inner limit o f the outer solution: lim (~l)outer = - - t ~ r 2/a(l -/~2) cos 2 t + 89 - l/2) n(t). (24) R---~ 0
The second term represents an unsteady but uniform stream with velocity H(t) along the axis of oscillation of the sphere. The function H(t) was obtained by Ockendon in terms of a convolution integral involving the velocity ofthe sphere relative to the fluid. The general form of this function and its behavior in the case of oscillating motion are discussed in the Appendix. Choosing the complementary function in equation (14) to match the second term in equation (24), one obtains for the inner solution ~'1='~
2r2-3r+l-
+
p(l-lt2)cos2t+ 88
2r2-3r+
(1-/a2).
(25)
Consider first the steady component of the stream function, obtained by averaging equation (25) over a period. Obviously ~ko has no steady component. Furthermore, one would expect that the time-average of H(t) should also be zero because of the antisymmetry of the second term in equation (25) about the equatorial plane (0 = 7r/2 o r / t = 0), which is inappropriate for acoustic streaming. In the Appendix it is shown that for an oscillating sphere the behavior of H ( t ) as t ~ co is indeed periodic, with zero mean value. Therefore the steady component of the stream function near the droplet is i~ = - ~ R (2r 2 _ 3r + 1 - l/r + 1/r2) sin2 0 cos 0, (26) from which the relations (4) yield the velocity components
3R/2_L+ L l+l 3cos20-1 r2-7
7/
i ( 3 l uo=~R 2 - ~rr -I-
r
l)
2r 3
2
r-4. sin0cos0.
(27) '
(28)
Note that these are proportional to R, which increases rapidly with the sound pressure level. The streamlines in one quadrant of the meridional plane are plotted in Figure 2 for the particular case R = I. The streamlines in the other quadrants are of course symmetric with respect to 0 = 0 and n/2.
Figure 2. Streamlines of steady acoustic streaming flow for R = 1. The dircction of the oscillating free stream is indicated by ~-~.
ATTENUATION IN A DROPLET-LADEN GAS
225
The drag on the droplet can be calculated to order R from equations (l l), (12) and (25). The first term in equation (25) contributes no net force, because of the symmetry of its velocity components about 0 = 0 and n/2. The drag is due to the Stokes stream function @o and the second term in @1, and since the latter has the same dependence on r and 0 as the former, the result is Fx = 6rratll 1Iol [cos t +RHCt)], (29) where '1 is the viscosity of the gas-vapor mixture. From equation (A19) of the appendix, as t--* co, RH(t)... Mcost. Thus the first-order correction to the Stokes drag formula for oscillatory flow about a sphere is O(M), instead of O(R) as in the Oseen formula for steady flow [17]. The physical reason is that the additional drag in steady flow is due to the wake formed by the slow convection and diffusion of vorticity far from the sphere [20]. In an oscillating flow such a wake cannot form, owing to the continual reversals of the free stream. For the small droplets considered in this paper, .~it 2 ,~ 1, and the correction to the Stokes drag can be neglected. This result suggests that Mednikov's treatment [21] of the absorption of large-amplitude sound in an aerosol is incorrect, because he assumes not only that the steady-state Oseen drag correction is valid for oscillatory motion, but also that it may be approximated by taking its maximum value. 3. ANALYSIS OF CONVECTIVE HEAT AND MASS TRANSFER BY ACOUSTIC STREAMING 3. I. FORMULATION OF THE PROBLEM AND METHOD OF SOLUTION
The steady flow obtained in the preceding section is used to determine the convective transport of heat and mass between the droplet and the fluid. The energy equation can be written in the dimensionless form V2 T = . R ( u,0T+-- u~ aT.I or 7 b-a-J '
(30)
where V2
1 a/ 2a'~ l ;(sin0a__~) -=~'-~r~ r ~r} q" r2sin----~
In equation (30) a = v/ct, is the Prandtl number, at is the thermal diffusivity of the gas-vapor mixture, and T represents the normalized temperature difference, obtained by subtracting the temperature T~0 at large distances from the droplet from the temperature, and dividing by the difference Ts - T~, where T~ is the surface temperature of the droplet. The boundary conditions are therefore T= 1 on r = l, (31) T --~ 0 as r -+ co. (32) The mass transfer problem is formulated in the same way, except that a is replaced by the Schmidt number v/D, where D is the coefficient of diffusion of vapor through the gas, and T by the normalized concentration difference. Although equation (30) is linear, the fact that it is non-separable for the flow described by equations (27) and (28) makes it unlikely that exact solutions can be obtained by analytical means. If R and a are small, a perturbation solution in these quantities is possible. Acrivos and Taylor [22] solved the corresponding heat and mass transfer problem for a sphere in Stokes flow by the method of matched asymptotic expansions, and Rimmer [23] extended the solution to higher Reynolds number by using the corrections to Stokes flow obtained by Proudman and Pearson [17]. In the present problem, however, the flow far from the sphere is not a uniform stream, which makes it difficult to obtain an outer solution valid for all angles.
