NASA Contractor Report 2951
Effects of Grazing Flow on theSteady-State Flow Resist:a andAcoustic Impedance of Thin Porous-FacedLiners
A. S. Hersh and B. Walker
CONTRACT NAS1 - 14310 JANUARY 19 78 I" I" - TECH LIBRARY KAFB, NM
NASA ContractorReport 2951
Effects of Grazing Flow on theSteady-State Flow Resistance andAcoustic Impedance of Thin Porous-FacedLiners
A. S. Hersh and B. Walker Hersh Acoustical Engineering Chatsworth, California
Prepared for Langley Research Center under Contract NAS 1- 14 3 lo
National Aeronautics and Space Administration Scientific and Technical Information Office
1978
I
TABLE OF CONTENTS SUMMARY ...... 1 1. INTRODUCTION ...... 1 DEFINITION OF SYMBOLS ...... 2 2 . REVIEW OF D.C. AND A.C. IMPEDANCE PREDICTION MODELS ... 5 2.1 D.C. Models ...... 5 2.1.1 Linear Models ...... 5 2.1.2 Nonlinear Models ...... 9 2.2 Acoustic Models ...... 10 3 . STEADY-STATE FLOW RESISTANCE TESTS ...... 17 3.1 Experimental Apparatus ...... 17 3.2 Test Results ...... 18 3.3 Derivation of Semi-Empirical Prediction Model .... 19 4 . ACOUSTIC IMPEDANCE TESTS ...... 22 4.1 Two-Microphone Method and Test Set-up...... 22 4.2 Test Results ...... 24 4.3 Derivation of Non-Grazing Flow Impedance Predic- tion Model ...... 26 5 . CONCLUSIONS ...... 29 5.1 Steady-State Flow Resistance ...... 30 5.2 Acoustic Impedance ...... 30 REFERENCES ...... 32 TABLES ...... 34 FIGURES ...... 40 APPENDIX A . Correction to Measured Incident Sound Pressure Level Due to Acoustic Radiation From the Test Specimen ...... 72
iii SUMMARY The results of an investigation of the effects of grazing flow on the steady-state flowresistance and acoustic impedance of seven Feltmetal and three Rigimesh thin porous-faced liners are presented. A state-of-the-art reviewof previous nongrazing flow studies is also presented. The steady-state flow resistance of the ten specimens were measured using standard fluid mechanical experimental techniques. The acoustic impedance was measured us- ing the two-microphone method. The principal findings of the study are (1) the effects of grazing flow were measured and found to be small, (2) small differences were measured between steady- state and acoustic resistance, (3) a semi-empirical model'was derived that correlated the steady-state resistance data of the seven Feltmetal liners and the face-sheet reactance of both the Feltmetal and Rigimesh liners. These findings suggest that nongrazing flow steady-state flow resistance tests could provide considerable insight into the acoustic behavior of porous liners in a grazing flow environment. 1. INTRODUCTION Sound absorbent treatments consisting of cavity-backed por- ous faced liners are currently being used to control internally generated machinery noise. For application to the inlet and ex- haust ducting of jet engines, the efficient design of the liner should incorporate the effects of both high grazing flow veloci- ties and intense sound pressure levels. Unfortunately, there have been few investigations of the effects of grazing flow and sound pressure on the relationship between the liner impedance and its construction (i.e., sheet thickness, porosity, fiber diameter, etc). The purpose of this report is to describe the results of a fundamental investigation of the effects of grazing flow and sound pressure on the steady-state flow resistance (here- in referred to for brevity as d.c. resistance) and the specific acoustic impedance (i.e., resistance and reactance) of ten cavity- backed porous-faced specimens. The acoustic measurements were made using the two-microphone method. This method was used by Hersh and Walker' to study the effects of grazing flow and sound pressure level on the'impedance of isolated orifices. Feder and Dean2 studied experimentally the effects of graz- ing flow and sound pressure on the d.c. resistance and acoustic impedance of a variety of acoustic liner configurations including porous-faced liners. The acoustic measurements were conducted using an impedance tube. They concluded that the d.c. and a.c. resistance increased and the reactance decreased with increasing grazing flow velocity and sound pressure level. The correlation between the d.c. and a.c. resistance for three porous-faced mat- erials at a fixed grazing flow speed and sound pressure level was found to be generally poor; the a.c. resistance values were con- sistantly higher (e.g., an a.c. value of 600rayls vs. a d.c. value of about 37ORayls for a Feltmetal specimen) than the correspond- ing d.c. values. Feder and Dean did not interpret their results physically. P. Dean3 used the two-microphone method to measure the effects of grazing flow on the impedance of cavity-backed per- forates and porous faced liners. His results were in contrast with Feder and Dean’s results in that the acoustic resistance of some materials did not change while other decreased with increas- ing grazing flow; the reactance increased with increasing grazing flow. Dean offered no explanation. Plumblee, et. a14 also in- vestigated experimentally the effects of grazing flow on porous- face materials using the two-microphone method. Since the graz- ing flow speeds never exceeded 15 m/sec., they found no signifi- cant change in impedance. None of these studies investigated the nonlinear behavior of porous-faced materials. The present understanding of the connection between porous material properties and a.c. and d.c. impedance for nongrazing flow is reviewed in Section 2. The results of an experimental and analytical investigation of the d.c. resistance and the a.c. impedance of ten porous-faced liners are described in Section 3 and 4 respectively. The major findings of the report are summar- ized in Section 5. DEFINITION OF SYMBOLS Symbol Definition A cross-section area, (meter2) a average spacing between fissures (meters) ; also constant b average slit height of fissure (meters); also constant C speed of sound (meters/second) Cf skin friction coefficient CO numerical coefficient (see Eq. 9) CD drag coefficient D diameter of porous specimen (meters); also drag (Newtons) f frequency (Hz) g gravitational constant (meters/seconds2) H height of duct containing specimen (meters) h complex wave number within porous material (meter<’) 2 Symbol Definition k material permeability constant (meters2) K material hydraulic conductivity constant (meter/se,cond) ; also compressibility mod- ulus (= - ap'/2pf-1/Newtons/meters2) Po L thickness of porous specimen (meters); also length scale characterizing interac- tion between cylinders (meters); also depth of cavity (meters) Lch length scale characterizing thickness of unsteady viscous layer (meters) m material structure factor (ratio of effec- tive density of the air in the pore space to the actual density po of the air) M material specific internal area (meters') N number of fibers or capillaries PS static pressure upstream of specimen (Newtons/meter2) static pressure in test section (Newtons/ meter') PT total pressure in test section (Newtons/ meter2) P material porosity P steady-state pressure (Newtons/meters2) P' acoustic pressure (Newtons/meters2) AP pressure drop across specimen (N/m2) Q volume flow, (meters3/second) R average hydraulic radius (meters) Rac acoustic resistance (Rayls-mks) Rdc steady-state flow resistance (Rayls-mks) Re Reynolds number r radial distance '(meters) S spacing between cylinders (meters) t time(seconds) T material tortuosity U longitudinal component of velocity (meters/ second) vm test section speed (meters/second) v volume of cavity (meters3)
3 Symbo 1 Definition VD average flow speed incident to porous mat- erial (meters/second) acoustic velocity (meters/sec) reactance (Rayls-mks) axial and/or longitudinal coordinate (meters) normal and/or lateral coordinate (meters) width of test section containing specimen (meters) ; characteristic (complex) impedance of porous material (Rayls-mks) constant defined by Eq. (46) constant; also attenuation constant (metergl) probability distribution function constant; also phase constant(degrees/meters ) average cylinder diameter (meters); also average pore diameter (meters) r parameter defined by Eq. (26) P, P’ steady-state and acoustic density (kilograms/ meters3) Y velocity potential (meters2/second) Y ratio of specific heat of air T material thickness (meter) wall shear stress (Newtons/meter2) sound wavelength (meters) phase change across specimen (degrees) fluid coefficient of viscosity (kilograms/ second/meter) V fluid kinematic viscosity (meters2/second) w radian frequency (Hz) IC effective compressibility of the air within the porous material (Newtons/meters2) Subscripts
( 10 denotes open quantity ( )c denotes closed quantity inc incident cav cavity S along stream tube direction t face-sheet 4 Superscripts Definition
( 1' denotes acoustic quantity 2. REVIEW OF D.C. AND A.C. IMPEDANCE PREDICTION MODELS In contrast to the sparse number of investigations of the effects' of grazing flow on the d.c. and a.c. impedance of porous- faced materials, there have been a large number of investigations of their nongrazing flow behavior. A review of the more impor- tant papers is given below to provide background information necessary to place in perspective the results of the present in- vestigation. The d.c. models are described in Section 2.1 followed by the a.c. models in Section 2.2. 2.1D.C. Models The review of the d.c. resistance models is divided into linear and nonlinear parts. The linear models represent varia- tions and/or refinements of Darcy's Law'. Darcy was the first to establish a linear connection between the deriving pressure gradient Ap across a porous specimen and the average flow speed VD approaching the specimen. For sufficiently high average speeds, the connection between Ap and VD is nonlinear due to a quadratic relation between Ap and VD. Most of the details of the various models described below are based on the excellent review by Bear6. 2.1.1 Linear Models In 1856 Darcy investigated the flow of water in vertical homogeneous sand filters in connection with the fountains of the city of Dijon, France. From his experiments, he concluded that the volume rate of flow Q through. thefilter was proportional to the product of the cross-sectional area of the filter A and the pressure difference across i.t (Ap) and inversely proportional to the filter length (L) leading to the relationship
where the constant K is called the material hydraulic conductiv- ity, p is the density of the water and g is the gravitational constant. Later researchers separated the influence of the ma- terial from that of the fluid by defining K = pgk/p where p is the viscosity of the fluid and k is the specific permeability of the porous material. Substituting this expression for K into Eq. (1) yields the modern form of Darcy's Law,
5 I
where VD is the average flow speed approaching the porous mater- ial. A particularly attractive and straight-forward application of Darcy's Law is to model a porous material as a collection of straight capillary tubes. Applying Hagen-Poisseuille's law to the flow through a capillary tube of radius 6, Scheidegger' de- rived the following connection between VD and Ap,
Comparing Eqs. (2) and (3), the permeability constant k = 62/32 for a capillary tube. If there are N tubes per unit cross- section of the model (normal to the direction of flow) then the porosity P = N(.rrS2/4) and the filter speed through the porous material is
What is important in Eq. (4) is that the filter speed is propor- tional to the product of the square of the average tube diameter 62 and the material porosity P; the coefficient 1/32 is obviously unimportant. Scheidegger generalized the capillary model described above by replacing the porous material with a distribution of capillar- ies of different diameters. He introduced the quantity a(6)dB to represent the fraction of capillaries having radii between 6 and 6+d6. Assuming that the volume occupied by these capillaries parallel, say, to the Xi direction, is 1/3PAxia(b)d6 per unit cross-section, the average filter speed VD is Xi
