DEMONSTRATION OF INADEQUACY OF FFOWCS WILLIAMS AND HAWKINGS EQUATION OF AEROACOUSTICS BY THOUGHT EXPERIMENTS. Alex Zinoviev 1

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1 ICSV14 Cains Austalia 9-12 July, 27 DEMONSTRATION OF INADEQUACY OF FFOWCS WILLIAMS AND HAWKINGS EQUATION OF AEROACOUSTICS BY THOUGHT EXPERIMENTS Alex Zinoviev 1 1 Defence Science and Technology Oganisation Maitime Opeations Division Edinbugh, SA, 5111, Austalia alex.zinoviev@dsto.defence.gov.au Abstact In 1952, Si James Lighthill published a theoy of sound adiation by a tubulent fluid flow. This theoy was extended by Cule in 1955 to a flow with immoveable igid boundaies. In 1969, Ffowcs Williams and Hawkings extended this theoy to a flow with moving igid objects and deived an equation, which has since become one of the majo tools of pediction of the flow induced noise. In 24, Zinoviev and Bies e-examined Cule s deivation and showed a theoetical inconsistency in his calculations. In this pape, the inadequacy of Ffowcs Williams and Hawkings equation fo solving poblems of sound adiation by a tubulent flow nea solid boundaies is demonstated by its application to simple thought expeiments. Fist, sound adiation by an oscillating solid sphee is compaed with sound adiation by a stationay solid sphee in a vaiable velocity field (votex steet). It is shown that, contay to the well-known equivalence of these situations, the Ffowcs Williams and Hawkings equation poduces diffeent esults in these cases. Second, it is demonstated that this equation pedicts significant acoustic adiation by a light solid object caied by a tubulent flow. Finally, sound adiation by an acoustically small igid plate vibating paallel to itself is consideed. It is shown that Ffowcs Williams and Hawkings equation pedicts significant but physically unjustified dipole acoustic adiation with the dipole axis paallel to the plate. 1 INTRODUCTION A majo goal of the aeoacoustical theoy is pediction of sound adiation by tubulent fluid flows. In 1952, Lighthill [1] poposed a theoy of sound adiation by a flow without boundaies. In his theoy, the mass and momentum consevation equations ae tansfomed to obtain a linea inhomogeneous wave equation fo the adiated sound whee the non-linea and viscous tems ae combined in the ight-hand pat in the fom of quadupole acoustic souces. In 1955, Lighthill s theoy was extended by Cule [2] to the case of a fluid flow with stationay igid boundaies. Accoding to Cule, the sound adiated by such a flow is

2 detemined, in addition to Lighthill s quadupole souces, by the distibution of total stesses, including the viscous stesses, ove the igid boundaies. The acoustic souces, detemined by the stesses, have dipole chaacteistics. Utilising a methodology diffeent fom that used by Cule, Ffowcs Williams and Hawkings [3] deived an equation detemining the sound adiation with a flow with moving sufaces. In this equation, Lighthill s and Cule s souce tems ae supplemented with the thid tem which depends on the nomal component of the velocity of the suface with espect to a stationay obseve. This tem can be intepeted as a distibution of the monopole souces on the sufaces. Fo a stationay object, Ffowcs Williams and Hawkings (FW-H) equation educes to Cule s equation. Since its deivation, the Ffowcs Williams and Hawkings equation has become the foundation fo one of the most fequently used methodologies of pediction of sound adiated by a fluid flow nea igid sufaces. A bief list of applications whee the FW-H equation is utilised includes the otating helicopte blades [4], otating fans [5], and a flow nea an aifoil [6]. This equation is also used in the pediction of noise adiated by moving ships and ship popelles. As mentioned by Faassat [7], this equation is the foundation of a helicopte noise pediction code, which is employed extensively by the helicopte industy. Zinoviev and Bies [8] have conducted a citical analysis of the histoically fist pape on sound geneation by a flow nea boundaies [2]. In that analysis, it has been shown that Cule s oiginal account neglects a non-zeo integal ove the igid suface. It has been also shown that, if this integal is taken into consideation, Cule s deivation leads to a diffeent equation, whee the geneated sound is detemined by Lighthill s quadupole tem as well as by Kichhoff s integals ove the igid sufaces [9]. The diffeences between Cule s equation and the obtained equation have been discussed. The goal of this pape is to apply the equation deived by Ffowcs Williams and Hawkings to fou cases of sound adiation. It will be demonstated that Ffowcs Williams and Hawkings equation leads to physically uneasonable esults in thee of these cases. 2 FFOWCS WILLIAMS AND HAWKINGS EQUATION Accoding to Ffowcs Williams and Hawkings [3], acoustic pessue fluctuations p ( x, t) = p( x, t) p in the acoustic wave adiated by a fluid flow nea a solid bounday, S, ae detemined by ( x, ) ( x, ) ( x, ) ( x, ) p t = p t + p t + p t = whee p(, t) quad mon dip 1 T 1 nv 1 n p d ds( ) ds( ), 4π x x y 4 t y 4 x y (1) ij j j j ij + ρ i j π π Vtot S i S x is the vaiable pessue in the fluid, p is the pessue in the fluid at equilibium, c is the speed of sound in the fluid at est, v i is the i-th component of the velocity of the bounday, n is the unity vecto nomal to the igid suface in the outwad diection, x = ( x1, x2, x3 ) is the coodinate of the obsevation point, y = ( y1, y2, y3 ) is the coodinate of the souce point, tenso p ij is the compessive stess tenso, = x-y, t is time, and squae backets indicate the dependence on the etaded time, τ = t x-y c. The 2 Lighthill s stess tenso, T = ρvv + p c ρδ whee δ ij is Konecke s delta function. ij i j ij ij,