226
r.A. LY.MAN
Therefore it was necessary to resort to numerical techniques. The series truncation method developed by Dennis et aL [24] was found to be particularly well suited to this problem. Substituting the expressions (27) and (28) for the velocity components into equation (30) and transforming to the variables ~ = lnr and tt = cos0, one obtains a 2T OT a f 0 T ] e -r
ae
(33)
e
where g = - ~ a R z,
d? = 89
2 - l ) f ( O e ~,
f ( ~ ) = 2e ~ - 3 + e -~ - e -2~ + e -a'.
The temperature is expanded in a series of Legendre polynomials T ( ~ , p ) = ~ t.(~) P.(p).
(34)
n=O
In the present case the symmetry of the acoustic streaming flow about the midplane it = 0 requires that T ( ~ , p ) also be symmetric about this plane, so that n is restricted to even integers. The functions t.(~) satisfy the boundary conditions t.(0) =
1, 0,
n = 0, n # 0,
(35)
t. ~ 0 as ~ ~ m. (36) By substituting the expansion (34) into equation (33) and using the properties of the Legendre polynomials, one obtains an infinite set o f ordinary differential equations t~, + (1 - A.) t,; - [n(n + 1) + B.] t. = S., (37) where in the present problem A.(~) = - - ~ e ~ . . f ( O ,
B . ( O = --lzefl.. g ( Q ,
o0
S.(~) = - 5 e ~o{~.~f({) t;,({) + fl.kg({) t~(r k•n
d g(O = e - e ~ - [eef(O] = 4e~ - 3 + e -2r - 2e -3r . The summation in S.(~) is over ~ e n integer values of k, excluding k = n, for which the corresponding terms have been included on the left side of equation (37). The constants ~.k and/~.k are defined by the following definite integrals: +1
ffnk
~--"
4
!. fj --1
+1
P.(p) Pk(pJ(3p a -- 1) dp,
flnk
=.,2n 4+
1
fj
--1
These are evaluated by using the recursion relations for Legendre polynomials, with the results n(n + 1) 3 (n + 1)(n + 2) 3 n(n - 1) Ct"k= (2n -- 1) (2n + 3) 6.k + -~ (2n + 3) (2n + 5) 6.+=, k + ~ (2n -- 3) (2n -- 1) &.-2.k, n(n - I)(n - 2) a ] ' 1 [ n(n + 1) (n + l)(n + 2)(n + 3)6.+= ~ ('~n'--'l)--~'--'~-"-2'~] /~"~ = ~- I.(2n --- l]-(2n + 35 a"~+ (2n + 3)(2n + 5) '
where 6.k equals 1 when k = n and is zero otherwise. Thus the differential equation (37) for t.({) contains on the right side only terms involving t._2({) and t.+=({) and their first derivatives. In practice, the series (34) is truncated at a finite number, n = n o . The finite system of equations is solved numerically, using central differences to approximate the derivatives. The corresponding finite-difference equations are of the t'orm al t.(i - 1) + bi t.(i) + ct t.(i + 1) = h z S.(i), (38)
A T T E N U A T I O N I N g D R O P L E T - L A D E N GAS
227
where tn(i ) -- tn(~,) denotes the value of the function at the ith mesh point ~, = ih, and a, = 1 - 89 [1 - A,(i)], b, = -{2 + h2[n(n + 1) + R~(i)]}, c, -- 2 - at. For each n and i = I t o / , the tridiagonal system of equations (38) is solved by standard techniques [25]. Inasmuch as Sn contains the function tn+2(~) and its derivative, and this function is unknown at the time tn(~) is being generated by solving the finite-difference equations, an iterative procedure is necessary. At each step in the iteration procedure