16.50 Lectue 36 Subjects discussed: Aicaft Engine Nise : Pinciples; Regulatins Nise geneatin in the neighbhds f busy aipts has been a seius pblem since the advent f the jet-pweed tanspt, in the late 1950's. Althugh pistnengined aicaft had caused sme cncens pi t that, it was the tubjet-pweed fist geneatin jets (707, Cmet, DC-8) with thei jet nise that led t wide public cncen, and t egulatins by sme aipt authities. With the cntinuing gwth f ailine tavel, the pblem has cntinued t expand, leading t ules pmulgated by the FAA that limit the nise that any individual aicaft can make at each f 3 measuing statins. Fm Keebck, Jack L. Aicaft Engines and Gas Tubines. 2nd editin. MIT Pess, 1992. Massachusetts Institute f Technlgy. Used with pemissin. Nise is the human ea's espnse t pessue fluctuatins in time, at the ea f the bseve. As such it has bth physical and psychlgical aspects. Thus, ne pesn's music can be anthe pesn's nise. Thee is geneal ageement f example that the exhaust nise f mtcycles is annying, but the bike will nt agee. We will fm this pint n avid the psychlgical and cncentate n the physical aspects f aicaft nise, withut meaning t imply that the latte is me imptant. Nise is tansmitted fm the suce (aicaft) t the bseve by sund waves. S let us begin with a bief eview f sund ppagatin. Cnsevatin f mass Cnsevatin f mmentum D! +!" u = 0 (1) Dt Du! = "#p (2) Dt Suppse the velcity and pessue t be the sum f a lage steady cmpnent and a small time-vaying ne 1
!! u = u + u! 0 ' (t) p=p + p'(t) ρ=ρ + ρ'(t) and futhe suppse that u! 0 = 0. Then t fist de in small quantities (1) and (2) becme!"'! +" 0 #. u ' = 0 (3)!t! "u '! 0 = #$p' (4) "t In the absence f heat cnductin and viscsity, ρ' and p' ae elated isentpically, i.e. It fllws that whee a 0 is the speed f sund. p 2! dp dp = ( 2 ) " p' "' # = " =! p 1! 1 p 0! 0 p 0 p 0 p p' = γ 0!' = "RT 0!' # a!' 2 (5)! 0 S in 1a Nw taking! (1a) -!.(2a) we have!t!"' 1!p' =!t a 2 0!t In ne dimensin (f a plane wave) 1! 2 p' 2 2 " # 2 p' = 0 (6) a 0!t 1! 2 p'! 2 p " = 0 a 2 0!t 2!x This is a Wave Equatin, satisfied in geneal by slutins f the fm p' = p'(x ± t), s that p' is cnstant f x = ± t. Thus the slutin f a plane wave wuld be f the fm p! = P csk(x " a 0 t) Excitatin We can nw ask hw such waves ae geneated in an engine. Sme f the main sund suces ae schematized belw: 2
Fm Keebck, Jack L. Aicaft Engines and Gas Tubines. 2nd editin. MIT Pess, 1992. Massachusetts Institute f Technlgy. Used with pemissin. Elementay suces. It is useful t examine the simplest acustic suces, which ae all cnfiguatins with an impsed pessue fluctuatin an impsed wall vibatin. The fist in a systematic seies f such suces ae shwn belw: 3
Fm Keebck, Jack L. Aicaft Engines and Gas Tubines. 2nd editin. MIT Pess, 1992. Massachusetts Institute f Technlgy. Used with pemissin. (a) Mnples Suppse nw the petubatin is due t a pulsating sphee, whse adius scillates by sme small! 0 at a fequency ω. This sets up spheically symmetic pessue field that scillates at ω as well, and ppagates as a sund wave tain. In spheical cdinates, the wave equatin is 1! 2 p " 1!!p" = ( 2 ) (7) a 2 2 2 0!t!! We expect acustic enegy t be cnseved, and since at least the cmpessin pat f this enegy vaies as p' 2, let us ty a slutin f the Mnple fm: p' = P( 0 )csk(! at) (8) It can be seen by diect substitutin that this des satisfy the wave equatin. The 2" quantity k= ω/a 0 is called the wave numbe, and the wavelength is! = = 2" a0. k # The velcity field (puely adial) can be calculated fm (4) and fm (8), afte integating in time,!u ' 1!p' = " (4a)!t #! 4