3 The vaiable V tot is the total volume of the fluid, wheeas the suface integals ae taken ove the igid bounday, S. Indices epeated in a single tem ae to be summed fom 1 to 3. Fo example, vk xk must be undestood as v1 x1 + v2 x2 + v3 x3. Eq. (1) is deived fo v c. The vaiable p quad ( x, t) denotes Lighthill s quadupole acoustic souces distibuted in the fluid volume. This tem is ignoed in the analysis below, as this pape is concened only p, t p x, t with sound geneation on the suface of a igid object. The vaiables ( x ) and ( ) in Eq. (1) denote sound pessue geneated by layes of elementay monopoles and dipoles on the igid suface espectively. It is impotant to emphasize that the stength of the dipole laye is detemined by the total stess tenso p ij, which includes both viscous and acoustic stesses, wheeas the stength of the monopole laye is detemined by the velocity v with espect to a stationay obseve. mon dip 3 APPLICATION OF FFOWCS WILLIAMS AND HAWKINGS EQUATION TO PROBLEMS OF SOUND RADIATION 3.1 Equivalence of an oscillating sphee and a stationay sphee in a vaiable velocity field An oscillating sphee Conside a igid sphee of adius, R, vibating with angula fequency, ω, without changing its fom (Figue 1). U e i ω t θ R If the sphee is acoustically small, so that ( ω ) kr = c R, and the amplitude of the 1 velocity of the sphee is U, the complex amplitudes of the nomal component of the velocity of the sphee, v n, and the total pessue field on the suface of the sphee, P tot, ae known to be detemined by [1] P Figue 1. Oscillating sphee. tot ( θ ) U cos, θ vn = (2) = R 1 θ = ρ cu ikrcos θ. (3) = R 2 ( )

4 Substitution of Eqs. (2) and (3) into the FW-H Eq. (1) leads to the following equations fo the monopole and dipole tems of the acoustic pessue in fa field of the oscillating sphee [11]: ρ U ω R e p = OS, mon 2 3 ik x cos θ, (4) 3c 2 3 ik ρuω R e ( ) ( ) p OS, dip x = cos θ, (5) 6c whee is the distance between the obsevation point and the cente of the sphee. As a esult, the total adiated field is detemined by ρ U ω R e p = OS ( ) 2 3 ik x cos θ, (6) 2c Since Eq. (6) is well-known [1], the FW-H equation leads to the coect pediction in this case A stationay sphee in a vaiable velocity field Landau and Lifshitz [1], in Section 78 that is devoted to sound scatteing by a stationay object, use the fomulae deived peviously in this efeence fo sound adiation by a moving object. As tanslated fom Russian by the pesent autho, they state that the diffeence is only in the fact that, instead of motion of an object in a fluid, we now deal with motion of a fluid with espect to an object. These two poblems ae, obviously, equivalent. This equivalence can be undestood as the dependence of the sound wave adiated by a igid object only on the elative motion of the object and the fluid (if the involved velocities ae much smalle than the speed of sound). Theefoe, it can be concluded that the poblem of sound adiation by the vibating sphee is equivalent to the poblem of sound adiation by a igid sphee in a vaiable velocity field of the same amplitude. This situation can be ealised, fo example, if the sphee is immesed into a votex steet (Figue 2) that consists of otating votices moving towads the sphee with the aveage flow velocity, V. V 2R Figue 2. A sphee in a votex steet.

