On the coexistence of particle and turbulence scattering

On the coexistence of particle and turbulence scattering C.C. Yang Communications and Space Sciences Laboratory, Department of Electrical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802

(Received 8 December 1985; accepted for publication 3 August 1986)

A theory is presented to take care of a mixed system in which scattering effects from small particles and turbulences coexist. In this theory, a correlation function for irregularities is successfully derived such that it covers the variations due to sharp boundaries between particles and the surrounding medium, particle distributions, and volume concentration fluctuations. Hence, the theory of turbulence scattering becomes applicable to this mixed system when the Rayleigh approximation is valid. Numerical results are compared with experimental data from other research groups. Also, the coupling effect of, and the dominance between, these two kinds of scattering are discussed.

PACS numbers: 43.20.Fn

INTRODUCTION

Although wave scattering in a random medium is usually classified into the categories of particle and turbulence scattering, in practical systems both kinds may be co- existent. This may happen.when the wavelength becomes so short that individual particles in turbulent flows are "visi- ble" to the incident waves. In other words, if the particle size becomes comparably large compared with the wavelength, the particle scattering contribution becomes important in a turbulent system. Also, in a particle-distributed system, if the spatial variation of the number density is significant, ex- tra scattering contribution due to this turbulencelike fluctuation can be comparable with that due to particles them- selves. Actually, in a recent publication,' experimental data have shown that the backscattered intensity from turbulences is at the same order of magnitude as that from particles, which are red blood cells in this ultrasonic scattering case. In order to compute the scattering effects in such a mixed system, the scattering theories in two categories have to be unified.

Based on another publication 2 by the same author, a scattering theory for such a mixed system is presented in this paper. In that publication, the author has introduced an al- ternative approach for particle scattering by applying the continuous scattering conception when the particle size is very small. Therefore, it becomes possible to take care of both kinds of scattering by a single kind of theory, i.e., the continuous scattering theory. In this paper, we add the fac- tor of turbulence and rederive the theory. According to this theory, not only two types of scattering in a mixed system can be distinguished, but also the coupling effects between them can be seen. This is very useful when the particle number and the flow rate are the quantities concerned, because the particle and turbulence scattering intensities are related to those two quantities, respectively.

One thing worth noting here is that the turbulence scattering may include two parts in a mixed system: the random distribution of the particle number density and the inhomogeneities of the background medium (some fluid). Both can exist when a laminar flow evolves into a turbulent one. In

this paper, however, we assume that the inhomogeneities of the background medium contribute nothing or negligibly lit- tle to turbulence scattering. Actually, experimental results also confirm this in the case of the aforementioned refer-

ence. •'3 Therefore, throughout this paper, the turbulence scattering is attributed to the random fluctuation of the particle number density or volume concentration.

The geometry of the problem is shown in Fig. 1. We are concerned with the average backscattered intensity from particles and turbulences in a volume V far away from the transmitter and receiver. For simplicity, only one-sized, one- shaped particles are considered. If there is no turbulence, the volume concentration or number density is expected to be uniform in V. Also, the turbulences in this mixed system are supposed to be statistically homogeneous and isotropic. In addition, the temporal variations of particle distribution in a period of our measurement are assumed to be negligible. Only the spatial variation is concerned. In Sec. I of this paper, essentially following the method in the earlier paper, 2 a proper expression for the correlation function of irregularities is derived. This correlation function is supposed to include the sharp discontinuities at the boundaries of particles, particle distributions, and the continuous fluctuations of the

., receiver r

FIG. 1. Geometry of scattering. The distance R between the receiver and some reference point inside the scatteringsvolume Vis assumed to be much larger than the size of V. The unit vectors i and b represent the incident and scattered directions, respectively.

1495 J. Acoust. Soc. Am. 80 (5), November 1986 0001-4966/86/111495-06500.80 ¸ 1986 Acoustical Society of America 1495

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volume concentration. The contribution of particles in the expression for the correlation function has been fully discussed in Ref. 2. For our later computations, they are discussed briefly in Sec. II. The part due to turbulences in the correlation function is discussed in Sec. III. Numerical re-

sults are presented graphically and discussed in Sec. IV. Fin- ally, conclusions are made in Sec. V. Because of the analogy between acoustic and electromagnetic wave scattering, the theory presented in this paper will be applicable to both cases.