" & ( " a ( (9)! a $ k ' P # ( ) u ' = sin a t %cs t) " The acustic pwe flux (enegy cssing unit aea pe unit time, aveaged ve ne cycle) is the aveage f the wk dne by p!: 1 2! = # " P 2 % ( p'u ' dt = ' * " 2$ & ) (10) 2" 2" whee! = =. # k 2 2 2 P The net acustic adiated pwe is Pm = 4" # = 2", which is seen t be! independent f, as it shuld. The mnple pwe adiated is independent f wave numbe k = 2! /", and nly dependent n pessue amplitude. A pssible physical implementatin f a mnple suce is a pulsating jet, such as pduced by a pulse-jet (like that in the V-1 missile), by the scillatins duing an engine suge. (b) Diples Cnside next tw mnples f equal stength P peating in cunte-phase t each the, and spaced a small distance d alng the x-diectin. If an bseve is lcated at a distance fm ne f them, and at an angle ϑ fm the x diectin, its distance t the the will be (appximately + d cs!. Then the pessue p' at the bseve s lcatin will vay as p' P = cs k(! t) P! cs k( + d cs"! a 0t) (11) + d cs" Expanding the secnd tem and assuming d t be much smalle than bth, the wavelength ( 1/k) and the bsevatin distance, P p' = (k d cs! )sin k( " t) + ( d cs#) P 0 cs k( " a0t) (11b) f the tw tems in (11b), the fist has the 1/ dependence that will ensue enegy flux cnsevatin at all distances, while the secnd will decay faste and will be negligible f distances >>1/k; this is the nea-field diple sund, which can be imptant nea the suce. We cncentate hee n the fa-field cntibutin (fist tem in (11b)). The calculatin f the fa-field pwe flux is as befe, emembeing that it is still a adial flux, even thugh thee is an angula dependence in ϑ. S we still have the fist equality in Eq. (10), and the calculatin is staightfwad. We btain 5
1 $ ' 2!(," ) = & P k dcs" ) (12) 2# % ( cs 2! This flux has nw a 2 distibutin (n sund at 90 t the diple axis, maximum alng the diple axis). The ttal adiated pwe is 2 (P k d) 2 P d = 2! #! "( $ ) 2 sin$ d$ =!, (13) 3 % Aside fm the 1 fact, this diffes fm Pm, the mnple pwe, by the fact 3 (kd) 2 = (2!d /") 2, which is, by assumptin a small numbe, and becmes smalle the lnge the wavelength is ( the lwe the fequency! = /k). An imptant bsevatin is that a physical diple equies applicatin f an extenal scillaty fce. A pssible physical implementatin f the adiating diple is any vibating cmpact bject, such as a fan a tubine blade. The diple axis is then the diectin f vibaty mtin, and the suunding ai is fced back and fth as it wuld between the hypthetical tw mnples in cunte-phase. (c) Quaduples Stngly tubulent flws, such as an engine exhaust jet, ae knwn t be stng suces f acustic adiatin. If the jet is steady and subsnic, thee is n pssibility f macscpic mnple (expansin/cntactin) type f adiatin, and since thee is n extenal fce in the bdy f the fluid, n diple suces eithe. Hweve, thee ae fluctuating pessues at diffeent pints (tubulent eddies), exeting fces n each the with ze net n the lage scale. The lwest de multiple with these featues is the Quaduple, which can be built up fm tw diples with a cmmn axis, and sepaated by 2d and with ppsing diectins. The detailed deivatin is simila t that f a diple. We calculate (in the fa field) P p' and f the acustic pwe flux, =!2 (k d cs" ) 2 csk(! t) (14) " P % 2 (kdcs( ) 4! = 2$ ' (15) # & ) 6
which integates f all diectins t a adiated pwe 8 (P ) 2 (kd) 4 P q =! (16) 5 " The quaduple adiat patten is me shaply diectinal alng the axis (cs 4! ), and is an additinal (kd) 2 weake than the diple, with an even stnge ate f incease with fequency. It shuld be nted that the cllinea-diple type f quaduple is nt the nly ne pssible, but all f them shae the (kd) 4 featue (althugh with diffeent angula pattens). 7
MIT OpenCuseWae http://cw.mit.edu 16.50 Intductin t Ppulsin Systems Sping 2012 F infmatin abut citing these mateials u Tems f Use, visit: http://cw.mit.edu/tems.