On the Isentropic Forchheimer s Sound Waves Propagation in a Cylindrical Tube Filled with a Porous Media

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1 5th WSEAS Int. Conf. on FLUID MECHANICS (FLUIDS') Acapulco, Mexico, January 5-7, On the Ientropic Forchheimer Sound Wave Propagation in a Cylindrical Tube Filled with a Porou Media H. M. Dwairi Civil Engineering Department, Faculty of Engineering, The Hahmaite Univerity, Zarqa, Jordan Hazim.dwairi@gmail.com Abtract: - A theory of ound wave propagation in porou media that include the nonlinear effect of the Forchheimer type with the nonzero radial velocity effect i laid out by uing variational olution. It i hown that the main parameter governing the propagation of ound wave are the hear wave number = R ρ ω / μ, the reduced frequency number = wr a, the poroity, the Darcy number Da = R K and the Forchheimer number C = C. The manner in which the flow influence the attenuation and the phae velocitie of the F forward and bacward propagating ientropic acoutic wave i deduced. It i found that the increaing of Darcy number and Forchheimer number increaed wave attenuation and phae velocitie for both forward and bacward ound wave, while the increaing of poroity decreaed attenuation and phae velocitie. The effect of increaing reduced frequency i found to increae the attenuation of the forward wave and decreae attenuation and phae velocitie of the forward and bacward ound wave. The effect of the teady flow i found to decreae the attenuation and phae velocitie for forward ound wave and enhance them for the bacward ound wave, repectively. Key-Word: - ound wave, porou medium, fluid flow Introduction The coutic problem find it application in many different ituation. For example if the acoutic improvement are retricted to interior pace uually mineral wool or open pore foam can be ued to olve the problem, for outdoor problem for intant acoutic noie barrier againt traffic noie, the aborption provided by granular material uch a porou concrete or imilar material, a they behave better with bad weather and other atmopheric phenomena. In porou material the aborption proce of the acoutic wave tae place through vicoity and thermal loe of the acoutic energy inide the micro tube forming the material. The problem of a propagation of ound wave in fluid contained in a plain medium i a claical one, to which famou name are connected lie Helmholtz [], Kirchhoff [] and Rayleigh []. A variational treatment of the problem of ound tranmiion in narrow tube i decribed by Cumming [] a an alternative to the more uual analytical procedure which i limited to mathematically tractable geometrie. A firt approximation to the effect of mean flow on ound propagation through cylindrical capillary tube i achieved by Peat [5]. A ound tranmiion in narrow pipe with uperimpoed uniform mean flow and acoutic modeling of automobile catalytic converter i done by Doumaci [6]. A numerical tudy on the propagation of ound through capillary tube with mean flow i achieved alo by Jeong and Ih [7] and finally an approximate diperion equation for ound wave in a narrow pipe with ambient gradient i done by Doumaci []. The problem of ound wave propagation in a tationary or flowing fluid in a porou medium i not addreed yet. An attempt i made in thi article to develop a implified nonlinear theory that predict the propagation characteritic of a tationary or flowing fluid in aturated porou media. Thi theory i an extenion of the claical plain medium theory, uing a modification to Darcy law due the Forchheimer effect and auming nonzero radial velocity effect. Analytical expreion for the propagation contant are obtained from variational olution. Comparion with previou wor in the limit of plain medium how an excellent agreement. Problem Formulation Conider a rigid tube filled with a aturated porou material, the fluid i aumed to be a tationary or movable inide the tube. The x- coordinate i meaured along the tube and the r- coordinate i meaured normal to the axial direction. Under the boundary layer approximation the baic equation which govern acoutic wave propagation in a rigid tube filled with a porou media are the continuity and momentum equation: ρ ρ ρ ( v v u u v ρ ) = () t x r r r x ISSN: Page ISBN:

2 5th WSEAS Int. Conf. on FLUID MECHANICS (FLUIDS') Acapulco, Mexico, January 5-7, u u ρ u t x v u r p = x () μ C F ρ u u u u μ / K K r r r p = () r Where u,v are the velocity component in the axial and normal direction; repectively. ρ and p are the fluid denity and preure, μ i the abolute vicoity and K i the permeability of the porou media and i the poroity of the porou medium. Since one i dealing only not with capillary tube the radial velocity might be expected to be not negligible. Thi effect of thi i to couple the continuity equation () and the momentum equation (5). Next, it i aumed that the flow through the capillary duct i a uperpoition of a fully developed laminar, incompreible, axial teady flow and a mall harmonic acoutic diturbance of frequencyω. The teady flow i taen to have contant denity ρ and a peed of ound a uch that the fluid variable can be expanded in the form: ( ( ) ) Γξ iω ρ = ρ αρ η e e ( ) t Γξ iωt u = a M ( η ) αu( η) e e, Γξ iω v = aαv( η) e e t, Γξ iωt p = ( ρ a γ ) ( p ( ξ ) αp( η) e e )(-7) Where α << and γ i the ratio of pecific heat. It i een that the teady flow variable p and Mach number M together with acoutic variable ρ, u, v and p are dimenionle. Now introduce the following variable in the tranformation: ξ = ωx / a η = r / R () R i the radiu of the capillary duct. The axial acoutic wave motion ha been aumed to have complex propagation contant Γ which can be expanded a: Γ = Γ i Γ (9) Where Γ repreent the wave attenuation per unit ditance and Γ repreent the phae hift over the ame ditance. The aumed form of the variable, equation (-7) are ubtituted into the governing equation () and ()-() and term of imilar order in α equated. It i found that for zeroth order, the teady flow olution, the equation of continuity and radial momentum are identically atified, while the axial momentum equation (6) become: dp d dm C = η Da M M () γ dξ ϕη dη dη Here = R ρ ω / μ i the hear wave number, = ωr / a i the reduced frequency parameter, Da = R / K i the Darcy number and C i the F Forchheimer number. Thi i the claical equation of Hagen-Poieuilli flow, the olution of which, with no-lip boundary condition, give a parabolic velocity profile: dp η M ( = = η ) γ ξ M () d Where M i the mean Mach number of the teady flow. The linearized acoutic equation follow from equating term of firt order in α in the governing equation, and are: iρ dv v [ Γu MΓ ( η ) ρ] = () dη η iu MΓ M [ ( η u] ηv = ( Γ / γ ) p d u (/ ) (/ η) du dη ( Da / ) u (CF Da / ) M u dη () Where Da = K / R and = wr / a i the Darcy number and the Forchheimer number repectively. The cae of = or Da = correpond to the plain medium without the preence of the olid matrix and any value of < < or Da > repreent a porou medium with different pore pace. For the cae of = and Da =, the governing equation () and () reduce to thoe obtained by Peat [5] for the cae of a pure plain medium. In the limit of zero teady flow, M =, thee equation are found to reduce to thoe for the reduced frequency olution of Tijdeman [9]. It will be aumed hat the tube are rigid which implie the no-lip boundary condition of the fluid velocity at wall: u = at η = () The olution of equation ()-() i greatly implified if one aume that the acoutic diturbance occur ientropically, ince then: p = ργ (5). Variational Solution The continuity equation (), with the aumption of ientropic diturbance and the given form of the trial olution of the axial velocity, equation () and (9),and Integrating thi expreion and uing the boundary condition that v = when η = give: ηv η i η p η η u = [ MΓ ( )] Γ ( ) (6) γ η Thi expreion can now be ubtituted into the full momentum equation () to give: iu MΓ M η i MΓ η p ( η ) u [ ( )] γ η u d du (7) ( )] = ( Γ / γ ) p ( ) ( )] η η η d d η η C DaM ( Da / ) u Equation (7) correpond to the minimum of the functional: ISSN: Page ISBN:

3 5th WSEAS Int. Conf. on FLUID MECHANICS (FLUIDS') Acapulco, Mexico, January 5-7, G = du iu η MΓ ( η / ) ( η ) ηu dη im p MΓ p η η u η ( ) u γ γ MΓ u Γp η ( η ) ηu η γ Da C Dau η(m ( η ) ηu dη () Now the aumed form of trial olution for u, equation (), i ubtituted into thi expreion and the minimum i found by etting: G C = (9) Which reult in an expreion for the contant C ; namely, i MΓ p Γ im M Γ = () C γ Da C Da 6 Subtitution of the ame trial olution for u into equation (6) and ue of boundary condition v = at η = lead to a econd expreion of the propagation contant, p i C = ( MΓ ) () Γγ Equation () and () enable C to be eliminated which reult in an expreion for the propagation contant: M i i i Da ( ) Γ i Da i C Dai = C Da MΓ () With a olution of the propagation contant of the form: i i Da i C Da M Γ = ± M ( ) () / i i Da i C Da M M i Da i C Dai ( ) M ( ) Note that when = or Da = the propagation contant, equation () i reduced to thoe obtained by Peat [5] for the cae of a pure plain medium. It i important alo to note that the M will reflect the effect of teady flow on the acoutic problem under conideration; the cae of M = correpond to the abence of mean flow velocity and to the acoutic problem in a tationary porou media.. Reult and Dicuion Comparion of variational olution with exact olution a given by Peat [5] in the limit of plain medium for = and Da = are hown in table.. Figure. i a plot of the modulu of wave attenuation per unit ditance, Γ and phae hift Γ for varying hear wave number and Mach number M =,.,.,. and for Da =, C =., =. and =.5π. It i clear that a the Mach number i increaed the attenuation i decreaed and the phae velocitie are increaed for the forward wave, while a the Mach number i increaed both the attenuation and phae velocitie are increaed for the bacward ound wave; thi i due to colliion effect of the forward ound wave and favorable vertical velocity effect in more damping of the bacward ound wave. Figure. how the effect of increaing Darcy number Da =,.,, 5, for M =., C =., =. and =. 5π, it i clear that a the Darcy number i increaed the attenuation and phae velocitie for both the forward and bacward ound wave; thi i due to favorable effect of the olid matrix in damping ound wave. Figure. how the effect of poroity =.5,.6,.7,.,. 9 on attenuation and phae velocitie for elected value of Da =, C =. M =. and =.5π, it i found that the increaing of poroity decreae the attenuation and phae velocitie for both the forward and bacward wave; thi i due to the mall effect of the olid matrix a moving toward the plain media limit. Figure. how the effect of Forchheiemr term C =.,, 5,on attenuation and phae velocitie for Da =, =., M =. and =. 5π, it i found that a the Forchheimer term i increaed the attenuation and phae velocitie are increaed for the forward and bacward ound wave; thi i due to favorable damping effect of the fluid inide the large ued pore of the olid matrix. Finally figure. 5 how the effect of increaing =.5π,.5π,. π,.π,.π,. 5π on the attenuation and phae velocitie for Da =, =., M =. and C =, it i found that a the reduced frequency i increaed the attenuation i increaed and the phae velocitie are decreaed for the forward ound wave and both the attenuation and phae velocitie are decreaed for the bacward ound wave; thi i due to higher ISSN: Page ISBN:

4 5th WSEAS Int. Conf. on FLUID MECHANICS (FLUIDS') Acapulco, Mexico, January 5-7, Shear wavenumber, Preent Peat [5] Table. comparion of variational olution with Peat [5] frequency of the impacted ound wave on the olid matrix, it i important to note that the ame effect i noticed for ound wave propagated in a plain medium. 5. Concluion - It i found that the effect of increaing Darcy number or Forchheimer number i to increae the attenuation and phae velocitie for both forward and bacward ound wave; thi i due to favorable role of olid matrix in damping ound wave. - It i found that the effect of increaing poroity or reduced frequency parameter i to decreae attenuation and phae velocitie for both forward and bacward ound wave; thi i due to abence of favorable role of porou matrix and high incident ound wave trength repectively. Reference: [] H. V. Helmholz, Verhandlung der Naturhitorih-Medizinichen Verein zu Heidelberg, Bd III, vol. 6, 6. [] G. Kirchhoff, Uber den Einflu der Waermeleitung in einem Ga auf den Schallbewegung, Proggendorfer Annalen, vol., 6, pp [] Lord Rayleigh, Theory of ound, volume II, London: The Macmillan Company econd edition, pp. 9-6, 96. [] A. Cumming, Sound propagation in narrow tube of arbitrary cro-ection, Journal of Sound and Vibration, vol. 6, no., 99, pp. 7-. [5] K. S. Peat, A firt approximation to the effect of mean flow on ound flow on ound propagation through cylindrical capillary tube, Journal of Sound and Vibration, vol. 75 (), 99, pp [6] E. Doumaci, Sound tranmiion in narrow pipe with uperimpoed uniform mean flow and acoutic modeling of automobile catalytic converter, Journal of Sound and Vibration, vol., no. 5, 995, pp [7] A.-W. Jeong and J.-G. Ih, A numerical tudy on the propagation of ound through capillary tube with mean flow, Journal of Sound and Vibration, vol. 9, no., 996, pp [] E. Doumaci, An approximate diperion equation for ound wave in a narrow pipe with ambient gradient, Journal of Sound and Vibration, vol. (),, pp [9] H. Tijdeman, On the propagation of ound wave in cylindrical tube, Journal of Sound and Vibration, vol. 6, 975, p Phae Shift Forward wave 5 Attenuation Bacward wave 5 Phae Shift Bacward wave Fig. Effect of Mach Number ISSN: Page ISBN:

5 5th WSEAS Int. Conf. on FLUID MECHANICS (FLUIDS') Acapulco, Mexico, January 5-7, Da=5 Da= 7 6 Da= 5 Da=. 5 Attenuation Forward wave Phae Shift Forward wave. 5 Phae Shift Forward wave... 5 Attenuation Bacward wave 6 5 Attenuation Bacward wave 6 5 Phae Shift Bacward wave Fig. Effect of Darcy Number 5 Phae Shift Bacward wave Fig. Effect of Poroity ISSN: Page ISBN:

6 5th WSEAS Int. Conf. on FLUID MECHANICS (FLUIDS') Acapulco, Mexico, January 5-7, Phae Shift Forward wave Phae Shift Forward wave 5 5 Attenuation Bacward wave Attenuation Bacward wave Phae Shift Bacward wave Fig. Effect of Forchheimer 5 Phae Shift Bacward wave Fig. 5 Effect of reduced frequency ISSN: Page 5 ISBN: