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Two causality correlation techniques applied to jet noise Rackl, Robert 1973-12-31

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.  '151'Kf  TWO  CAUSALITY CORRELATION  TECHNIQUES  APPLIED TO J E T NOISE  by  ROBERT RACKL Dipl.  Ing., Technische  Hochschule Graz,  Austria,  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF  in  PHILOSOPHY  the Department of  Mechanical  We a c c e p t  this  Engineering  thesis  required  as conforming standard  THE UNIVERSITY OF BRITISH April,  to the  1973  COLUMBIA  1969  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r  an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  and study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s .  I t i s understood t h a t copying or p u b l i c a t i o n  of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be allowed without my written  permission.  Department o f  H e ^ a m (  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Date  MrSU^  (O.  Columbia  /<?73  i'd^Tl'naj  i S u p e r v i s o r : D r . Thomas E .  Siddon  ABSTRACT The  thesis  experimentally  describes  the generation  recently  developed  Budapest  1971),  further aim  The  the exhaust  stimulate  distribution diagnostic general. a cold  the sound  of  E x p e r i m e n t s were c o n d u c t e d  method u s e d ,  i n the  far  source  sources  in  the  technique  as w e l l ;  The pressure  this  a  field. strength  density.  sources,  and  o f about  the  j e t . The method  mechanisms,  to  the  provide  chamber  cross correlates the r a d i a t e d  deduction  location  of  which of  i s shown t o be s e l f  i n the investigation  using  0.32.  surface area  approximate  The  of turbulent flows i n  i n an a n e c h o i c  unit  t o reduce  finding  c l o s e to the j e t with  per  o f the  by  enables  ICA,  aircraft.  research  t h e image t e c h n i q u e ,  This  using a  7th  by t h e need  surveys  number  surface  c a n be u s e d  the  i n turn  the  sound  consistent.  of supersonic  jets  i s where  i t sreal  power may l i e .  s e c o n d method  directly  c r o s s c o r r e l a t e s the hydrodynamic.  fluctuation  allowing  strength  (Siddon,  j e t transport  noise  f o r making n o i s e  on  investigating  by t u r b u l e n t j e t s  motivated  theoretical  and c h a r a c t e r  first  for  i n f o r m a t i o n on n o i s e g e n e r a t i o n  future  tools  is  n o i s e o f modern  g i v e s an i n d i c a t i o n  sound  work  new  pressure  acoustic  The  of n o i s e  model a i r j e t a t a Mach  The  techniques  method o f c r o s s c o r r e l a t i o n  i s to provide  to  two  per  the unit  A specially  the  pressure  the  jet for radiation  i n the t u r b u l e n t j e t flow deduction volume  and  designed  fluctuations.  of  foil  the  the type  with  local  acoustic  associated sensor  A distribution  the f a r f i e l d  power  i s used  of source  source spectral  t o measure  strength  a t 45° t o the j e t a x i s i s o b t a i n e d ,  over  and i s .  ii  not  unexpectedly,  shear. by  The  axial  radially  similar  to the  distribution  integrating  qualitative  agreement  However, d i f f i c u l t i e s  with are  consistent,  i . e . , the  over  sources  a l l the  the  i n the  Reasons  t o overcome i t a r e  A Fourier transform  i n the  turbulence.  reflected The-  number  estimated number  by  turns  between two given  by  technique  Thus,  as of  as  jet results i t  other  i n making  widely  the  from  discrepancy  is  in  researchers. method  p r e d i c t e d by  allows  the  self  integrating  the are  the  directly given  deduction  radiated pressure  although  and  of  the  a  unit  from  c h a r a c t e r of the  the  above  sound  difficulties  well. independent the  out  of  to  of  the s p e c t r a l  calculating  widely  other  of  j e t i s obtained, here  results  this  the  velocity  proposed.  to the spectrum  sources are  for  of  mean  strengths;  jet differs  solutions  of  source the  of  a "slice"  r a d i a t e d sound  sound.  volume  from  encountered  measured  contribution  distribution  be  different  researchers.  acoustic sources average  the  order  estimates  i n the  jet  correlationvolume. of  100  of about  t o 200; 3 and  this  about  was The lies 2500  iii  ACKNOWLEDGEMENTS  This  work  supervision patience  of  and  for  helping  fibreglassing This of  to  the  also the  research  Canada, under  pressure  sensor  Siddon  summer  anechoic  with  Thanks  T.E.  whose e x p e r i e n c e  go  the  undertaken  Dr.  on  Thanks construct  was  the  who the  and  N.  to  Mr.  John Smith  and  heavily  with  helping  K.F.  Richmond,  the  relied.  for  S a n s o n and  of  under  research  Michailoff  t o R.  acquisition.  Phoon  BC,  for  nozzle.  was  sponsored No.  by  the  67-7106.  r e c e i v e d support No.  the  author often  data  jet  suggestion  guided  students  facility,  Grant  Canada u n d e r G r a n t  at  66-9603.  by  the  National The  Research  development  Defense  Council of  Research Board  the of  iv  NOTATION  (Special  n o t a t i o n s of  the  appendices are  A  axial  flow  error  coefficient  B  cross flow  error  coefficient  c  D  ambient  Cp  static  C  C  P u  p  speed  of  pressure  error  f o r u-component  coefficient  only  probe c o n t a m i n a t i o n  D  nozzle  db  decibels  h  distance  j e t a x i s to s u r f a c e S  L  integral  length scale  /j  unit  m  meters  mm  millimeters  H  Mach  n  number o f  P  total  ratio  diameter  vector  perpendicular  incoherent  static  fluctuating o  sources  pressure  ambient  p a r t of  II p s e u d o s o u n d "  p(i)  acoustically  P  measured  Pj  true  p^  surface  p  static  i n the j e t  ( s u b s c r i p t s have same meaning  case  static  p's) pressure  pressure  pco)  m  to s u r f a c e S  number  below f o r l o w e r  p  included)  sound  C  p  not  pressure propagating  p a r t of  pressure  pressure pressure pressure  at  jet exit  the  pressure  p  as  V r  r=|x-yj;  rms  root  5  surface  Sj  imaging  S  source  strength  per u n i t  area  source  strength  per u n i t  volume  U A  Sy  V  or r a d i a l  distance  mean s q u a r e = J  j e t center  \  surface  time  \.  retarded  Tjj  Lighthill's  u  component o f v e l o c i t y  Uj  velocity  U  nozzle  time stress  tensor in axial  direction  vector  exit  velocity  6  mean  V  volume o f s o u r c e  V.  correlation  velocity region  volume  Vj f  volume o f s o u n d p r o d u c i n g  v,w  components o f v e l o c i t y  x  x=|x|  Xj,x  space c o o r d i n a t e (often i n d i c a t i n g d e t e c t i o n i n the f a r f i e l d )  x  distance  e  Q  y.  space  1  from  nozzle  Kronecker  6  angle  perpendicular  exit  t o a x i s o f probe  the  to f a r f i e l d  between  p  density  p  ambient  T  time  t  retarded  point  j e t a x i s and f a r f i e l d  density  delay time  delay  of  sound  microphone the source  Delta  frequency  o  j e t turbulence  coordinate (usually used in and/or as i n t e g r a t i o n v a r i a b l e )  8.. •J  v  line  2  4  t  0  from  direction  region  vi T..  viscous  £  space separation  •J  V  2  a 2  stress  /a 2  tensor  or d %  y  vector x  (£j,£ ,£ ) 2  3  2  •  a d o t on a quantity i n d i c a t e s d e r i v a t i o n with respect t o r e a l time an o v e r b a r d e n o t e s t i m e averaging  [...]  replace i n t h e expression within the brackets t by t = t r / c , o r T by f=T+r/c whichever i s appropriate. Q  The  jet Strouhal  number i s d e f i n e d  o f  by  i/D/U o  vii  TABLE OF CONTENTS  I.  Foreword  and H i s t o r i c a l  II.  Present Trends  and  to Sections III.  IV.  Background  Introduction  I I I and IV  Image T e c h n i q u e 3.1 3.2 3.3 3.4 3.5 3.6  ..... 1  ..10 .........16  I m a g i n g System Cross C o r r e l a t i o n Formalism Experiments Nondimensionalization Results Check C a u s a l i t y C o r r e l a t i o n Technique  ..16 ......19 21 23 .-24 29 ........31  4.1 4.2  Introduction .............31 Theory ...32 4.2.1 F l u i d D i l a t a t i o n s 32 4.2.2 C a u s a l i t y C o r r e l a t i o n s . . . . . . . . . . . 35 4.2.3 L o c a l S o u r c e S t r e n g t h and S p e c t r u m . . . . . . . . . . . . . . . . . 37 4.2.4 C o r r e l a t i o n Volume 38 4.3 D e v e l o p m e n t o f t h e P r e s s u r e S e n s o r ......42 4.3.1 L i m i t a t i o n s on C o n v e n t i o n a l C y l i n d r i c a l P r o b e s . . . . . . . . . . . 42 4.3.2 A i r f o i l P r o b e 44 4.3.3 S t a t i c P r e s s u r e C a l i b r a t i o n ......47 4.3.4 The C o m p l e t e P r e s s u r e S e n s o r .....50 4.4 E x p e r i m e n t s ..............51 4.4.1 A i r S u p p l y , S e t t l i n g Chamber, and N o z z l e ..51 4.4.2 The F a r F i e l d M i c r o p h o n e 51 4.4.3 S i g n a l P r o c e s s i n g .....52 4.5 D a t a R e d u c t i o n ..........53 4.5.1 S o u r c e S t r e n g t h D i s t r i b u t i o n .....53 4.5.2 Check, C l o s u r e D i f f i c u l t y .55 4.5.3 S p e c t r a 58 4.5.4  V.  Number  Summary  of Incoherent Sources  Conclusions,  60  Recommendations ...63  References App. A: M a t h e m a t i c a l D e t a i l s C o n c e r n i n g Eq.(4.7) T h r o u g h Eq. (4.11)  .66 ...71  viii  App. App.  B: C:  Nondimensionalization of Correlation Functions Prediction  of the  Shape o f  Correlation App.  D:  Figures  Probe  Contamination ..  the  Function  Ratio  ...74  Cross 77 83 90  ix FIGURES  Fig.3- 1 Fig.3- 2 Fig.3- 3 Fig.3- 4 Fig.3- 5 Fig.3- 6  Image g e o m e t r y Jet noise spectrum with and w i t h o u t adjacent surface Mounting o f 1/4-inch microphone i n sheet o f 1/2inch p l e x i g l a s s Typical c r o s s c o r r e l a t i o n between s u r f a c e p r e s s u r e and r a d i a t e d p r e s s u r e ; a l s o s h o w i n g p r e c u r s o r Relationship between surface pressure and exit velocity Distribution of source strength over surface; r e f l e c t i o n path. Equal source strength contours f o r 0=90° Equal source strength contours f o r 0=60° Zone o f i n f l u e n c e 3 t y p i c a l p p (r) D i s t r i b u t i o n o f r o o t mean s q u a r e d s u r f a c e p r e s s u r e " S l i c e w i s e " i n t e g r a t e d source s t r e n g t h Experimental setup of Causality Correlation Technique Geometry D e f i n i t i o n of c o r r e l a t i o n l e n g t h C l a s s i c a l c y l i n d r i c a l probe c o n f i g u r a t i o n C o r r e l a t i o n f u n c t i o n c o n t a m i n a t e d by d i p o l e n c i s e P r i n c i p l e o f a r r a n g i n g f o i l probe i n t h e f l o w L o c a t i o n o f p r e s s u r e s e n s i n g h o l e s on a i r f o i l P r e s s u r e d i s t r i b u t i o n on s u b s o n i c airfoil Contaminated and uncontaminated correlation f u n c t i o n s and t h e i r F o u r i e r transforms V e l o c i t y v e c t o r s on a i r f o i l Probe shapes t r i e d o u t but d i s c a r d e d The f o i l t y p e p r e s s u r e sensor S t a t i c pressure c a l i b r a t i o n s Pressure distribution over foil with d i p near trailing edge F o i l t y p e s e n s o r mounted on p r e a m p l i f i e r Frequency r e s p o n s e o f f o i l type sensor Schematic of experimental setup Variation of pressure f l u c t u a t i o n with d i s t a n c e from j e t S i g n a l flow D i s t r i b u t i o n of source s t r e n g t h i n a subsonic j e t f o r 45° to the j e t a x i s "Slicewise" distribution of source stren.gth; comparison with o t h e r r e s u l t s Some t y p i c a l c r o s s c o r r e l a t i o n f u n c t i o n s p<o)p Some t y p i c a l c r o s s s p e c t r a ( F o u r i e r transforms of f u n c t i o n s shown i n F i g . 4 - 2 2 a ) Peak f r e q u e n c i e s v e r s u s downstream p o s i t i o n Cross c o r r e l a t i o n p<o>p and f i t t e d to i t the predicted function Sears* f u n c t i o n E f f e c t i v e c h o r d l e n g t h on c y l i n d r i c a l p r e s s u r e probe S p e c t r a l d e n s i t y o f t h e " p s e u d o s o u n d " p<o) a t 4D 0  Fig.3Fig.3Fig.3Fig.3Fig.3Fig.3Fig.4-  7 8 9 10 11 12 1  Fig.4- 2 Fig.4- 3 Fig.4- 4 Fig.4- 5 Fig.it- 6 F i g . it- 7 Fig.4- 8 Fig.4- 9 Fig.4Fig.4Fig.4Fig.4Fig.4-  10 11 12 13 14  Fig.4Fig.4Fig.4Fig.4-  15 16 17 18  F i g . 4 - 19 F i g . 4 - 20 F i g . 4 - 21 F i g . 4 - 22a F i g . 4 - 22b F i g . 4 - 23 F i g . 4 - 24 F i g . 4 - 25 F i g . 