I. CORRELATION FUNCTION

In an acoustic system, the irregularities in elastic prop- erties of the medium cause random scattering. The sound •peed is given by the square root of the ratio of the bulk modulus of elasticity and the mass density. In our notation system, the sound speed in the particle material is assumed to be Cv and that in the surrounding medium is Cs. In the following, instead of the fluctuation of sound speed, that of the inverse square of sound speed, denoted by •', will be discussed. Because, in a statistically homogeneous random medium with suspended particles, a point is either inside some particle or outside any particle, the nonzero-asymptote correlation function for •' is given by

b• (r' -- r" ) • (•(r')•(r") ) 2

= • •,• (P [•(r') = •, •(r") -- • ])•. •=1

(1)

Here, •'1 = ( C s 2 -- Cp 2 )flOp 2 is the •' value when the point is inside some particle; •'2 = 0 is that when the point is outside any particle. The notation (-) stands for an ensemble average, and (')v means an ensemble average with respect to the volume concentration v. Also, P( -,. ) is the joint probability specified by the arguments. Since it is a function ofv(r') and v(r" ), the ensemble average with respect to v must be taken in Eq. (1). Starting with this equation and following the same procedure as that in Ref. 2, we obtain

b• (r' -- r")

--/•2v(r') fffd3ri fffd30p(r• r•,r")) , V• V• (r') V• (r") v

(2)

where Vl(r') and V1 (r") represent two particle volumes centered at r' and r", respectively. The notation p(rj- ri,r") is the conditional probability density of the event that'a particle is centered at rj when there is a particle centered at ri. Notice thatp is a random function ofv(r" ) as signified by its second argument. This probability density function can be separated into two parts2:

p(r•- r•,r") =6(r•--r•) + n(r") g(r• -- r•,r") . (3)

Here, n is the particle number density and is related to the volume concentration v by v = n V1, with V1 denoting the Volume of a single particle. As mentioned in Ref. 2, the delta function corresponds to the case where r' and r" are inside the same particle; the term with the pair correlation function

g represents the case where those two points are allocated in two different particles. For our convenience later, this pair correlation function is further series expanded around v = 0:

g(rj - ri,r" ) - go(rj -- r i ) q- v(r" )gl (r• -- ri )

q- v2(r")g2(rj -- ri ) q- '". (4) Putting Eqs. (3) and (4) in (2) and truncating the terms with order higher than 1 in the series, we produce

b• (r' - r" ) = •' • • (v)• (r' -- r" ) + (v(r')v(r"))[B•o(r'--r") + 1]

+ (v(r')v2(r"))•p• (r'- r")), (5) with

- l fff fff B. (r' -- r" ) = • d 3r i d 3 5 6(r• - r• ) , V• (r') V• (r")

Bpo (r'--r")

(6)

and

_- 1 fffd"r, fffd'r, [go(r• -- ri, -- 1] , (7, V• (r') V• (r")

•pl (r'-- r") = V-• V, (r') V,(r")

(8)

This truncation represents that Eq. (5) is a good approximation when v 3 { 1 in the order of magnitude. By comparing Eq. (5) in the case of no turbulence with •. (15) in Ref. 2, we can see that

Bp(r'--r")= • vmBpm(r'--r"), (9) m•0

where Bv is defined in Ref. 2 and is given by

- l fffa3r, fffa3rg(r_r,). r") -- V, (r') V, (r")

(10)

For further computations in Eq. (5), we assume that v is statistically Gaussian distributed. This assumption is usually acceptable when the turbulent fluctuations are not extremely strong. 4 Hence, in the third term of Eq. (5),

(v(r')v2(r")) = (v)(v 2) + 2(v) (v( r')v(r'))- 2(v) 3 . (11)

