- Email: [email protected]
Acoustic propagation in porous media with internal mean flow
Journal of Sound and Vibration (1987) 114(3), 565-581
A C O U S T I C P R O P A G A T I O N IN P O R O U S MEDIA W I T H INTERNAL M E A N F L O W t A. CUMMINGS AND I.-J. CHANG Department of Mechanical Engineering, University of Missouri-Rolla, Rolla, Missouri 65401, U.S.A. (Received 11 March 1986, and in revised form 2 July 1986) A parallel-fibre model is presented here, describing sound propagation in rigid-framed porous materials with internal mean fluid flow. Expressions for the bulk acoustical properties of the material are obtained in terms of geometrical parameters--volume porosity, tortuosity and effective fibre radius--and in terms of the mean internal flow speed; fluid properties and acoustic frequency are, of course, also involved. Empirically determined steady-flow viscous and inertial flow-resistive coefficients are required in these expressions. Experimentally measured bulk properties are compared to predicted data both with and without mean flow and generally very good agreement is noted. 1. INTRODUCTION Porous sound-absorbing materials are widely used in devices designed to attenuate noise in flow ducts. Glass fibre wall liners in air conditioning ducts, splitter type silencers in ventilating systems and glass fibre expansion chamber liners in the exhaust silencers of internal combustion (IC) engines are all examples o f these various applieations. "Bulk reacting" absorbers in jet engine inlet ducts have been studied, though they are not in common use. Naturally, there are gradients of static pressure in flo w ducts; these would seem to be of roughly similar magnitude in the inlet diffusers of jet engines and in IC engine exhaust mufflers, and would generally be smaller in air-conditioning systems, but could have locally high values. Any layers of porous acoustical absorbent that are built into the duct will, of course, also be exposed to the static pressure gradients. There will therefore be some degree of mean fluid flow within the porous material itself, to an extent that depends upon the pressure gradients and on the flow resistivity of the porous material. Although the convective effects of this internal mean flow on the bulk acousticproperties of the material are likely to be quite small, it might be anticipated that inertial effects on the flow resistivity (similar to those occurring in fluid flow through an orifice plate) would be noticeable at relatively low fluid velocities, and that the bulk properties could be modified accordingly. Surprisingly, the possible effects of internal mean flow on the bulk acoustic properties of porous material do not appear to have received any attention in the literature, apart from a simple treatment by Beeckmans and Sen-Gupta [1] in the context of filter beds. The objective of the work presented in reference [1] was, however, to investigate variations in the mean static pressure gradients in porous media brought about by the presence o f intense sound waves. The authors obtained quantitatively reasonable agreement between measurements and a theoretical treatment based on a quasi-steady Darcy's law for the t This work was first presented at the AIAA 10th Aeroacoustics Conference, Seattle, Washington, U.S.A. 9-11 July, 1986. 565 0022-460X/87/090565+ 17 S03.00/0 O 1987 Academic Press Inc. (London) Limited
566
A. C U M M I N G S
AND
l.-J. C H A N G
flow resistivity of the porous medium. Kuntz [2] and Nelson [3] reported investigations concerning high amplitude sound propagation in porous materials with no internal mean flow, and the principal effects were caused here, too, by inertial flow losses. The details of the flow interaction at high amplitude are, however, rather different from those which are involved in the case of internal mean flow, and include the generation of harmonics in an initially sinusoidal signal by non-linear processes. The customary treatments of bulk-reacting liners in flow ducts (see, for example, references [4] and [5]) involve the tacit assumption that there is no mean flow in the porous material or--perhapsmthat if there is, then its effects are negligible. There is an apparent lack of supporting experimental data to substantiate this assumption. The practical importance of internal mean flow effects in porous materials is not yet clear, but preliminarytests and calculations show that, at least in IC engine silencers, the internal mean flow velocities can approach 1 m/s which, as will later be demonstrated, is sufficient to cause substantial alteration of the bulk acoustic properties of the material. Rough calculations on porous absorbers in jet engine ductwork indicate possible internal mean flow speeds as high as 1.5-2 m/s, which could produce even greater changes in bulk properties. The motivation for the present investigation was the likelihood that internal mean flow effects in porous absorbents in s i t u could be important in many situations of interest. The results do, indeed, show that these effects a r e likely to be significant, thereby providing justification for the study. In this paper, a theoretical model for the bulk properties of rigid-framed porous acoustic absorbents in the presence of uniform internal mean fluid flow is pi'esented, for the particular case of one-dimensional propagation in which the mean flow velocity is parallel to the acoustic particle velocity, and experimental data on an open-celled polyether foam are compared to theoretical predictions. It is assumed that the acoustic particle velocity within the material is sufficiently small for high amplitude effects to be absent. 2. THEORETICAL MODEL In this first approach to the problem, it is assumed that the solid frame of the porous material is rigid and that the pores are filled with a gas. This is not generally an unduly restrictive assumption; structural wave motion tends to be of importance at rather low frequencies for most porous materials. Three microstructure models have been examined in the present investigation: first, a very simple "phenomenological" model that is essentially similar to that described by Morse and Ingard [6], secondly a "Rayleigh type" model embodying parallel capillary tubes, and thirdly a parallel-fibre model essentially similar to those of Burns [7] and Mechel [8]. The number of empirically determined parameters in these three models decreases in the order of mention. This is directly related to the degree of resemblance between the assumed and actual microstructures. The Morse/Ingard model implies no assumptions at all about the microstructure and hence contains parameters that have to be found from acoustic measurements (or guessed). Chief amongst these are the effective compressibility of the fluid in the pores of the material and the effective viscous flow resistivity of the material. The former quantity may be assigned a value somewhere between the figures appropriate to reversible isothermal and adiabatic processes, and the latter may be assumed equal to the steady flow resistivity of the material (which may readily be measured). At audiofrequencies, however, both of these quantities take on complex values and this Simple model does not give entirely satisfactory results, etiher with or without internal mean flow. The Rayleigh model gives better results because it makes allowance for the presence of viscous and thermal i
POROUS MEDIA WITH INTERNAL
FLOW
567
boundary layers in the fluid adjacent to the surface of the solid frame. The effective compressibility and flow resistivity are both complex and, apart from the empirical determination of effective pore radii (which are different in the cases of the compressibility and flow resistivity), can be made to give fairly satisfactory results. Even so, the disparity between the assumed and actual microstructures brings about a residual lack of correspondence between prediction and measurement in the present case (though the Rayleigh model might be appropriate to other geometries, such as granular media). The parallel-fibre models o f Burns [7] and Mechel [8] (which are essentially similar) are an improvement over the Rayleigh model for both fibrous materials and open-called reticulated foams (which are basically fibrous in structure), to the extent that the solid frame is modelled as an array of fibres. The inclusion of an effective fibre radius is less ambiguous than it is in the case of the Rayleigh model, and a complex compressibility and flow resistivity are a part of these models. Attenborough [9] gave an excellent review of porous material models, including those previously mentioned. Comparison between predictions based on the three models and measured data revealed the parallel-fibre model to be superior to the other two, and so the discussion here will be restricted to this model. Although the development of the model follows partly along the lines described by Mechel [8], it will be described here in detail because of the inclusion o f internal mean flow effects and also because viscous effects are incorporated in a different way from that described in reference [8]. 2.1. GEOMETRY The model will be discussed with particular reference to the polyether foam used in the experimental tests. Figure 1 shows an electron micrograph of a sample approximately two cells thick. The longer dimension of the white rectangle is 100 I.tm. The structure of the foam consists essentially of triangular section strands (or fibres) connected at their ends; the solid frame is what remains of the interstices of the original foam after the bubbles have burst. The width of the strands is typically 30-100 ~m. An idealized geometrical structure o f a single cell would be a regular dodecahedron. It should be noted that a fairly small proportion ofthe inter-cell holes in the foam contain unburst membranes or "closed windows". In modelling a foam (or fibrous) material with internal mean flow, one must consider the nature of the fluid stresses at the surface of the solid frame, and the Reynolds' number (based on the hydraulic diameter of a fibre) for a crossflow at the mean internal flow
Figure 1. Electron micrograph of polyether foam.
568
A. C U M M I N G S
AND
I.-J. C H A N G
speed is an appropriate parameter. For an internal flow speed of 0.5=2 m/s and a fibre width of 50 ~.m (approximately equal to the hydraulic diameter), the Reynolds' number is about 1.5-6-5, which is in the range where inertial stresses can be as large as, or greater than, viscous stresses. The regions where the fibres join are of greater width and will be associated with higher Reynolds' numbers, as will be the closed windows. The mean flow in the parallel-fibre model will be assumed to be parallel to the fibres, and thus, for laminar mean flow, no inertial stresses would be present. To introduce these inertial stresses, flow obstacles will be made a part of the model, in the (arbitrarily chosen) form of discs attached to the fibres as shown in Figure 2; the fibres are assumed to have a circular cross-section (of radius a) and to be identical. The diameter and positioning of the discs are unspecified: their sole effect is to introduce inertial stresses in the mean flow. The fibres are assumed to be arranged in a hexagonal array as shown and each may be regarded as being surrounded by a "cylinder of influence" of radius R (see reference [7]). The fact that these cylinders must actually be hexagonal prisms, in order to fit together, is of little practical consequence. The volume porosity, ~, of the material is given by = 1 -a2/R 2 (1) and the volume of the discs is neglected. In the polyether foam studied, the fibres are roughly triangular in section. A good approximation of their shape is that of the interstice between three equal, touching circles as shown in Figure 3. The cross-sectional area is 0.1613 ro2, ro being the radius of each circle. A hydraulic fibre radius may be found as (2)
ah = g - 2 3 / P ,
where A is the area of the space between the fibre and the boundary of the cylinder of influence and P is the total perimeter of this space (including the fibre). From equation (2), it is found that ah = R{1-211 - O . 0 5 1 3 ( r o / R ) 2 ] / [ 2 + ( r o / g ) ] } .
(3)
In what follows, t/h will be used as the equivalent fibre radius, and denoted a.
F i i ~ , ~Dis~ci ;
Cylinderofinfluence
R
~
\
x
~
Directi onofpropagat meanflowion and sound Figure 2. Geometry of parallel-fibre microstructure.
~'"
POROUS
MEDIA
Sf
WITH
%
INTERNAL
SS
,, ~1~ u f
,
f I
569
FLOW
S, ~ ro I
~,~
Fibre
%.