where VD (6) is the average speed in the capillaries of diameter 6 given 6$ Eq. (3). Comparing Eqs. (5) and (2) yields for the permeability constant
6 Scheidegger also generalized the above model by permitting -the capillary tubes to be tortuous (i.e., non-straight) and of var- iable diameter. He proposed the following expression for k,
Here T is the "tortuosity" factor defined as the ratio of the length of the actual flow to the thickness of the porous material. Irmay' applied the capillary model to predict the resistance of narrow fissures (i.e., slits of constant width), a fractured rock being the closest application of his approach. Irmay derives the following expression Eor the permeability,
k = a' P3/12 (1 -P)'=Pb2/12 where b is the average slit height of the fissure and a is the average spacing between fissures. Here the material porosity P = b/(a+b). In 1927, Kozenygapplied the concept of hydraulic radius R to connect the permeability constant k to the material properties. Defining R as the ratio of volume of a conduit to its wetted sur- face area, he derived R = P/M, where M is the specific surface area of the porous material. Using this definition, R may be interpreted as the average hydraulic radius of quite complicated flow channels. By solving the Navier-Stokes equations for all channels passing through a cross-section normal to the flow, Kozeny derived the following expression for the material permea- bility constant k,
where co is a numerical coefficient called Kozeny's constant (co = 0.5, 0.562, 0.597, 0.667 for a circle, square, equilateral trlangle, and rectangle respectively). Recently theoretical estimates of the permeability constant have been derived using a drag model. The idea here is that the pressure drop through a material is related to internal shear (tangential) and normal (pressure) stresses by assuming that a porous material can be modelled as a collection of closely packed 7 spheres.For example, Rumer andDrinker" and later Rumerll derivedthe following expression for the material permeability constant,
where d is theporous material average diameter, X is a coeffi- cientthat takes into account the effects of neighboring spheres (for a singlesphere in an infinite fluid X = 37r) and B is a shape factor (6 = 7~/6for a sphere). Iberall12assumed that highly porous materials could be modelledas a collection of randomlyorientated circular cylindri- calfibers of the same diameter. He thenderived an expression forthe material permeability byassuming thatthe pressure drop necessaryto overcome the viscous drag is linearlyadditive for all ofthe fibers. The ideahere is thatthe average speed is sufficiently low, thepressure drop across the material is equal tothe viscous drag of thefibers. Since Iberall ignored inter- actionsbetween fibers, his analysis is restricted to largese- parationdistances between fibers and hence to high porosity materials. Iberall assumed thatgiven n fibersof unit length per unitvolume, n/3 of them will beorientated perpendicular to the localupstream velocity VD. UsingLamb's13 calculationof the dragper unit length for a cylinderperpendicular to a stream, Iberallderived the following expression for thepressure drop across a specimen
(11) where 6 is theaverage radius of the fibers. ComparingEqs. (11) and(21) the permeability k is
k= 3Pd2 . 2- Ln (Re) (12) 16 (1 - P) 4 - ln (Re)
Equation(12) is unusualbecause it shows thatthe material per- meability is notonly a function of the material properties but is also a functionof the fluidinertia through the Reynolds number.Scheidegger7 suggests that the drag theory may represent a generalization of Darcy's Law.
8
I 2.1.2 NonlinearModels
Forthe linear relationship between Ap and V to bevalid, theReynolds number characterizingthe flow througg the porous material mustbe extremely small. As usedhere, the Reynolds number is definedas Re = VDG/PV where VD/P is theaverage velocity inthe material and 6 theaverage pore diameter of thematerial. The Reynoldsnumber can be interpreted as a measureof the rela- tiveimportance between inertia and viscous forces. To show this,consider flow over an isolated cylinder of radius 6. By nondimensionalizingall length changes by the cylinder diameter 6 and all velocity changes by the Zoea2 upstreamvelocity VD/P, it followsthat the order-of-magnitude of a typicalnonlinear termpuau/ax is (p/6) (VD/P)~and that the order-of-ma nitude of a typical shear stress term is pa2u/ay2 - pvD3/VP Q . The ratio of thenonlinear inertial to viscousterms is VD&/PV=Re.
It is clearthat the flow within a porousmaterial cannot beaccurately modelled by a collectionof straight, constant area, capillarytubes. A more realistic modelwould have toincorpor- ateboth twisting fully three-dimensional motion with the fluid acceleratingand decelerating within the material. Recognizing thisScheidegger believes that the onset of nonlinearity first occurs due to Zaminar flowturning and fluidaccelerations and decelerations. At higherspeeds, nonlinear losses arise due to turbuZenee generatedwithin the porous material. According to thereview by Bear, initial deviations from Darcy's linear law occurfor Reynolds number as low as Re = 0.1. The deviations increasefor increasing Reynolds number.These deviations are believed to arise fromthe onset of laminar nonlinearity as dis- cussedabove. The onsetof turbulence generated nonlinearity doesnot occur until Reynolds numbers between 60 to 150. In general,the following relationship between Ap and VD character- izethe d.c. resistance,
Here a is proportionalto 1-1 andindependent of the fluid density p and b is proportionalto p andindependent of 1-1. Blick14analyzed the nonlinear behavior of porous materials in a ratherinteresting way. His model consistedof a bundle of capillarytubes with orifice plates spaced along each tube, sep- arated by a distanceequal to the mean porediameter. Assuming thefluid is homogeneous,Newtonian and one-dimensional, Blick derivesthe following expression relating p and VD.
9 where cf is the skin friction coefficient of the flow along the tube walls of constant diameter 6, Re=VD6/vP is the diameter based Reynolds number and CD is the drag coefficient of the ori- fice plate. Blick’s model 1s important because it shows quali- tatively the nature and form of nonlinear pressure losses. For a detailed discussion of other models, the reader is referred to the review by Bear. This review of flow nonlinearity concludes by quoting from Bear’s review. “Cheaveteauand Thirri~t~~performexperiments in severaZtwo-dimensionaZ modeZs that permit visua2 ovservations of streamZinesand streakzines. In a typicaZmodel, which has the shapeof a conduitthat diverges and converges aZternateZy, they observestreakzines. For Re<2, the fZow obeysDarcy’s law and thestreamZines remain fixed. As Re increases,streamZines start to shift and fixededdies begin to appear in thediverging areas ofthe modeZ.They become Zarger as Re increases. At Re=75 turbuZenceappears and starts to spread out as Re increases. TurbuZencecovers some 50% of thefZow domain at Re=llS and 100% of it at Re=180.The deviationfrom Darcy’s Zaw is observedat Re=2-3.Thus the deviation from Darcy’s law as thevelocity in- creases is associatedwith gradual shifting of streamZinesdue tothe curvature of the microscopic soZid waZZs of thepore space. I’ 2.2 Acoustic Models Despite widespread application of porous materials as sound absorbers, surprisingly little fundamental research has been con- ducted in the past twenty years to understand their acoustic be- havior. Most of the recent research has been applications orien- tated in the sense that existing parameters used to define the important physical properties, namely structure factor, bulk mod- ulus of compressibility and dynamic flow resistance are measured or assumed known. The results of three investigations are re- viewed below representing the pioneering work of Scott16, Zwikker and Kosten17 and Morse1*. These three are believed to adequately summarize the state-of-the-art. None of the investigators consid- ered the acoustic nonlinear behavior of porous materials. Scott was one of the first to be concerned with the propa- gation of acoustic disturbances in porous materials. He assumed the porous material to be homogeneous, isotropic and sufficiently large in extent so that reflected waves were negligible. He 10 further assumed that the porous material consisted of very many small fibers or particles packed together so as to leave between them interconnected air-spaces or pores of irregular shape. De- fined in this way, the linearized equations of motion for sound waves propagating through the pore space are obtained. The lin- earized conservation of momentum equation becomes
where m is the ratio of the effective density of the air in the pore space to the actual density po of the air (called the struc- ture factor by other 'investigators), p' is the acoustic pressure, Ra.c. is a dynamic resistance constant analogous to the d.c. resistance used in Darcy's law, P is the material porosity and VD is the vector velocity approaching the porous material. The corresponding linearized mass conservation equation is given by
By introducing the acoustic velocity potent'ial vD=grad I$, Scott derives the following linearized wave equation
with
The quantity h is the complex propagation constant within the mat- erial and K is the effective compressibility of the air in the pore spaces. By introducing the complex density p' defined as
p'r -h2 K PW2
11 Scottderived the wave equationin a formsimilar to the wave equationin free space. The remainderof his paper describes theuse of a probetube to measure the real andimaginary parts ofthe propagation constant h. Writing h=D-ia, the decay of intensity with distance provides a measurementof the attenua- tionconstant a. The correspondingchange in phase of the acousticpressure with distance provides a measuremntof the wavelength,and therefore of 6. With h specified,the complex density p' followsimmediately from Eq. (19). Scottcompares his theory with measurements of a variety of rock-woolporous samples of different thicknesses. At f=200 Hz, heconcludes that for a rock woolsample, the effec- tivedensity of the air trapped within the pores was roughly twice its freeair value; it decreased to about 1.32 at f=4000 Hz. His measurementsalso showed thatat f=200 Hz, the dynamic resistance was equal to its d.c.value. The speed of soundvaried from co/y at low frequencies to co atthe higher frequencies,thus indicating that lowfrequency sound propagat- esisothermally and highfrequency adiabatically. Scott's paper is obviouslyimportant. His desire to recastthe sound propaga- tionequations in a formequivalent to that in free space is understandable.Unfortunately, in the process of forcing the analogy,he as well as Zwikkerand Kosten may haveinduced work- ersin the field into believing that the effective mean density ofsound waves propagating in porous materials is a functionof thestructure factor my porosity P, flowresistance Rdec, and frequency w. Inreality the mean density of theair is constant sincethe air can be considered incompressible. While my P, r and w affectthe sound pressure, density and velocity, they do notaffect mean density.This is most evidentat low frequencies where thea.c. resistance is equalto the d.c. flow resistance. At the low particlespeeds, generated by thesound waves, the incidentsound driving pressure cannot increase the air-density - clearlythe process is incompressible. Thus interpretations otherthan increases in the fluid mean densityare required.
In 1949, Zwikkerand Kosten published their famous book entitled, Sound AbsorbingMaterials, which dealt in part with the connectionbetween the quantities h and p' andthe material struc- turalproperties. Zwikkerand Kosten (herein referred to Z 6 K) introducedthe following assumed solution describing the propaga- tionof a planesound wave in a homogeneous isotropic medium ex- tending to infinity,
-ax+i (ut-ax) p' (x, t)=Ae
12 where a is theattenuation constant of thesound wave, B=w/c is thecorresponding phase constant (c is thephase velocity of the sound wave inthe material) and A is anarbitrary amplitude. A secondconstant is defined,called W, thecharacteristic (complex) impedance of the infinite extent material, W=p' (x,t)/vx' (x,t) . Herevx'(x,t) is thesound particle speed in the x - direction withinthe material. W is definedas -where K=(ap'/apt)/po is related to the complexspeed of sound inthe material. W is com- plex due to internalmaterial dampingof thepropagating sound wave; it denotesthat acoustic density variations within the mat- erial are not in phase with the driving pressure variation. Thereare two limitingcases in which p' and pt vibratein phase withother. In the first limit, thesound frequency is suffi- ciently low that the transmission of heat to and from the solid frame is rapidenough so thatthe enclosed air is keptat constant temperature(isothermal case). At highfrequencies, the trans- missionof heat is so slow that the air vibrates adiabatically. Between these limits hysteresis,occurring between the instantan- eousheating and cooling of the air bythe sound field and the subsequenttransfer of heat to and from the material, results in a frequencydependent complex propagation constant. The continuityand momentum conservationequations govern- ingthe motion of the soundwaves are, respectively,
where P is thematerial porosity, Rac thematerial a.c. resistance and m thestructure constant. For low frequencyoscillations (-<<1) Ra.c.ERd.c. The quantityRdac. follows fromPoise- uille's lawgoverning the behavior of steady-stateflow in capil- larytubes, namely
where 6 is theaverage material pore radius. For highfrequency oscillations (->>I) , -
13 !
- . -.- , " Z 6 K offerthe following interpretation of the term m/P thatmultiplies the density po ofthe free air in the momentum equation. They splitthe factor into two parts, m and 1/P. The quantity m is thoughtto consist of two parts,one of which
is relatedto the internal friction of the air, theother to the I orientationof the material pores. A thirdpart also exists, the inertia ofthe structural material, but Z 6 K feel that in mostcases this effect is small. Of the various contributors to m thepore orientation is themost important. Z 6 K show that it containsthe factor (l/cos20).For randomly orientated pores m-3. Accordingto ex- periments, m hasvalues lying between 3 and 7. By introducing theapparent (complex) density defined as
theequations of motion for a traveling wave is broughtinto the universalform,
where
Zwikkerand Kosten’s book is anaccurate state-of-the-art summary ofthe acoustical understanding of porous materials through 1949. FollowingScott, the authors chose to castthe form ofthe equations of motion into a universalform - i.e.,similar in form to the equations of motiondescribing the propagation of sound in air. Thus theirinterpretations suffer the same limita- tionsas those of Scott. Morse extendsthe approach adopted by Zwikker 6 Kosten (firstglresented by Rayleigh)” and by Scott, Morseand Bolt2’ , Beranek 6 othersto model thepropagation of sound in granular media. He assumesthat all of thegrains have roughly the same diameter. He derivesthe following equations governing the sound motion in the granular material, 14
.. . Here z1 is volume of fluidper unit area in direction pressure gradient p',po arethe acoustic and equilibrium fluid density respective- 1Y p' is theexcess instantaneous pressure m is thestructure factor (explained below) P is thematerial porosity Rdc is thedynamic flow resistance Morse definesthe structure factor m asarising because of the way in whichthe solid material constrains the motion of the fluid. Not allthe fluid isain the direction of thepressure gradient VP'. To explainthis, consider small tubes inclined at anangle 0 to thepressure gradient,
Inthe S-direction (along the tube), the momentum equation is
Now us'cosO=uxland scos~=x~~/~s=cose~/ax.Thus substitution yields
15 Averagingthe tubes over all possible angles assuming that all thetubes are orientated uniformly in direction, the probability that a giventube lies between the spherical angles e, e+deand $, @+d$is unity,then
Thus, m=3. Morse doesnot allow for inertial coupling between the motion ofthe porous material and its effecton the air. Assuming a plane wave solution of the formp' (x,t)-edmt-kx) Morseshows that k=(m/co) Jm-iRdcP/Pow. Assuming furtherthat RdcP/Poc Thus theabsorption (a) andsound propagation speed (c) withinthe material follows immediately to be a = Rd.C. P / 2 p, CO2 G- / Clearly,the structure factor m(m>l) asdefined above results in a reducedsound propagation speed. Obviously m includesheat transfereffects (isothermal, adiabatic) not explicitly discussed by Morse. Morse proceedsto connect the dynamic resistance Rdc to the known acousticbeha\ior in small tubes. At low frequencieswhere theacoustic Reynolds number Ra=-< Table I summarizesthe manufacturers' predicted acoustic properties.Specimens #1-7 were made fromFeltmetal fiber metals andspecimens #8-10 fromRigimesh stainless steel wire cloth. Photographsenlarged to illustratethe detailed structure of typicalFeltmetal and Rigimesh specimens are shown inFig. 1. 3.1 Experimental~~ Apparatus The effectsof grazing flow were studied experimentally in the HAE wind tunnel shown inFigure 2. The testsection, made from .0127 metersthick transparent acrylic sheet, has a remov- ableside-wall containing the test specimen. The poroustest material was backedby a cylindricaltube of diameter .0508 meters; it was securedand flush mounted to the test section side wall bymeans of threeset screws. Flow throughthe side branch was controlled bymeans of blowersused in the suction mode (i.e.,from the test sectioninto the side branch) or blowing mode (i.e. , fromthe side branch into the test section). The sidebranch volumeflow rate was monitoredby means of sharp- edgeorifices. By accuratelymeasuring the pressure drop across the orifices with Charles Merriam Company calibrated slant mano- meters, volumeflow rates as low as 8.5 x lG5 cubic meters per second were recorded. Testsection speeds up to 91.44 meters/secondwere measured. The velocityprofile for Vm=85.6 meters/second, is shown inFigure 3. As shown, theboundary layer is turbulent - measurements 17 showed it to beturbulent for all test speeds considered. For distances less than.00051 meters from the tunnel wall, the data indicatesthat the flow is separateddue to interaction between theboundary layer probe and wall. The datamatches closely the classical1/7th velocity law profilefor .00051 meters The pressuredrop was measureddirectly from a calibratedCharles Merriam slant manometer. The averagespeed was calculatedusing standard ASME techniques23from the pressure drop measurements acrossthe orifices (orifice diameters of .00635 meters, 0127 metersand .01905 meterswereused to extendthe useful range of VD from a minimum of .04 to a maximum of 10 meters/second). 3.2Test Results Figures 4-9 summarize theeffects of grazing flow on the d.c.flow resistance of the ten specimens operating in both the suction andblowing mode. Inagreement with the discussion pre- ceeding Eq. (13),the data can be accurately presented by the empiricalcurve fit The variousmeasured values of a and b aretabulated in Table 11. Inspectionof Table I1 shows that,in general, there are only smalldifferences between the blowing and suction data. Further, theeffects of grazing flow arealso shown to besmall. Most of thedifferences are of the order of 20% orless. These differ- encesare considered small relative to theeffects of grazing flow on thed.c. resistances of orificesas measured by Federand Dean. The measurednegligible differences in flow resistance for thespecimens operating in the blowing and suction modes indicate thatthe material permeability is isotropicin the axial direction. Flow nonlinearity is shown to become importantfor VD'Z meters/ secondfor specimens 2, 3, 4, 8, 9 and 10 forV~>.30 meters/second forspecimens 1, 5, 6 and 7 (recallfrom Table I thatthese specimens should be equivalent). This is reflected in the various values of the coefficient b summarized in Table'II. Since b is a measure of the importance of flow nonlinearity, following the arguments of Scheidigger, it is a function of the area changes associated with the various pore diameters. The idea here is that relatively large pressure gradients are re- quired to balance the relatively large convective velocity gradients which arise from abrupt changes in internal pore diameter cross-sectional areas. Specimen 1, 5, 6 and 7 should, in principle, exhibit roughly the same steady-state flow resistance behavior. Clear- ly , the data characterizing specimens 5 and 6 are reasonably close - the coefficients a and b are within 20%. The corres- ponding coefficients characterizing specimens 1 and 7, however, are quite different. These differences are probably associated with the variability inherent with their manufacture. The de- tails of this variability are not well understood nor documented. The negligible changes in flow resistance with grazing flow require special comment. A small positive pressure drop APs=Ps-Pm induced across the specimen by the grazing flow was observed. The flow control valves across the side branch were closed so that there was zero mean flow across the specimen. According to the investigations by first Beavers and Joseph24 and later Taylor25, the observed pressure drop arises because of a local recirculating (vortical) flow induced across the speci- men. A sketch showing the vortical flow pattern is shown in Fig. 10. The recirculating flow is driven and maintained by steady-state shear stresses transferred across the porous sur- face. Figure 11 shows the effect of correcting the data by subtracting this pressure drop from the measured blowing data (adding for the suction data) for the data of Specimen #4. With- out this correction, the flow resistance would increase without limit as VD+O as indicated in Fig. 11 (in the suction mode, it would decrease without limit). Table 111 summarizes the measured pressure drops APs in meters of water across the ten specimens at a grazing flow speed of Vcp=79.6 meters/second. The measure- ments showed that (APs) is linearly proportional to the test sec- tion head (PT-PJ where PT is the test section total pressure and Pm is the test section static pressure. 3.3 Derivation of Semi-EmDirical Prediction Model A simplified prediction model of the nongrazing linear d.c. resistance of thin porous-faced liners is derived. The intent here is to correlate the linear d.c. resistance data. It is under- stood because of the variations inherent in the manufacturing pro- cess that only an approximate correlation is possible. The photo- graphs of the Feltmetal specimens,shown in Figure 1, suggest that 19 they can be modelled as a collection of small diameter cylinders. A model of the Rigimesh specimens w.il1 not bederived because of the very limited number of specimens supplied by the manufac- turer. To simplify the model, the following assumptions are made. The cylinders have mean diameter 6. They are aligned perpendicular to the incident flow VD as shown in Fig. 1. The specimen has zero thickness, that is, all the cylinders are as- sumed to lie on a plane surface - this treats the specimen sur- face as a momentum sink in a manner similar to that proposed by Zorumski and Parrott. The cylinders are separated by an average center-to-center spacing S as shown schematically in Figure 1. Here the specimen is placed in a duct of width W and height H with incident flowVDas shown. The parameters H, N and 6 are related as follows H N NS for N >>1 (33) Consistent with the liner zero thickness assumption, the porosity P is defined as the ratio of the open area A, to the total area AT, Substituting Eq. (33) into Eq. (34) and assuming N large yields the following approximate estimate of the porosity, P N 1 - d/s (35) The d.c. resistance is calculated by assuming that the cy- linders act as a momentum sink. From momentum balance considera- tions, the following relationship exists between the pressure drop Ap across the specimen and the drag force due to the cylin- ders, Ap. HW = ND (36) where N is the total number of cylinders (of width W) and D is the drag force on each cylinder. Now assume for very slow upstream average speeds VD, that the cylinder drag is proportional to the 20 skin friction shear stresses T~ where L J Here (t/P) is the local pore velocity incident to the cylinder and Lch is a measure of the thickness of the viscous shear layer normal to the cylinder surface. For very low Reynolds number flows, the characteristic length Lch for an isolated cylinder is the diameter 6. For collections of closely spaced cylinders, how- ever, Lch must be a function of the separation parameter S. Thus for S/6>>l, Lch=6and for S/6<1, Lch=f(S/6). Here S/6 may be thought of as an interaction parameter. The following simplified empirical relationship is suggested, For S/6>>1, Lch'6. For S/6<<1, Lch=aS where a is a constant that must be determined from experimental data. Hopefully, a will be independent of specimen geometry. Combining Eq. (37) and Eq. (38) and substituting into Eq. (35) yields Substituting Eqs. (33) and (34) into Eq. (39) and solving for the ratio AP/VD yields The following form of the d.c. flow resistance model is written as where CY and 6 are unknown constants. The results of applying Eq. (41) to determine the constants a and 6 are summarized in the fifth column of Table IV. The best 21 fit tothe data occurred with a=0.8; the average value of @av=1864. The model correlatesthe Feltmetal data quite well over a widerange of porosities (0.45 to 0.97), cylinderdiameters (4~10'~ to10" meters)and d:c. resistances (115 to 890 Rayls-mks). No particularsignificance is attachedto the values of c1 and f.3. The porosityvalues shown inTable IV wereestimated bycomputing the relative densitiesof the specimens. This in- volvedsubtracting out the weight of the screens and weighing thespecimens and measuring their volume. The agreementbetween manufacturersestimated porosities and measured porosities are quiteclose. No attempt was made to model thenonlinear behavior of the specimens. The reasonfor this is thatspecimens #2, 3 and 4 were justbarely entering the nonlinear regime at thehigher test filterspeeds. Basedon the flow visualization tests performed byChauveteau and Thirriot (see Section 2.1), nonlinear devia- tionsfrom Darcy's law wereobserved at values of Reynoldsnumber Re=VD6/Pv between 2 and 3. Usingthese values of Re, the last column inTable IV shows thevalues of filter speedvDcorrespond- ingto Re=2 and 3. Comparing these valueswith the observed values of VD representingthe onset of nonlinearity (the sixth column inTable IV) indicatefairly goodagreement with Chauve- teauand Thirrot's criteria. Accepting Chauveteau and Thirriot's criteria, it is clear thatthe onset of nonlinearity can be ex- tendedto very high average speeds VD byusing porous materials made ofvery small fiber diameters. 4. ACOUSTIC IMPEDANCE TESTS Inthe previous section, the effects of grazingflow on the d.c.resistance of ten porous-faced materials were measured and found to benegligible. In this Section, the two-microphone method is usedto measure the effects of grazing flow,sound pres- surelevel and frequency on the acoustic resistance and reactance of the same tenspecimens. The testapparatus and the two-micro- phonemethod is describedin Section 4.1. The testresults are summarized inSection 4.2. A non-grazingflow semi-empirical impedanceprediction is derivedin Section 4.3. Themodel is usedto correlate the nongrazing test data of Section 4.2. 4.1 Two-MicrophoneMethod and Test Set-up The acousticresistance, total reactance, and face-sheet reactanceof the test specimens were measured using the two mic- rophonemethod described by Dean3. The method requiresthe specimens to becavity-backed. Figure 12 shows a schematic of the test set-upand the instrumentation used. The acoustic resistanceand reactance of cavity-backed specimens can be 22 described by thefollowing expressions SPL (inc)- SPL (cav) sin4 20 1 sin(%) whereSPL(inc)-SPL(cav) represents the sound pressure level dif- ference(in dB) betweenthe incident and cavity sound fields and $I representsthe corresponding phase difference. The quanties w, L and c representthe sound radian frequency, cavity depth and fluid soundspeed respectively. The abovemeasurements are ob- tained by flushmounting one microphone at the cavity base and theother flush with the side walls containing the orifice as shown inFigure 12. It is importantto locate the incident mic- rophonesufficiently far from the specimen to avoid near field effects. The microphoneshould be located sufficiently close, however, so thatthe separation distance is smallrelative to the incidentsound wavelength. This is necessaryto insure accurate measurementsof the incident sound wave amplitudeand phase. A discussionof the errors associated with using the two-microphone method is containedin Appendix A. A schematic of theinstrumentation used to conduct the ex- periments is shown inFigure 13. To generateincident sound pressurelevels up to 150 dB, a JBL type 2480 drivercapable of producingin excess of 10 watts ofrelatively “clean” acoustic power is usedas the sound source. . The .OS08 meterdiameter driverthroat is coupledto the test section bymeans of a .0508 meter to .lo16 metersdiameter exponential expansion, JBL type H-93. Sound pressurelevels in excess of 150 dB exceedthe input capabilityof the GR-1560-P42preamp. A 10 dB