4 - 26 F i g . 4 - 27  s  X  Fig.4-28 Fig.B-1  The  d i r e c t l y measured and t h e a t 4 5 ° i n the f a r f i e l d Image geometry t o s c a l e  integrated  spectrum  1  I.  In t h i s with  FOREWORD  section,  emphasis  a brief  on s u b s o n i c  work. S e c t i o n I I w i l l undertaken  As  experiments historical  It  say  and e s t a b l i s h  effort.  AND HISTORICAL  with  review  on  i t s proper  was  they  primarily  why  the  research  up t o t h e p r e s e n t present  work  of  science,  f o r reasons  was  research  theory  of c l a r i t y  and  i n the  are separated.  t h e advent  methods a t t h e end o f W o r l d  lead  noise  place i n the o v e r a l l  disciplines  go hand i n hand. Here, development,  of aerodynamic  jet noise w i l l more  most  BACKGROUND  War I I  o f new, v e r y -  the  noisy propulsion  j e t engine  and t h e  rocket  - t h a t s t i m u l a t e d t h e d e v e l o p m e n t o f a new d i s c i p l i n e , t h e  study  o f sound  of  generated  aerodynamically.  a Nonhomogeneous Moving  thorough  account  of  Medium,"  into  the  dynamics  for  literature,  English only  Russian.  show  English  10  Lighthill  years  References probably  after  (1946)  to this  work a r e  perturbations:  of  i t was  first  an i n h o m o g e n e o u s form  fluids  disturbed  a  rare  b e c a u s e i t was t r a n s l a t e d published  ( c o n s e r v a t i o n o f mass and momentum) i n s u c h  case  gave  between a c o u s t i c and  (1952) combined t h e b a s i c e q u a t i o n s  how t o f o r m u l a t e the  Blokhintsev  the interrelationship  aerodynamic motions o f a f l u i d . in  I n h i s book " A c o u s t i c s  by  larger  of f l u i d  a way a s  o f t h e wave than  in  to  equation acoustic  where: p =  density,  T-,j =  a  t=time, c = s p e e d o f sound,  tensor  ( e x t e r n a l from  The  representing  externally  to  Eg. (1.1)  is  p o t e n t i a l s [ s e e Eq. ( 1 . 2 ) ] . L i g h t h i l l  found  that  should and  applied,  the  considered for  jet  replacing  with  noise  models  1954).  large that  terms  of  Proudman  model  Ribner  the f a r f i e l d by  a  Lighthill  gradient.  mainly This  subsonic  region,  Curie  surfaces  in  such  2  a  guadrupoles, at  rest  the properties of assuming  Gaussian  i s , however,  being  because  the f i r s t  can  viewed  (1955) surfaces  an the  isotropic  was a l s o be  He  model  (see  1972), i s c h a r a c t e r i z e d  f r o m any f l o w  of observation.  8  j e t noise  f a r from  Proudman  direction  (U D -Law).  and s u g g e s t e d  estimated  flow  velocity  i n an a m b i e n t f l u i d  processes.  other  aerodynamic  acoustic  turbulence,  1965 o r F u c h s  flow;  retarded  many  the theory:  t h e components o f t h e v e l o c i t y  the  of  and t h e n  dimension  convected  for  noise  form  free  of noise  (1952)  stochastic  mean v e l o c i t y  generated  as a source  i n the mixing  instance  stresses  power o f some c h a r a c t e r i s t i c  by i s o t r o p i c  the  oversimplified  for  from  t h e s t r e s s e s i n the f l u i d ,  for  turbulence  in  the  himself  o f some c h a r a c t e r i s t i c  turbulence  generated  in  and s i m p l i f i e d  intensity  the eighth  square  (Lighthill noise  enlarged  t h e sound  vary  applied  t h e p o i n t o f view o f t h e a c o u s t i c i a n ) .  solution  workers  Xj=space variable,  c  by a  t o show  as  being  f l u c t u a t i o n s i n the  included could  the be  effect  real  of  or just  3  hypothetical  (e.g.,  so  as  to  exclude  excessively  complicated  region  most g e n e r a l  s o l u t i o n o f Eg. ( 1 . 1 ) :  d u P  I  d  1 — r - J  T..  or  He o b t a i n e d t h e  dS(y)  n  l  r r  2  uninteresting  the f i e l d ) .  r r  a  l ——<r  +  from  an  i  d S  (y)  (y — d V  (1.2)  where: x p o i n t s t o the p o i n t of S=surface, variable  =vector in  p=pressure, generating time  quasi an  flow,  stress  to  S  at  y,  r=|x-2l# tensor,  y=space  Uj=velocity,  V=volume o f sound  [...] means r e p l a c e t i m e t by  aerodynamic sound  the  are  source  heat  admissible.  dimensions  conduction  Additional  equation  Eg. ( 1 . 2 )  earlier  the source; away;  i f  simplifying  the  receiver  is  many  the motions a t the source a r e  move i n  In the g e n e r a l  unison  from  over  are negligible; case,  a  certain  the flow i s  however, E q . ( 1 . 2 ) i s  h a s n o t been s o l v e d arise  approximately  may b e : t h e r e c e i v e r i s many  and v i s c o s i t y  which  difficulties  of  Such  i . e . , the p a r t i c l e s  incompressible.  integral  p r o b l e m s c a n be s o l v e d  application  w a v e l e n g t h s away f r o m  "coherent", region;  region,  density,  o  c  assumptions  typical  source  Tjj =viscous  straightforward  typical  perpendicular  p =ambient  t-r/c .  Many by  the  observation,  In  closed  form.  t h e i n d e t e r m i n i s t i c , random  character  of  Powell noise  (1964) s u g g e s t e d  generation:  similar  A  a different  vortex  flow  s t r e a m l i n e p a t t e r n . He  equivalent view  turbulence.  dipole field.  mechanisms  impact  Perhaps the Ribner  on  this  vortices  (Tennekes 5 L u m l e y  a significant  1972,  j e t noise  at  first  in  for  aerodynamic  a c o u s t i c d i p o l e have a  the  flow  theory  field  by  an  seems a t t r a c t i v e  model o f  turbulent  s e c t i o n 3.3),  in  transfer  i t has  not  had  developed  by  research.  simplest concept  (1962),  an  replaces  Although  of T a y l o r ' s i n t e r a c t i n g  and  model  of flow  1958,  noise  was  elaborating  on  ideas also  by  arrived  at independently  regarded  as  an  a r r a y of d i l a t i n g  into  each o t h e r .  first  r e p r e s e n t i n g the  second  the  where time be  p  is  the  the  derivative evident  radiating net  of  sound,  source  such  field  eddies  into  is  bumping  three parts,  the the  pressure  fluctuation  (or  the  pressure the  Ribner  and  pressure.  this the  disturbance  fluid.  