On the equivalence of the Ffowcs Williams - Hawkings and the non-uniform Kirchhoff equations of aeroacoustics

Transcription

1 Pape Numbe 48, Poceedngs of ACOUTIC Novembe 011, Gold Coast, Austala On the equvalence of the Ffowcs Wllams - Hawkngs and the non-unfom Kchhoff equatons of aeoacoustcs Alex Znovev Matme Opeatons Dvson, Defence cence and Technolog Ogansaton, Ednbugh, 5111, Austala ABTRACT The pupose of ths stud s to compae the well-known Ffowcs Wllams and Hawkngs (FW-H) equaton of aeoacoustcs wth the non-unfom Kchhoff equaton deved pevousl b the autho The pupose of ths compason s the clafcaton of the ssue of possble equvalence of the two equatons, as the poduce equvalent pedctons fo adated sound n man cases despte havng dffeent appeaance The compason s done b econsdeng the ognal devaton b Ffowcs Wllams and Hawkngs It s shown that one set of condtons wthn the medum nsde the gd bod leads to the FW-H equaton, wheeas anothe set of condtons leads to the non-unfom Kchhoff equaton As the condtons wthn the gd bod can be chosen abtal, t s concluded that the two equatons ae equvalent and the non-unfom Kchhoff equaton epesents a new fom of the FW-H equaton INTRODUCTION The development of et acaft n the 1950s caused sgnfcant nteest fom eseaches and engnees n the ablt to pedct the nose adated b flud flow nea sufaces of fastmovng obects James Lghthll, n hs well-known pape (Lghthll 195), acheved a beakthough n undestandng the mechansm of sound adaton b tubulent flow He showed that ths mechansm can be descbed b a nonunfom wave equaton whee the souce tem s detemned b Lghthll s stess tenso whch ncludes all non-acoustc stesses n the flud Lghthll also showed that the adated sound had quadupole chaactestcs Cule (1955) extended Lghthll s theo to flud flows n the pesence of gd boundaes He showed that, n addton to Lghthll s quadupole sound, flud flow nea a gd bounda adates dpole sound whch was detemned b the foce actng upon the flow fom the bounda Ffowcs Wllams and Hawkngs (1969) extended Cule s theo to the case of movng boundaes The showed that the sound adated b a movng bounda s descbed b monopole souces on the bounda The stength of these souces s detemned b the veloct of the bounda wth espect to a statona obseve Fo a statona obect, Ffowcs Wllams and Hawkngs (FW-H) equaton s educed to Cule s equaton nce ts devaton, the Ffowcs Wllams and Hawkngs equaton has become the foundaton fo one of the most fequentl used methods of pedcton of sound adated b flud flow nea gd sufaces A bef lst of applcatons whee the FW-H equaton s utlsed ncludes otatng helcopte blades (Bentne, Faassat 003), otatng fans (Moon et al 003), and flow nea an afol (Howe 1999) Ths equaton s also used n the pedcton of nose adated b movng shps, submanes, topedoes, and shp popelles Despte beng wdel used, the FW-H theo has lacked accuate expemental vefcaton, as the obtaned expemental esults showed a dscepanc of at least a few decbels wth theoetcal pedctons (see Znovev (010) fo efeences) In vew of ths fact, Znovev and Bes (004) conducted a ctcal analss of the devaton of Cule s equaton, whee the showed that the coect applcaton of theoems of vecto analss lead to a dffeent equaton Late the obtaned equaton has been gven the name the non-unfom Kchhoff equaton b the pesent autho, as ths equaton contans Kchhoff ntegals (tatton 1941) togethe wth the souce tem whch detemnes Lghthll s quadupole sound The clam b Znovev and Bes (004) that the devaton of Cule s equaton contans eos caused obectons fom some membes of the aeoacoustcal communt (Faassat 005, Faassat & Mes 006), to whch the authos povded the esponses (Znovev and Bes 005, Znovev & Bes 006) Although the pesent autho now accepts that some ctcsm egadng examples of applcaton of FW-H and Cule s equatons s ustfed, he emans convnced that hs ognal analss of Cule s equaton s coect As both FW-H and the non-unfom Kchhoff equatons ae deved fo the same poblem wth the same assumptons, thee exsts a possblt that the ae equvalent, e the ae, n fact, dffeent foms of the same equaton The fst attempt to answe the queston whethe these equatons ae equvalent was made b the pesent autho (Znovev 010) He showed that the two equatons wee ndeed equvalent f the sum of two ntegals contanng Lghthll s stess tenso ove the gd suface was zeo These two ntegals ae equvalent to the acoustc feld adated b souces detemned b Lghthll s stess tenso and ts spatal devatves on the bounda The equvalence of the two equatons was consdeed fo a few stuatons b Znovev (010) Fst of all, t was shown that the sum of the ntegals was zeo whee Lghthll s stess tenso vanshed altogethe, fo nstance, fo lnea acoustcal waves n an deal flud Also, t was shown that, fo a weakl non-lnea flow n an deal flud nea an nfnte plane the equatons ae equvalent f the plane s statona If the plane was vbatng o fo flow n a vscous flud, the sum of the Acoustcs 011 1

2 -4 Novembe 011, Gold Coast, Austala Poceedngs of ACOUTIC 011 two ntegals could be, n geneal, non-zeo, and a moe detaled nvestgaton was shown to be equed fo a defnte concluson As Cule s and the FW-H equatons ae deved b dffeent methods, t has become necessa to consde the equvalence of the FW-H and the non-unfom Kchhoff s equatons sepaatel fom the analss conducted n Znovev (010) fo Cule s equaton In ths pape, the devaton of the FW-H equaton s econsdeed wth the pupose to vef the possble equvalence of ths equaton and the non-unfom Kchhoff equaton Ths pape has the followng stuctue Fst, the ognal devaton of the Ffowcs Wllams and Hawkngs equaton s consdeed econd, t s shown that the bounda condtons on the nne sde of the gd suface can be chosen abtal Thd, t s demonstated that a dffeent set of bounda condtons leads not to the FW-H equaton, but to the nonunfom Kchhoff equaton Last, t s concluded that these two equatons ae equvalent THE DERIATION OF THE FFOWC WILLIAM AND HAWKING EQUATION To deve the equaton, Ffowcs Wllams and Hawkngs (1969) consdeed a volume of flud,, bounded b a suface, (Fgue 1) Due to the movng bounda, fo each egon the ate of change of mass equals the sum of mass flows though the sufaces and ( 1,) ρ d ( 1,) () ( ) ( 1, ) ( ) ( 1, ρ ) u ld u v nd, In Equaton (), u ae components of the vecto of flud veloct, whee the ndex 1,,3 efes to one of the axes n the thee-dmensonal Catesan space ummaton ove epeatng ndces s assumed The ate of change of the total flud mass wthn the volume can now be wtten as ρd ( ρu) ld+ (3) ( ) ( ) ρ( ) () 1 u v nd u v nd Accodng to the dvegence theoem, the flux of a vecto ove a closed suface equals the ntegal of the dvegence of the vecto ove the volume bounded b the suface B applng the dvegence theoem to the fst tem on the ght n Equaton (3) the followng can be obtaned ρ + ( ρu ) d x (4) ( ) ( ) ρ( ) () 1 u v nd u v nd l 1 v n Rewtng Equaton (4) fo an elementa volume of the flud leads to the mass consevaton law fo ths volume ρ ( ) ( ) ( + ρu ) ρ u v nδ ( ) x (5) ( ) () 1 u v nδ ( ) whee δ() s Dac s delta-functon detemned fo the suface δ δ xx x (6) ( ) ( ),, 0 0 Fgue 1 Laout of the flud volume,, wth the suface of dscontnut, The volume s dvded nto egons 1 and b a closed suface,, movng towads the egon wth the veloct, v Note that the suface s a suface of dscontnut, e souces of mass and momentum can be located on, so that the vectos of mass and momentum flows can become dscontnuous Assume that l s the outwad nomal to, and n s the nomal to dected fom the egon 1 towads the egon As n the ognal devaton b Ffowcs Wllams and Hawkngs, n the analss that follows the supescpts 1 and efe to values n the coespondng egons, and an oveba mples that the vaable s defned n the total volume If ρ s the flud denst and t s tme, then the ate of change of mass n the volume equals the sum of ates of change of mass n the two egons () 1 ( ) ρd ρ d + ρ d (1) The momentum consevaton law fo the elementa volume can be deved analogcall It has the followng fom ( ρu) + ( ρuu + p) x ( ) ( ) (7) p ρu u v + nδ ( ) ( ) () 1 p ρu u v + nδ ( ) Hee p s the compessve stess tenso (Lghthll 195) u u u k p pδ + µ + δ, (8) x x 3 xk whee µ s the coeffcent of vscost, p s the pessue, and δ s Konecke s delta Note that Equatons (5) and (7) ae deved fo a geneal flud volume The onl assumptons used dung the devaton ae the valdt of mass and momentum consevaton laws evewhee n the flud except on the suface To deve an equaton fo a gd suface n the flud, t s necessa to make addtonal assumptons egadng the bounda condtons on the gd suface Acoustcs 011

3 Poceedngs of ACOUTIC Novembe 011, Gold Coast, Austala Followng Ffowcs Wllams and Hawkngs (1969), assume that the egon 1 s the egon nsde the gd suface Due to the gdt of the suface t can be consdeed mpenetable fo the suoundng flud Theefoe, t s logcal to consde the fst bounda condton to be the equalt of the nomal component of the flud n the outsde egon and that of the suface ( ) u v (9) n n In addton, Ffowcs Wllams and Hawkngs (1969) assume that the flud nsde the gd obect (e n the egon 1) s at est and, as a esult, all themodnamc vaables n that egon have the equlbum values Ths leads to the followng bounda condtons fo the egon 1 ( 1) ρ ρ0 const, (10) ( 1) p 0, (11) whee p s ntepeted as the dffeence of the stess tenso fom ts mean value Afte the substtuton of Equatons (10) and (11) nto Equatons (5) and (7), the latte two equatons take the followng fom ρ + ( ρu) ρ0 vδ ( ) n, (1) x ( ρu ) + ( ρuu + p) pδ ( ) n (13) x B elmnatng ρ u fom Equatons (1) and (13) one can obtan Ffowcs Wllams and Hawkngs equaton n the dffeental fom c ( ρ ρ0 ) x (14) T ( pδ ( ) n ) + ( ρ0 vδ ( ) n ), xx x whee T s Lghthll s stess tenso T ρuu + p c ρ ρ δ (15) ( ) 0 Equaton (14) s a non-unfom wave equaton fo flud denst fluctuatons whee the ght-hand pat detemnes the acoustc souces n the flud The soluton of such an equaton can be wtten va the ntegal of the souce tem ove the flud volume Wth the assumpton of low Mach numbes (v/c<<1), such a soluton fo Equaton (14) takes the followng fom (Ffowcs Wllams and Hawkngs 1969) 4 πc ρ ( x, t) T ( t c) d ( ) xx (16) p ( t c) n d ( ) + x vn ( t c) ρ0 d ( ), whee ρ ( x, t) ρ( x, t) ρ ae denst fluctuatons, 0 x s the dstance between the souce pont and the obsevaton pont x Equaton (16) s Ffowcs Wllams and Hawkngs equaton n the ntegal fom The fst tem n the ght-hand pat detemnes Lghthll s quadupole souces n the flud volume, the second tem s esponsble fo dpole souces on the suface due to foces actng between the flud and the suface, and the thd tem descbes the monopole souces due to the suface moton ARBITRARINE OF BOUNDARY CONDITION ON THE INNER IDE OF THE RIGID URFACE Fom the stat of the devaton, the bounda s assumed to be the suface of dscontnut, and ndeed, Equatons (9), (10) and (11) consdeed togethe mpl dscontnuous vectos of mass and momentum flows on, as the nomal components of these vectos ae, n geneal, dffeent on both sdes of At the same tme, the condton of the mpenetable bounda detemned b Equaton (9) s, n fact, the defnton of the gd bounda and consdeng ths condton onl s suffcent to solve an poblem of sound adaton and scatteng b a gd bounda The condtons detemned b Equatons (10) and (11) on the nne sde of the gd bounda ae dffeent n ths espect The souces of mass and momentum, whch ae allowed on the suface, can account fo an pessue and veloct felds on the nne sde of and ths wll not affect the bounda condton on the oute sde of detemned b Equaton (9) It s possble to sa that the flud nsde s effectvel shelded fom the flud outsde b the souces of mass and momentum on ouces of mass u n () 0 v n 0 u n (1) 0 1 Fgue De-couplng of the veloct felds on both sdes of the suface due to souces of mass on The ndependence of the veloct felds n the egons 1 and s demonstated n Fgue 1 The Fgue shows that the nomal veloct of the suface and that of the flud n the egon (the oute sde of ) ae equal accodng to Equaton (9) Note that these veloctes ma not be zeo, as shown n Fgue, but the must be equal At the same tme, the nomal veloct n the egon 1 (the nne sde of ) s not zeo Ths dscontnut n the nomal veloct can be attbuted to the souces of mass n the egon 1 These souces of mass affect onl the feld n ths egon, as an flow of mass fom ths souces towads the egon s pohbted b the condton of the gd bounda (Equaton (9)) A smla agument can be put fowad about the momentum flows, o, whch s the same, the stess felds n the flud An dffeence n stess felds on both sdes of can be compensated b foce actng fom the suface on the flud Acoustcs 011 3

4 -4 Novembe 011, Gold Coast, Austala Poceedngs of ACOUTIC 011 Ths agument leads to the nevtable concluson that the two egons should be consdeed as decoupled and nonnteactng, whch means that the bounda condton nsde should have no beang on the feld outsde As a esult, the choce of bounda condtons on the nne sde of the bounda should not affect the esult of ths devaton, e the equaton fo the sound wave adated b the flud flow nea ths bounda ALTERNATIE BOUNDARY CONDITION AND THE NON-UNIFORM KIRCHHOFF EQUATION Consde, fo nstance, the bounda condtons on the nne sde of the gd suface, whch mpl contnuous nomal component of veloct and the compessve stess tenso acoss the suface uch condtons ae commonl used at boundaes between two meda, as, n man cases, thee ae no souces of mass and momentum on such boundaes The contnuous bounda condtons on the suface can be wtten as () 1 ( ) u u v (17), n n n () 1 ( ) p p (18) Equatons (17) and (18) wll now take place of the Equatons (10) and (11) Afte the substtuton of Equatons (17) and (18) nto Equatons (5) and (7), the ght-hand pats of these equatons vansh, and the followng equatons wll take place of Equatons (1) and (13) ρ + ( ρu ) 0, (19) x ( ρu ) + ( ρuu + p) 0 (0) x B elmnatng, as above, ρ u fom Equatons (19) and (0) t s possble to obtan the followng non-unfom wave equaton fo denst fluctuatons T c ( ρ ρ0 ) (1) x xx Accodng to well-known theoems of the potental theo (Kon 1971), the soluton of such an equaton n the pesence of gd boundaes can be wtten as 4 πc ρ x, t xx c ( ) ( ) ( t c) 1 ρ n ( ) ( ) ( ) ( ) c t c d T t c d ρ n d + () Note that, dung the devaton of Equaton (), devatves n the fst tem n the ght-hand pat have been ntechanged due to the popet of such ntegals mentoned b Ffowcs Wllams and Hawkngs (1969) Equaton () s the non-unfom Kchhoff equaton It s clea that t dffes n ts appeaance fom the FW-H Equaton (16) Wheeas the fst tems n the ght-hand pats of these equatons, whch ae esponsble fo Lghthll s quadupole sound, ae dentcal, the second and the thd tems ae dffeent In the FW-H equaton, the second and thd tems ae detemned b the total foce and the total nomal veloct on the bounda espectvel In the non-unfom Kchhoff equaton, the second and thd tems ae detemned b the denst fluctuatons and the nomal devatve on the bounda o, n othe wods, b acoustc components of the foce and the nomal veloct Despte the dffeent appeaance, the FW-H and the nonunfom Kchhoff equatons ae deved wth equvalent bounda condtons Theefoe, the do not contadct each othe and, n fact, ae dffeent foms of the same equaton Ths statement s the man esult of ths pape Beng equvalent, both equatons can be used fo pedctng the nose adated b gas o flud flows nea sold boundaes As stated above, examples of such flows ma be found n vaous cvlan and mlta applcatons n a and wate Wheeas n dffeent applcatons an of these equatons can be moe convenent fo use than the othe, an nvestgaton of ths queston fo an specfc examples s consdeed to be outsde of the scope of ths wok CONCLUION In ths pape, the ognal devaton of the Ffowcs Wllams and Hawkngs equaton s consdeed It s shown that the bounda condtons used b these authos on the nne sde of the gd bounda can be chosen abtal, as the egons outsde and nsde the gd bounda ae shelded fom each othe b the bounda It s also demonstated that, f the bounda condtons mplng contnuous themodnamc vaables acoss the bounda ae used, the devaton leads to the non-unfom Kchhoff equaton As a esult, the concluson about the equvalence of the two equatons s made Due to the equvalence, these two equatons can be used fo pedctng flow nose paametes n smla ccumstances The queston as to whch of the equatons s moe applcable to an patcula flud flow eques futhe nvestgaton REFERENCE Bentne, K & Faassat, F 003, 'Modelng aeodnamcall geneated sound of helcopte otos', Pogess n Aeospace cences, vol 39, no -3, pp Cule, N 1955, 'The Influence of old Boundaes upon Aeodnamc ound', Poceedngs of the Roal ocet A, vol 31, pp Faassat, F 005, 'Comments on the pape b Znovev and Bes On acoustc adaton b a gd obect n a flud flow ', Jounal of ound and baton, vol 81, pp Faassat, F & Mes, MK 006, 'Futhe comments on the pape b Znovev and Bes, On acoustc adaton b a gd obect n a flud flow ', Jounal of ound and baton, vol 90, pp Ffowcs Wllams, JE & Hawkngs, DL 1969, 'ound geneaton b tubulence and sufaces n abta moton', Phlosophcal Tansactons of the Roal ocet of London A, vol 64, pp Howe, M 1999, 'Talng edge nose at low Mach numbes', Jounal of ound and baton, vol 5, pp Kon, GA & Kon, TM 1971, Mathematcal handbook fo scentsts and engnees, McGaw-Hll, New Yok Lghthll, MJ 195, 'On sound geneated aeodnamcall I Geneal theo', Poc Ro oc A, vol 1, pp Moon, YJ, Cho, Y & Nam, H 003, 'Computaton of unstead vscous flow and aeoacoustc nose of coss flow fans', Computes & Fluds, vol 3, pp Acoustcs 011

5 Poceedngs of ACOUTIC Novembe 011, Gold Coast, Austala tatton, JA 1941, Electomagnetc theo, McGaw Hll, New Yok Znovev, A & Bes, DA 004, 'On acoustc adaton b a gd obect n a flud flow', Jounal of ound and baton, vol 69, no pp Znovev, A & Bes, DA 005, 'Autho s epl to: F Faassat, comments on the pape b Znovev and Bes On acoustc adaton b a gd obect n a flud flow', Jounal of ound and baton, vol 81, pp Znovev, A & Bes, DA 006, 'Authos epl', Jounal of ound and baton, vol 90, pp Znovev, A 007, 'Demonstaton of nadequac of Ffowcs Wllams and Hawkngs equaton of aeoacoustcs b thought expements', Poceedngs of IC14, Cans, Austala Znovev, A 010, 'On condtons of equvalence between Cule s and non-unfom Kchhoff equatons of aeoacoustcs', Poceedngs of ICA 010, dne, Austala Acoustcs 011 5