S I
I I I
I i i
'S.
is,
l
e
Figure 3. Assumed cross-sectionalgeometry of "triangular" fibre. 2.2. FLUID COMPRESSIBILITY TO find the effective compressibility of the fluid in the pores of the material, one must solve the energy equation relating pressure and temperature fluctuations. The linearized form of this equation is ~CpoO T'/ D t - 7"flDp'/ O t =/~V: T'+/St/,'
(4)
for an ideal gas with constant viscosity, where Cpo is the ideal gas specific heat of the fluid at constant pressure, T is temperature, t is time, p is fluid density,-fl is the coefficient of thermal expansion, p is pressure, k is the thermal conductivity of the fluid and 9 is the dissipation function; the overbar denotes a time average and the prime denotes a perturbation. Figure 4 shows the co-ordinate system; the presence of the discs is neglected. Equation (4) m a y b e simplified ifit is assumed that a( )/ax = 0, a( )/a0 = 0 (valid provided radial derivatives >>axial derivatives, for axisymmetric disturbances) and that the fluid velocity u is purely axial. Then /5~'= i.t(dff/dr)(0u'/ar), where p. is the coefficient o f shear viscosity. This term may be shown to be small compared to the other terms in the equation, for typical numerical data, and it will be ignored. The simplified version of equation (4), for harmonic time variation of radian frequency to, and a uniform pressure perturbation is d 2T6/d(ar) 2+ {1/(ar)} d T~/d(ar) + T~ = p~/Cpo~,
/ !
I
V__
! ! ! !
R,
ja
I
Fibre
/ Cylinder of influence
Figure 4. Co-ordinate system of a single fibre.
(5)
570
A. C U M M I N G S A N D I.-J. C H A N G
where T ' = T~(r) exp (itot), p ' = p~ exp (iwt) and a = ( - iwCpofi/k)i/2. It can be seen that the only influence of mean flow is in the fluctuating part of the dissipation function q~ and that if this is negligible, the solution to equation (4) will be the same with and without [tow. The general solution to equation (5) is
T~( r) = ( p'/ ~Cpo) + CiJo( ar) + C2Yo( ar),
(6)
lo( ) and Yo( ) being zero order Bessel and Neumann functions respectively, and CI and C2 arbitrary constants. Appropriate boundary conditions are
T~(a) = O,
(dT'o/dr),=R = 0.
(7a, b)
(The first of these conditions actually breaks down at frequencies that are sufficiently low for the thermal boundary layer inside the fibres to be significantly thick; ultimately the solid and gas are in thermal equilibrium. For practical purposes, however, equation (7a) will suffice.) Applying conditions (7a, b) to equation (6) yields
T~(r) =p'{1 -Jo(ar)/S(a)+[Jl(otR)/Yl(aR)]Yo(ar)/S(ot)}/~Cpo,
(8)
S(a) = Jo(aa) - J , ( o t R ) Y o ( a a ) / Y l ( a R ) .
(9)
where
In the customary manner, the effective fluid compressibility K(to) m a y b e defined as K(to) = (1 - gcfi(T')/p')/~,
(10)
where ( ) denotes a space average over the region a ~< r ~< R and R e is the gas constant. Equations (8), (9) and (10) yield, finally, K(to)=[l+2(l_~
) (y-1)cta Q ( a ) ] / T / ~ =
F,,/Tfi
(11)
where y is the specific heat ratio of the fluid and
Q(a) = { J l ( a R ) Y l ( a a ) - J l ( a a ) Y l ( a R ) } / { Y l ( a R ) J o ( a a )
-Jl(aR)Yo(ota)};
(12)
the compressibility function F~ is defined by equation (11). 2.3. fLOW RESISTIVITY The flow resistivity of the material embodies two parts: that associated with the (parallel) mean and fluctuating flow past a fibre (a purely viscous resistance), and that associated with flow past a disc (an inertial resistance). These will first be considered separately, before being combined together. Expressions for the viscous flow resistivity for mean and fluctuating flows are found as follows. Subject to the assumptions made in the previous subsection (except Op/Ox = 0) the Navier-Stokes equations yield
Ou/Ot = - (1/~) Op/Ox + v{t92u/t~r2+ (1/r) Ou/Or},
(13)
where p and u are the total (mean+fluctuating) pressure and velocity respectively and / ~ is the kinematic fluid viscosity. This equation is linear, and thus the solutions for mean and fluctuating flows may be superimposed. With harmonic time variation, the general tsolution to equation (13), for the velocity perturbation, is
J
u~(r) = - (Op~/Ox)/ito~ + DlJo(sCr) + D2Yo(~r),
(14)
POROUS M E D I A WITH I N T E R N A L FLOW
571
where u'-- u~(r) exp (itot), st = ( - i t o / v ) i / 2 and Di, D2 are arbitrary constants. The boundary conditions are u~(a) = 0,
(du'o/dr),=R = 0.
(15a, b)
Equations (14) and (15) then yield u~(r) = -(Op'o/OX){[1 -Jo(str)/S(st)]+[Jn(stg)/Yl(stR)]Yo(str)/S(st)}/ito~,
(16)
where S( ) is given by equation (9). A "shear impedance" z's may be defined as z'~ = T'~/(U'),
(17)
T" being the fluctuating shear stress at the surface of the fibres and (u') the space-average of u'. Since T'w=--/.t(0u'/0r) . . . . equations (9), (16) and (17) give z' = - # s t
st(R-~-- a2 ) 4- [Jt(~a)Yl(stR) - J~(StR)Y~(Sta)J J"
The solution to equation (13) for the mean flow velocity, together with boundary conditions analogous to those in equations (15), gives = - (a2/41~)(dff/dx)[1 - ( r / a ) 2 + 2 ( R / a ) 2 In ( r / a ) ] ,
(19)
and a shear impedance ~s (= fw/(t~)) may be found as above, zs =
- (/~/a)(1 - a 2 / R 2 ) / [ { l n ( R / a ) / ( 1 - a 2 1 R 2 ) } - 3 + ( a 2 / 4 R 2 ) ] .
(20)
An effective viscosity function F~, may be defined as F , = z'~/~.,,
(21)
expressing the ratio between the effective fluid viscosity in the acoustic wave and the viscosity coefficient #. As one would expect, F,, --, 1 as st--, 0. The "viscous" flow resistivity in the acoustic wave, o-v(~o), is now given as o%(to) = o's,F,,
(22)
where o-s, is the steady flow resistivity (pressure drop/unit distance/unit flow speed) of the porous material, defined at low flow velocities where viscous stresses predominate. The flow speed in the definition of o'~ and o's, is volume velocity/cross-sectional area of bulk material, in the usual way. At acoustic frequencies, o'~ is usually determined in part by inertial forces (in shear waves exhibited by the boundary layer adjacent to the fibre) as well as by viscous forces, but these inertial forces are linearly proportional to (u'). It must be noted that cr~ also describes the steady flow viscous resistivity, since F,, = 1 for to=0. Clearly, o-,, could be calculated from equation (20), with appropriate modifications, but one could not expect this procedure to give very good results because of the lack of close resemblance between the parallel-fibre model and the actual microstructure. Instead, an accurately measured value of cry, is utilized, and it is assumed that F,, (calculated from the idealized model) will still give a reasonable description of the frequency dependence of o-~. This is essentially a compromise measure but gives tolerably good results, as will later be seen. The inertial component of the flow resistivity can be included as follows. If it is assumed that there are no viscous forces acting on the discs and that pressure drag forces dominate, then the drag force acting on each disc may be written, for the mean flow, as FD = C o ~ ( ~ ) I ( ~ ) I A a / 2
(23)
572
A. C L I N i M I N G S A N D l.-J. C H A N G
(the absolute value of (fi) being inserted to allow for negative flow velocities), where Co is a drag coefficient that can be assumed to be approximately constant, and Aa is the frontal area of the disc. For a quasi-steady perturbed flow, (23) becomes
vo + F'o = CoAd~l(a) l((a)+ 2(u'))/2,
(24)
where the quadratic terms in perturbations are assumed to be relatively small and the term p'(tT) 2 is taken to be of order (fi)/Co compared to the second term in parentheses, and therefore negligible (Co is the adiabatic speed of sound). It is assumed here that the inertial flow resistivity is quasi-steady and that equation (24) applies. On this basis, and together with equation (22), a modified Darcy's law may be written for the perturbed flow, embodying the combined viscous and inertial drag forces acting on the fluid in the bulk material,
(Op/OX)arag=
--O's,((l]) + F t , ( u ' ) ) -
l(a)+ 2 I(a)I (u'))
(25)
The angle brackets henceforth denote a space average of velocity equal to volume velocity/cross-sectional area of bulk material, rather than the average velocity within the open spaces of the porous structure and o'i is an empirically determined inertial flow resistivity coefficient (note that it does not have the same dimensions as or,t). The mean and fluctuating components of equation (25) may be equated separately by first taking a time average of the equation and then subtracting this from the original equation:
(Off/OX)drag=--O',t(a)--IT,[(•)l(a),
(Op'/OX)drag=--ITslF•(ut)--2ortl(a)l(U').
(26a, b)
2.4. DEVELOPMENT OF A GOVERNING WAVE EQUATION The bulk properties of the porous material in the presence of mean flow are found from solutions to the governing wave equation, which is derived from the linearized continuity and momentum equations for the fluid in the pores. It is assumed for the time being that one-dimensional motion occurs parallel to the fibres of Figures 2 and 4. The continuity equation may be written
FI Op/at + O(p(u))/Ox = 0,
(27)
and the linearized version of this, for acoustic perturbations, is
Op'/Ot+~ cO(u')/~x = 0,
(28)
where the term (t~) Op'/Ox is of order (~)]eo compared to the second term on the left side of equation (28) and has been discarded. A one-dimensional momentum equation for the perturbed flow may be derived from (26a) if the convective acceleration term in D(ld)/Dt (which is of order (t~)] Cocompared to the local acceleration) is neglected. The net pressure gradient may be written
Op'/ Ox = ( Op'/ OX) drag+ ( Op'/ OX)i,e~ti, ,
(29)
where the second term arises because of the mass of fluid contained in the pores of the material. The inertial contribution to the pressure gradient may be expressed in terms of the local fluid acceleration as (t~q2/g2) O(u')/Ot (see the paper by Smith and Greenkorn [10]), where q is the tortuosity factor (see the book by Carman [11]) and q2 is identified with the Smith and Greenkorn inertial factor tensor, being otherwise often known as the structure factor. Equation (29) may now be written
Op'/Ox = -cr,,F~,(u')- 2o-~[(a) ]( u ' ) - (~q2/ O ) 3(u')/Ot.