microphoneattenu- ator, GR Type 1962-3200has been added, which extends the 23 measurementrange accordingly. The signalgenerated by theHeath 1G-18 audiogenerator is amplified by theMcIntosh MC2100 100 watt/channel power amplifier to power the JBL driver. The audiogenerator provides a tracking signal for the AD-YU SynchronousFilter and phase meter system. The 1036system filters the two microphoneinput signals to the trackingsignal frequency with a bandwidthof 5 Hz. The AD-YU Type 524A4 PhaseMeter reads phase angle between the signals independentof signal amplitudes. The phaseangle output is displayed on the AD-YU Type 2001 digitalvolt meter. A General Radio-1565 1/10 octavefilter together with a Heath Type 1M2202 DVM is usedto record the output signals from eachof the two microphones.Also the two signalsare observed on a TEK 533 Oscilloscopeto visually note approximate phase and distortion effects. The outputof the incident microphone channel of thesyn- chronousfilter is usedas a controlvoltage for an automatic levelcontrol amplifier. This control amplifier adjusts the drive levelto the power amplifierin such a way asto keep the incident levelconstant, independent of frequency and amplitude response irregularitiesin the loudspeaker and tunnel. As a convenience, a triple ganged 5 dB perstep ladder attenuator is usedto simultaneously increase the power amplifier drivelevel and decrease the synchronous filter input signals so thatthe control loop of the automatic level control amplifier alwayshas the same gain.This has the added advantage keeping thelevels at the AD-YU Filterinput constant for alltesting levels.Since the AD-YU Filter displays a small amplitude-phase dependency,this automatic level control improves accuracy as well asspeed of data acquisition. A phasetracking test of both mi- crouhonesmounted flush in the wind tunnel wall indicated accurate phasetracking within +O. 2' over a soundpressure level range of 70-150 dB. 4.2 Test Results Figures(14)-(16) summarize the effects of grazing flow and soundpressure level on theacoustic resistance of specimens #1-10. Grazing fZow is shown toincrease onZy negZigibZythe zero grazing flow resistancedata. Only grazingflow data correspond- ing to incidentsound pressure levels greater than 110-120 dB are presentedto insure an adequate signal-to-noise ratio. In agree- ment withthe steady-state flow resistance data, the zero grazing flowdata show thatnonlinearity effects are important only for theFibermetal FM 134porous material - specimens #1, 5, 6 and 7. Nonlinearityeffects are unimportantfor specimens #2, 3, 4, 8, 9 and 10. 24 The effects of grazing flow and sound pressure level on the acoustic reactance of specimens #1-10 are summarized in Figures (17)-(25). In all cases, the porous face-sheet reac- tance and total system reactance (i.e., face-sheet and cavity reactance) are either unaffected or changed by only small (but measurable-like 10% differences) amounts from their zero graz- ing flow, low sound pressure level values. The frequency at which the reactance measurements were conducted requires special comment. For each specimen tested, the frequency of the inci- dent sound field was selected so that at zero grazing flow and at an incident sound pressure level of 90 dB, the phase shift across the specimen face-sheet was ninety degrees. Under these coditions, the total reactance of each specimen was zero corres- ponding to resonance. Figures (26) -(31) summarize the acoustic response of speci- mens #1, 2 and 7 to variations of frequency. In general, meas- urements show that the acoustic resistance is independent of frequency as shown, for example, in Figure 26. Specimens #2 and 7 show, however, an interesting dip in resistance in the neighbor- hood of 700-750 Hz. The dip occurs regardless of the amplitude of the incident sound or the speed of the grazing flow. Impact measurements showed that mechanical resonance corresponding to the first resonant clamped mode occurred at 1550 Hz for specimen #1, 730 Hz for specimen #2, and 360 Hz for specimen #7. This accounts for the observed resistance dips for specimens #2 and 7- it corresponds to the excitation of the simple-harmonic fundamen- tal mechanical resonance of the porous face-sheet material of specimen #2 and possibly to a higher harmonic of specimen #7. The dip in resistance physically corresponds to the sound par- ticle field converting less mechanical energy into heat within the porous material. The corresponding variation with frequency of the face- sheet and total reactance of specimens #1, 2 and 7 are shown in Figures (29) thru (31). Figure 29 shows that both the face-sheet and total reactance of specimen #1 to increase almost linearly with frequency. Figures 30 and 31, however, show the face-sheet reactance of specimens #2 and 7 to increase abruptly at the fre- quency corresponding to mechanical resonance. The total reac- tance of these specimens were only slightly perturbed at mechani- cal resonance. This is more clearly shown in Figure 32 which shows the sound pressure amplitude ratio across the specimen to increase significantly for specimen #2 with virtually no increases for specimen #1 and 7. Figure 32 also shows that specimen #2 experiences a large phase change between 700-750 Hz; however, specimens #1 and 7 do not. To support the nongrazing flow two-microphone test results, the impedance of the ten porous samples were measured in a 1.83 meter long, .OS08 meter diameter impedance tube. Since the 25 impedancemethod is well known it will notbe described herein. The impedancetube test dataare comparedwith the nongrazing flowd.c. resistance and two-microphone impedance test data in Table V. Here thetwo-microphone data are corrected for acoustic radiationfrom the specimens using the method described in Appin- dix A. Comparison of thed.c. and a.c. resistance is generally quitegood. To first order,d.c. and a.c. resistances are equal. Comparisonof the reactance data between the two-microphone and impedancetube measurement methods is onlyfair. It is extremely difficultto measure accurately using an impedance tube the reac- tanceof a specimenwhose resistance is closeto PC. Thus the two-microphonemethod of measuring reactance is believed to be superiorto the impedance tube method. 4.3Derivation of Non-Grazing Flow Impedance~- Prediction Model A- prediction model is derivedof the impedance of thin porous-facedcavity-backed liners. Since the liner thickness is verysmall compared to the sound wavelength, the model will be derivedusing the concepts of lumpedelements (i.e. , theslug- massmodel). This method was used by Rayleighl'inderiving the impedanceof the Helmholtz resonator. Rayleigh's approach is non-fluidmechanical but gives the actual acoustic impedance characteristicsfor lowsound pressure levels (i.e., the "linear" impedanceregime) when anempirical end correction is added to theslug mass. Rayleighderived the following expression for the impedance of a singleorifice of diameter d andthickness T backedby a cavityof length L, orif ice orificeinertiaicecavity stiffness orif resistance reactance reactance reactance resistance where P is definedas the ratio of orificearea to cavitycross- sectionalarea. The impedancedefined byEq. (44) hasbeen re- ferencedto the cavity cross-sectional area. The impedancede- fined byEq. (44)can be used to model the impedance of the porousmaterials by replacing the orifice a.c. resistance with thed.c. resistance of Eq. (41)with cc=O.8 and f3=1865 (the datadescribed in Section 4.2 showed this to be a good approxi- mation)and by rewriting d=@D (forthe cylindrical cavity shown inFigure 12, P=(d/D)2)leads to thefollowing estimate, 26 ". Here theconstant 0.85 in Eq. (44) is replacedwith an unknown (hopefully)constant E tobe determined from experimentaldata. The variousvalues of E computedusing Eq. (45) andthe two- microphonemeasured values of face-sheet reactance are summar- izedin Table VI. With theexception of specimens #5, 6 and 7, E is seento be constant with average value ~~~'0.52indepen- dentof specimen properties for boththe Feltmetal and Rigimesh specimens. The lack of agreementbetween specimens #5, 6 and 7 and theremaining specimens cannot be explained. The excellent agreement among theremaining specimens is encouragingand suggeststhat specimens #5, 6 and 7 (whichshould be equivalent to specimen#1) may beimproperly fabricated by themanufacturer. The finalexpression for the face-sheet reactance, valid forall ten specimens, is The substitutionfor the Feltmetal specimens of the d.c. forthe a.c. resistance is validonly for low frequencies. A physicalexplanation for the low frequencyrestriction is des- cribedbelow. The steadystate model describedin Section 3.3 will now beextended to unsteady flows. What is required is a reasonable estimateof the fluctuating drag of a cylinder immersed in a soundpressure field. It is assumed that most of theforces exerted by anoscillating sound field on a cylinderoccur in the formof fluctuating wall shear stresses T~.Thus, the drag force