Department of Mechanical Engineering, University of Adelaide, Adelaide, SA 5005, Australia.

Transcription

1 APPLICATION OF FFOWC WILLIAM AND HAWKING EQUATION TO OUND RADIATION BY VIBRATING OLID OBJECT IN A VICOU FLUID: INCONITENCIE AND THE CORRECT OLUTION. Alex Znovev Department of Mechancal Engneerng, Unversty of Adelade, Adelade, A 55, Australa. Emal:alex.znovev@mecheng.adelade.edu.au. 1 INTRODUCTION The ablty to predct the ampltude of a sound wave radated by a sold object n a flud flow s one of the most sgnfcant goals n aeroacoustcs. Lghthll 1 made an mportant step n achevng ths goal n 195, when he developed a theory, whch determnes that sound radated by turbulent flow n a flud wthout sold boundares has quadrupole characterstcs. hortly afterwards, Curle extended Lghthll s theory to a flow where mmoveable sold objects are present. Accordng to Curle, a sound wave radated by a flow n the presence of a sold object s the sum of the Lghthll s quadrupole sound and an acoustc wave generated by the dstrbuton of dpole acoustc sources over the surface of the object. Curle also showed that the strength of the dpole sources s proportonal to the total force per unt area on the surface. Curle s equaton can be smplfed for an acoustcally small object, for whch the ampltude of the radated sound wave s proportonal to the total force actng upon the flow from the object. Many practcal approaches to the sound radaton problem are based on an equaton derved by Ffowcs Wllams and Hawkngs 3. Ths equaton s more general than Curle s equaton and descrbes flow around a sold object, whch moves at an arbtrary speed. Unlke Curle s equaton, Ffowcs Wllams and Hawkngs (FWH) equaton contans a monopole term, whch depends on the velocty of the object wth respect to a statonary observer. At the same tme, the man conluson of Curle about the dpole characterstcs of the radated sound remans unchanged n the Ffowcs Wllams and Hawkngs theory, and for an mmoveable object the FWH equaton reduces to Curle s equaton. However, despte beng wdely accepted, the Curle Ffowcs Wllams Hawkngs theory has never been relably verfed by experments. Earler attempts 4,5 showed dscrepances of a few db between expermental data and theoretcal predctons, whch was attrbuted to mperfectness n carryng out the experments. On the other hand, two recent experments 6,7 demonstrated that expermental dependence of the total radated sound power on the flow speed has a dfferent slope compared wth the dependence predcted by Curle s equaton. Dfferences n slopes were

2 observed for varous sets of expermental parameters, thus makng t unlkely that the dscrepancy was smply a lack of expermental accuracy. To explan the observed dscrepancy, the author and hs co-author carred out a detaled analyss of Curle s equaton 8. The analyss showed that n Curle s calculatons the contrbuton of the dscontnuty of hydrodynamc stresses on the rgd surface to the radated sound has been erroneously omtted. By takng ths dscontnuty nto proper consderaton, an alternatve equaton was obtaned, whch dffers from Curle s equaton n two mportant aspects. Frst, the strength of the dpole sources on the surface on the object depends on the acoustc component of pressure rather than on the total pressure. econd, contrary to Curle s equaton, the obtaned equaton contans a monopole term, for whch the strength s determned by the relatve moton of the object and the flud. To demonstrate the dfference between the two equatons, they were appled to two well-known problems of sound scatterng and radaton by a rgd sphere. It was shown that Curle s equaton gves predctons, whch are n dsagreement wth results known from the lterature, whereas predctons gven by the obtaned equaton concde wth the known results. In the present work a detaled reconsderaton s gven to the more general FWH equaton. It s shown, that the algorthm used for obtanng ths equaton, n fact leads to an equaton, whch concdes wth the equaton derved by n reference 8. Dfference between the obtaned equatons and the FWH equaton s demonstrated by an example of a thn plate vbratng n ts own plane n a flud. New methodologes of actve nose control, whch can be based on the obtaned equaton, are brefly dscussed.