(30)
The one-dimensional wave equation for harmonic time dependence may be derived by eliminating (u') from equations (28) and (30), and utilizing the relationship K(w) = p'/p'~
POROUS
MEDIA
WITH
INTERNAL
FLOW
573
(see equation (10)) to yield
oEp'/Ox2-t-fiKq2to2p'--itoflK(Ors,Fa +2o', I(a) l)p'
= 0.
(31)
This wave equation embodies no convective mean flow effects (the mean flow being too slow to be significant), but does contain a quasi-steady mean flow resistive term. 2.5. B U L K A C O U S T I C P R O P E R T I E S Substitutlnn of a travelling wave solution
p ' ~ e -rx+i~t
(32)
F = +iqco (/SK)!/2[-1 - ifl (cr,,F~ + 2o', [(fi) [)//~q2t0] ,/2.
(33)
into equation (31) gives
The positive root represents a positive travelling wave and the negative root a negative travelling wave. Where the positive root is taken, F is identified with the bulk propagation coefficient. Combination of equations (30), (32) and (33) yields an expression for the characteristic impedance za.(= p'/(u') in a positive travelling wave),
za = ( q / ~ )(/5 / K )1/2[ 1 -- if2 (cr~,F~ + 2cri[ ( fi) [)/ fiq2to ],/2.
(34)
The two quantities F and za are the primary characterizing parameters of the bulk material. Equations (33) and (34) are generally applicable to any open-celled, rigid porous material with one-dimensional motion of a gas in its pores. Several observations may be made, concerning equations (33) and (34). First, there are three parameters--q, o's, and o-,--which are best determined by experiment for any realistic microstructure. The tortuosity q is readily found by an electrical conductivity method as described by Carman [11], and tr, t and o'i are easily determined from steady flow pressure drop measurements over a range of flow velocities. The porosity Y2 may be found very simply from the bulk density of the material and the density of the solid matrix. Section 3.1 gives details of these "static" properties. Secondly, K and F,~ may be determined for a fibrous or foam material by using the microstructure model described in sections 2.1-2.3, provided a representative average hydraulic fibre radius can be found. One also notes here that for a fairly slow internal mean flow; neither F~, nor K is affected by the flow, at least on the basis of the assumed model. Thirdly, it can be seen that the direction of the mean flow appears to be immaterial in determining the change in bulk properties brought about by flow. An increase in the real p a r t o f the "acoustic flow resistance" tr,,F~, is predicted, for mean flow travelling in the same direction as, or in the opposite direction to, the sound wave. Fourthly, any modification in the bulk properties caused by mean flow is dependent principally on the relative magnitudes of tri and o'~t. One would expect that materials containing a high density of structural discontinuities or blockages (such as the closed windows in foams or "shot" in fibrous materials) would have relatively high values of tr~ and would hence exhibit quite large mean flow effects whereas, at the opposite extreme, structures (such as arrays of parallel, purely cylindrical, fibres) containing no discontinuities at all would show little or no mean flow effects on their bulk properties. 3. MEASUREMENTS The measurements carried out in the present investigation fall into two categories: measurements of the steady flow coefficients cry, and cr~ together with the essentially geometrical features s q and ro, and measurements of the bulk acoustic properties of the material.
574
A. C U M M I N G S A N D l.-J. C H A N G
3.1. G E O M E T R I C A L FEATURES
The volume porosity is easily found from the measured bulk density pb of the porous material and the density of the solid frame ps, as 1-1 = 1 - p b / m .
(35)
The bulk density is determined by weighing a sample of the material and the density of the frame may be found by independent means, usually from published or quoted data. In the ease of the polyether foam used in the present series of tests, pb was found to be 46-4 kg/m 3 and ps was (according to the manufacturers of the foam) about 1100 kg/m 3. The porosity was thus found to be 0.958. The tortuosity was measured in this investigation by using an electrical conductivity method cited by Carman [11]; the experimental procedure is as follows. The electrical resistance of a particular volume (of rectangular parallelepiped shape) of saline solution was measured in a conductivity cell, by means of an AC bridge with a frequency of 100 kHz. This rather high frequency was used to minimize the capacitive component of the electrical impedance, since a pure resistance was used to balance the bridge. Then a piece of foam, cut to be of the same shape and size as the volume of solution, was saturated with the solution whilst in the cell and its resistance was measured by the same means. Then the tortuosity q is given (see the book by Carman [11]) by
q = ORI/Ro,
(36)
where Ro is the electrical resistance of the saline solution alone and Rf is the resistance of the saturated foam. The value of tortuosity found from the present tests was 1.167. The dimension ro is the radius of the circle defining the fibre shape in Figure 3, but is also the distance between the "points" of the fibre's cross-section and will be referred to as the "fibre width". An approximate, average, value of ro was found--by direct measurements on a series of electron mierographs like that in Figure 1--to be 50 p.m. The viscous and inertial steady flow resistivity coefficients were found by passing a steady airflow through a plug of the foam and measuring the pressure drop across the plug as well as the volume flow through it. The measured pressure gradient d~/dx was plotted against (t~) (see equation (26a)) and a quadratic curve was fitted (by means of the least squares method) to the points, thereby defining o's, and tri. Figure 5 shows the measured data and the fitted curve. It can :be seen that the quadratic curve fits the data very well, thereby justifying this form of Darcy's law. The values of the flow resistivity coefficients corresponding to the quadratic curve are trs, = 5425 N s/m 4 (SI rayl/m) and ~ri = 3071 N s2/m 5. 3.2. ACOUSTICAL PROPERTIES The bulk acoustical properties of the foam were measured by using the standing wave tube shown in Figure 6, which had provision for passing a steady airflow through the foam. Half of the standing wave tube was filled with cylindrical pieces of foam, and the total depth of foam tested was 0.96 m. The normal impedance of this thickness of foam was measured by the usual standing wave meihod. Over the frequency range of the tests (500-2000Hz), acoustic reflections from the open end of the tube were sufficiently attenuated at the opposite face of the sample that the impedance measured was effectively that of a semi-infinite thickness of foam--that is, the characteristic impedance; this was verified by plotting the axial variation of sound pressure level. Both the velocity profile of the mean airflow within the foam and the "entrance length" (the distance from the inward-facing end of the plug of foam to the point where the flow became fully developed) were of considerable interest in this experiment, since these features of the flow would
POROUS MEDIA WITH INTERNAL FLOW
/
/
~2 X
E
g.
/
1
/.o/
575
e/e
O0
jl /I /I
l" I 0/0
-000
0
0
Im/sl
Figure 5. Steady flow pressure gradient vs. flow velocity for polyether foam; O, measurements; curve fit.
I
Sinewove generolor
, quadratic
I Frequency counler
S7 =:
I:::"
7;ram Probe,
"i":: 41k
I!=
0-96m O.99m
Acousticdriver
T~-
0-99m
Tometer,
'~'% J- I
('7)-n-~.~ "L-
Seal
Stole
Microphone
J
-I
Figure 6. Standing wave tube apparatus for the measurement o f bulk acoustic properties.
determine the extent to which the assumption of uniform flow was realized in practice. Approximate numerical calculations were carried out, by using a finite difference approximation to a simplified version of the Navier-Stokes equations, with added flow-resistive terms but with no radial velocity component. The results showed the fully developed flow to be virtually uniform, with an extremely thin boundary layer of less than 1% of the pipe radius in thickness; additionally the entrance length was shown to be very short, typically less than an eighth of a pipe radius. Though the approximate form of the momentum equation would no longer be valid under these circumstances (and, of course, the continuity equation would not even be approximately satisfied), an upper bound was
76
A. C U M M I N G S
AND
1.-J. C H A N G
mt on the entrance length. It was quite clear from these results that the uniform flow pproximation should be good, since beyond a thin surface layer of the foam plug, trtually uniform flow would be expected to prevail. It was not, unfortunately, possible carry out experimental tests to confirm these calculated data. The propagation coefficient of the foam was found by plotting the level and relative aase of the acoustic wave within the porous layer in a direction from the surface nearer ie source, into the material, over a sufficient distance for the rates of change of sound vel and phase--with distance--to be accurately determined. Only the linearly varying trts of these curves were used, and the method of least squares was used in fitting raight lines to the measured data. The attenuation coefficient a of the sound wave is ~e real part of F and the phase coefficient/3 is the imaginary part. The presence of airflow in the standing wave tube produced a certain degree of more 9 less random noise, and it was thought desirable to filter the microphone output in "der to obtain an accurate measure of the amplitude and phase of the sinusoidal signal .at was generated by the loudspeaker (this procedure also eliminated the effects of any ~ssible "wind noise" at the probe hole). Accordingly, a Hewlett-Packard type 3582A ,ectrum analyzer was used to give a spectrum of the microphone voltage and thereby , separate the signal and noise; the analyzer also yielded the phase of the signal. In the standing wave analysis, the effects of mean airflow were taken into account, :ough since the mean flow Mach number was very low, these effects were small. From a knowledge of F and za in the absence of mean flow, it is possible to infer dues of K(to) and o'stF~,. Equations (33) and (34) may be combined to give the relationtips
r,(toi = F/itol2za,
os, Re (F~,) = Re (Fz~),
o's, Im (F~,) = Im (Fz,)- q2pta/O.