r a d i a t e d sound  i s c o n s i s t e n t with as  compressing  flow  time,  the  sight,  The  independent of  pseudosound  first  acknowledged  and  (1958):  pressure  through  far field  Ford  i s separated  ambient  third  of the  at  monopole t y p e  the  pressure  waves t r a v e l i n g  Dilatations" been  The  hydrodynamic  "pseudosound"), acoustic  Heecham &  theory  guadrupole  (Appendix  necessarily  B  of  changes  a g u a d r u p o l e does n o t .  p<°>  i s the  second  i t may  "Sound f r o m concept  and  Lighthill its  In a f r e e  c h a n g e i n volume o f a l l m o n o p o l e s c o m b i n e d  to  shows t h a t :  Although of  due  not Fluid has  1963). A  volume  when  turbulent  flow  is  zero,  the  5  net be  radia'tion thought of At  well  is  a s m a k i n g up  moderate s u b s o n i c  approximated  turbulence. evaluated.  turbulent  jets  1958)  the  8  further  these  two  in  similar  regions that  is  inverse  exit  spatial  on  source  approximately rapidly fully  the  developed  evidence,  (see  confirmation  i n the  region.  of  extent  of  Hach numbers.  the  (the the  the  of  the  square  and  then  sources of  i t was  "slices"  mixing  d i s c u s s i o n i n s e c t i o n IV) present  far  (based  is  field  show a  on  0.2  exit  and  section  the  and  was  decrease  predicted of  region  there  i t  With r e s p e c t  jet  0.3 5),  to  that  falls  power o f a x i a l  Although  between  measured s p e c t r a  radiation.  5  developed  density  1963,  of  4 to  Thus  Kartucelli  1958;  properties  transition  between  for  Lilley  fully  number  be  Other  first  defined).  the  attempt.  confirmed  frequency)  seventh  i s l a c k i n g . The  an  (Ribner 1958;  peak S t r o u h a l  &  strength  inverse  and  with  and  sound  such  region  clearly  very  could  preservation  of frequency;  direction of  self  spectral  Kolpin  constant  with  yet  l o c a t e the  distribution  acoustic  (The  square  the  j e t noise  nozzle)  increase  diameter  subsonic  the  is  Eg. (1.3)  describes  mixing  power  first  (Mollo-Christensen, depending  not  the  s h a p e and  velocity,  of the  can  fluctuations within  been e x p e r i m e n t a l l y  using the  pseudosound  measured.  paper  has  downstream,  should  the  be  laws d e a l i n g w i t h  were d e r i v e d  flow  with  of t h i s  pressure  o v e r a wide r a n g e o f  d i a m e t e r s downstream  pressure  could  2  monopoles  quadrupole.  static  U D -Law  turbulence  predicted  the  IV  a  four neighbouring  Kach numbers t h e  latter  Section  semi-empirical Powell  by  I f the  Lighthill's  of  not. Thus,  the the  remains off  distance  very in  the  some  supporting  conclusive  experimental  research  not  only  attempts  6 to  make t h i s  confirmation  way  to  measure  over  the  j e t volume.  In  the  surveying found  the  that  lower  pattern  the  frequency  to  that,  detail,  radiation  field  that the  this  d i p on  mentioned  above  prediction  was  were not  of  with  data.  step  The  concentrated  on was  is  radiated  more  or  less  frequencies  follow  downstream d i p of  axis -  the  turbulence aim  of  discovered,  but an  a  heart  the  higher  Parallel  were s t u d i e d  intensities,  predicting of  a  heart.  j e t turbulence  to  first It  properties  possible  distribution  jets.  the  the  strength  show a  turbulent  the  the  f u r t h e r and  from  sound  velocities,  lengths,  from  field  more p r o n o u n c e d  mean  one  efforts  1  higher  properties  as  correlation  and  go  a c o u s t i c source  frequency  with  the  the  such  will  r a d i a t e d sound  omnidirectionally shaped  local  experimental  the  but  self  the  in and  noise  preservation  quantitative  acceptable  noise  degree  of  confidence. Although properties have  to  tried  pioneering.  theoretical the to  convert  it  pressure  i n the  velocity  radiated verify  Using into  formalisms  the a  sound  those  abound, v e r y theories.  third  integral  form  such  that  depended  on  far f i e l d  relating  space-time c o r r e l a t i o n .  He  few  Chu's  o f Eq. (1.2) the the  the  turbulence  experimenters (1966)  he  was  mean s q u a r e d  work  was  able  to  radiated  f o u r t h d e r i v a t i v e of  measured  this  correlation  a at  * A l i s t o f r e f e r e n c e s o f e a r l y e x p e r i m e n t a l and t h e o r e t i c a l work can be found i n L i g h t h i l l (1962) o r R i b n e r ( 1 9 6 4 ) . The l i s t o f r e f e r e n c e s o f t h e p r e s e n t p a p e r a t t e m p t s t o i n c l u d e many of the experimental papers published since 1963 i n t h e open E n g l i s h l i t e r a t u r e ; t h e c i t a t i o n s a r e p r e c e e d e d by a * " . A more recent list of references of t h e o r e t i c a l • work can be f o u n d i n Doak (1972). M  7  a point 4 diameters radiation  and  pressure; for  two  the  agreement  to  correlators derivative a  estimated  reasons:  consuming  downstream  1)  survey  the  o f an  the  the  source  basic  resulting two  sound  sound  signals  strength  flow  Siddon  has  and  jet  totally  (modern  If  i t  far field  giving  rise  b a s i s f o r the  be  &  Ribner  was  in this  the  present  work  for  tying  implementing  can  cause  be  Any  viewed  and  The  (the  the the  technique  1970,  word  source  (the r a d i a t e d  small a i r f o i l s  Siddon  with  manner, the a c o u s t i c  to  measurement. The  to  simultaneously,  p r o c e s s of the  1969;  varied  measured  the  from  subject  quantitatively.  effect  fourth  1971b).  (1.3)  be  time  channel  t a k i n g the  means  or  can  deduced  acoustic radiation  (Clarke  a  i s monitored  cross correlated;  because  multi  ( S i d d o n 1970,  i n E q . (1.2)  acoustic satisfying  the r e s u l t s  introduced  technique"  sound  prohibitively  t u r b u l e n c e p r o p e r t i e s by  generator.  be  to the  not  been  of  squared  a t t h e t i m e ) ; 2)  appearing  i s used  the  angles  mean  but  have  complete  t h e s o u r c e s can  fluctuation)  applied  would  used.  i n the  may  of  "causality"  provides  reasonable  fitting  correlation  fluctuation  as t h e  overall  of u n c e r t a i n t y , as  far field  "causality  different  experimentally obtained function  method o f c u r v e  together  the  were n o t a v a i l a b l e  More r e c e n t l y ,  a  It  c o n s i d e r a b l e degree  the  was  for  sound)  has  been  embedded i n  1973a) and  to j e t  9  noise in  by  Lee  (1971) and  the  (see f u r t h e r  discussion  section I I ) . Although  give  clear  the cut  mathematical guidance  theories  for  substantial  noise  applying  U D -Law. Reductions  the  8  2  reductions  designing have  been  of the  have not quiet  jet  achieved,  jet exit  been a b l e  to  engines, mainly  velocity  by with  8  corresponding  increase  o f the n o z z l e diameter  to the e v o l u t i o n of high noise  reductions  resulted. devices of  A  not  large  were p r o p o s e d  the  device  with  very  a suppressor  bypass r a t i o  number  and t r i e d  decreased  overall  generated  increased  The  additional  to  be  aerodynamic the  acceptable,  penalty r a t i o .  multitube  expected  from  terms  one  weighted  designed  the notion of reducing  or  t h e amount  smoother, also exact that  mainly  the  by  on t h e i n s i d e  refractive  may  is  be used  noise  improvement  of such  may  with the net  c o n f i g u r a t i o n s were acoustic benefit/ nozzle  noise PNdb  in  and  than the terms  of  o r dbA. They a r e  the i n t e n s i t y  of  turbulence  i . e . , t o make t h e m i x i n g  process  on t h e o u t s i d e b u t  up t h e m i x i n g  region.  Their  subject to speculation. I t i s f e l t play a s i g n i f i c a n t  as w e l l .  as t o o l s  generation  as  but  minimal  corrugated  defines  such  breaking  still  shielding  behaviour  shear,  turbulence  that  e n t r a i n i n g a i r not only  thereby  functioning  acoustical paper  of v e l o c i t y  the i n s e r t i o n  however, p r o d u c e l e s s  subjectively with  suppression  of t h e i r  These a r e t h e  indices  noise  of the flow  two s u p p r e s s o r  i f  performance  the j e t  d i p o l e n o i s e such  in  Significant  of  o u t . I n many c a s e s  n o z z l e . Even t h e s e ,  noise reduction,  passive  interaction  n o i s e r e d u c t i o n r e s u l t e d . Only found  engines.  efficiency  of  the noise  noise.  fan  and mass f l o w l e d  both  process  role  in  The two methods p r e s e n t e d t o f u r t h e r our and  noise suppression  as  aids  devices.  their  in this  understanding  of  i n t h e d e s i g n and  9 Although has  been  decades  t h e n o i s e from  reduced  people  by s l i g h t l y  more t h a n  jet  transport  increases.  c o n t i n u e and t h e r e f o r e c a n n o t  aircraft  2 0 PNdb i n t h e p a s t two  c o n t i n u e t o be a n n o y e d i n t h e v i c i n i t y  a s t h e number o f f l y o v e r s to  commercial  We c a n e x p e c t  stop trying  of a i r p o r t s  this  to reduce  increase j e t noise.  10 II.  PRESENT TRENDS  INTRODUCTION TO  Present of 1970 of  the  theoretical  Aerodynamic  (Fisher the  understood  and  refraction understand and  are  all and and  He  infers was  will  The  types  and  convection  are  adequately  form.  g e n e r a t i o n and  Effects  in  which  propagation  new  was  discussed  aerodynamics,  t a k e us a n o t h e r  big step  Symposium a  i s confirmed  ..."  (Fisher  statement  was  based  on  Meeting  Lau,  Fisher  & Fuchs,  d i v e r g e w i d e l y on  that  go  i . e . , the  turbulent  back  to  works o f  the  Stokes, of  these  forward.  degree & Lowson  "...  of  i t appears  order  1971,  in  jet  p. 5 9 4 ) .  This  presented  by Crow & Champagne.  this  avoid  t h e o r i e s by Crow, Doak  experimental evidence and  are  a big step  to d e l i b e r a t e l y  also concluded  significant  in  t o Doak*s  approaches  theory  designed  of  have t o t r y t o  reader i s r e f e r r e d  ...".Recent  turbulence  still  field  Lighthill's  are  findings  of  R a y l e i g h . I n Doak's o p i n i o n , a c o m b i n a t i o n  the e x i s t e n c e of  opinions  Effects  sound  The  of motion  Loughborough  by  the  t h e i n t e r d e p e n d e n c i e s among a c o u s t i c ,  of a c o u s t i c s  theories  1971).  specifically  (among o t h e r s )  and  by  t o d e a l w i t h : We  (1972) that  guided  mathematical  layer.  article  "...  thermal  Kirchhoff  into  difficult  shear  q u e s t i o n s of  origins  that  a  but  Lilley  Lowson  IV.  a t L o u g h b o r o u g h i n September  radiated  be c a s t  more  review  suggested: forward,  the  I I I AND  seem t o be  t h e mechanisms o f s o u n d  through  excellent  1971;  on  can  efforts  N o i s e Symposium  & Lowson  sources  SECTIONS  AND  s u b j e c t . The  at  the  However,  supporters  of  11  the their  orderly view  evidence 1972;  structure by  hypothesis  publishing  techniques coherent  S  Fuchs  structures  in  the  jet  from  laminar  the  number i n t h e  behind  a  Karmann V o r t e x contribution  decay  of  i s of  spectrum.  a  wide  the  may  behave  quantity  many  is  may  is  so  shows a l s o of  the  a  Reynolds  similarly; show  an  in  time  i s thereby  the  periodic number  band  the  a  is  spectrum,  scales. shape  of  The the  a discrete periodic  orderly  direct  noise  for  Reynolds  strong  turbulence  velocity structure  and  well  structure  i n a broad  well established  second  (after  paper.  the  of  exist  the  This  m o d e r a t e R e y n o l d s numbers  When  of  that  diameters  not  but  turbulence  does not  necessarily  the  indicate  peaked energy d i s t r i b u t i o n i s  controls  d e p e n d s on  Visualization  order  in this  range  (Fig.4-27)  It  the  for  Fuchs  identities.  i t  develops, r e s u l t i n g i n  convecting that  fluctuation  which  S  R e y n o l d s numbers  layer);  c y l i n d e r : At  a broadly  noise.  periodic) radiation  of  i t i s not  the  5  consider  though . t h e r e  turbulence  small  p e r i o d i c i t y gets buried  spectrum but  may  the  in a jet  Even  radiated  one  Street  characteristic  component,  shear  Fisher  for  their  case  existential  seem t o  R e y n o l d s number i n c r e a s e s  bluff  to  this  turbulence  at  experiments described  analogy,  Lau,  convect losing  recognizable  the  as  before  of  1971).  scale  R e y n o l d s numbers o v e r 2 * 1 0  pressure  Michalke  large  is easily  increased  1971;  c o r r e l a t i o n s indeed  recognizable  wake  Davis  space  transition  By  1971;  amounts  and  downstream coherence  1972;  making a s t r o n g  substantial  (Crow C Champagne  Fuchs  are  evident.  structure  consequence that  emission. derivative biased  to  to  i t i s the  (whether  in jet  rate  turbulent In  of  fact, the  higher  the of or the  source  frequency  12 content  than  present  the t u r b u l e n c e  work,  for  discovered  o f an  radiation,  but  large  forward  by  small  the  jets,  structure  results  Scharton  not  and  the r e s u l t i n g jet  detailed  noise  causality also  the b a s i c  Lee  t o be  measuring  the  to perform  the  whose  sound  found  increasing from  uses  yielded  originating  the from  band  relative  noise  been  the  o f more o r draws  some  postulating  nozzle  exit,  present author  that  put  (of t h e o r d e r o f  i t  does  not  a  3). then  feels  for prediction  that  of  the  provide  IV. implemented  the v e l o c i t y was  from  far field.  that I t was  in  of  the  of Lee  intensity  volume o f a  S Ribner and  j e t (35  as hot  phase response  was  microphone  t h e r e f o r e necessary  narrow  of  Lee  with a  frequency  c o v e r i n g a l l f r e q u e n c i e s of t h e amount  by  fluctuations  measured  dependent  correlations  abstract  unit  to  velocity  frequency  substantially  the  jet  also  h y p o t h e s i s by  technique as  1972)  i n the  of i n a broad  paper  in section  different  cross  evidence  t h e n o i s e g e n e r a t i n g mechanisms. A more  correlation  quite  the  be s u i t e d  but  e m i t t e r o f s o u n d . The  suspected  Citing  be  S Ribner  anemometer  instead  close  model may  on  the  the idea  pressure sources  radiation,  can  While  downstream. The  empirical  critique  The  film  convect  information  thorough  (1971,  scale  to  no  the  sources.  (1972).  structure  presumed t o d e v e l o p  fluctuate  overall  large  o n l y was  and  Model of J e t N o i s e " has  White  orderly  (1971)  to a c o n t r a r y h y p o t h e s i s of a  sources i s retained  the  number o f  5  Lee's  pertinent  point  P r e s s u r e Source  from  These are  "...  orderly  independent  support  thus  speed  In  number o f i n d e p e n d e n t l y r a d i a t i n g  A "Simple  less  low  proper.  interest,  experimental (1972):  spectrum  of  locations)  bands  (Their the and  work. work) noise  received  13 at  a  far  relative  field  point  emission  distance  x  agreement  was  of s u c c e s s i v e " s l i c e s "  over  the  found  for  each s l i c e  by  Powell  (r=96D, Q = 4 0 ° ) ;  measurement  with  Ribner's  I t was  range  5  in  White  sharp  (1972).  incoherent  In s e c t i o n  There,  i t i s also  noise  and  apparent  White  (1972)  staggered In  both  associated well  The noise  flows  theoretical used  as  penalty  and  do  the  in  not  of  jet  aim  i s on  the  i n surveying designing  noise  1972)  is  Scharton of  five  of the j e t .  however,  reduction) rather  the  i s not  high.  their objective  the  generating  some  of  and  core  the  ambiguity.  consisting  have a s  200.  decrease  reported,  t o be  to  of  electronic  Arndt  potential  noise  o t h e r hand  diagnostic tools  phenomena  here  Rather, of  the  is  100  area  &  (e.g. t h r u s t  maybe p r o v i d e the  the  i s expected  presented  work; on  i n c r e a s e or  suppressor  reduction  knowledge and  of  of  Scharton  t h e number  a c o u s t i c or  (Barefoot  a  effective  the order  paper  been made i n t h e  filling  indeed  r e d u c t i o n as s u c h . the  extraneous  peaks  as p r e d i c t e d  total  the o r d e r  perturbing  tested  some n o i s e  techniques  turbulent  noise  have  performance  increase  be  screens  documented, and  present  t o be on  has  Qualitative  number 3 r e p o r t e d by  the  Arndt  c o n c e n t r i c tubes  cases  essentially  the  axial  spectral  number, i n t r o d u c i n g a s o u r c e  devices. with  the  e d d i e s i s on  substantially  recent progress  experimenting  of  d i s c u s s e d how  value of t h i s  suppression  6  IV  the  i s estimated  bandwidth can  Little  to  versus  (1D<x<7D).  a l s o reported that the  contrast  sources  jet  x°-Law, and  number o f u n c o r r e l a t e d n o i s e - p r o d u c i n g 2500,  i n t u r n l e d to  of a  were l o c a t e d i n f r e q u e n c y  (1959)."  this  one  hand  to  mechanisms  in  stimulation techniques  for  future  developed  more  complicated  and  optimizing  can flow  noise  14  suppressors. The  image  correlates far  field  the  in  location an  undisturbed  deduction  This  of the a c o u s t i c a l  axial  sense.  The  The t e c h n i q u e  noise  flows  suppressors.  and  can  supersonic insert  a  be  flows  causality  (1971)  work.  fluctuating  Instead  more a t t r a c t i v e  Difficulties  arise  fluctuations  as t h e  pressures  loss  of the  jet  at  r  the flow  of  in  spatial  the  to cold, of  the  source  c o n s i s t e n t and  t o use t o o l  at  technique  thesis.  may  design subsonic  hot  and/or  i f not i m p o s s i b l e t o  being  local probe,  are taken  at d i f f e r e n t  and  to  the  experimenter  tries  t o measure  velocity thus  field  creatinq type  causality  correlations  downstream volume  and  locations. the  fluctuations  than  velocity.  turbulent  sensor  pressure  additional  a pressure  error.  as  a  i s then  by s c a n n i n g  power  The  means used t o  the  Thus the l o c a l  associated  noise  quantity, i s i n  produces  pressure  extends  as t h e b a s i c  e r r o r s i s r e p o r t e d . The s e n s o r  the pressure  unit  complements  a nondirectional scalar  when one  the  avoiding those  per  It  d e s c r i b e d i n s e c t i o n IV  of the t u r b u l e n t v e l o c i t y  development of a unique a i r f o i l  strength  the  having  investigation  turbulent pressures  principle  radially  in  o f f against a  the  correlation  Pressure,  perform  cross  surface  of  i t i s quite d i f f i c u l t  emitters.  of  sources  advantage  and e a s y  in  f o r m s t h e main body o f t h i s  the  III  probe.  The  Lee's  the  A l s o , i t i s not l i m i t e d  used  where  of  i s shown t o be s e l f  have a p p l i c a t i o n s as a q u i c k of  section  i n t u r n g i v e s an i n d i c a t i o n  by a p r o b e i s t r a d e d  resolution.  in  on a s u r f a c e c l o s e t o t h e j e t w i t h  a l l o w i n g the  distribution.  approximate  described  pressure  noise  strength  least  technique  jet  source  spectral  15  d e n s i t y a r e deduced. The c o r r e l a t i o n s a r e done i n b r o a d both  transducers  characteristics. technique with  self  are  Difficulties consistent,  the r e s u l t s of Lee The t h e s i s  condenser  microphones with matched  are  encountered  in  although results agree  making  as  phase the  qualitatively  (1971) and o t h e r w o r k e r s .  hopes t o e s t a b l i s h  t h a t t h e two methods  have e x p e r i m e n t a l advantages a l l u d e d t o be f u r t h e r d e v e l o p e d .  band  t o above and a r e  presented worthwhile  16  III.  This  technique  p r o b e s o f any is  placed  the  sound  noise  kind  in  this  A.  Powell  reflected  from  the  from  layer  noise  "cause"  or  the  imaging  source  of  representative  of  extent  will  per  of  which  unit  s u r f a c e area  footprint" on  the  the  flow  distribution  flow.  plane  It will  "image" f l o w derivation the  surface  be shown  behind  the  f o l l o w s the  imaging  work e m p l o y s t h e r a d i a t e d sound  that the  surface ideas  principle  causality  i n the  surface. This pressure the  sound  in  limited  of  with  will  be  the  with  i s thought  of  sense  acoustic  deduced. Looking  the  surface allows  noise sources  i n the  principle  far f i e l d  r e g i o n i n the  d i s c u s s e d . The  l e a v e s on of  the  a rigid  inserting  i n mind. H e r e , i t i s a p p l i e d t o j e t n o i s e .  a certain be  Instead,  developed  present the  a d e s i r e to a v o i d  s u r f a c e ' i s e x a c t l y e q u i v a l e n t to  identical  who  correlating on  of  were removed. The  (1960)  pressure  an  from  flow.  field  In a d d i t i o n , t h e cross  the  near  surface  boundary  into  born  the  radiation  if  was  IMAGE TECHNIQUE  that  the as a  i t  j e t flow,  source  by  is the  strength  at the " a c o u s t i c some  image f l o w  conclusion behind  the  surface. 3.1 equation  Imaging.  System^  He  f o r aerodynamic noise  start  with  Lighthill's  [ E q . (1.1) r e w r i t t e n ] :  (1952)  17  V  p  c where  the isentropic P(x)  (which  at  2  2  <'> 3  i n the a c o u s t i c f a r f i e l d  first  J  v  o indicate  integral  integral  i s  the S  (  integral  |x-H/c  the  solution  for  If S  this  (  reflected  from from  indicated  i s not the case  the j e t ; the the  i s t h e outward  were v i b r a t i n g  [as  time  e  contribution  pressure.  appear  but  and n e g l e c t i n g  I e v a l u a t i o n at the e a r l i e r  i s the d i r e c t  surface  would  region  s  r e p r e s e n t s t h e sound  (Fig.3-1).  Eg. ( 1 . 2 ) ],  stresses,  2  becomes:  t -  surface  and i n t h e s o u r c e  i f e n t r o p i c f l u c t u a t i o n s a r e not present)  where t h e b r a c k e t s  s  Vj  c  i  second  ay; a  2  assumption  t h e s u r f a c e a l l but t h e p r e s s u r e  The  0  *  3  0  the r a d i a t e d sound  P  c  I  T ;J . • ( p - c ^ ) 8 ;J  -  Uj  H—  p" 3  lpi*)-p ) Z  =  i s good  as w e l l , at  c  -T,j = />Uj  Using  — =  — o  -  surface.  normal t o t h e  or translating  in  the  more  f o r the r i g i d  a third general  w a l l assumed  here. In o r d e r the  exact  The  instantaneous  surface  rigid adding  to e s t a b l i s h  S  (  i s  t h e image s y s t e m  we add a j e t w h i c h i s  m i r r o r image o f t h e f i r s t  the plane  o f symmetry.  s u r f a c e or a h y p o t h e t i c a l plane of the instantaneous  Whether S  does n o t m a t t e r  m i r r o r image  with  S,  (Fig.3-1).  jet (  i s a real  because t h e  removed  will  18  produce  exactly  S,  place  in  radiated  system  If  as a r e a l  from  reflected  t h e same boundary rigid  from S, . W i t h  x i s large  by p r i m e s  (far field  (see L i g h t h i l l  1952  can s a y t h a t  reflected  radiated  from  information  T h e r e f o r e t h e sound equal to the  field  sound  field  w i t h t h e image  write:  approximation)  terms  by  like  - - ^  2  f  [•"]  (3*5)  1964)  the  the second  integral  the s u r f a c e  on  the  image  and  jet  i s good valid  space  noise  follows  i t  s o u r c e s i n t h e image  spectral  character)  either  is  for the r e a l  isolated  can  jet  of the s u r f a c e  o f t h e j e t (see s e c o n d  sound how  j e t (approximate be  obtained.  j e t . However t h e r e s u l t s  for a single  the  shown  i s an i n s t a n t a n e o u s m i r r o r image  as t h e i n s e r t i o n  change the s t r u c t u r e  represents  S, , o r [ b y Eq. (3.4) ] t h e  t h e image j e t . I n what  identically  unbounded  ^1  o  from  distribution  information be  can  as with  becomes:  where we  Since  we  or R i b n e r  v  spatial  by symmetry,  the q u a n t i t i e s a s s o c i a t e d  aire r e p l a c e d  - ^ • / " M - l  sound  boundary.  t h e image j e t i s e x a c t l y  denoted  Eq. (3.3)  conditions  t h e same may  radiating S,  may  paragraph of  not into  slightly section  19 3.2 C r o s s multiplied t , 1  Correlation  by t h e f a r f i e l d  sound  and t h e n a t i m e a v e r a g e  statistical absolute & Hark  stationarity  time  and depend  1963, s e c t i o n  Formalism.  only  (3.6)  pressure p at a d i f f e r e n t  i s performed. the  Equation  result  Under  will  the assumption of  on t h e t i m e d e l a y T = t - t * a time  T  where, a g a i n , t h e s e c o n d  integral  the  the  contribution  from  could  image.  of the  (Crandall average:  ^/[f^]p dV^M^]pdS  4.pp( ) =  time  be i n d e p e n d e n t  1 . 