Putting Eq. (11 ) in (5) and after some computations, we can obtain an expression for the correlation function B•-

B• (r' -- r • ) = b• (r' -- r" ) -- b• ( o• )

= • •2 [ (v)•, (r' -- r") + (V2).•p0 (r' -- r") + (3(v) (v2) -- 2(v) 3) •pl (r'--r") q- { (O 2) -- (O)2)• v (r'-- r" ] . (12)

Here, •v (r'--r"is the normalized correlation function of v; i.e., •o(r'-r") = [(v(r')v(r")) - (v)2]/((v 2)

1496 d. Acoust. Soc. Am., Vol. 80, No. 5, November 1986 C.C. Yang: Particle and turbulence scattering 1496


-- (0)2). In obtaining Eq. (12), we have used the approxi- mations

Bv (r' - r")Bpo (r' - r")

=By (0)B•o (r' - r") - B•o (r' - r") (13) and

B• (r' - r")B• (r' - r") =B•(O)Bv• (r'-r") -Bv• (r'--r"). (14)

These a.p.pproxim_ations are valid because the characteristic sizes of_Bro and B• are at the order of the particle size • and that of B• is much larger since it describes the variation of the number density. In the extreme case of no turbulence, Eq. (12) reduces to Eq. (16) in Ref. 2, as expected. The coupling effect between particle and turbulence scattering can be seen in the second and third terms on the right-hand side of Eq. (12).

The Fourier transform of Eq. (12) gives us an expression for the power spectral function of •; i.e.,

% (.) = [ (.) + (.) + (3(v) (v 2) -- 2(v)3)•pl + ((v) 2 -- (v)2)•v (•) ] . (15)

Here, •, •vo, •, and (I)%are_the _counterp?ts in the Fourier transform domain of B•, B•o, Bp•, and Bo, respectively. Also, •c is the spatial frequency.

II. PARTICLE SCATTERING CONTRIBUTION

From now on, the discussion will be confined to the case of spherical particles with a radius a. Explicit forms for B• (r) and (I), (;) have been computed in Ref. 2 and (I)• (;) given by

• (;) = ( 12•r/tc•a •) [sin(tea) -- (tea)cos(tea) ]:. (16)

Also, Bp (r) was evaluated analytically or numerically under the assumption of two pair correlation models. In this section, we review these results and distinguish the parts of B•o and B• inB•.

Since, in general, the pair correlation g(r,r") is zero in the range Irl •<2a and no less than unity otherwise, for any pair correlation model, we can set

o, Irl •<2a, go(r) = 1, Irl (17) Higher-order terms such as g• depend on the particular pair correlation model. In the case of "well-stirred" hard spheres (WS), 5 g•(r)•g:(r)•...=0. However, in the Percus-Ye- vick (P-Y) 6.7 model, g: (r) •g3 (r) •...•0, with

0•

g•(r)= 8 1 • [- , 128a3/

0,

Irl<2a,

2a < Irl<4a, (18)

Therefore, for the WS model, Bp• (r)•Bp2 (r)_=...=0; for the P-Y model, Bp: (_r) =Bp3 (r)=...=0. Although B•o (r) for both models and Bp• (r) for the P-Y model can be evalu-

ated analytically, the quantities concerned are •o (K) and ("). Fourier transforming Bpo in Eq. (7) after inserting the

expression for go (r) in Eq. ( 17 ) produces

•po (K) -- • ('•)Go(•), (19) where Go (•) is

Go ( • ) = - ( 3/a'3a 3 ) [ sin ( 2•ca ) -- 2•ca cos ( 2•ca ) ] . (20)

For the P-Y model, (I)• (K) is given by •p• (•) = (I), (•)G•(•) , (21)

where

G•(•) = (3/4•c6a 6) [ -- 24•ca sin(4•ca)

+ ( 24•c2a 2 -- 6) cos(4•ca )

-[- ( 8K3a 3 -[- 12•ca ) sin (2•ca)

q- (20 K4a 4 q- 12•c:a 2 + 6) cos(2•ca) ] . (22)

Equations (16) and (19)-(22) form the bases for comput- ing the scattering contribution from particles.

III. TURBULENCE SCATTERING CONTRIBUTION

Now we turn our attention to the correlation function

related to turbulences, i.e., By. The behavior of By is quite different rfrom those of B•o and B.e • , which have only finite range. The correlation function B• (r) for the variation of the volume concentration is expected to be decreasing as is increased and vanishing asymptotically. The exact form of B• depends upon the structure of the turbulent flow and cannot be determined easily. Nevertheless, the following argument can he_lp in obtaining, at least approximately, an expression for B•. Since the correlation function for the fluctuations of dielectric permittivity e has been widely discussed, 4'8'9 that for the fluctuations of volume concentration v can be obtained if a relation between e and v is derived. This

derivation is given in the following paragraph. Let us consider the asymptotical expression for B• (r),

the correlation function for the fluctuations of effective di-

electric permittivity, when the particle size a approaches to zero. In this limit, each particle can be regarded as a dipole in an applied electrical field just like a polarized molecule in a uniform dielectric material. From this microscopic view- point, the effective dielectric constant er of our particle system is given by lø

•r : 1 q- 47r){ e q- 47r Z Xei , (23) i

with Xe standing for the electric susceptibility due to the small particles and Xe• representing that due to one kind of molecule in the surrounding medium. From the Clausius- Mossotti equation, •ø the electric susceptibility is

/•ei : Si•i/[ 1 - (4•/3)NiYi ] . (24) Here, •'• is the molecular polarizability defined as the ratio of the average molecular dipole moment to the applied electric field, and N• is the number density of that kind of molecule. For those small suspended particles, an equation similar to Eq. (24) can be written without the subscript i, i.e., in terms

1497 J. Acoust. Soc. Am., Vol. 80, No. 5, November 1986 C.C. Yang: Particle and turbulence scattering 1497


of y and N, representing the particle polarizability and number density, respectively. The dipole moment m in one individual particle is given by lø

m ---- (3/4•r) [ (el -- 1)/(el + 2) ]viE. (25)

Here, E is the total electric field and v l is the volume of one particle. Also, el is the dielectric permittivity of particle material relative to that of the surrounding medium. From the definition of y, we can easily see y = (3/•)V• (e•- 1 )/ (e• + 2). Hence, from Eq. (24), X, becomes

3 (e•- 1)/(e• + 2)

4• 1--(e•--l)/(e•+2)

3 e I -- 1 3 ffl • •v= v, (26) 4• e1+2 4• •1+3

where the equality v = NV1 has been used and v< 1 has been recognized. Notice that 4•Xe in Eq. (23) represents the fluctuation part (due to particle distribution) of e or er. There- fore, a linear relation between the fluctuation part of e and v is obtained when the particle size is extremely small.

By using Eqs. (23) and (26), the correlation function for the fluctuations of effective permittivity can be derived:

B• (r' -- r") = [3•1/(•1 + 3)]: ((v :) -- (v):) • (r' -- r") ß (27)

The result in Eq. (27) implies that, when the particle size is extremely small, the medium can be regarded as a purely continuous random medium with relative permittivity fluctuation 3•1v/(•1 + 3). This reflects the fact that, in the situ- ation of small but finite particle size, Bois reasonably approximated by the same form as that of the correlation function of permittivity fluctuation in a turbulent medium without suspended particles. Having this approximation, we can use the Kolmogorov spectrum 4 for •. In this spectrum, • (•) possesses essentially a power law of •. For analytic computations, several compact forms for • have been introduced. 11 Here, for our numerical computations in Sec. IV, we introduce the Bessel-type power spectrum 8 as fol- lows:

•v(K) = (2•r)3/2(Køli)(•'-3)/2l•g•'/2(li 4K2 q-K2ø ) (28) g,p_ 3,/2 (Ko/i) [/i ( K2 q- •o ),/2]•,/2

where Kv (.) is a v-order Bessel function with imaginary arguments. In the range between the inner scale li and the outer scale lo -- 2•r/%, q•v (•c) has a power-law form •c -p with a power index p.

IV. NUMERICAL COMPUTATIONS AND DISCUSSIONS

Having the power spectral function given in Eq. (15) and the results for (I),, (I)•o, (I)p•, and (I)v in the preceding sections, we are ready to compute the backscattered intensity by using the continuous scattering theory. As mentioned earlier, we consider only the case when ka is much smaller than unity, i.e., in the limit of Rayleigh scattering, with k representing the wavenumber in the surrounding medium. In this limit, the total field u in the expression of the scattering amplitudef (•,b), given by 9

4•r •(r') u(r')exp( -jkr'.b)d3r ' , v

(29)

is approximated by

u(r') ---- [ 3/(•1 q- 3) ] e •"• (30) if a plane wave with unit amplitude is incident. Here, the notation V stands for the scattering volume. In Eqs. (29) and (30), • and b are the directions of incident and scattered waves, respectively. Notice that, in using Eq. (29), we also assume that the distance between any two successive scattering points is long enough such that farfield approximation is valid. 2'9 Starting with Eqs. (29) and (30) and following a standard procedure, 2'9 we can derive an expression for the backscattered cross section cr as

ty= (k4/16•r 2) [3/(•'1 q- 3)]2 (I)g(2k). (31) To present the numerical results, we introduce a normalized scattering cross section r/as

r I = 16cra•/(v) [3•l/(•l q- 3)]2

= (k4a/(1))•'2•) (I)g(2k) . (32)

The value for r/can be computed once (I)g (2k) is replaced by the expression given in Eq. (15) and the P-Y model is assumed for pair correlation.

The parametric values for our numerical computations are chosen to describe the scattering system in Ref. 1. In this system, the ultrasonic wave is scattered by bovine erythro- cytes suspended in a saline solution. The following parametric values are assumed: The bovine erythrocyte compress- ibility----35.7 X 10 -12 cm2/dyn; the saline compress- ibility = 44.3 X 10 -12 cm2/dyn; the bovine erythrocyte density = 1.099 g/cm3; the saline density---- 1.005 g/cm3; the volume of bovine cells = 53 •m3; and the ultrasonic wavelength = 200 pm. Actually, in our computations for the normalized cross section, only the last two parameters are concerned. By assuming a spherical shape for bovine cells, we can obtain ka -- 0.073. In Fig. 2, the values for •/as a function of the turbulent strength, crv•[((v 2) - (v)2)/ (v)•] •/2, are shown for four cases. The dashed curve represents the case without turbulences, i.e., when (v 2) = (v) • and (v 3) = (v) 3; therefore, it is flat. Those three solid curves are the results with turbulences existent when the power index p in Eq. (28) is set at 4, the inner scale 1• is fixed at the diameter of a particle; however, the outer scale 1o is assigned to be 3000, 5000, and 7000 times the particle radius, respectively. In this computation, the average volume concentration (v) is set to be 0.2. Apparently, because of the existence of turbulences, the scattering cross section is increasing al- most quadratically as the root-mean-square value of the volume concentration •r,becomes larger. Also, we can see that shorter outer scale corresponds to stronger turbulence scattering. Similar results are given in Fig. 3 except that the outer scale is fixed at 5000 times the particle radius and three power indices are given: p ---- 3, 4, and 5.

In a comparison between our theoretical results and the experimental data in Ref. 1, similar trends can be observed although there seems to be an insufficient number of experi-



-4

,•o=5000 •

.

7OO0•

.

, I I I I 15 0.1 0.5 O.

FIG. 2. Normalized scattering cross section r/as a function of the root- mean-square volume concentration fluctuations tr v . The flat dashed line stands for the contribution from particles when there is no turbulence. Three solid curves represent the total scattering effect, including the turbulence contribution for three different outer scale values: lo - 3000, 5000, and 7000 a, with a representing the radius of particles. The power indexp is fixed at 4, and the mean volume concentration (o) is 0.2.

mental data points. In Figs. 6 and 7 of Reft 1, the abscissa is the flow rate of the solution with suspended particles. This flow rate is supposed to be linearly proportional to the square root of the kinetic energy of the laminar flow before turbulences are formed. This kinetic energy is partially con- verted into turbulent kinetic energy when the laminar flow evolves and becomes a turbulent flow. 4 Since the turbulence strength cr ois also linearly dependent on the square root of the turbulent kinetic energy, the flow rate is expected to be linearly proportional to try. Therefore, the curve formed by small squares in Figs. 6 and 7 of Reft 1, i.e., the scattering

,

coefficient from a mixed system, must be increasing quadratically.

The criterion for particle or turbulence scattering to be dominant can be estimated by ignoring the second- and third-order terms of v in Eq. (15). A ratio • is defined by

-3

1.0

0.5

0.0'

p=3

p=5

I i I • 0 • o.I 0.3 .5

v

FIG. 3. Same as in Fig. 2 except that the outer scale is fixed at 5000 times the particle radius and three power index values, 3, 4, and 5, are assigned.

•= ((v) 2-- (v)2)•v(2k)/(v)•,•(2k). (33) If •>> 1, the scattering intensity in the backward direction mainly comes from turbulence scattering. On the other hand, if •,• 1, particle scattering becomes dominant. To obtain just a simple picture, we consider only the extreme case when ka ,• 1. From Eqs. (16) and (28), Eq. (33) can be reduced to

3• (v) a 1 •'=• . (34)

8 l o (ka) 4

Usually, • (o) ,• 1 and a '•/o; however, a significant • value can be obtained if ka is small enough. For instance, when (o) = 0.1, cr o = 0.3 and lo = 1000a; if kay<0.042, we have •>• 1; i.e., the turbulence scattering is stronger than the particle scattering.

The last point worthy of discussion is the coupling effect between two kinds of scattering. The importance of this effect can be reflected by a quantity/• given by

((v 2) -- (v) 2) •vo (2k) + (3(v) (v 2) -- 3(v) 3) •pl (2k)

(35)

The numerator in Eq. (35) is obtained by subtracting • (2k) value when there is no turbulence from the first three terms on the right-hand side of Eq. ( 15 ). Once again, we consider the extreme case when ka,• 1 in which Eq. ( 35 ) becomes

/z = -- 8• (v) + 102• (v) 2 , ,, (36)

when the P-Y model for the pair correlation is assumed. In a numerical case with crv = 0.3 and (v) = 0.2, it is 0.22, which means that about 20% of the particle scattering contribution has been increased due to the coupling effect.

V. CONCLUSIONS

In this paper, we introduced a theory applicable to a mixed scattering system in which both particles and turbulences exist. The most crucial part in this theory is to derive a correlation function to describe properly the irregularities in the medium. This has been done successfully by introducing an infinite series of the volume concentration. For the WS

and P-Y pair correlation models, this series can be truncated and becomes a closed form. This closed form includes main-

ly two parts: those from particles and those from turbulences. However, the coupling effect can be eminently seen. Later, from numerical results, this effect was shown to be significant in some situations. Having the correlation function and the power spectral function, after a Fourier transform, we can compute the scattering effects by using the theory of continuous scattering when the Rayleigh scattering approximation is valid. Numerical results were compared with the experimental data in Ref. 1. The same trends can be observed. The criterion for particle or turbulence scattering to be dominant was also discussed. Comparable contributions from these two kinds of scattering are often expected.

Although some results are achieved, the theory is by no means complete. The first problem that can be raised is that a



better description of the fluctuations of volume concentration must be pursued. Also, theories ofc9ntinuous scattering need to be modified properly before they can be applied to a mixed system, especially when the particle size is at the same order of magnitude of the wavelength. The third problem is relevant to the anisotropy of turbulences. Since the turbulences produced in the experimental system of Ref. 1 are contained in a narrow pipe, they are very likely statistically anisotropic. If this is true, the theory introduced in this paper requires modifications.

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On the coexistence of particle and turbulence scattering