(37a, b) (37c)
hus the compressibility function F~ (see equation (11)) and the effective viscosity function may be found from measurements o f F and zo. The above equations are also valid the presence of mean flow in the case of K and Im (F~,); Re (F~,) may be found by tbtracting the inertial flow resistivity component from the right side of equation (37b). 4. MEASUREMENTS OF ACOUSTICAL PROPERTIES AND COMPARISON WITH THEORY Figure 7 shows theoretical curves of Re ( F ~ ) - I and - I r a (FK), from equation (11); ,gether with measured data obtained in the absence of mean flow as described in Section 2. The independent variable is A = aPr~/2(to/v) ~/2, where Pr is the Prandtl number of e fluid in the pores o f the material. It can be seen that the measured and predicted e (F~) data are in good agreement. The data on the imaginary part of F~ do not agree well, particularly above A = 0.6, though below that point the agreement is fair. The ~rcentage accuracy in measuring Im (F~) is not high and this may explain the fact that :e measured absolute values of this quantity exceed the theoretical values above A = 0.6. rver most of the frequency range, however, IIm (FK)] is no greater than about 10% of e ( F , ) and in the range where Im (FK) has its maximum absolute value, the predictions ?e at their best. Overall, the theory would seem to give fairly satisfactory predictions of '~, especially when one considers the real versus assumed microstructures. It is noted hat for A--)0, an isothermal thermodynamic process is predicted in the fluid wave, hereas as A ~ 0% an adiabatic process is forecast. Although, as previously mentioned, e boundary condition T'(a) = 0 is likely to break down at low frequencies, there is no ddence of this occurring in the frequency range embraced by the present results.
577
POROUS MEDIA %VITH INTERNAL FLO%V 0"4 Re ( F ~ ) - I
0-3
0-2 c
o
o
oo oo
0.1
0
I
I
0
I
i
I
1
3
2 .4
Figure 7. Compressibility function F~ of polyether foam for {,~)=0; O, measured Re ( F ~ ) - 1; O, measured theory (equation (11)).
- l m (FK); - - ,
Measured data on FK in the presence of flow--not shown in Figure 7 for the sake of clarity--agree almost as well with the theoretical curves as the no-flow data. This constitutes experimental justification for neglecting the dissipation function in equation (4). Figure 8 shows predicted curves of the real and imaginary parts of F . , obtained from equation (21), together with measured data for zero mean flow (see Section 3.2). The independent variable here is A = a(to/u) ~/z. Agreement between experimental and theoretical values of Re (F~) is good, and even Im ( F . ) appears to be predicted fairly well by the theory, though the measured data are consistently rather higi{er than the predicted data. As expected, a quasi-steady behaviour is predicted as A ~ 0, where F. ~ 1. Measured data on F,. in the presence of mean flow also agree reasonably well with predictions, thereby justifying the use of equation (13). It is, perhaps, surprising that F~ is predicted as accuratelylas it is by the theory, since the assumed parallel flow past an array of parallel fibres would seem to bear only a passing resemblance to the actual, very complex, flow situation. A contributory factor in
i
i
I
I
I
i
I
I
I
2
co ~ oOO
0
1
2 x
Figure 8. Effective viscosity function F~ of polyether foam for (~)= 0; O, measured Re (F~,); O, measured Im (F~,); , theory (equation (21)).
578
A. C U M M ' I N G S
AND
l.-J. C H A N G
this is no doubt that the measured value of o-, was used in finding F, from the acoustical data (see equations (37b, e)). It is Worth noting that ~r,,, inferred from equation (20) (with appropriate corrections for porosity and tortuosity), is only about 50% of the measured value. This would be expected, since the majority of fibres would be exposed to a cross-flow velocity component, which would bring about an increased viscous drag, over and above that corresponding to parallel flow. There would seem, on the other hand, to be a reasonable possibility o f the predictions o f F~ being acceptable since it is the pressure fluctuation that drives the acoustic temperature field and this should be reasonably insensitive to the fibre orientation. A gratifying feature of the data shown in Figures 7 and 8 is that the hydraulic radius of the fibres appears to be a suitable parameter to use in the circular-fibre model, and no empirically determined effective radius need be found, as it would have to be in the Rayleigh model. A comparison between measured and predicted data on the real and imaginary parts of F is shown in Figure 9, for zero flow and for (~) = 0.82 and 1.96 m/s. The most striking feature o f these results is the large effect that even these low flow velocities have on the
50
J
!
J
40
50
E C~ 20
~ /3 ~//~,~ ~s>
10 o~O--O--D ~cl
~--1:] - D
0 0
I 05
I i Frequency (kHz)
I 15
2
Figure 9. Propagation coefficient o f polyether foam; El, Ig, measured, predicted a or fl for (~) = 0; A, A, measured, predicted a or fl for (~)=0.818 m/s; O, Q, measured, predicted a or 13 for (~)= 1.956 m/s.
579
POROUS MEDIA WITH INTERNAL FLOW
attenuation coefficient o~. At the intermediate flow velocity, a increases by about 50%, over and above its no-flow value, and at the highest flow velocity, an increase of roughly 100% is apparent. This increase in a occurs because of the enhanced dissipation o f acoustic energy into heat, brought about by the inertial fluid drag on the fibres. The phase coefficient/3 is only slightly affected by the flow, a small increase being apparent. The agreement between predicted and measured values of a and/3 is generally very good for all three flow velocities. Small discrepancies are present at (t~) = 1.96 m/s in the phase coefficient but these are probably caused largely by experimental error, and are of no great consequence. In Figure I0, the predicted and measured data on Re (za) and Im (z,) (denoted r, and x, respectively) are shown in pc units, again for ( ~ ) = 0 , 0.82 and 1.96 m/s. Mean flow is seen to affect za~too, and brings about an increase in both r, and the magnitude of xa. Generally, the percentage changes are smaller than they are in the case of a, though at high frequencies, the relative change in x, is of a similar order. Again, the theoretical predictions are in good agreement with measurements, and only modest discrepancies between experiment and theory are evident.
{ ?"'< i
o
~ a m - -
IB--
I
i
i
!
I _ _ i
--
t - - i - -
I--
I --In
- -
I - - i
--
mm- -
i
--mmii
d
0
_.~, /i' -I
I
05
J'~-'--" ,
-1
"~" t
I
,
I
t-5
Z
Frequency (kHz)
Figure 10. Characteristic impedance of polyether foam; IS}, i , measured, predicted ra/~c o or Xo/~Co for (a)=0; A, A, measured, predicted rQ/fico or xo/~c o for (~)=0.818 m/s; O, Q, measured, predicted r~/~co or x J ~ c o for (ti) = 1.956 m/s.
The data shown in Figures 9 and 10 are all for airflow travelling in the same direction as the acoustic wave. Tests were carried out with reverse flow, but various difficulties of a mechanical nature were experienced in these tests and had not, at the time of writing, been satisfactorily resolved. Preliminary data were, however, similar to those with forward flow and showed a corresponding increase in a (particularly) and also in/3, ro and xa; these d a t a are in keeping eith equations (33) and (34). 5. DISCUSSION The foregoing discussion has indicated that the principal effect of internal mean flow on the properties of rigid-framed porous materials is to increase the effective flow resistivity by a quasi-steady inertial process. The chief manifestation of this in terms of the bulk
t0
A. C U M M I N G S A N D l.-J. C H A N G
coustical properties is an increase in the plane-wave attenuation coefficient. The .aodification of the bulk properties by mean flow is quite marked in the present case, ;en for internal flow speeds of less than 1 m/s. The theoretical approach described here ~pears to g i v e q u i t e accurate predictions of the observed bulk properties, and may be ore generally applicable 9 In the context of IC engine exhaust silencers, the modification of the properties of the isorbent by internal mean flow could be a desirable feature--particularly since it is the sistive properties that i n c r e a s e - - a n d it is possible that in many cases an overall improveent in performance could be brought about by the flow. O f course, any quantitative formation on this must await studies of the effects of internal mean flow on porous aterials" in situ. ,The results presented in this paper cover only the situation in w h i c h the mean flow ~d the acoustic particle velocity are parallel. Nonetheless, the data should be applicable the lowest m o d e of propagation in a duct having a bulk-reacting liner, since here both oustic motion and the internal mean flow velocity should be largely axial, at least for w frequencies. Further investigation would be in order concerning the case wherein the trticle velocity and the mean flow are not parallel; it is likely that, here, the effects o f can flow would be somewhat reduced. Studies of mean flow effects in fibrous materials, and also the use of other microstructure odels would be of value. In particular, detailed modelling of non-circular section fibres ossibly by numerical methods; see the p a p e r by Craggs and Hildebrandt [ 12]) is relevant the context of foams and metal wools 9 The inclusion Of f r a m e flexibility is also of .urse a refinement that should be examined, particularly in the low frequency region. It may well be that numerical computations of mean fluid flow within porous materials various situations would be in order, to provide the internal flow information for use finding the bulk acoustical properties, which could then be used in calculating the oustical behaviour of the porous material in situ. ACKNOWLEDGMENT This investigation was carried out with financial support from the University of Missouri eldon Springs Fund. REFERENCES 9 J. M. B E E C K M A N S and P. SEN-GUPTA 1971 Canadian Journal o f Chemical Engineering 49, 721-726. Flow through porous media in the presence of sound. 9 H. L. KUNTZ 1982 University o f Texas at Austin Applied Research Laboratories Report ARL-TR82-54. High intensity sound in air saturated fibrous bulk porous materials. (Also University o f Texas at Austin Ph.D. 771esis 1982 and N A S A CR-167979.) 9 D. A. NELSON 1984 University o f Texas at Austin M S Thesis. Propagation of finite-amplitude sound in air-filled porous materials. 9 m. H. N A Y F E H , J. SUN and D. P. TELIONIS 1984 American Institute o f Aeronautics and Astronautics Journal 12, 838-843. Ettect of bulk-reacting liners on wave propagation in ducts. 9 S.-H. Ko 1975 Journal o f Sound and Vibration 39, 471-487. Theoretical analyses of sound attenuation in acoustically lined flow ducts separated by porous splitters (rectangular, annular and circular ducts). ~. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill. I. S. H. BURNS 1971 Journal o f the Acoustical Society o f America 49, 1-8. Propagation constant and specific impedance of airborne sound in metal wool. F. MECHEL 1976/77 Acustica 36, 53-64, 65-89. Eine Modelltheorie zum Faserabsorber Teil I: Regul~ire Faserordnung, Tell II: Absorbermodell aus elementarzellen und numerische Ergebnisse.
POROUS MEDIA WITH INTERNAL FLOW
581
9. K. ATYENBOROUGH 1982 Physics Reports 82, 179-227. Acoustical characteristics of porous materials. 10. P. G. SMITH and R. A. GREENKORN 1972 Journal of the Acoustical Society of America 52, 247-253. Theory of acoustical wave propagation in porous media. 11. P. C. CARMAN 1956 Flow of Gases through Porous Media. London: Butterworth. 12. A. CRAGGS and J. G. HILDEBRANDT 1984 Journal of Sound and Vibration 92, 321-331. Effective densities and resistivities for acoustic propagation in narrow tubes.