perunit cylinder length is balanced by a shearforce per unit cylinderlength given by Here 1-1 is thefluid coefficient of viscosity, 6 is thecylinder diameter, Lch is theextent to whichthe unsteady shear stresses diffuselaterally from thecylinder surface and vxf is theampli- tudeof the fluctuating sound particle velocity. An order-of-magnitudeestimate of the high frequency solu- tionto Eq. (47) is describedbelow. The estimatestarts with theboundary-layer approximations to the momentum equationfor anunsteady flow past a stationary cylinder, 27 The boundary-layer assumptions require that the lateral region of unsteady diffusion of vorticity be small w.r.t the cylinder radius. Physically this requires high frequency sound oscilla- tions. Linearizing Eq. (48), and solving for the pressure grad- ient yields a PI av I a2U; -=-P x_ + P - (49) dX at dY Far from the surface of the cylinder (see sketch), that is for sufficiently large values of y, the particle velocity must satis- fy the following boundary conditions, where v' is the amplitude of the sound particle velocity field. The corresponding sound pressure field becomes, Since the pressure is transmitted through the boundary-layer with- out change (this is one of the major benefits of boundary-layer theory), Eq. (49) may be rewritten as The solution to Eq. (52) that satisfied the boundary condition given by Eq. (50) and the no-slip wall boundary conditions, vx' = 0 at y=O, is 28 (53) An estimateof the fluctuating drag Dl followsfrom Eq. (53) upon substitutioninto Eq. (47) toyield ComparingEq. (54) and(47), the characteristic thickness Lch of thefluctuating shear stress is For Eq. (55) to be valid,the characteristic length Lch< Substitutingthe maximum values of B=10-4meter(Specimens #1, 5, 6, 7) fromTable IV into Eq. (56), f mustexceed the minimum value(for the ten specimens) of f>>240 Hz. Sincethe frequencies at whichspecimens #1, 5, 6 and 7 were testednever exceeded f=1200 Hz, theeffects of frequency should be small in accord with themeasurements shown in Figure 26. Thus inlight of the above analysis,the agreement between a.c. and d.c. resistances is to be expected. At veryhigh frequencies, however, the a.c. resistance should become much largerthan the d.c. resistance. 5. CONCLUSIONS The resultsof this study have immediate application to the controlof internally generated noise within jet engines. Steady- state andacoustic tests were performed on sevenFeltmetal and threeRigimesh porous lining materials. The principalfindings ofthis study are summarized interms of thematerial steady-state andacoustical behavior. 29 5.1 Steady-State Flow Resistance (1) The effects of grazing flow on the resistance of thin porous-faced liners were measured and found to be generally small except for specimen #7. (2) The data for the seven Feltmetal specimens were corre- lated in terms of a semi-empirical fluid mechanical model that treated the porous material as a collection of interacting small cylinders. The correlation, shown below, is not general but does indicate the qualitative behavior of the resistance in terms of the material and medium (air) properties. (3) The nonlinear behavior of the specimens was correlat- ed in terms of a fiber diameter based Reynolds number Re=VDb/Pv. In general, the onset of nonlinearity occurred when Re ranges from 2 to 3. Both the data and the correlation indicate that small diameter fibers delay nonlinearity to higher specimen through flow speeds. 5.2 Acoustic Impedance (1) The effects of grazing flow on the resistance and reactance of thin porous-faced liners were measured and found to be generally small. (2) A simple lumped element model was derived which correlated the face-sheet reactance of the ten specimens. The correlation, shown below, was derived by treating the porous material as an equivalent orifice of area PA where A is the cavity backing area and P is the material porosity, A simple prediction expression for the impedance of the seven Feltmetal specimens was derived based on the substitution for sufficiently low frequencies of the a.c. resistance by the d.c. resistance. An analysis presented in Section 4.3 showed that the a.c. and d.c. resistances are approximately equivalent for frequencies f< 30 An importantfinding of thisstudy is that a greatdeal can belearned about the acoustic properties of porouslining mater- ials in a jet engineduct environment by the much simplerstudy of theirnongrazing flow, steady-state flow resistance behavior. 31 REFERENCES 1. Hersh, A. S. andWalker, B., "The AcousticBehavior of HelmholtzResonators Exposed to High Speed Grazing Flows", AIM Paper No. 76-536, AIM 3rdAeroAcoustics Conference, July, 1976. 2. Feder, E. andDean, L. W., "Analyticaland Experimental Studiesfor Predicting Noise Attenuation in Acoustically TreatedDucts for Turbo Fan Engines", NASA CR-1373, 1969. 3.Dean, P. D., r'An InSitu Method of Wall AcousticImpedance Measurements in DuctFlow", Journal of Soundand Vibration, 1974, Vol. 34,97-130. 4. Plumblee, H. E. Jr., Dean, P. D., Wynne, G. A., andBurrin, R.H. , "Sound Propagationin and Radiation from Acoustically Lined Flow Ducts: A Comparison of Experimentand Theory", NASA CR-2306, Oct. 1973. 5. Darcy, H. "LesFountaines Publiques de la Ville de Dijon" Dalmont, Paris, 1856. 6. Bear, J. , "Dynamics of Fluidsin Porous Media", American ElsevierPublishing Company, N.Y., N.Y., 1972. 7. Scheidegger, A. E. ThePhysics of FZow Through Porous Media, 2nd ed.,University of TorontoPress, Toronto, 1969. 8. Irmay, S., "On theTheoretical Derivation of Darcyand ForcheimerFormulas", Trans. Amer. Geophys.Union. No. 4. , V-39, 702-707 (1958). 9. Kozeny, J., "Uker KapillareLeitung des Wassers in Boden", Sitzungober, Ahod. Wiss. Wien., V. 136, 271-306 (1927). 10. Rumer, R. R. andDrinkar, P. A., "Resistance to Laminar Flow through PorousMedia", Proc. Amer. SOC. Civil Eng. NO. HY5, V. 92, 155-164 (1966). 11. Rumer, R.R., Resistance to FZow Through Porous Media in FZow ThroughPorous Media (R. J. M. dewiest,Ed), Academic Press, New York, 1969. 12. Iberall, A. S., "Permeability of Glass Wool andOther High- lyPorous Media", J. Res.Nat. Bus. Stand, V45, 398-406 (1950). 13. Lamb, H., Hydrodynamics, 6thEd., Cambridge Univ. Press, London (1932). 32 14. Blick, E. F., "Capillary Orifice Model for High Speed Flow Through Porous Media I f, EC" Process Design and Develop- ment, No. 1, V. 5, 90-94 (1966). 15. Cheavetau, G. and C. Thirriot, "Regimes d'e'coulement en milieu poreux et limite de la loi de Darcy", LaHouille Blanche, No. 1, V. 22, 1-8 (1967). 16. Scott, R. A., 1946 Proceedings of the Physical Society 58, 165-183, "The Absorption of Sound in a Homogeneous Porous Medium", also 1946 Proceedings of the Physical Soci.ety 58, 358-368, "The Propagation of Sound Between Walls of Porous Material". 17. C. Zwikker and C. W. Kosten, Sound Absorbent Materials, Elsvier Publishing Company, Inc., New York, 1949. 18. Morse, R. W., "Acoustic Propagation in Granular Media", J. Acoust. SOC. Am. V. 24, NO. 6, 696-700 (1952). 19. Rayleigh, Theory of Sound, The Macmillan Company, New York, VOl. 11. 20. Morse, R. W. and Bolt, R. H., "Sound Waves in Rooms", Revs. Modern Phys. V. 16, N. 2, 69-150 (1944). 21. Beranek, L. L., "Acoustical Properties of Sound Absorbing Materials", J. Acoust. SOC. Am., V. 19, No. 4, 556-568 (1947). 22. Zorumski, W. E. and Parrott. T. L.. "Nonlinear Acoustic Theory for Rigid Porous Materials"; NASA TND-6196, June 1971. 23. "Fluid Meters", American Society of Mechanical Engineers, Part 1 - 4th Edition, New York, 1937. 24. Beavers, G. S. and Joseph, D. D., "Boundary Conditions at a Naturally Permeable Wall", J. Fluid Mech., V. 39, 197-207 (1967) . 25. Taylor, G. I., "A Model for the Boundary Condition of a Porous Material, Part I", J. Fluid Mech., V. 49, Part 2, 319-326 (1971). 33 TABLE I. SUMMARY OF MANUFACTURERS PREDICTED ACOUSTIC PROPERTIES Spec. Trade Name (Nominal)NLF* Average Porosity Average # or Predicted 500/20 Thickness Pore Identifica- Resistance (meters) Diameter t ion(meters) (Rayls-mks) Feltmetal Materials 1 FM 134 [email protected]/sec4.7 8.9~10-~0.59 5. 9x1ti5 2 347-10-20-100 11 2.0 4.1 0.97 4.8 AC3A-A 3 12581-4580 11 NA** 9.1 0.80 3.2 4 FM-122 500 I1 1.87.6 0.91 3.9 11 5 FM-134 320 4.7 8.9 'I 0.59 5.9 6 FM-134 320 11 11 11 11 11 11 11 7 FM-134 320I1 I1 11 I! 11 Rigimesh Materials 8 A0300S0204L [email protected]/sec NA 3.8 It NA 9 A0300S0304L 300 11 NA 3.8 NA 10 A0300S0404L 400 I1 NA 3.8 NA * Material Nonlinear Factor defined as the ratio of flow resis- tances at flow velocities of 5.0 m/sec and 0.2 m/sec. ** Information not available 34 Table 11. EMPIRICAL FIT OF D.C. FLOW RESISTANCE DATA [R = a+bVD (Rayls-mks)] /# Blowing Suction Specimen vm=0 Vm=79.6 m/s vm=o Vm=79.6 m/s 1 2 3 4 5 6 7 8 9 10 35 Specimen # Corrected APs(m-H20) at VD=O V =56.39 Vm=79.55 m meters/sec meters/sec (mks 1 (mks 1 ~- ~ - -~ . ~ 1 .OOll m-HnO .0023 m-HzO 2 .oozo .0038 .0020 .0041 .0020 -0043 negligible negligible ,0013 .0025 .0013 .0025 8 .0018 .0036 9 negligible negligible 10 .0005 .OOlO 36 TABLE IV. CORRELATION OF D.C. FLOW RESISTANCE MEASUREMENTS - Spec. Rd.c PorosityFiber Dia. Corr.Para- Obs. VD Values of # (Rayls- (P) (meters) meter* (6) at Onset V.D where mks) of NL Re=2; 3 (meters/ (meters/ s ec) sec) 1 398 10 .51 -4 1784 .3-. 