Sn is calculated by using the function tn+2(~) generated during the previous iteration, but the latest values of the function tn-2(O. This iterative, procedure was followed to solve equations (38) on a digital computer for = 0.1, 0.5, 1.0, 1-5 and 2.0. For a Prandtl number tr = 0-72, this corresponds to Reynolds numbers R = 0-86, 1-92, 2.72, 3.33 and 3.85. For most ofthe calculations a step size h = 0.025 was used, and the calculations were carried out to ~l = 5.0, corresponding to a radial distance of 148.4 times the radius of the sphere. In order to ensure convergence, it was found expedient to use the "interpolation" procedure suggested by Dennis et al. [24]. At the completion of t h e j t h iteration, the functions tt~J)(~) have been calculated. On the next iteration, as each function, which will be denoted fn(~), is calculated, the new ( j + l) approximation to tn is obtained by interpolating between in and t~J), as follows: t~+~'(~) = E~(~) + (1 - E)ttJ'(~). The parameter E was selected by trial to prevent divergence of the iterative procedure. The value E---- 0.05 was used for most of the calculations. For e = 0.1, the convergence was obtained at values of E up to 0.2, however. The iterative procedure was considered to have converged when It,'~+-(O- t'.J'(r < 10-4 for all ~ and n ~
to(~j) = -t;(O) e -~ + ]-6
f ( O t2(O d~,
(39)
r
where co
t6(0 ) =--1 +~-0 ~ f ( ~ ) t2(~)d~.
(40)
0
Therefore, once t2 is determined, to can be found by simple quadrature. Thus it was necessary only to calculate those tn for n ~> 2 by solving equations (38) by the iterative method, t~(~) was eliminated from the first of these equations 0l = 2) by using relation (39). To start the procedure it was assumed that tn = 0 for n >/4, and the initial t2 was obtained by the finite-difference solution of the n = 2 equation. The iterative procedure was then carried out for all equations 2 ~
228
F.A.
LYMAN
region near It = 0 (the temperature "wake"), and because the validity of the asymptotic representation is open to question, its development, described in reference [26], will not be given here. The numerical results indicated that t2 varies as e x p ( - 2 0 for large 4, regardless of the actual boundary condition applied at ~z-This behavior is also predicted by an asymptotic analysis of the differential equation for t2 when t+ is neglected [26].
3.2.
CALCULATED
RESULTS
The rates of heat and mass transfer from the droplet are the quantities of main interest. Since these rates are derived in an identical manner, only the calculation of the former will be described. The heat transfer per unit area, per unit time through a point on the surface ot the droplet located at angle 0 to the oscillating flow is
q ( O ) = - k T s - T= 0(~_) a
~
(411
=0'
where k is the thermal conductivity of the surrounding gas-vapor mixture. Thecorresponding local Nusselt number based on droplet radius is
Nu.(O) = k(Ts- T.~)= -
~=o = -
.=o ~" t;,(0) P.(cos 0).
(42)
The average value of this over the surface of the droplet is 71"
Flu. = 89J Nu.(O) sin 0 dO = -t~(0).