A C O U S T I C P R O P A G A T I O N IN P O R O U S MEDIA W I T H INTERNAL M E A N F L O W t A. CUMMINGS AND I.-J. CHANG Department of Mechanical Engineering, University of Missouri-Rolla, Rolla, Missouri 65401, U.S.A. (Received 11 March 1986, and in revised form 2 July 1986) A parallel-fibre model is presented here, describing sound propagation in rigid-framed porous materials with internal mean fluid flow. Expressions for the bulk acoustical properties of the material are obtained in terms of geometrical parameters--volume porosity, tortuosity and effective fibre radius--and in terms of the mean internal flow speed; fluid properties and acoustic frequency are, of course, also involved. Empirically determined steady-flow viscous and inertial flow-resistive coefficients are required in these expressions. Experimentally measured bulk properties are compared to predicted data both with and without mean flow and generally very good agreement is noted. 1. INTRODUCTION Porous sound-absorbing materials are widely used in devices designed to attenuate noise in flow ducts. Glass fibre wall liners in air conditioning ducts, splitter type silencers in ventilating systems and glass fibre expansion chamber liners in the exhaust silencers of internal combustion (IC) engines are all examples o f these various applieations. "Bulk reacting" absorbers in jet engine inlet ducts have been studied, though they are not in common use. Naturally, there are gradients of static pressure in flo w ducts; these would seem to be of roughly similar magnitude in the inlet diffusers of jet engines and in IC engine exhaust mufflers, and would generally be smaller in air-conditioning systems, but could have locally high values. Any layers of porous acoustical absorbent that are built into the duct will, of course, also be exposed to the static pressure gradients. There will therefore be some degree of mean fluid flow within the porous material itself, to an extent that depends upon the pressure gradients and on the flow resistivity of the porous material. Although the convective effects of this internal mean flow on the bulk acousticproperties of the material are likely to be quite small, it might be anticipated that inertial effects on the flow resistivity (similar to those occurring in fluid flow through an orifice plate) would be noticeable at relatively low fluid velocities, and that the bulk properties could be modified accordingly. Surprisingly, the possible effects of internal mean flow on the bulk acoustic properties of porous material do not appear to have received any attention in the literature, apart from a simple treatment by Beeckmans and Sen-Gupta [1] in the context of filter beds. The objective of the work presented in reference [1] was, however, to investigate variations in the mean static pressure gradients in porous media brought about by the presence o f intense sound waves. The authors obtained quantitatively reasonable agreement between measurements and a theoretical treatment based on a quasi-steady Darcy's law for the t This work was first presented at the AIAA 10th Aeroacoustics Conference, Seattle, Washington, U.S.A. 9-11 July, 1986. 565 0022-460X/87/090565+ 17 S03.00/0 O 1987 Academic Press Inc. (London) Limited
566
A. C U M M I N G S
AND
l.-J. C H A N G
flow resistivity of the porous medium. Kuntz [2] and Nelson [3] reported investigations concerning high amplitude sound propagation in porous materials with no internal mean flow, and the principal effects were caused here, too, by inertial flow losses. The details of the flow interaction at high amplitude are, however, rather different from those which are involved in the case of internal mean flow, and include the generation of harmonics in an initially sinusoidal signal by non-linear processes. The customary treatments of bulk-reacting liners in flow ducts (see, for example, references [4] and [5]) involve the tacit assumption that there is no mean flow in the porous material or--perhapsmthat if there is, then its effects are negligible. There is an apparent lack of supporting experimental data to substantiate this assumption. The practical importance of internal mean flow effects in porous materials is not yet clear, but preliminarytests and calculations show that, at least in IC engine silencers, the internal mean flow velocities can approach 1 m/s which, as will later be demonstrated, is sufficient to cause substantial alteration of the bulk acoustic properties of the material. Rough calculations on porous absorbers in jet engine ductwork indicate possible internal mean flow speeds as high as 1.5-2 m/s, which could produce even greater changes in bulk properties. The motivation for the present investigation was the likelihood that internal mean flow effects in porous absorbents in s i t u could be important in many situations of interest. The results do, indeed, show that these effects a r e likely to be significant, thereby providing justification for the study. In this paper, a theoretical model for the bulk properties of rigid-framed porous acoustic absorbents in the presence of uniform internal mean fluid flow is pi'esented, for the particular case of one-dimensional propagation in which the mean flow velocity is parallel to the acoustic particle velocity, and experimental data on an open-celled polyether foam are compared to theoretical predictions. It is assumed that the acoustic particle velocity within the material is sufficiently small for high amplitude effects to be absent. 2. THEORETICAL MODEL In this first approach to the problem, it is assumed that the solid frame of the porous material is rigid and that the pores are filled with a gas. This is not generally an unduly restrictive assumption; structural wave motion tends to be of importance at rather low frequencies for most porous materials. Three microstructure models have been examined in the present investigation: first, a very simple "phenomenological" model that is essentially similar to that described by Morse and Ingard [6], secondly a "Rayleigh type" model embodying parallel capillary tubes, and thirdly a parallel-fibre model essentially similar to those of Burns [7] and Mechel [8]. The number of empirically determined parameters in these three models decreases in the order of mention. This is directly related to the degree of resemblance between the assumed and actual microstructures. The Morse/Ingard model implies no assumptions at all about the microstructure and hence contains parameters that have to be found from acoustic measurements (or guessed). Chief amongst these are the effective compressibility of the fluid in the pores of the material and the effective viscous flow resistivity of the material. The former quantity may be assigned a value somewhere between the figures appropriate to reversible isothermal and adiabatic processes, and the latter may be assumed equal to the steady flow resistivity of the material (which may readily be measured). At audiofrequencies, however, both of these quantities take on complex values and this Simple model does not give entirely satisfactory results, etiher with or without internal mean flow. The Rayleigh model gives better results because it makes allowance for the presence of viscous and thermal i
POROUS MEDIA WITH INTERNAL
FLOW
567
boundary layers in the fluid adjacent to the surface of the solid frame. The effective compressibility and flow resistivity are both complex and, apart from the empirical determination of effective pore radii (which are different in the cases of the compressibility and flow resistivity), can be made to give fairly satisfactory results. Even so, the disparity between the assumed and actual microstructures brings about a residual lack of correspondence between prediction and measurement in the present case (though the Rayleigh model might be appropriate to other geometries, such as granular media). The parallel-fibre models o f Burns [7] and Mechel [8] (which are essentially similar) are an improvement over the Rayleigh model for both fibrous materials and open-called reticulated foams (which are basically fibrous in structure), to the extent that the solid frame is modelled as an array of fibres. The inclusion of an effective fibre radius is less ambiguous than it is in the case of the Rayleigh model, and a complex compressibility and flow resistivity are a part of these models. Attenborough [9] gave an excellent review of porous material models, including those previously mentioned. Comparison between predictions based on the three models and measured data revealed the parallel-fibre model to be superior to the other two, and so the discussion here will be restricted to this model. Although the development of the model follows partly along the lines described by Mechel [8], it will be described here in detail because of the inclusion o f internal mean flow effects and also because viscous effects are incorporated in a different way from that described in reference [8]. 2.1. GEOMETRY The model will be discussed with particular reference to the polyether foam used in the experimental tests. Figure 1 shows an electron micrograph of a sample approximately two cells thick. The longer dimension of the white rectangle is 100 I.tm. The structure of the foam consists essentially of triangular section strands (or fibres) connected at their ends; the solid frame is what remains of the interstices of the original foam after the bubbles have burst. The width of the strands is typically 30-100 ~m. An idealized geometrical structure o f a single cell would be a regular dodecahedron. It should be noted that a fairly small proportion ofthe inter-cell holes in the foam contain unburst membranes or "closed windows". In modelling a foam (or fibrous) material with internal mean flow, one must consider the nature of the fluid stresses at the surface of the solid frame, and the Reynolds' number (based on the hydraulic diameter of a fibre) for a crossflow at the mean internal flow
Figure 1. Electron micrograph of polyether foam.
568
A. C U M M I N G S
AND
I.-J. C H A N G
speed is an appropriate parameter. For an internal flow speed of 0.5=2 m/s and a fibre width of 50 ~.m (approximately equal to the hydraulic diameter), the Reynolds' number is about 1.5-6-5, which is in the range where inertial stresses can be as large as, or greater than, viscous stresses. The regions where the fibres join are of greater width and will be associated with higher Reynolds' numbers, as will be the closed windows. The mean flow in the parallel-fibre model will be assumed to be parallel to the fibres, and thus, for laminar mean flow, no inertial stresses would be present. To introduce these inertial stresses, flow obstacles will be made a part of the model, in the (arbitrarily chosen) form of discs attached to the fibres as shown in Figure 2; the fibres are assumed to have a circular cross-section (of radius a) and to be identical. The diameter and positioning of the discs are unspecified: their sole effect is to introduce inertial stresses in the mean flow. The fibres are assumed to be arranged in a hexagonal array as shown and each may be regarded as being surrounded by a "cylinder of influence" of radius R (see reference [7]). The fact that these cylinders must actually be hexagonal prisms, in order to fit together, is of little practical consequence. The volume porosity, ~, of the material is given by = 1 -a2/R 2 (1) and the volume of the discs is neglected. In the polyether foam studied, the fibres are roughly triangular in section. A good approximation of their shape is that of the interstice between three equal, touching circles as shown in Figure 3. The cross-sectional area is 0.1613 ro2, ro being the radius of each circle. A hydraulic fibre radius may be found as (2)
ah = g - 2 3 / P ,
where A is the area of the space between the fibre and the boundary of the cylinder of influence and P is the total perimeter of this space (including the fibre). From equation (2), it is found that ah = R{1-211 - O . 0 5 1 3 ( r o / R ) 2 ] / [ 2 + ( r o / g ) ] } .
(3)
In what follows, t/h will be used as the equivalent fibre radius, and denoted a.
F i i ~ , ~Dis~ci ;
Cylinderofinfluence
R
~
\
x
~
Directi onofpropagat meanflowion and sound Figure 2. Geometry of parallel-fibre microstructure.
~'"
POROUS
MEDIA
Sf
WITH
%
INTERNAL
SS
,, ~1~ u f
,
f I
569
FLOW
S, ~ ro I
~,~
Fibre
%.