5 If the velocity fluctuations in the votex steet ae much smalle than the sound speed, the pessue fluctuations in such a flow can be neglected. In addition, if the diamete of the sphee is much smalle than the size of the votex, the incident flow velocity can be consideed appoximately the same on the whole suface of the sphee at any given moment of time. Due to the equivalence of this case to sound adiation by an oscillating sphee consideed above, the pessue on the suface of the sphee in this case is also detemined by Eq. (3), and the dipole component of the acoustic field is equal to the dipole component of the field in the case of the oscillating sphee (Eq. (5)). Howeve, since the monopole tem in Eq. (1) is detemined by the velocity of the sphee with espect to a stationay obseve and the sphee is stationay, the monopole tem vanishes, so that the total adiated acoustic field is detemined by SS 2 3 ik x = SS, dip x = cos θ. (7) 6c ( ) p ( ) p ρ U ω R e Eqs. (6) and (7) show that, accoding to the FW-H equation, the amplitude of the sound wave adiated by a stationay sphee in a vaiable velocity field is thee times smalle than the amplitude of the sound wave adiated by an oscillating sphee in a stationay fluid. This diectly contadicts the well-known notion that these two cases ae equivalent [1]. 3.2 A solid object embedded in a tubulent flow Conside a solid object suspended in a tubulent flow. If the object is light and the foce of gavity is neglected, the object will be caied by the flow. If the size of the object is smalle than the scale of velocity fluctuations in the flow, such an object can be said to be embedded in the flow (Figue 3) so that the velocity of the object is equal to the velocity of the flow. It has to be noticed that this object diffes fom a fluid paticle in that espect that it cannot change its shape and its bounday is impenetable. Figue 3. A light solid object embedded in a tubulent flow. The application of the FW-H equation to this case shows that this equation pedicts significant noise adiation fom such an object. Indeed, as the velocity of the object is not zeo and may expeience shap changes, the monopole tem in Eq. (1) detemines adiation of acoustic noise of consideable amplitude. At the same time, the foce on the object fom the flow is negligibly small and the dipole tem in Eq. (1) vanishes. Theefoe, the total sound adiation fom the object is not zeo accoding to the FW-H equation. The absence of acoustic adiation fom an embedded object can be demonstated in the following way. Since the velocity of the object is equal to the velocity of the flow, the object is not an obstacle fo the flow, and thee is no foce which could compess the fluid on the suface of the object thus leading to adiation of sound. Theefoe, the FW-H equation leads to incoect pedictions fo the adiated sound amplitude in the case of a light solid object embedded in a tubulent flow.

6 3.3 Oscillations of a thin plate in a viscous fluid Deivation In this section, the FW-H equation is applied to a thin ound igid plate of adius, R p, oscillating in a viscous fluid in paallel to itself (Figue 4). It will be shown below that the FW-H equation leads to pediction of dipole acoustic adiation with the axis of the dipole paallel to the plate. U e i ω t 2R p θ x Since the nomal component of the plate velocity is zeo, the monopole tem in Eq. (1) vanishes. By substituting the hamonic tempoal dependence, e iωt, to the FW-H Eq. (1) this equation can be educed to ik 1 e p ( x) = ( ) ds( ), 4π P y y (8) S whee P( y ) is the vecto detemining the total foce pe unit aea on the suface of the plate, and the opeato = ( x, x, x ). The plate is consideed to be acoustically small, so that Figue 4. Oscillating thin plate in a viscous fluid In fa field, the following appoximation can be made: kr 1. (9) p ik ik ik ik e e e e = ik ik, 2 (1) whee is the unity vecto in the diection of the vecto = x y. Accoding to Landau and Lifshitz [1], the foce on the vibating plate is tangential and its value pe unit aea is detemined by