(43)
0
The calculation oft6(0) was carried out by quadrature, equation (40). For n ~> 2, t,~(0) was calculated by using a three-point Lagrangian formula. The local Nusselt number is plotted as a function o f 0 in Figure 3 for various values of the parameter e. As expected, the heat transfer is largest at the pole 0 = 0 and decreases monotonically towards the equator. 0"5
I
I
I
]
I
I
I
I
I
F
I
~
[
I
0-4
T
0-3
0-2
0.1 I
I0
20
30 40
50 60 70
80 90
e Figure 3. V a r i a t i o n o f local Nusselt number at droplet surface w i t h p o l a r angle 0 measured f r o m axis o f oscillation.
The average Nusselt number and Sherwood number are in this analysis a function of the parameter ~ = :-~aR z alone. For heat transfer, ais the Prandtl numberofthegas-vapor mixture, while for mass transfer it is the Schmidt number. The increase in the average Nusselt or Sherwood number due to acoustic streaming is plotted as a function ore in Figure 4. It is seen that at ~ = 2.0 the increase is about 30 percent.
ATTENUATION IN A DROPLET-LADEN o
.
3
GAS
229
~
"7
"7
0.2
0.1 z o
0
I
Convectionparameter,
2 _ 3
~r-
T~o-R2
Figure 4. Increase in Nusselt and Sherwood numbers due to acoustic streaming. The convection parameter E depends on the droplet Reynolds number R = [ Vo]a/v and a, which for heat transfer is the Prandtl number, and for mass transfer the Schmidt number.
4. EFFECT OF ACOUSTIC STREAMING ON SOUND ATTENUATION The effect of acoustic streaming on sound attenuation is due to enhanced heat and mass transfer. As pointed out at the end of section 2, acoustic streaming does not affect the drag on a droplet, which to a good approximation is given by the Stokes formula. Since convective heat and mass transfer from a droplet are governed by the linear equation (30), the corresponding transfer rates are proportional to the temperature or concentration differences (e.g., see equations (41) and (42)). Thus the heat and mass transfer from a single droplet are Q = 47rakN~vu(T~ -- To~), (44) M = 4naDFlsh(P~ -- Pv). (45) These are of the s a m e form as the expressions used by Cole and Dobbins [4] in their analysis of the attenuation of small-amplitude sound waves in an aerosol, except here the average Nusselt and Sherwood numbers are greater than unity. Since these parameters are independent of x and t, however, the analysis of sound attenuation can be carried out just as in reference [4], by incorporating N,~u and/Vsh as follows in the expressions for the relaxation times for temperature and concentration equilibration: z, = ma cp/4r~akNuu = Pl a2/3oq PNu,,,
(46)
rc = ma/ 4rcapD ~l~n = Pt a2/3p D EI~h (47) where ma = (4n/3)Pt a3 is the droplet mass, Pl is the liquid density, and cp and p are respectively the constant-pressure specific heat and density of the gas-vapor mixture. One point deserves mention: the ratio zc/rt is in the Cole and Dobbins theory equal to the Lewis number, NLe = o~t/D, whereas according to equations (46) and (47) zd
,=
(
Nt.e.
(48)
230
F.A. LYMAN
Therefore the Lewis number has to be multiplied by (.K[A.Jfitsh) whenever it appears in the Cole and Dobbins results. The Nusselt and Sherwood numbers depend on the droplet Reynolds number given by equation (8), and it is necessary to determine the latter as a function of the sound pressure level and frequency. For this purpose the amplitude of the relative velocity V is required. Since the drag on the droplet can be adequately represented by the Stokes formula, the equation of motion of the droplet is d U d / d t = (l/ra)(U~ -/./4). (49) where Tn = ma/6rr~q = 2pl a2/9pv (50) is the relaxation time for velocity equilibration. The result of solving equation (49) for the oscillating free-stream velocity of equation (I) is I Vol/Uo = cord/V'I + (cora)2,
(51)
where Uo is given by equation (2). According to equation (51), at low frequencies (cora < l) the amplitude of the flow velocity relative to the droplet is significantly smaller than the peak velocity induced by the sound wave. This effect tends to limit the effect of acoustic streaming on sound attenuation to high frequencies: i.e., those for which cord > I. Another implication of equation (51) is that the Reynolds number R defined by equation (8) is frequency-dependent, which makes the Nusselt and Sherwood numbers functions of frequency as well as sound pressure level and droplet radius. The fact that R depends on a both directly and implicitly, through cord, precludes the possibility of presenting, as in reference [4], curves of attenuation per wavelength versus cord which are independent ofa. The foregoing modifications were all incorporated in the calculation of the attenuation coefficient]" a according to equation (24) of Cole and Dobbins' paper [4]. Figure 5 displays typical results of attenuation as a function of frequency for selected sound pressure levels and droplet radii. For droplets of radius I lam, the increase in sound attenuation due to acoustic streaming is perceptible only for sound pressure levels approaching 180 dB and frequencies above 16 kHz, where it amounts to about I dB/m (Figure 5(a)). This is so because corn for I lam droplets is about I at 16 kHz; much below that frequency equation (51) predicts that the droplets follow the oscillations set up in the air-vapor mixture by the sound wave. For larger droplets the increase in attenuation due to acoustic streaming appears at lower frequencies and somewhat lower sound pressure levels, but the amount is too small to be significant. This is illustrated in Figure 5(b) for 2/am radius droplets, for which the increase in attenuation is slightly less than 0.2 dB at a sound pressure level of 170 dB. Although the results in Figure 5 were calculated by using the complete expression for attenuation derived by Cole and Dobbins, it is worthwhile to give the following approximate expression, which more readily displays the effect of acoustic streaming: CWo I [A+B(flcor,) 2"1
1 [ (cora)2 .]
C. -~.t i T(~r~) z J + ~ _ i + (cora)2J'
(52)
where Co is the speed of sound in the air-vapor mixture in the absence of mass, momentum and energy transfer with the droplets, and Cm is the droplet mass fraction (mass of droplets per unit mass of the air-water vapor mixture). The coefficients~ A, B and p depend mainly on t Defined in the conventional manner by l(x) = l(0)exp(--~x), where l(x) represents the sound intensity1 after the plane wave has propagated a distance x in the medium. Here 0~has units of m-x. The attenuation in( dB/m, equal to 4.34z, is the quantity presented in the figures. In terms of the constants Ct ..... C8 definedby Cole and Dobbins [4], A = (C4 - Cs)IC2, B = (Cl - C6)[Cs ? - l + (NrsJlqLeNtu~)f, and fl= C5[C2, where ~,is the ratio of specific heats of the air-water vapor mixture, and ~" PdP is the vapor mass fraction. =
231
ATTENUATION IN A DROPLET-LADEN GAS 18
(o) i
i
1
l
i
i
i
i
1
i
i
i
16
,r
//
14 12
io 8 6 4 II)
2 L
0
~
I
I
I 4k
I Elk
I
I
I
I [ 16k 32k
I 64k
4
(b) 3
2
I
0
f
I l 16 31-.,5 621
I [ I 12.5 2.~0 , , ~
I Ik
l 2k
12Bk
FrequenCy (Hz)
Figure 5. Effect of acoustic streaming on attenuation due to droplets, for various sound pressure levels (SPL). (a) D r o p l e t radius 1 lam: , SPL < 140 dB; . . . . . , SPL = 180 dB; (b) droplet radius 2 lam: , SPL < 140 dB; . . . . , SPL = 160 dB; - - - - - , SPL = 170 dB. Temperature 23~ droplet mass fraction, 0-01.
the thermodynamic and transport properties of air and vapor, and are only weakly dependent upon the frequency and sound pressure level, the latter dependences being introduced by the modification given in equation (48). Equation (52) is valid when C,, ,~ 1 and (ogr,)2 ~ C,,, except that unsteady effects neglected in the relaxation model also impose an upper frequency Iimit such that the parameter M defined in equation (7) must be small [2, 27]. Acoustic streaming affects only the first term in equation (52), and that primarily through the Nusselt number dependence ofz,. In fact, since acoustic streaming is only important when corn and tort are of order 1 or greater, the following estimate of the maximum increase in attenuation is obtained from equation (52) by assuming tort >> I : Co(A~).,~, B v-1
cm
~
(~,)o
(~TN~ - 1) ~ 5---:-, ( ~ , , ~ - 1),
(53)
trt)o where (r~)oiS given by equation (46) with -~.u = 1. Thus the maximum increase in attenuation is proportional to the increase in the Nusselt number, which can be determined from Figure 4 at any SPL. Also, since xt is proportional to the square of the droplet radius, the amount of increase is much greater for smaller droplets.
5. CONCLUDING REMARKS The calculated results indicate that the effect of acoustic streaming is not sufficient to provide an explanation for the anomalously high sound attenuation observed in experiments with duct and nozzle flows. Although the predicted magnitude of this effect alone is perceptible
232
F . A . LYMAN
(1 dB/m) for 1 pm radius droplets at the rather high sound pressure level of 180 dB, other effects ignored in the present analysis would likely mask it under such conditions. For one thing, non-linear acoustics effects, such as the growth of higher harmonics [28, 29], are important at these amplitudes. Despite much recent research on the non-linear acoustics of aerosols [28, 29], no way has yet been found to analytically determine the total attenuation~ only the decay of the fundamental and growth of the second harmonic have been determined, although much has been learned about the evolution of the waveform. Thus it is difficult to make a meaningful comparison of streaming and amplitude distortion effects. Another phenomenon of possible importance at high amplitudes is droplet-droplet collisions, leading to shattering and/or coalesence. This would most likely affect attenuation by altering the droplet size distribution although there does not appear to be a significant direct energy loss in collisions at the low droplet number densities considered here. The results obtained here for the enhancement of heat and mass transfer from droplets by acoustic streaming may also prove to be useful in other applications, such as the evaporation and combustion of liquid droplets in the intense sound fields set up in combustion chambers. The analysis presented in sections 2 and 3 is valid for droplet sizes and frequencies such that 1)12 ,~ 1, which according to Figure l is true for droplets of radius less than l0 llm at all audible frequencies. For larger droplets or higher frequencies, such that M 2 > I, the flow near the droplet is quite different from that shown in Figure 2, the most conspicuous feature being a shear layer near the droplet of thickness O(M -t) in which the vorticity generated at the surface is confined [15]. This would lead to a much larger increase in heat and mass transfer than in the caseconsidered here. Finally, it should be noted that the continuum approach of the present work excludes from consideration very small droplets (radii less than 0-1 lam). Such droplets have been shown [8, 10] to cause extraordinarily large attenuation when they are formed by sudden condensation from a supersaturated gas-vapor mixture undergoing rapid expansion through a nozzle. Since the radii of these droplets are smaller than the mean free paths of the gas and vapor molecules, the mass, momentum and energy transfer rates must be appropriately modified, as in reference [I0]. But if the flow about the droplets were in the free-molecular or transition regimes, it is doubtful whether acoustic streaming would even occur.-]" Also, the very small droplets are extremely short-lived. They rapidly grow or evaporate (depending on whether the mixture is super-saturated or unsaturated); hence one would expect them to be important mainly in rapid expanding nozzle flows but not so in low-speed duct flows, especially where the presence of foreign nuclei (dust particles, etc.) would not allow significant supersaturation. ACKNOWLEDGMENT This work was supported under National Science Foundation Grant ENG 73-04257. REFERENCES 1. P. S. EPSrEINand R. R. CARHAR'r1953 Journal of the A coustical Society of America 25, 552-565. The absorption of sound in suspensions and emulsions. I. Warm air in fog. 2. S. TEr,fKIN and R. A. DOBBINS1966 Journal of the Acoustical Society of America 40, 317-324. Attenuation and dispersion of sound by particulate-relaxation processes. 3. F. E. MARBLEand D. C. WOOtEN 1970Physics of Fhdds 13, 2657-2664. Sound attenuation in a condensing vapor. t In the continuum flow regime, the acoustic streaming velocity is proportional to the Reynolds number, as in equations (27) and (28), and therefore proportional to the density.
ATTENUATION IN A DROPLET-LADEN GAS
233
4. J. E. COLE, III and R. A. DOBBINS 1970 Journal of the Atmospheric Sciences 27, 426--434. Propagation of sound through atmospheric fog. 5. G. A. DAVlDSON 1975 Journal of the ,4 tmospheric Sciences 32, 2201-2205. Sound propagation in fogs. 6. S. TEMKIN and R. A. DOBBINS 1966 Journal of the Acoustical Society of America 40, 1016-1024. Measurements of attenuation and dispersion of sound by an aerosol. 7. J. E. COLE, III and R. A. DOBBINS 1971 Journal of the Atmospheric Sciences 28, 202-209. Measurements of the attenuation of sound by a warm air fog. 8. W . J . HILLER, M. J'AESCHKEand G. E. A. MEIER 1971 Journal of Sound and Vibration 17, 423--428. The influence of air humidity on pressure and density fluctuations in transonic jets. 9. K. W. BUSnELL 1974 Private communication regarding unpublished Rolls-Royce data. 10. M. JAESCHI,ZE,W. J. HlLLER and G. E. A. MHER 1975 Journal of Soundand Vibration 43, 467-481. Acoustic damping in a gas mixture with suspended submicroscopic droplets. 11. F . E . MARBLEand S. M. CANDEL 1975 American htstitute of Aeronautics and Astronautics Journal 13, 634--639. Acoustic attenuation in fans and ducts by vaporization of liquid droplets. 12. A. I. IVANOVSKY 1958 Soviet Physics--Acoustics 4, 142-152. On the connection of acoustical streaming with sound absorption. 13. S. GOLDS'rEIN(editor) 1938 l~[odern Developments bt Fhdd Dynamics. Oxford: Clarendon Press. 14. C. -Y. WANG 1965 Journal of Sound and Vibration 2, 257-269. The flow field induced by an oscillating sphere. 15. N. RILEY 1966 Quarterly Journal of Mechanics attd Applied ~,Iathematics 19, 461-472. On a sphere oscillating in a viscous fluid. 16. E. P. MEDNIKOV 1969 Soviet Physics--Acoustics 14, 487--492. Theory of acoustic streaming about very small spherical obstacles. 17. I. PROUDMANand J. R. A. PEARSON 1957 Journal of FluidAfechanics 2, 237-262. Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. 18. M. VAN DYKE 1964 Perturabtion Methods in Fhdd Mechanics. New York and London: Academic Press. 19. J. R. OCKENDON 1968 Journal of Fluid)~lechanics 34, 229-239. The unsteady motion of a small sphere in a viscous liquid. 20. G. K. BA'rCHELOR 1967 An Introduction to Fhdd Dynamics. Cambridge University Press. 21. E . P . MEDNIKOV 1970 SovietPhysics-Acoustics 15, 507-509. Absorption and dispersion of sound in aerosols at large particle velocity amplitudes. 22. A. ACRIVOSand T. D. TAYLOR 1962 Physics of Fhdds 5, 387-394. Heat and mass transfer from single spheres in Stokes flow. 23. P . L . RIMMER 1968 Journal of FhddMechanics 32, 1-7. Heat transfer from a sphere in a stream of small Reynolds number. 24. S . C . R . DENNIS, J. D. A. WALKERand J. D. HUDSON 1973 Journal of FluM Mechanics 60, 273-283. Heat transfer from a sphere at low Reynolds number. 25. F. B. HILDEBRAND 1974 bttroduction to Numerical Analysis. New York: McGraw-Hill Book Company, Inc., second edition. 26. F. A. LYMAN 1976 Syracuse University Department of Mechanical and Aerospace Engbteerhtg Report 3IAE-5192-TI. Attenuation of high-intensity sound in a droplet-laden gas. (Available from U.S. National Technical Information Service, Springfield, Virginia; Accession number PB-252985.) 27. C. L. MORrEY 1968 Jottrnal o f Sound attd Vibration 8, 156-170. Sound attenuation by small particles in a fluid. 28. G. A. DAVIDSONand D. S. SCOTT 1973 Journal of the Acoustical Society of America 53, 17171729. Finite-amplitude acoustics of aerosols. 29. G. A. DAVlOSOy 1976 Journal of Sound and Vibration 45, 473--485. A Burgers' equation for finite amplitude acoustics in fogs. 30. M. ABRAMOWITZ and I. A. STEGtrN (editors) 1964 Handbook of Afathematical Fttnctions. Washington, D.C.: U.S. Government Printing Office. APPENDIX T h e function H(t) derived b y O c k e n d o n [19] for a sphere accelerating from rest with velocity U(t) can be written as follows:
H ( t ) = Hi(t) + H2(t),
(AI)
234
F. A. LYMAN
where
Hi(t) = -(x/n) '/2 f (d U/dO (t
-
T) -1]2
dr,
(A2)
0 Z(t)
//2 -- 3(~r '/2 j G(X)T-'12 y-2 d Y,
(A3)
0
T = t - 3, r = Z ( t ) - Z(z), ~cZ'(t) = U(t),
(A4) (A5) (A6)
G(X) = (nm/2) X - ' erf(X) - exp ( - X 2) - 2X213,
(A7)
X 2 = tr Y2](4T). (A8) The constant r:, defined in equation (21), is of order one. The behavior of H(t) as t ~ oo for U(t) a periodic function of time will be deduced here. In this limit the transients associated with the start-up of the sphere will have died out, leaving only the steady-state response. In the present case U(t) = -V(t)]l Vol, (A9) because V(t) has been defined in equation (3) as the dimensional velocity of the fluid at infinity relative to the sphere, whereas Ockendon's U(t) is dimensionless and is the velocity of the sphere relative to the fluid at rest at infinity. Also, t is the dimensionless time variable (~ot'), in terms of which V(t) = Voexp(it) = I Vol exp [i(t - 6)], where 6, the argument of Vo, is related to the phase lag between the sphere and the flow. Since its precise value is immaterial to the present discussion, any convenient value may be chosen, such as 6 = n/2 so that
U(t) =
- s i n t,
(AI0)
which satisfies Ockendon's requirement that U(0) = 0. For this U(t) the integral in equation (A2) becomes
H,(t) = (2~) v" [C(V'2-~) cos t + S(V/2t/n) sin t] = (hi2) m [(cos t + sin t) - 2g( 2 v ' ~ ] , (AI 1) where C(z) and S(z) are the Fresnel cosine and sine integrals, respectively, and g(z) is an auxiliary function expressible in terms of them (see equations 7.3.1, 2, 6 of reference [30]). As t -~- oo g(2~/~t]n) = O(t-a/2); CA12) hence in this limit H,(t) is a simple sinusoid. In the present case I
Z(t) =_If U(z)dr =--I(I - cost), KJ
(Al3)
K
and 1
Y =--(cos t - cos z),
(A 14)
N
which greatly complicate evaluation of the integral (A3) for all t. To obtain the behavior for large t, however, we expand the integrand'of equation (A3) as y 2 T,12 -
2Ta/Z ,_~xn!(2n + 3) ~-'~--/"
(AI5)
Since in general [19]
T=xY/U(t)+O(Y
z)
as
Y-+0,
(AI6)
ATTENUATION IN A DROPLET-LADEN GAS
235
the integrand behaves as y-1/2 near Y = 0 and is thus integrable. The following estimate of /42(0 for large t is obtained from the leading term of equation (A15): ZCt)
3K5/2
H2(t) N - 40~ 1/2
t
y2dy 1 [ ( 1 - cost)3 5_f (COSt-- cosz)3dt1 Tst2-=4OX/-~K~( t ,'7l" 2.1 (t--z) w2 /"
o
(A17)
o
This indicates that H2(t) = O(t -5/2)
as
t --+ co.
(A18)
Therefore, as t --+ ~o, only the first two terms in equation (A11) remain, and after combining these terms and disregarding the phase difference, we have H(t) = ~0/2cos t. (A19)