S I
I I I
I i i
'S.
is,
l
e
Figure 3. Assumed cross-sectionalgeometry of "triangular" fibre. 2.2. FLUID COMPRESSIBILITY TO find the effective compressibility of the fluid in the pores of the material, one must solve the energy equation relating pressure and temperature fluctuations. The linearized form of this equation is ~CpoO T'/ D t - 7"flDp'/ O t =/~V: T'+/St/,'
(4)
for an ideal gas with constant viscosity, where Cpo is the ideal gas specific heat of the fluid at constant pressure, T is temperature, t is time, p is fluid density,-fl is the coefficient of thermal expansion, p is pressure, k is the thermal conductivity of the fluid and 9 is the dissipation function; the overbar denotes a time average and the prime denotes a perturbation. Figure 4 shows the co-ordinate system; the presence of the discs is neglected. Equation (4) m a y b e simplified ifit is assumed that a( )/ax = 0, a( )/a0 = 0 (valid provided radial derivatives >>axial derivatives, for axisymmetric disturbances) and that the fluid velocity u is purely axial. Then /5~'= i.t(dff/dr)(0u'/ar), where p. is the coefficient o f shear viscosity. This term may be shown to be small compared to the other terms in the equation, for typical numerical data, and it will be ignored. The simplified version of equation (4), for harmonic time variation of radian frequency to, and a uniform pressure perturbation is d 2T6/d(ar) 2+ {1/(ar)} d T~/d(ar) + T~ = p~/Cpo~,
/ !
I
V__
! ! ! !
R,
ja
I
Fibre
/ Cylinder of influence
Figure 4. Co-ordinate system of a single fibre.
(5)
570
A. C U M M I N G S A N D I.-J. C H A N G
where T ' = T~(r) exp (itot), p ' = p~ exp (iwt) and a = ( - iwCpofi/k)i/2. It can be seen that the only influence of mean flow is in the fluctuating part of the dissipation function q~ and that if this is negligible, the solution to equation (4) will be the same with and without [tow. The general solution to equation (5) is
T~( r) = ( p'/ ~Cpo) + CiJo( ar) + C2Yo( ar),
(6)
lo( ) and Yo( ) being zero order Bessel and Neumann functions respectively, and CI and C2 arbitrary constants. Appropriate boundary conditions are
T~(a) = O,
(dT'o/dr),=R = 0.
(7a, b)
(The first of these conditions actually breaks down at frequencies that are sufficiently low for the thermal boundary layer inside the fibres to be significantly thick; ultimately the solid and gas are in thermal equilibrium. For practical purposes, however, equation (7a) will suffice.) Applying conditions (7a, b) to equation (6) yields
T~(r) =p'{1 -Jo(ar)/S(a)+[Jl(otR)/Yl(aR)]Yo(ar)/S(ot)}/~Cpo,
(8)
S(a) = Jo(aa) - J , ( o t R ) Y o ( a a ) / Y l ( a R ) .
(9)
where
In the customary manner, the effective fluid compressibility K(to) m a y b e defined as K(to) = (1 - gcfi(T')/p')/~,
(10)
where ( ) denotes a space average over the region a ~< r ~< R and R e is the gas constant. Equations (8), (9) and (10) yield, finally, K(to)=[l+2(l_~
) (y-1)cta Q ( a ) ] / T / ~ =
F,,/Tfi
(11)
where y is the specific heat ratio of the fluid and
Q(a) = { J l ( a R ) Y l ( a a ) - J l ( a a ) Y l ( a R ) } / { Y l ( a R ) J o ( a a )
-Jl(aR)Yo(ota)};
(12)
the compressibility function F~ is defined by equation (11). 2.3. fLOW RESISTIVITY The flow resistivity of the material embodies two parts: that associated with the (parallel) mean and fluctuating flow past a fibre (a purely viscous resistance), and that associated with flow past a disc (an inertial resistance). These will first be considered separately, before being combined together. Expressions for the viscous flow resistivity for mean and fluctuating flows are found as follows. Subject to the assumptions made in the previous subsection (except Op/Ox = 0) the Navier-Stokes equations yield
Ou/Ot = - (1/~) Op/Ox + v{t92u/t~r2+ (1/r) Ou/Or},
(13)
where p and u are the total (mean+fluctuating) pressure and velocity respectively and / ~ is the kinematic fluid viscosity. This equation is linear, and thus the solutions for mean and fluctuating flows may be superimposed. With harmonic time variation, the general tsolution to equation (13), for the velocity perturbation, is
J
u~(r) = - (Op~/Ox)/ito~ + DlJo(sCr) + D2Yo(~r),
(14)
POROUS M E D I A WITH I N T E R N A L FLOW
571
where u'-- u~(r) exp (itot), st = ( - i t o / v ) i / 2 and Di, D2 are arbitrary constants. The boundary conditions are u~(a) = 0,
(du'o/dr),=R = 0.
(15a, b)
Equations (14) and (15) then yield u~(r) = -(Op'o/OX){[1 -Jo(str)/S(st)]+[Jn(stg)/Yl(stR)]Yo(str)/S(st)}/ito~,
(16)
where S( ) is given by equation (9). A "shear impedance" z's may be defined as z'~ = T'~/(U'),
(17)
T" being the fluctuating shear stress at the surface of the fibres and (u') the space-average of u'. Since T'w=--/.t(0u'/0r) . . . . equations (9), (16) and (17) give z' = - # s t
st(R-~-- a2 ) 4- [Jt(~a)Yl(stR) - J~(StR)Y~(Sta)J J"
The solution to equation (13) for the mean flow velocity, together with boundary conditions analogous to those in equations (15), gives = - (a2/41~)(dff/dx)[1 - ( r / a ) 2 + 2 ( R / a ) 2 In ( r / a ) ] ,
(19)
and a shear impedance ~s (= fw/(t~)) may be found as above, zs =
- (/~/a)(1 - a 2 / R 2 ) / [ { l n ( R / a ) / ( 1 - a 2 1 R 2 ) } - 3 + ( a 2 / 4 R 2 ) ] .
(20)
An effective viscosity function F~, may be defined as F , = z'~/~.,,
(21)
expressing the ratio between the effective fluid viscosity in the acoustic wave and the viscosity coefficient #. As one would expect, F,, --, 1 as st--, 0. The "viscous" flow resistivity in the acoustic wave, o-v(~o), is now given as o%(to) = o's,F,,
(22)
where o-s, is the steady flow resistivity (pressure drop/unit distance/unit flow speed) of the porous material, defined at low flow velocities where viscous stresses predominate. The flow speed in the definition of o'~ and o's, is volume velocity/cross-sectional area of bulk material, in the usual way. At acoustic frequencies, o'~ is usually determined in part by inertial forces (in shear waves exhibited by the boundary layer adjacent to the fibre) as well as by viscous forces, but these inertial forces are linearly proportional to (u'). It must be noted that cr~ also describes the steady flow viscous resistivity, since F,, = 1 for to=0. Clearly, o-,, could be calculated from equation (20), with appropriate modifications, but one could not expect this procedure to give very good results because of the lack of close resemblance between the parallel-fibre model and the actual microstructure. Instead, an accurately measured value of cry, is utilized, and it is assumed that F,, (calculated from the idealized model) will still give a reasonable description of the frequency dependence of o-~. This is essentially a compromise measure but gives tolerably good results, as will later be seen. The inertial component of the flow resistivity can be included as follows. If it is assumed that there are no viscous forces acting on the discs and that pressure drag forces dominate, then the drag force acting on each disc may be written, for the mean flow, as FD = C o ~ ( ~ ) I ( ~ ) I A a / 2
(23)
572
A. C L I N i M I N G S A N D l.-J. C H A N G
(the absolute value of (fi) being inserted to allow for negative flow velocities), where Co is a drag coefficient that can be assumed to be approximately constant, and Aa is the frontal area of the disc. For a quasi-steady perturbed flow, (23) becomes
vo + F'o = CoAd~l(a) l((a)+ 2(u'))/2,
(24)
where the quadratic terms in perturbations are assumed to be relatively small and the term p'(tT) 2 is taken to be of order (fi)/Co compared to the second term in parentheses, and therefore negligible (Co is the adiabatic speed of sound). It is assumed here that the inertial flow resistivity is quasi-steady and that equation (24) applies. On this basis, and together with equation (22), a modified Darcy's law may be written for the perturbed flow, embodying the combined viscous and inertial drag forces acting on the fluid in the bulk material,
(Op/OX)arag=
--O's,((l]) + F t , ( u ' ) ) -
l(a)+ 2 I(a)I (u'))
(25)
The angle brackets henceforth denote a space average of velocity equal to volume velocity/cross-sectional area of bulk material, rather than the average velocity within the open spaces of the porous structure and o'i is an empirically determined inertial flow resistivity coefficient (note that it does not have the same dimensions as or,t). The mean and fluctuating components of equation (25) may be equated separately by first taking a time average of the equation and then subtracting this from the original equation:
(Off/OX)drag=--O',t(a)--IT,[(•)l(a),
(Op'/OX)drag=--ITslF•(ut)--2ortl(a)l(U').
(26a, b)
2.4. DEVELOPMENT OF A GOVERNING WAVE EQUATION The bulk properties of the porous material in the presence of mean flow are found from solutions to the governing wave equation, which is derived from the linearized continuity and momentum equations for the fluid in the pores. It is assumed for the time being that one-dimensional motion occurs parallel to the fibres of Figures 2 and 4. The continuity equation may be written
FI Op/at + O(p(u))/Ox = 0,
(27)
and the linearized version of this, for acoustic perturbations, is
Op'/Ot+~ cO(u')/~x = 0,
(28)
where the term (t~) Op'/Ox is of order (~)]eo compared to the second term on the left side of equation (28) and has been discarded. A one-dimensional momentum equation for the perturbed flow may be derived from (26a) if the convective acceleration term in D(ld)/Dt (which is of order (t~)] Cocompared to the local acceleration) is neglected. The net pressure gradient may be written
Op'/ Ox = ( Op'/ OX) drag+ ( Op'/ OX)i,e~ti, ,
(29)
where the second term arises because of the mass of fluid contained in the pores of the material. The inertial contribution to the pressure gradient may be expressed in terms of the local fluid acceleration as (t~q2/g2) O(u')/Ot (see the paper by Smith and Greenkorn [10]), where q is the tortuosity factor (see the book by Carman [11]) and q2 is identified with the Smith and Greenkorn inertial factor tensor, being otherwise often known as the structure factor. Equation (29) may now be written
Op'/Ox = -cr,,F~,(u')- 2o-~[(a) ]( u ' ) - (~q2/ O ) 3(u')/Ot.
(30)
The one-dimensional wave equation for harmonic time dependence may be derived by eliminating (u') from equations (28) and (30), and utilizing the relationship K(w) = p'/p'~
POROUS
MEDIA
WITH
INTERNAL
FLOW
573
(see equation (10)) to yield
oEp'/Ox2-t-fiKq2to2p'--itoflK(Ors,Fa +2o', I(a) l)p'
= 0.
(31)
This wave equation embodies no convective mean flow effects (the mean flow being too slow to be significant), but does contain a quasi-steady mean flow resistive term. 2.5. B U L K A C O U S T I C P R O P E R T I E S Substitutlnn of a travelling wave solution
p ' ~ e -rx+i~t
(32)
F = +iqco (/SK)!/2[-1 - ifl (cr,,F~ + 2o', [(fi) [)//~q2t0] ,/2.
(33)
into equation (31) gives
The positive root represents a positive travelling wave and the negative root a negative travelling wave. Where the positive root is taken, F is identified with the bulk propagation coefficient. Combination of equations (30), (32) and (33) yields an expression for the characteristic impedance za.(= p'/(u') in a positive travelling wave),
za = ( q / ~ )(/5 / K )1/2[ 1 -- if2 (cr~,F~ + 2cri[ ( fi) [)/ fiq2to ],/2.
(34)
The two quantities F and za are the primary characterizing parameters of the bulk material. Equations (33) and (34) are generally applicable to any open-celled, rigid porous material with one-dimensional motion of a gas in its pores. Several observations may be made, concerning equations (33) and (34). First, there are three parameters--q, o's, and o-,--which are best determined by experiment for any realistic microstructure. The tortuosity q is readily found by an electrical conductivity method as described by Carman [11], and tr, t and o'i are easily determined from steady flow pressure drop measurements over a range of flow velocities. The porosity Y2 may be found very simply from the bulk density of the material and the density of the solid matrix. Section 3.1 gives details of these "static" properties. Secondly, K and F,~ may be determined for a fibrous or foam material by using the microstructure model described in sections 2.1-2.3, provided a representative average hydraulic fibre radius can be found. One also notes here that for a fairly slow internal mean flow; neither F~, nor K is affected by the flow, at least on the basis of the assumed model. Thirdly, it can be seen that the direction of the mean flow appears to be immaterial in determining the change in bulk properties brought about by flow. An increase in the real p a r t o f the "acoustic flow resistance" tr,,F~, is predicted, for mean flow travelling in the same direction as, or in the opposite direction to, the sound wave. Fourthly, any modification in the bulk properties caused by mean flow is dependent principally on the relative magnitudes of tri and o'~t. One would expect that materials containing a high density of structural discontinuities or blockages (such as the closed windows in foams or "shot" in fibrous materials) would have relatively high values of tr~ and would hence exhibit quite large mean flow effects whereas, at the opposite extreme, structures (such as arrays of parallel, purely cylindrical, fibres) containing no discontinuities at all would show little or no mean flow effects on their bulk properties. 3. MEASUREMENTS The measurements carried out in the present investigation fall into two categories: measurements of the steady flow coefficients cry, and cr~ together with the essentially geometrical features s q and ro, and measurements of the bulk acoustic properties of the material.
574
A. C U M M I N G S A N D l.-J. C H A N G
3.1. G E O M E T R I C A L FEATURES
The volume porosity is easily found from the measured bulk density pb of the porous material and the density of the solid frame ps, as 1-1 = 1 - p b / m .
(35)
The bulk density is determined by weighing a sample of the material and the density of the frame may be found by independent means, usually from published or quoted data. In the ease of the polyether foam used in the present series of tests, pb was found to be 46-4 kg/m 3 and ps was (according to the manufacturers of the foam) about 1100 kg/m 3. The porosity was thus found to be 0.958. The tortuosity was measured in this investigation by using an electrical conductivity method cited by Carman [11]; the experimental procedure is as follows. The electrical resistance of a particular volume (of rectangular parallelepiped shape) of saline solution was measured in a conductivity cell, by means of an AC bridge with a frequency of 100 kHz. This rather high frequency was used to minimize the capacitive component of the electrical impedance, since a pure resistance was used to balance the bridge. Then a piece of foam, cut to be of the same shape and size as the volume of solution, was saturated with the solution whilst in the cell and its resistance was measured by the same means. Then the tortuosity q is given (see the book by Carman [11]) by
q = ORI/Ro,
(36)
where Ro is the electrical resistance of the saline solution alone and Rf is the resistance of the saturated foam. The value of tortuosity found from the present tests was 1.167. The dimension ro is the radius of the circle defining the fibre shape in Figure 3, but is also the distance between the "points" of the fibre's cross-section and will be referred to as the "fibre width". An approximate, average, value of ro was found--by direct measurements on a series of electron mierographs like that in Figure 1--to be 50 p.m. The viscous and inertial steady flow resistivity coefficients were found by passing a steady airflow through a plug of the foam and measuring the pressure drop across the plug as well as the volume flow through it. The measured pressure gradient d~/dx was plotted against (t~) (see equation (26a)) and a quadratic curve was fitted (by means of the least squares method) to the points, thereby defining o's, and tri. Figure 5 shows the measured data and the fitted curve. It can :be seen that the quadratic curve fits the data very well, thereby justifying this form of Darcy's law. The values of the flow resistivity coefficients corresponding to the quadratic curve are trs, = 5425 N s/m 4 (SI rayl/m) and ~ri = 3071 N s2/m 5. 3.2. ACOUSTICAL PROPERTIES The bulk acoustical properties of the foam were measured by using the standing wave tube shown in Figure 6, which had provision for passing a steady airflow through the foam. Half of the standing wave tube was filled with cylindrical pieces of foam, and the total depth of foam tested was 0.96 m. The normal impedance of this thickness of foam was measured by the usual standing wave meihod. Over the frequency range of the tests (500-2000Hz), acoustic reflections from the open end of the tube were sufficiently attenuated at the opposite face of the sample that the impedance measured was effectively that of a semi-infinite thickness of foam--that is, the characteristic impedance; this was verified by plotting the axial variation of sound pressure level. Both the velocity profile of the mean airflow within the foam and the "entrance length" (the distance from the inward-facing end of the plug of foam to the point where the flow became fully developed) were of considerable interest in this experiment, since these features of the flow would
POROUS MEDIA WITH INTERNAL FLOW
/
/
~2 X
E
g.
/
1
/.o/
575
e/e
O0
jl /I /I
l" I 0/0
-000
0
0
Figure 5. Steady flow pressure gradient vs. flow velocity for polyether foam; O, measurements; curve fit.
I
Sinewove generolor
, quadratic
I Frequency counler
S7 =:
I:::"
7;ram Probe,
"i":: 41k
I!=
0-96m O.99m
Acousticdriver
T~-
0-99m
Tometer,
'~'% J- I
('7)-n-~.~ "L-
Seal
Stole
Microphone
J
-I
Figure 6. Standing wave tube apparatus for the measurement o f bulk acoustic properties.
determine the extent to which the assumption of uniform flow was realized in practice. Approximate numerical calculations were carried out, by using a finite difference approximation to a simplified version of the Navier-Stokes equations, with added flow-resistive terms but with no radial velocity component. The results showed the fully developed flow to be virtually uniform, with an extremely thin boundary layer of less than 1% of the pipe radius in thickness; additionally the entrance length was shown to be very short, typically less than an eighth of a pipe radius. Though the approximate form of the momentum equation would no longer be valid under these circumstances (and, of course, the continuity equation would not even be approximately satisfied), an upper bound was
76
A. C U M M I N G S
AND
1.-J. C H A N G
mt on the entrance length. It was quite clear from these results that the uniform flow pproximation should be good, since beyond a thin surface layer of the foam plug, trtually uniform flow would be expected to prevail. It was not, unfortunately, possible carry out experimental tests to confirm these calculated data. The propagation coefficient of the foam was found by plotting the level and relative aase of the acoustic wave within the porous layer in a direction from the surface nearer ie source, into the material, over a sufficient distance for the rates of change of sound vel and phase--with distance--to be accurately determined. Only the linearly varying trts of these curves were used, and the method of least squares was used in fitting raight lines to the measured data. The attenuation coefficient a of the sound wave is ~e real part of F and the phase coefficient/3 is the imaginary part. The presence of airflow in the standing wave tube produced a certain degree of more 9 less random noise, and it was thought desirable to filter the microphone output in "der to obtain an accurate measure of the amplitude and phase of the sinusoidal signal .at was generated by the loudspeaker (this procedure also eliminated the effects of any ~ssible "wind noise" at the probe hole). Accordingly, a Hewlett-Packard type 3582A ,ectrum analyzer was used to give a spectrum of the microphone voltage and thereby , separate the signal and noise; the analyzer also yielded the phase of the signal. In the standing wave analysis, the effects of mean airflow were taken into account, :ough since the mean flow Mach number was very low, these effects were small. From a knowledge of F and za in the absence of mean flow, it is possible to infer dues of K(to) and o'stF~,. Equations (33) and (34) may be combined to give the relationtips
r,(toi = F/itol2za,
os, Re (F~,) = Re (Fz~),
o's, Im (F~,) = Im (Fz,)- q2pta/O.
(37a, b) (37c)
hus the compressibility function F~ (see equation (11)) and the effective viscosity function may be found from measurements o f F and zo. The above equations are also valid the presence of mean flow in the case of K and Im (F~,); Re (F~,) may be found by tbtracting the inertial flow resistivity component from the right side of equation (37b). 4. MEASUREMENTS OF ACOUSTICAL PROPERTIES AND COMPARISON WITH THEORY Figure 7 shows theoretical curves of Re ( F ~ ) - I and - I r a (FK), from equation (11); ,gether with measured data obtained in the absence of mean flow as described in Section 2. The independent variable is A = aPr~/2(to/v) ~/2, where Pr is the Prandtl number of e fluid in the pores o f the material. It can be seen that the measured and predicted e (F~) data are in good agreement. The data on the imaginary part of F~ do not agree well, particularly above A = 0.6, though below that point the agreement is fair. The ~rcentage accuracy in measuring Im (F~) is not high and this may explain the fact that :e measured absolute values of this quantity exceed the theoretical values above A = 0.6. rver most of the frequency range, however, IIm (FK)] is no greater than about 10% of e ( F , ) and in the range where Im (FK) has its maximum absolute value, the predictions ?e at their best. Overall, the theory would seem to give fairly satisfactory predictions of '~, especially when one considers the real versus assumed microstructures. It is noted hat for A--)0, an isothermal thermodynamic process is predicted in the fluid wave, hereas as A ~ 0% an adiabatic process is forecast. Although, as previously mentioned, e boundary condition T'(a) = 0 is likely to break down at low frequencies, there is no ddence of this occurring in the frequency range embraced by the present results.