4 .15; .23 2 117 .97 8x10 -6 2026 3-4 3.63;5.44 3 865 8x10 .80 -6 1538 .7-1.1 3;4.5 4 598 4x10 .92 -6 1737 2.5-3.8 6.92;10.4 5 493 10 .48 -4 1915 .35-.40 .15;.23 540 .46 10 .466 540 -4 .16;.24.25-.35 1907 637 .45 10 .457 637 -4 .16;.25-15--40 2145 8 292 i 37 4x10 -5 - .7-2 .47;. 71 9 558 .29 4x10 -5 - 1-1.5 .53; .80 10 298 .43 4x10 -5 - .7-2 .43; .64 *The values of B tabulated above are determined from Eq. (41) with a=O. 8 37 TABLE V. COMPARISON OF NON-GRAZING FLOW STEADY STATE, TWO MICROPHONE AND STANDING WAVE TUBE MEASUREMENTS FaceSheet Reactance S pec ific ResistanceSpecific (Rayls-mks) (Xf/ PUD) Specimen # Steady Flow Two-Mic* Impedance Two-Mic*Tube Impedance Tube 400 405 395 .80 .72 115 141 130 .59 .28 890 931 906 .69 .40 600 684 710 .54 .77 500 515 548 1.14 .81 540 553 512 1.13 .79 650 676 687 1.32 .74 8 280 321 35 9 .79 1.16 5 60 585 9 560 528 .98 .90 28 0 235 10 280 285 .64 .68 *data corrected by methodof Appendix A TABLE VI. EXPERIMENTAL DETERMINATION OF INERTIAL CONSTANT E 39 P 0 RlGlMESH Sheet Enlarged Section x5 Magnification Photograph of Rigimesh Porous Material STAINLESS STEEL TYPE 430 Surfaceappearance at X 24 of 430 stainlesssteel FELTMETAL fiber metal.The material is 15% denseand employs Type E46 fibers. Photographs of Feltmetal Porous Materials FIGURE 1. ENLARGED PHOTOGRAPHS OF SPECIMEN INTERIOR AND MODEL GEOMETRY Orifice Meter Side Branch Blower CentrifugalBlower Screen 20 H.P. Motor Orifice SupplyPressure \ Static PressureTap Side BranchFlow Grazing FlowVelocity 0 - 79.25 mlsec ,- ,- Suction I Blowing Test Specimen FIGURE 2. SCHEMATIC OF STEADY-STATE FLOW RESISTANCE MEASUREMENT SYSTEM , . .... 1.250" 1.125- 1.000- v,= 85.65 meterslaec 0 Data 0.875" 0.750 -- 0.625" 0.500.- 0.375 .. / d 0.250- / B 0.125.. Velocity Profile 0- 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Non-dimensionalized Velocity Profile - V(y)/V, FIGURE 3. TEST SECTION BOUNDARY LAYER VELOCTIY PROFILE 42 I5O0 t 0 0 00 0 OO 0 0 0 00 V 0 V vv V V vv nv 0 vv 0" 0 0 00 0. [AAAMAAAAAA A AAAAAA : I 8 I 0 8 v4 ~ 1000.. 0 0 00000~~OO V V 5 00.. 0 00 0 0 O vm=ofo vv vvvv~8i V ;v 01 AAAAAAAAAAAAAAAAAAAI I .01 .02 .04 .06 .0-8.10 .20 .40 .60 .80 1.00 2 .OO 4. 00 6.00 8. 00 Average Specimen Through Flow-VD(meters/sec) FIGURE 4. EFFECT OF GRAZING FLOW ON THE STEADY-STATE RESISTANCE OF SPECIMENS 1-4 OPERATING IN THE BLOWING MODE P W I P P -A?s Rayls 0 (mks) "D 8 1000.. 0 on000on 0 OO 0 VV v vwg x7 vvV 500" V, = 79.55 meters I sec 0 00 0 AAAAA A A A AA 0. 8O 1000.. o 0 n uouo~~~0 Qov v V 5 00.- v, =o v~mvvvvvg00 00000 A AAAAAAAAAAL\AAAA 0. .01 .02 .04 .06 .08 .10 .20 .40 .60.80 1.00 2. 00 4.00 6.00 8.00 Average Specimen Through Flow - VD(meters/sec) FIGURE 5. EFFECT OF GRAZING FLOW ON THE STEADY-STATE RESISTANCE OF SPECIMENS 1-4 OPERATING IN THE SUCTION MODE 0 A 1500 0 0 A T 0 0 Voo = 79.55 meters I sec o 1000 { A 8 Ao 0 50 0 0 A0 0 1500 0 A 0 0 A 1000 0 5 00 .01 .02 .04 .06 .08 .lo .20 .40 .60 .80 1.00 2.00 4.00 6.00 8.00 Average Specimen Through Flow -VD(meters/sec) FIGURE 6. EFFECT OF GRAZING FLOW ON STEADY-STATE RESISTANCE OF SPECIMENS 5-7 OPERATING IN THE BLOWING MODE 0 Rayls " V (m ks) D 1500. A 0 0 v, v, = 79.55 meters I sec A 0 1000- A0 0 000 A 8 14 500. 1 I I, I I 1500. 0 A Ao 0 A 0 v, v, = 0 0 1000- Ao 0 Ao 0 500.. ,E !&f$0 00 8 i 8 0 1. I 'I I I I1 I I .01 .02 .04 .06.lo .08 .20 .40 .60 .80 LOO 2 .oo 4.00 61)O 8.00 Average Specimen Through Flow - VD(meters/sec) FIGURE 7. EFFECT OF GRAZING FLOW ON STEADY-STATE RESISTANCE OF SPECIMENS 5-7 OPERATINGI IN THE SUCTION MODE 1500 0 .c .o. .02 .04 .06 .OSLO .20 .40 .60 .80 1.00 2.0 0 4.00 6D0 8.00 Average Specimen ThroughFlow - VD(meters/sec) FIGURE 8. EFFECT OF GRAZING FLOW ON STEADY-STATE RESISTANCEOF SPECIMENS 8-10 OPERATING IN THE BLOWING MODE I I I I, 1 OJ I .Ol .02 . 04 .06 .08 .10 .20 .40 .60 .801.00 2.0 0 4.00 6.00 8.00 I Average Specimen Through Flow - V~(meters/sec) FIGURE 9. 'EFFECT OF GRAZING FLOW ON STEADY-STATE RESISTANCE OF SPECIMENS 8-10 OPERATING IN THE SUCTION MODE AP, = It vC0 Recirculating region induced by shear stresses from grazing flow causes (P,-P~)>o. Test iSection FIGURE 10. SCHEMATIC OF VORTICAL VELOCITY INDUCED ACROSS SPECIMEN BY GRAZING FLOW SHEAR STRESSES 49 I m 0 3000 -- 2000" 79.55 VV V Grazina Flow dP, vV 1000 V 900 1. V 800 I_ 700 -. 600 -- 500" 400 .. SPECIMEN #4 200 -- 10 y I .01 .02 .o;l .. . b6 ' '.i .2 :4 ' .6 ' - ''1 2 4 6 AverageSpecimen Through Velocity - VD(meters/sec) FIGURE 11. EFFECT OF GRAZING FLOW ON STEADY-STATERESISTANCE OF SPECIMEN-BLOWING Horn Driver Coupler Porous Face Sheet- - .c Digital Synch. Ocillator Power Phase Filter inc. Analyser 9: 9: Mtr. el < FIGURE 12. SCHEb.IATIC OF TWO-MICROPHONEMEASUREMENT SYSTEM 51 120 Watt-DriverJBL 2480 Horn JBL H-93 lest Section 3 t AD - YU 'Dual Ch. S ynch. SynchronousAD-YU Synch. C onv erter Filter Type Filter Converter 1034 Volt Type 1036 I ' Meter I I 1 c PowerAmpl. - Audio V Y 2 McIntosh Generator AD -YU Phasemeter HP- 200CD MC 2100 Type 524 A4 v- Frequency Counter Heath IB -1100 Automatic LevelControl < Control (40.5 KHz) FIGURE 13. TWO-MICROPHONEMETHOD INSTRUMENTATION 52 1000 -- A 0 0 0 0 900 -. 800- 700 -- 0 0 AQ0 600.. Q Specimen #1 500 .. J A 8 A 0 400- 0 0 0 0 300.- 200" 0 0 0 A 100- 0 I I 90 100 120110 130 140 Incident Sound PressureLevel - dB FIGURE 14. EFFECTS OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON THE ACOUSTICRESISTANCE OF SPECIMENS #1-4 A 0 800 A Specimen it 7 AA A0 0 7 00 0 0 0 0 0 600 i a, u 700 A AA~O 600 0 Specimen #6 5 00 1 Sym.1 vm I (meters /sec)] 7001 79.55 A A' 600 AAA 0 500 ~ I- ~ I- 90 110100 120 140130 Incident Sound Pressure Level - dB FIGURE 15. EFFECTS OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON THE ACOUSTIC RESISTANCE OF SPECIMENS #5- 7 54 800 700 n v) ,x E 7 v v) 600 4 0 x ‘cd p: 500 400 Open Symbols, v, Specimen #8 a) p: 8 0 300 0 f A 1Closed Symbols, Specimen #10 I 200 100 ” 1- I I I I 0 I 90 100 110 120 130 140 Incident Sound Pressure Level - dB FIGURE 16. EFFECTS OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON THEACOUSTIC RESSITANCE OF SPECIMENS #8-10 55 0.8 0.7 V I I 0.6 I 90 100 110 120 130 y40 150 0.5 0.4 0.5 o-6 I 90 100 110 120 130 140 150 0.8 @...om 84 0.6 Specimen 0 0 0 0000 00 0.4 - 9’0 bo lio ~io 1io 180 1-50 Incident Sound Pressure Level - dB FIGURE 17. EFFECT OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON FACE-SHEET REACTANCE OF SPECIMENS #1-4 56 I 20 0 n - 20 20 0 0 0 0 A A 0 Specimen #3 A 0 A 20OI 20 I J - 20OI 85.65 A 0 0 0 0 0 Specimen #/ 1 0 J -20 0 I I I 1 2l0 90 100 110 120 130 140 IncidentSound Pressure - dB FIGURE 18. EFFECTSOF GRAZING FLOWAND SOUND PRESSURE LEVEL ON THETOTAL REACTANCE OF SPECIMENS #1-4 57 100 90 V V vv M a, a I c, a, a, c . c13 I .. 92 a, Specimen V cd Fr, 0 0 0 88 91 Specimen 90 A A 89 1 98 ...... 1 Open symbols, V,= 0 Closed symbols, V, = 85.65 meters /sec Specimen 94 ~~ ~ I 0 0 0 0 0000 90 J IncidentSound Pressure Level - dB FIGURE 19. EFFECT OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON PHASE ANGLE CHANGE ACROSS THE FACE- SHEET OF SPECIMENS #1-4 58 1.3 -. 0 o a 1...= Specimen #7 0 1.2 -- 0 I 90 100 120110 130 150140 Open symbols, V,=O Closed symbols, V,= 85.65 meterslsec A AA A c Specimen #6 A AAA 1.0 A A A 7 1.0 J I I I I I I I I I 1 I I 90 100 110 120 130 140 150 Incident Sound Pressure Level - dB FIGURE 20. EFFECT OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON FACE-SHEET REACTANCE OF SPECIMENS #5-7 59 0 0 0 Specimen #7 0 ooOoJ AA AAAA~ 0 0 0 0 Specimen #5 OOOO IncidentSound Pressure Level - dB FIGURE 21. EFFECT OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON THE TOTAL ACOUSTIC REACTANCE OF SPECIMENS #5-7 60 9 1.. 90" M 0 Specimen #7 a 0 0 0 I 8 9.. c, 0 Q) Q) I A I I cn 90 100 110 120 130 I 140 150 a, U cd F4 c a, E .A U Spec imen # 6 a, !a cn 90 v) A A A AAAA r/) 0 k U I 4 90 100 110 120 130 140 150 Q) Do c cd c U 9 1" ..aaa a, v) Open symbols, V, =O cd c Closed symbols, V, = 85.65 meterslsec a Specimen #5 90" 0 00 0 0 8 9.- 000 1 I I 90 110100 120 140130 150 Incident Sound Pressure Level - dB FIGURE 22. EFFECT OF GRAZING FLOW AND' SOUND PRESSURE LEVEL ON THEPHASE ANGLE CHANGE ACROSS THE FACE-SHEETOF SPECIMENS #5-7 61 .70 .' . ..