3 DERIVATION OF AN EQUATION FOR THE RADIATED OUND AMPLITUDE To obtan an equaton for the ampltude of a sound wave, radated from a movng body n a flud flow, an analytcal approach s utlsed here, whch s analogous to the approach used by Ffowcs Wllams and Hawkngs 3. A fxed volume of flud, V, enclosed by a surface, Σ, s consdered (see Fgure 1). The volume V s dvded nto regons 1 and by a closed surface of dscontnuty,, movng nto regon wth velocty, v. The velocty of the flud s represented by u. The outward normal from V s l, and normal to drected from regon 1 to regon s n. The subscrpts 1 and refer to the two regons, and an overbar mples that the varable s regarded as a generalsed functon vald throughout V. Σ V 1 v l Fgure 1. n If ρ denotes the flud densty, the mass conservaton law for the volume V can be formulated as follows: ρdv = ( ρu) ldσ + ρ( u v) n ( ), V d ρ u v n 1 d Σ (1) = 1,,3. By means of the dvergence theorem, the ntegral over the closed surface Σ can be represented as a volume ntegral over V, and equaton (1) takes the followng form: ρ ( ) ( ) ( ). V + ρu dv = ρ u v nd ρ u v nd 1 () It s necessary to emphasze that, for obtanng equaton (), no assumptons have been made about the composton of the momentum feld ρu. Generally, however, ths feld can be separated nto a ental, ( ρu ), and a solenodal, ( ρu ) sol, components whch can each be represented respectvely as sol ( ρu) = ψ, ( ρu) = A, (3) where ψ and A are the scalar and vector entals respectvely. Potental and solenodal felds can be also understood usng the concept of streamlnes 9. These are lnes such that the tangent to a streamlne n any pont gves the drecton of the velocty at ths pont. Accordng to equatons (3), curl of the ental component s zero and, therefore, ts ntegral over a closed contour s zero as well. Consequently, the streamlnes for the ental component cannot be closed and must have the begnnng and the end at acoustc sources or n nfnty. Conversely, for the solenodal component the dvergence s zero, the ntegral over a

4 closed contour s not zero and, consequently, the streamlnes for the solenodal component are closed lnes. As the solenodal component s a curl of a vector, ts flux through the closed surface s zero. Consequently, equaton () s ambguous wth respect to the feld ρu; t s vald not only for the total feld ρu, but also for the ental component of the feld, ( ρu ). Ths ambguty becomes mportant at the next step of dervaton. The authors of reference 3 concluded that expressons under the ntegral n equaton () can be made equal for every pont of volume V, leadng to the followng equaton: ρ + ( ρu ) ( ( ) ( ) ) ( ) = ρ u v ρ u, v l 1 δ (4) where δ( ) δ( ) r r s s a three-dmensonal delta-functon, and s r s a radus vector of a pont belongng to the surface. However, t can be proven that equaton (4) s vald only for the ental component, ( ρu ), and cannot nclude the solenodal component, ( ρu ) sol. On the one hand, the solenodal component, the second term n the left-hand part of equaton (), whch s the dvergence of the vector ( ρu ) sol, s zero at every pont of the volume V. On the other hand, the expresson n the rght-hand part s zero under the ntegral only and may dffer from zero at dfferent ponts of the surface. Therefore, equaton (4) should be, n fact, wrtten as ρ + ( ρu ) ( ( ) ( ) ) ( ) = ρ u v ρ u, v l 1 δ (5) If regon 1 s an absolutely rgd body, the followng boundary condtons are satsfed 3 : sol u l, 1 = u l 1 = (6) ρ = ρ, p = p, (7) [ ] [ ] 1 1 sol +, = u u l vl (8) where ρ and p are the flud densty and pressure n the state of equlbrum. Wth the use of equatons (6), (7) and (8), equaton (5) takes the followng form: ρ sol + ( ρu ) ( [ ] ) ( ) = u ρ + ρ v l. 1 δ (9) Let the analyss be restrcted to a lnear approxmaton, for whch fluctuatons of densty and velocty are small. In ths case the followng condtons are satsfed: ρ ρ ρ, << (1) sol u << c, u << c, (11) where c s the sound speed n the flud. Takng account of equatons (1) and (11) and droppng the ndex, equaton (9) s reduced to the followng equaton: ρ + ( ρu ) ( ) = ρ u lδ, (1)