7 ωηρ P( y ) = ( i 1 ) U, (11) 2 whee η is the viscosity of the fluid. Theefoe, the expession unde the integal in Eq. (8) can be educed to ik ik ik ik e e e e P( y) = P( y) ik P( ) ik cos U ωηρ = y θ = ( i + 1) k cos θ. (12) 2 If Eq. (12) is substituted into Eq. (8), the following equation fo the sound pessue adiated by the plate is obtained: ik 1 2 ωηρ e p PL = UkRp ( i+ 1) cos θ. (13) Discussion Equation (13) shows that, accoding to the FW-H equation, the plate vibating paallel to itself in a viscous fluid is a dipole souce of sound with the dipole axis paallel to the plate. Compaison of Eq. (13) with Eq. (6) fo an oscillating sphee shows that the sound adiation by the plate is not negligible. Fo example, simple calculations show that in the ai at nomal conditions at the fequency of 1 Hz an oscillating plate 8 cm in diamete should, accoding to the FW-H equation, adiate sound of the same intensity as a sphee 1 cm in diamete oscillating with the same amplitude. To confim that the sound adiated by such a sphee, and consequently by the plate, is significant, the following values have been substituted into Eq. (6): ρ = 1.23 kg / m, U =.628 m / s, ω = 2π f 628 s, c = 33 m / s 3 1 R = m = m k = c m = 1.5, 7, ω 1.9, θ. (14) Fo the above paametes, Eq. (6) gives the amplitude of the sound equal to 21.3 db e 1µPa. This sound is not negligible and should be audible by a peson. Note that the sound will be loude at smalle distances fom the sphee. The value of = 7 m has been chosen to make sue that the obsevation point is located in fa field ( k 1) so that Eq. (6) could be applied. The actual absence of sound adiation in this case is clea fom the fact that stesses existing at the suface of the plate ae tangential to the plate. The oscillations of the plate do not ceate any egions of compession and aefaction, which could popagate as an acoustic wave. Theefoe, it can be concluded that the Ffowcs Williams and Hawkings equation gives eoneous pedictions in the case of a thin plate oscillating in a viscous fluid.

8 4 CONCLUSIONS In this pape, the Ffowcs Williams and Hawkings (FW-H) equation has been applied to fou cases of sound geneation by oscillating igid objects in a fluid. It has been shown that the FW-H equation leads to incoect pedictions in thee cases. Fist, sound adiation by an oscillating igid sphee has been consideed. It has been confimed that the FW-H equation gives coect pediction of adiated sound in this case. Second, the FW-H equation has been used to calculate sound adiation by a stationay sphee in a vaiable pessue field. Such a sphee has been modelled as a sphee located in a votex steet. It has been shown that, wheeas this case is equivalent to the fist one, the FW-H equation leads to sound pessue amplitude thee times smalle in this case. Thid, the FW-H equation has been applied to a light igid object embedded in a tubulent flow. This equation has been shown to pedict that this object adiates acoustic noise of significant amplitude. The absence of such noise in eality has been demonstated by physical consideations. Finally, oscillations of a igid thin plate in paallel to itself have been consideed. Detailed calculations have shown that, accoding to the FW-H equation, such oscillations adiate sound of significant amplitude. It has been also demonstated that, in eality, such oscillations cannot lead to sound geneation fom the suface of the plate, as they do not poduce compession and aefaction of the fluid. Oveall, it has been shown that the Ffowcs Williams and Hawkings equation is not adequate fo solving eal-life poblems of sound adiation by fluid flows nea solid boundaies. REFERENCES [1] M.J. Lighthill, On sound geneated aeodynamically, Poc. Roy. Soc. A 211, (1952). [2] N. Cule, The influence of solid boundaies upon aeodynamic sound, Poc. Roy. Soc. A 231, (1955). [3] J.E. Ffowcs Williams and D.L. Hawkings, Sound geneation by tubulence and sufaces in abitay motion, Phil. Tans. Roy. Soc. A 264, (1969). [4] K.S. Bentne and F. Faassat, Modeling aeodynamically geneated sound of helicopte otos, Pogess in Aeospace Sciences 39, (23). [5] Y.J. Moon, Y. Cho and H. Nam, Computation of unsteady viscous flow and aeoacoustic noise of coss flow fans, Computes & Fluids 32, (23). [6] M.S. Howe, Tailing edge noise at low Mach numbes, Jounal of Sound and Vibation 225, (1999). [7] F.Faassat, Acoustic adiation fom otating blades the Kichhoff method in aeoacoustics, Jounal of Sound and Vibation 239, (21). [8] A. Zinoviev, D.A. Bies, On acoustic adiation by a igid object in a fluid flow, Jounal of Sound and Vibation 269, (24). [9] J.A. Statton, Electomagnetic theoy, McGaw Hill, New Yok, [1] L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Volume 6 of Couse of Theoetical Physics, Pegamon Pess, Oxfod, [11] W.K. Blake, Mechanics of flow-induced sound and vibation, Academic Pess, Olando, 1986.