577
POROUS MEDIA %VITH INTERNAL FLO%V 0"4 Re ( F ~ ) - I
0-3
0-2 c
o
o
oo oo
0.1
0
I
I
0
I
i
I
1
3
2 .4
Figure 7. Compressibility function F~ of polyether foam for {,~)=0; O, measured Re ( F ~ ) - 1; O, measured theory (equation (11)).
- l m (FK); - - ,
Measured data on FK in the presence of flow--not shown in Figure 7 for the sake of clarity--agree almost as well with the theoretical curves as the no-flow data. This constitutes experimental justification for neglecting the dissipation function in equation (4). Figure 8 shows predicted curves of the real and imaginary parts of F . , obtained from equation (21), together with measured data for zero mean flow (see Section 3.2). The independent variable here is A = a(to/u) ~/z. Agreement between experimental and theoretical values of Re (F~) is good, and even Im ( F . ) appears to be predicted fairly well by the theory, though the measured data are consistently rather higi{er than the predicted data. As expected, a quasi-steady behaviour is predicted as A ~ 0, where F. ~ 1. Measured data on F,. in the presence of mean flow also agree reasonably well with predictions, thereby justifying the use of equation (13). It is, perhaps, surprising that F~ is predicted as accuratelylas it is by the theory, since the assumed parallel flow past an array of parallel fibres would seem to bear only a passing resemblance to the actual, very complex, flow situation. A contributory factor in
i
i
I
I
I
i
I
I
I
2
co ~ oOO
0
1
2 x
Figure 8. Effective viscosity function F~ of polyether foam for (~)= 0; O, measured Re (F~,); O, measured Im (F~,); , theory (equation (21)).
578
A. C U M M ' I N G S
AND
l.-J. C H A N G
this is no doubt that the measured value of o-, was used in finding F, from the acoustical data (see equations (37b, e)). It is Worth noting that ~r,,, inferred from equation (20) (with appropriate corrections for porosity and tortuosity), is only about 50% of the measured value. This would be expected, since the majority of fibres would be exposed to a cross-flow velocity component, which would bring about an increased viscous drag, over and above that corresponding to parallel flow. There would seem, on the other hand, to be a reasonable possibility o f the predictions o f F~ being acceptable since it is the pressure fluctuation that drives the acoustic temperature field and this should be reasonably insensitive to the fibre orientation. A gratifying feature of the data shown in Figures 7 and 8 is that the hydraulic radius of the fibres appears to be a suitable parameter to use in the circular-fibre model, and no empirically determined effective radius need be found, as it would have to be in the Rayleigh model. A comparison between measured and predicted data on the real and imaginary parts of F is shown in Figure 9, for zero flow and for (~) = 0.82 and 1.96 m/s. The most striking feature o f these results is the large effect that even these low flow velocities have on the
50
J
!
J
40
50
E C~ 20
~ /3 ~//~,~ ~s>
10 o~O--O--D ~cl
~--1:] - D
0 0
I 05
I i Frequency (kHz)
I 15
2
Figure 9. Propagation coefficient o f polyether foam; El, Ig, measured, predicted a or fl for (~) = 0; A, A, measured, predicted a or fl for (~)=0.818 m/s; O, Q, measured, predicted a or 13 for (~)= 1.956 m/s.
579
POROUS MEDIA WITH INTERNAL FLOW
attenuation coefficient o~. At the intermediate flow velocity, a increases by about 50%, over and above its no-flow value, and at the highest flow velocity, an increase of roughly 100% is apparent. This increase in a occurs because of the enhanced dissipation o f acoustic energy into heat, brought about by the inertial fluid drag on the fibres. The phase coefficient/3 is only slightly affected by the flow, a small increase being apparent. The agreement between predicted and measured values of a and/3 is generally very good for all three flow velocities. Small discrepancies are present at (t~) = 1.96 m/s in the phase coefficient but these are probably caused largely by experimental error, and are of no great consequence. In Figure I0, the predicted and measured data on Re (za) and Im (z,) (denoted r, and x, respectively) are shown in pc units, again for ( ~ ) = 0 , 0.82 and 1.96 m/s. Mean flow is seen to affect za~too, and brings about an increase in both r, and the magnitude of xa. Generally, the percentage changes are smaller than they are in the case of a, though at high frequencies, the relative change in x, is of a similar order. Again, the theoretical predictions are in good agreement with measurements, and only modest discrepancies between experiment and theory are evident.
{ ?"'< i
o
~ a m - -
IB--
I
i
i
!
I _ _ i
--
t - - i - -
I--
I --In
- -
I - - i
--
mm- -
i
--mmii
d
0
_.~, /i' -I
I
05
J'~-'--" ,
-1
"~" t
I
,
I
t-5
Z
Frequency (kHz)
Figure 10. Characteristic impedance of polyether foam; IS}, i , measured, predicted ra/~c o or Xo/~Co for (a)=0; A, A, measured, predicted rQ/fico or xo/~c o for (~)=0.818 m/s; O, Q, measured, predicted r~/~co or x J ~ c o for (ti) = 1.956 m/s.
The data shown in Figures 9 and 10 are all for airflow travelling in the same direction as the acoustic wave. Tests were carried out with reverse flow, but various difficulties of a mechanical nature were experienced in these tests and had not, at the time of writing, been satisfactorily resolved. Preliminary data were, however, similar to those with forward flow and showed a corresponding increase in a (particularly) and also in/3, ro and xa; these d a t a are in keeping eith equations (33) and (34). 5. DISCUSSION The foregoing discussion has indicated that the principal effect of internal mean flow on the properties of rigid-framed porous materials is to increase the effective flow resistivity by a quasi-steady inertial process. The chief manifestation of this in terms of the bulk
t0
A. C U M M I N G S A N D l.-J. C H A N G
coustical properties is an increase in the plane-wave attenuation coefficient. The .aodification of the bulk properties by mean flow is quite marked in the present case, ;en for internal flow speeds of less than 1 m/s. The theoretical approach described here ~pears to g i v e q u i t e accurate predictions of the observed bulk properties, and may be ore generally applicable 9 In the context of IC engine exhaust silencers, the modification of the properties of the isorbent by internal mean flow could be a desirable feature--particularly since it is the sistive properties that i n c r e a s e - - a n d it is possible that in many cases an overall improveent in performance could be brought about by the flow. O f course, any quantitative formation on this must await studies of the effects of internal mean flow on porous aterials" in situ. ,The results presented in this paper cover only the situation in w h i c h the mean flow ~d the acoustic particle velocity are parallel. Nonetheless, the data should be applicable the lowest m o d e of propagation in a duct having a bulk-reacting liner, since here both oustic motion and the internal mean flow velocity should be largely axial, at least for w frequencies. Further investigation would be in order concerning the case wherein the trticle velocity and the mean flow are not parallel; it is likely that, here, the effects o f can flow would be somewhat reduced. Studies of mean flow effects in fibrous materials, and also the use of other microstructure odels would be of value. In particular, detailed modelling of non-circular section fibres ossibly by numerical methods; see the p a p e r by Craggs and Hildebrandt [ 12]) is relevant the context of foams and metal wools 9 The inclusion Of f r a m e flexibility is also of .urse a refinement that should be examined, particularly in the low frequency region. It may well be that numerical computations of mean fluid flow within porous materials various situations would be in order, to provide the internal flow information for use finding the bulk acoustical properties, which could then be used in calculating the oustical behaviour of the porous material in situ. ACKNOWLEDGMENT This investigation was carried out with financial support from the University of Missouri eldon Springs Fund. REFERENCES 9 J. M. B E E C K M A N S and P. SEN-GUPTA 1971 Canadian Journal o f Chemical Engineering 49, 721-726. Flow through porous media in the presence of sound. 9 H. L. KUNTZ 1982 University o f Texas at Austin Applied Research Laboratories Report ARL-TR82-54. High intensity sound in air saturated fibrous bulk porous materials. (Also University o f Texas at Austin Ph.D. 771esis 1982 and N A S A CR-167979.) 9 D. A. NELSON 1984 University o f Texas at Austin M S Thesis. Propagation of finite-amplitude sound in air-filled porous materials. 9 m. H. N A Y F E H , J. SUN and D. P. TELIONIS 1984 American Institute o f Aeronautics and Astronautics Journal 12, 838-843. Ettect of bulk-reacting liners on wave propagation in ducts. 9 S.-H. Ko 1975 Journal o f Sound and Vibration 39, 471-487. Theoretical analyses of sound attenuation in acoustically lined flow ducts separated by porous splitters (rectangular, annular and circular ducts). ~. P. M. MORSE and K. U. INGARD 1968 Theoretical Acoustics. New York: McGraw-Hill. I. S. H. BURNS 1971 Journal o f the Acoustical Society o f America 49, 1-8. Propagation constant and specific impedance of airborne sound in metal wool. F. MECHEL 1976/77 Acustica 36, 53-64, 65-89. Eine Modelltheorie zum Faserabsorber Teil I: Regul~ire Faserordnung, Tell II: Absorbermodell aus elementarzellen und numerische Ergebnisse.
POROUS MEDIA WITH INTERNAL FLOW
581
9. K. ATYENBOROUGH 1982 Physics Reports 82, 179-227. Acoustical characteristics of porous materials. 10. P. G. SMITH and R. A. GREENKORN 1972 Journal of the Acoustical Society of America 52, 247-253. Theory of acoustical wave propagation in porous media. 11. P. C. CARMAN 1956 Flow of Gases through Porous Media. London: Butterworth. 12. A. CRAGGS and J. G. HILDEBRANDT 1984 Journal of Sound and Vibration 92, 321-331. Effective densities and resistivities for acoustic propagation in narrow tubes.