5 whch s a generalsed equaton of contnuty. An equaton, representng the momentum conservaton law, for the case under consderaton takes the followng form: ( ρu) ( ρuu j pj) dv V + + j (13) ( ρ ( ) ρ ( ) ) =, pj + u uj vj pj + u uj vj ljd 1, j = 1,,3, where a compressve stress tensor, p j, s determned by 9 : u u j u k pj = pδ j + µ + δj, xj x 3 x k and µ s the vscosty of the flud. By means of an argument analogous to the argument used n the dervaton of equaton (1), t can be proven that equaton (13) must nclude only the ental component of the tensor p j, f equaton (13) s wrtten for every pont of volume V. Thus, neglectng the nonlnear terms n the rght-hand part, equaton (13) can be reduced to the followng: ( ρu ) ( ) ([ ] [ ] ) ( ) + ρuu j + pj = p p l. 1 jδ (15) j Wth the use of equaton (7), equaton (15) takes the followng form: ( ρu ) ( ) ( ) ( ) + ρuu j + pj = p p lδ, (16) j whch represents a generalsed momentum equaton. Excludng ρ u from equatons (1) and (16), one can obtan the followng nhomogeneous wave equaton: Tj ( ( ) c ) ( ( ) pl ), ρ = δ + ρ u lδ (17) j where T j s Lghthll s stress tensor, determned by Tj = ρuu j + pj c ρδj, (18) and ρ and p are understood as perturbatons of the densty and the pressure from the state of equlbrum. The soluton of equaton (17) can be wrtten as the followng ntegral equaton: Tj [ p] u 4πcρ = dv( ) ld ( ) + ρ ld ( ), V r j r r (19) r r r r r r where r s the observaton pont, r s the source pont, and the quanttes n square brackets are taken at retarded tmes, t r r c. Equaton (19) represents the contrbuton of the present paper. (14)

6 3 COMPARION OF THE OBTAINED EQUATION WITH FFOWC WILLIAM AND HAWKING EQUATION Ffowcs Wllams and Hawkngs (FWH) equaton n ts ntegral form wthout nonlnear terms takes the followng form 1 : Tj pj [ v ] 4πcρ = dv ( ) ld ( ) + ρ ld ( ). V r j r r () r r r r r r Comparson of the obtaned equaton (19) and the FWH equaton () shows, that both equatons nclude dentcal volume ntegrals, whch determne Lghthll s quadrupole sound generated by turbulence n the volume V. At the same tme the second and the thrd terms n the rght-hand parts of both equatons dffer. The thrd term, whch determnes sound generated by a layer of monopoles on the surface, s dscussed n detal n reference 8. However, t may be noted brefly that equaton () cannot descrbe a smple reflecton of sound from an mmoveable object. The ncdent sound wave would have a non-zero normal component on the surface, whch, due to boundary condtons on an mmoveable surface, must be cancelled by the velocty n the reflected wave. On the one hand, ths stuaton s descrbed perfectly by equaton (19), where the strength of the monopole sources s proportonal to the normal velocty n the scattered wave, whch s equal to the normal velocty n the ncdent wave wth opposte sgn. On the other hand, n the FWH equaton () the thrd term vanshes at the mmoveable surface, and there wll be no monopole sources at all. An example 8 demonstrates, that wthout the monopole term t s not possble to determne correctly the ampltude of the reflected sound. Consder a stuaton where the sold object s a sphere of radus, R, the volume contanng turbulence s small, and ts dstance from the sphere s large n comparson wth the acoustc wavelength. In these crcumstances, the sound radated by the quadrupole sources can be consdered as a plane wave near the sphere, and the problem under consderaton reduces to the problem of a plane wave scatterng. It s well-known 9, that a monopole component wll be present n the sound feld scattered by the sphere. Obvously, equaton () cannot descrbe the monopole component, as the lowest multpole n ths equaton s the dpole. On the contrary, for the reasons mentoned above, the thrd term n equaton (19), whch descrbes the monopole component, s dfferent from zero. The second term n equatons (19) and () determnes the ampltude of sound radated by a layer of dpoles on the surface. However, the strength of the dpole sources dffers n both equatons. In the FWH equaton () the strength s determned by the compressve stress tensor n ts general form as defned n equaton (14), whle n the obtaned equaton (19) only the acoustc pressure s essental. The dfference between the dpole terms n both equatons can be demonstrated by the followng example. Let the surface be the surface of a thn rgd plate, vbratng n ts own plane n a vscous flud. If all plate dmensons are much smaller than the acoustc wavelength, the surface ntegral n the FWH equaton () can be smplfed, so that the dpole sound ampltude wll be proportonal to the total force actng upon the flud from the object. Due to the vscosty of the flud there wll be force actng upon the plate n the drecton parallel to the plate. Therefore,

7 accordng to the Ffowcs Wllams and Hawkngs theory, such a plate wll be a source of dpole radaton wth the dpole moment parallel to the plate. On the contrary, accordng to the obtaned equaton (19), there wll be no radaton from the surface of the plate, because such moton of the plate cannot cause pressure fluctuatons 9. olvng the boundary value problem drectly can prove the absence of the acoustc radaton from such a plate. Indeed, n the absence of external pressure and velocty felds the radated sound feld must satsfy the condton of zero normal velocty on the surface of the plate that, n turn, leads to the absence of sound radaton from the plate. It s to be noted, that the volume of the flud, surroundng the plate, wll stll radate sound waves. These sound waves, however, are caused by the dffuson of vortcty to the flud 9 and descrbed by Lghthll s quadrupole term rather than by the surface ntegrals n equatons (19) and (). It s also mportant to note that the FWH equaton () and the obtaned equaton (19) can gve dentcal results n some cases. However, ths concdence can be shown to be purely accdental. For example, a known formula for the ampltude of sound radated by a transversely oscllatng sphere n a vscous flud can be obtaned on the bass of the Ffowcs Wllams and Hawkngs equaton 11. At the same tme t can be shown that the obtaned equaton results n the same formula. Analyss shows that the predctons of both formulas concde only due to the sphercal symmetry of the object and, therefore, the concdence s fortutous. The obtaned equaton (19) also can be compared wth the equaton derved n reference 8. uch comparson shows that both equatons concde, although they have been derved for a movng and mmoveable object respectvely. Ths can be explaned by the fact that, at least n the lnear approxmaton, only the normal component of the velocty of the radated acoustc wave s ncluded drectly nto the equatons, whle the velocty of the object s taken nto account n the boundary condtons. 4 IMPLICATION OF THE OBTAINED EQUATION FOR NOIE CONTROL The Ffowcs Wllams and Hawkngs equaton ncludes two terms related to the sound generaton by a surface, but control of nose generaton can be based only on one of them, whch s proportonal to the stress tensor. Therefore, the FWH equaton allows only one strategy n nose control, namely, to mnmse the force actng upon the surface from the flow. Ths also makes a study of the flow structure near the surface unmportant for the purpose of nose control. Conversely, the obtaned equaton (19) has two terms determnng sound radaton by a surface even for an mmoveable object. Ths leads to a possblty to mnmse aerodynamc nose radaton by developng strateges, whch would utlse mutual cancellaton of both terms n far feld. In addton, knowledge of the velocty feld near the radatng surface becomes sgnfcant for effectve nose control.

8 5 CONCLUION In the present paper an equaton has been derved, whch determnes the ampltude of the sound wave generated by a movng rgd object n a flud flow. The argument used n the dervaton s analogous to the argument used n the dervaton of the Ffowcs Wllams and Hawkngs (FWH) equaton. It s shown that f the velocty and pressure felds n the flud are represented as a sum of ental and solenodal parts, the obtaned equaton ncludes only the ental parts of the felds. The dfference between the obtaned equaton and the FWH equaton s demonstrated by the example of sound radaton by a thn plate vbratng n ts own plane n a vscous flud. It s shown that the FWH equaton leads to the predcton of dpole radaton wth the dpole moment parallel to the plate, whle the obtaned equaton predcts the absence of such radaton, whch can be also proven by drect soluton of the boundary value problem. Implcatons of the obtaned equaton for actve nose control are dscussed. It s shown that the equaton allows new strateges of nose control, based on mutual cancellaton of dpole and monopole terms n the far feld. REFERENCE 1. M.J.Lghthll, 195, On sound generated aerodynamcally, Proc. Roy. oc. A, 11, N.Curle, 1955, The nfluence of sold boundares upon aerodynamc sound, Proc. Roy. oc. A 31, J.E.Ffowcs Wllams and D.L. Hawkngs, 1969, ound generaton by turbulence and surfaces n arbtrary moton, Phl. Trans. Roy. oc. (London) er. A, 64, P.J.F. Clark and H..Rbner, 1969, Drect Correlaton of Fluctuatng Lft wth Radated ound for an Arfol n Turbulent Flow, J. Acoust. oc. Am., 46, H.H.Heller and.e.wdnall, 197, ound Radaton from Rgd Flow polers Correlated wth Fluctuatng Forces, J. Acoust. oc. Am., 47, D.A.Bes, J.M.Pckles and D.J.J.Leclercq, 1997, Aerodynamc nose generaton by a statonary body n a turbulent ar stream, Journal of ound and Vbraton, 4, D.J.J.Leclercq and M.K.ymes,, Dense compact rgd object n a turbulent flow: Applcaton of Curle's theory, 8th AIAA/CEA Aeroacoustcs Conference, June, Breckenrdge CO, UA. 8. A.Znovev and D.A.Bes, On acoustc radaton by a rgd object n a flud flow, submtted to Journal of ound and Vbratons. 9. L.D.Landau and E.M.Lfshtz, 1959, Flud Mechancs. Volume 6 of Course of Theoretcal Physcs, Oxford: Pergamon Press. 1. M..Howe, 1998, Acoustcs of flud-structure nteractons, Cambrdge Unversty Press. 11. A.D.Perce, 1989, Acoustcs: An Introducton to Its Physcal Prncples and Applcatons, New York: Acoustcal ocety of Amerca.