= 0 Specimen #10 R .60 -- 0 3 Q 00 \ n 0 0 o w 1 I x .so - I I u 90 100140130 120 110 150 1 a, ~~ u A c Opensymbols, V,= 0 cd c, 1.04 Closed symbols, V,=85.65 metershec A 0 cd a AA Specimen #9 p: A c, 1.00 a, a, c [/) .90 u cd c4 I I L 1 I I I I I I I I 1 a 150140 13090 120 110 100 a, N .r( d I cd 0 .I+ vr 0 c Specimen #8 .70 .r( 0 0 0 0 a go c 0 0 .66-741 I I I I I ~""I A I 1 I I 90 150100140 130 120 110 Incident Sound Pressure Level - dB FIGURE 23. EFFECT OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON THE FACE-SHEET REACTANCE OF SPECIblENS #8-10 62 10 J= 5 .. Specimen #8 0 -- 0 0 0 0 A - 5” -lo* 10 1 -5 -10 40 A A A 20” #10 0 0 0 0 ” 0 0 0 I I I I I I I I 90 100 110 120 130 140 Incident Sound Pressure Level - dB FIGURE 24. EFFECT OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON THE TOTAL REACTANCEOF SPECIblENS #8-10 63 94 0 Specimen # 10 92 0 00 90 0 0 0 I I I I I I I I I 1" I 90 100 110 120 130 140 150 91 A I ~~ ~ Open symbols, V, = 0 Closed symbols, V, = 85.65 meters/sec A AA Specimen#9 90 A A A A AAAAA I 89 J 90 100 110 120 130 140 150 92 0 0 91 0 0 Specimen#8 0 0 90 0 0 0 0 00 IncidentSound Pressure Level - dB FIGURE 25. EFFECT OF GRAZING FLOW AND SOUND PRESSURE LEVEL ON PHASE ANGLE CHANGE ACROSS THE FACE-SHEET OF SPECIMENS #8-10 64 al V d cd Incident Sound Frequency - Hz FIGURE 26. EFFECT OF VARIATION OF FREQUENCYON THE ACOUSTICRESISTANCE OF SPECIbIEN #1 65 AA A 140.. AA 8 0 0 0 120” 0 80 SPL (inc) Sym. v, looI dB (meters/sec) 0 90 0 40 A 135 0 A 135 86.26 I I I I 1 I I I 0 400 500 600 700 800 900 1000 Incident Sound Frequency - Hz FIGURE 27. EFFECT OF VARIATION OF FREQUENCY ON THE ACOUSTIC RESISTANCE OF SPECIMEN #2 66 Sym. SPL(inc) "ea dB (meters /sed A 0 90 0 A 135 0 A 135 86.26 381360 A 8 A 0 0 0 460 50'0 60b ' 7b0 800 960 1000 Incident Sound Frequency - Hz FIGURE 28. EFFECT OF VARIATION OF FREQUENCY ON THE ACOUSTIC RESISTANCE OF SPECIMEN #7 67 94=~ ' SpecimenFace-Sheet and 200t b B Cavity Reactance V rd A c:a, 0 i; 4 400 cd " w A SPL(inc) V, Syrn. b0 t dB (meters /sec 0 90 0 n A 135 0 ul' A 135 86.26 d rj w u rj 100 c: 0 0 Specimen Face-Sheet Reactance c, a, AA a I I I 1 s AA I I I I cn 0 I I I 4010 500 6b0 700 800 900 1000 Q) V rd IncidentSoundFrequency - Hz L FIGURE 29. EFFECT OF VARIATION OF FREQUENCY ON THE ACOUSTIC REACTANCE OF SPECIMEN #1 200 0 800 900 1000 200 SpecialFace-Sheet and 0 CavityReactance 0 400 0 AA 160t SPL(inc) v, Sym. dB (meters/sec) 0 90 0 A '1 35 0 135 86.26 A A 100- A A oA aA0 s 0- a - A i 0 SpecimenFace-Sheet Reactance 60t 00 400 500 700600 800 900 1000 Incident SoundFrequency - Hz FIGURE 30. EFFECT OF VARIATION OF FREQUENCY ON THE ACOUSTIC REACTANCE OF SPECIMEN #2 69 4: E v 200" A 0 I 1I 04dPA 400 5007 -800 900 1000 - 20 0" Face-Sheetand Cavity Reactance Sym. SPL(inc) v, dB (rneters/sec) A - 0 90 0 160- 0 Ao a A 135 0 A 135 86.26 0014 140- 0 A OA* 0 120- A A A AA A 0 100- Bo ReactanceFace-Sheet "A 2 OA 8 0- A A 0 0 I I I I I I 400 500 600 700 900800 1000 Incident SoundFrequency - Hz FIGURE 31. EFFECT OF VARIATION OF FREQUENCY ON THE REACTANCE OF SPECIMEN t7 70 ." A 4 " AA AA 3 " A A A A A 2 -- A A 1" 400 500 140 A A 120 8 I3 AA 0 A 0 100 0 80 O o13 Ou A 0 0 A 60 0 0 0 A 0 40 0 A ~1 I A I I 1 I I I I I -0 -0 700 800 900 1000 Incident Sound Frequency - IIz FIGURE32. EFFECT OF FREQUENCY VARIATION ON THE PRESSURERATIOS AND PHASECHANGES ACROSSSPECIMENS 1, 2 G 7 71 APPENDIX A CORRECTION TO MEASURED INCIDENT SOUNDPRESSURE LEVEL DUE TO ACOUSTIC RADIATION FROM THE TEST SPECIMEN To measurethe acoustic impedance of a sheet of porous materialbacked by a cavityusing the two-microphone method, it is necessaryto accurately determine the difference in sound pressurelevel and phase between a microphonemounted flush with thesurface of the test specimenand another mounted atthe back ofthe resonator cavity. As hasbeen pointed out in Section 4.1, it is necessaryto mount the"incident" microphone suffi- cientlyfar fromthe test specimen so that it is notinfluenced by thehydrodynamic near field. The typicalbehavior of highly resistive specimens was asfollows. The frequency of theincident sound field was ad- justed to resonancewhich corresponds to a relativephase across thespecimen $ = 90 degrees.During this adjustment the incident soundpressure level was set at 90 dB; $ was observedto de- creasemonitonically with increasing sound pressure level. This was theusual response of highly resistive porous specimens. However, forparticularly low resistancespecimens, $ was ob- servedto initially increase by several degrees with increasing soundpressure level before decreasing. Additionally, it was notedthat in the range of this anomalous phase behavior, in- creasingthe drive level to the loudspeaker by 5 dB would re- sult in somewhat greaterthan 5 dB increasein the measured inci- dentsound pressure level. Thiscan be explained in terms ofacoustic radiation from thetest sample as follows: Let Q bethe inward volume velocity inthe orifice or porous sample, PC,, be thecavity sound pres- sure, Pint bethe incident sound pressure, (ZD) = (RD) + (XD) be thespecific acoustic impedance of the sample, normalized to thecavity diameter D, L be thecavity depth, and r bethe distancefrom the acoustic center of thespecimen to the "inci- dent"measuring location. Forsinusoidal excitation Pint = Pi ceiut,where hence- forththe eiut will beassumed. We there?orehave, for the volume velocity 72 - ~- and from mass conserlfation and adiabatic compr ession, . 4pc2Q Qdt -I Pcav 21 - = TD~LO Combining Eqs. (A-1) and (A-2) , The radiated pressure Pr for a small source in a plane surface is given by POQ e-~c. or Pr(r)=-i - 2 7rr (A-4) Combining Eqs. (A-1) and (A-4) Therefore, the total sound pressure Pt at the incident microphone will be given by Unfortunately, this otherwise convenient expression includes RD and XD, which we are attempting to measure, and an iterative calculation is required to correct for the effect. Transposing we have 73 or where A= (A- 9) and (A- 10) Thus, sp~~= SPLt - 10 log (1 + A* -F 2 AcosB) (A-11) The iterativecalculation proceeds as follows: 1. Use SPLt-SPLiand @t-@itodetermine on initialvalue of RD and XD. 2. Use thesevalues of RD and XD in Eqs. (A-8) thru (A-11) to determine a first iterationvalue of SPLiand @. 74 3. Calculate new valuesof RD and XD. 4. Repeatsteps 2 and 3 untilthe answers from two successful iterationsare equal within the desired accuracy (0.1% re- quiresonly 3-4 iterations). Results of thiscorrection procedure for low sound pressure leveldata fromporous samples 1 through 10 are shown inTable A-1. Note thatfor each case, ASPL is lowerthan measured, I$ is higherthan measured, R/pc is slightlyhigher than measured and X/pc is more positivethan measured. 75 TABLE A-1 SUMMARY OF CORRECTED IMPEDANCES TO TEN POROUS SPECIMENS Vo3 = 0; SPL(inc) = 90 dB Sample Meas Meas Init Init CorrCorrCorrCorr # ASPL @ic R/ PC X/ PC ASPL @ ic R/PC X/PC (dB) (deg) (ded (dB) .58 90.1 .9794 .0017 .37 92.31.0026 .0397 10.07 90.0 .3219 .OOOO 9.35 96.47.3474 .0394 -6.90 90.4 2.2791 .0159 -7.00 91.372.3045 ,0551 -4.01 90.0 1.6503 .OOOO -4.26 91.531.6983 ,0455 - .82 89.9 1.2453 -.0022 -1.03 91.751.2748 .0389 -1.84 89.9 1.3400 -.0023 -1.97 91.411.3595 .0335 -3.02 89.8 1.6500 -.0058 -3.16 91.171.6770 .0342 8 2.83 90.1 -7773 .0014 2.60 92.68 .7971.0373 9 7.63 90.0 .4725 .OOOO 7.01 94.82 .5059.0427 10 5.56 89.7 .5621 -.0029 5.23 93.30 .5829.0336 76 1. Report No: 2. Government Accession No. 3. Recipient’s Catalog No. NASA CR-2951 I and Subtitle 4. Title and I 5. Report Date Effects of Grazing Flowon the Steady-State Flow January 1978 Resistance and Acoustic Impedance of Thin Porous-Faced 6. PerformingOrganization Code Liners ___- 7. Author(s1 8. PerformingOrganlzation Report No. A. S. Hersh & B. Walker _~_.- . .. - .. -_..- 10. Work UnitNo. 9. Performing Organization Name and Address Hersh Acoustical Engineering 11. Contract or Grant No. 9545 Cozycroft Avenue NAS1-14310 Chatsworth, CA 91311 13. Type ofReport and Period Covered 12. Sponsoring Agency Name and Address NationalAeronautics and Space Administration Contractor Report 14. Sponsoring Agency Code Washington, DC 20546 1 -~ 15. Supplementary Notes FinalReport, Project Manager, Tony L. Parrott,Acoustics Branch, Acoustics and Noise Reduction Division, NASA LangleyResearch Center, Hampton, VA 23665 16. Abstract The results of an investigation of the effects of grazingflow on thesteady- state flowresistanceand acoustic impedance of sevenFeltmetal and three Rigimesh thin porous-facedliners are presented. A state-of-the-art review of previous nongrazingflow studies is alsopresented. The steady-state flow resistance of the ten specimens are measured using standardfluid mechanicalexperimental techniques. The acoustic impedance was measured usingthe two-microphone method. The principalfindings of the study are (1) the effects of grazingflow were measured and found to be small, (2) small differences were measured between steady- state and acousticresistance, ‘(3) a semi-empirical modelwas derivedthat correlatedthe steady-state resistance data of the seven Feltmetalliners and theface-sheet reactance of both theFeltmetal and Rigimesh liners. These findingssuggest that nongrazing flow steady-state flow res i stance tests couldprovide considerable insight into theacoustic behavior of porous 1 i ners in a grazingflow environment. . ”~. 7. Key Words (Suggested by Author(s)l 18. DistributionStatement Acoustic Impedance Unclassified - unlimited Grazing Flow Effects STAR Category 71 Measurements 9. Security Classif. (of this report) 20. SecurltyClarrif. (of this page) Un classified UnclassifiedUnclassified 78 $6 .OO * For sale by the National Technical Information Servlce. Sprtnefleld. Vlrglnla 22161 NASA-Langley, 1978