5 ) . The o v e r b a r d e n o t e s  is  (3-7)  be r e p l a c e d by  This  i s  the  part  we a r e  interested i n :  pp <*>,„«,.,„„ where Xj / j = - x c o s 0 differentiation converted into  the  processes  1963,  section  retarded  respect  Its  to time  made under  of  mean  -  9)  fact  to time  stationary now  the  (3  that  t h e time average can  with respect  brackets  delay  i f  ( C r a n d a l l & Mark, evaluation  at  the  ( S i d d o n 1973a) X  that  is  statistically  1.6)» The  time delay  Notice  Use  differentiation  are  where y i n d i c a t e s  -i—ril-A^}^  (Fig.3-1).  with  be  =  = T =  the point  where p  the L i g h t h i l l  complicated structure  lx-jj|/c  does  g  stress  (3. 10)  0  i s detected.  t e n s o r Tjj  has d i s a p p e a r e d .  n o t c o n c e r n us h e r e .  20  Writing  Eg. (3.9)  dp  g i v e s the  associated  with  s  is  source  describe source  a  portion  of  We  area can  method  of  three  are  content  in  It  flow  Only  Eg. (3.7)  one  time  errors  whether t h e  as  from  the  a t the  image  jet.  the  m e a s u r i n g an  _ •  point  Section  actual  _  image j e t  a measure o f t h e  where "local"  IV  will  distribution  approximation resolution  derivative  two  time  Ho  flow,  radiated  further the  sound  for  supersonic  or  of  o f T|j .  but  gain  is  the  and  is  by  in  a  of  i n the  assumption  far field.  The  investigation  jets.  than  At  higher  between t h e  a i r , the  far field  low  work o f  the  derivative  digital  methods.  the nature  therefore  o f n o i s e from  subsonic  before  p e r t a i n s only to could  hot  the  microphone  would  source  fluid  have t o be  of the be  and/or  Hach numbers o r  d e n s i t i e s of  Chu  correlation.  circuitry  method  i n s t a n c e i n the  Eq.(3.11);  analog  (3.2)  the  sufficient.  the  of  order  i s done by afterward  s  involved  derivative  with  p  device into  a s s u m p t i o n s have been made a b o u t  isentropic  differences  ambient  a probing  derivatives  fourth  increase  process  to i n s e r t  of s u r f a c e p r e s s u r e  differentiation  correlating  3)  the  this  l o s s of s p a t i a l  measurement  indicates  Numerical  large  x=0:  #  s u r f a c e S,  the  necessary  (1966) r e q u i r e d t a k i n g a  used  setting  the r a d i a t i o n  the  by  and  ——]  determining  with a  i s not  a simple 2)  the  form  advantages:  1)  the  of  d  regard  strength  s t r e n g t h i n the  H e r e we  flow,  f  a unit  detected.  acoustic  -COSfl  2  Eq. (3.11)  p  in differential  with and  placed  21 such  that  the  sound  m i c r o p h o n e would would  be  not pass a g a i n  refracted  supersonic  reflected  exit in  mm  was  as  larger  noise.  expected  with  other. are  completely  of  That  (b)  their  The  contains and  the  ground  i t with  should  from  be  close  to  enough  away  the flow  the  dragged  along  fluid  of  the between  by e n t r a i n m e n t ,  nozzle pressure  one f o r t h e j e t  behind  alone  t h e j e t ( b ) . They to  about  t o be due t o t h e  the d i r e c t  reflected  more t h a n  and t h e  one p r o n o u n c e d  reflected  sound  waves  peak i s will  t o enhance or c a n c e l arising  line  requirements:  On t h e o t h e r  because  The j e t  (b) i s o f  the j e t center  resolution.  the tendency  Fig.3-1.  an  at a l l frequencies,  - tend  spatial  from  in  when  jet  each  engines  plane.  c o m p r o m i s e between c o n f l i c t i n g surface  in  (a) i s t h o u g h t  both  frequency  a hard  distance  conducted  (corresponding  T h i s i s a w e l l known d i f f i c u l t y  t e s t e d near  i.e.,  0.2  anechoic  as i t r e p r e s e n t s  on  shown  the wall i n s t a l l e d  as the d i r e c t  depending  t  were  at #=45° to the j e t axis;  other  not being  jet  where  shows s p e c t r a o f t h e a c o u s t i c  700 H e r t z ) . The "bumpy" n a t u r e  course  field  (1.5 i n c h e s ) , t h e Mach number a t t h e  peak a t a S t r o u h a l number o f a b o u t  room  the flow  experiments  setup  0.32. F i g . 3-2  the f a r f i e l d the  The  The  was 38.1  about  (a) ,  through  s u r f a c e towards the  jets.  room.  diameter  the  o r even s h i e l d e d , a s c o u l d be t h e c a s e  3.3 E x p e r i m e n t s . anechoic  from  the flow hand  i n order jet  to  the  to the s u r f a c e On t h e one hand  i n order  was a the  to i n c r e a s e the  t h e s u r f a c e has t o  be  to avoid  effect;  attach jet  decreasing  and  to  t h e Coanda nearby  far  surfaces  the s u r f a c e i s being  the p r e s s u r e  and  thereby  22 drawing also its  the j e t towards the s u r f a c e .  interfere developed  single with  with  the surface  noise  unimportant  [this  recognition  of  surface  Host  the assumption assumption  these  5 nozzle  of  line  a  short  frequency small  S  inch  in  high  that  direct  of the flow  interaction  would  introduce  that  surface  implicit  in  was c h o s e n  of  a  unwanted  shear s t r e s s e s are Eg. (3.3) ].  f a c t o r s the d i s t a n c e  In  between t h e  t o be 197  mm  (little  c o n s i s t e d o f a 0.5 i n c h t h i c k  (  (type  p  s  was measured  4 136)  connected  diameter  hole  resonator  formed  of the Helmholtz  sufficiently  may  s u b s t a n t i a l l y from  pressure  microphone  0.020  cavity  surface  diameters).  the surface  1/4-inch  close  this  is  limiting  p l e x i g l a s s . The s u r f a c e  & Kjaer by  must be a v o i d e d ;  and t h e j e t c e n t e r  more t h a n  of  j e t . Furthermore,  by v i o l a t i n g  very  the entrainment of a i r i n t o the j e t so that  s t r u c t u r e may d i f f e r  isolated  &  front  of  the  using  a Bruel  to the  surface  ( F i g . 3 - 3 ) . The by t h e h o l e  microphone  above t h e f r e q u e n c i e s  sheet  of i n t e r e s t  resonant and  diaphragm in  the was  the j e t  noise.  The  far  field  microphone  was  placed  for  most  the  about  1.9  experiments at 45° to the j e t a x i s at a  distance  meters  Some d a t a  was a l s o o b t a i n e d  was measured  a t many p o i n t s  from  the j e t o r i f i c e .  of  of  a t 60°  and 9 0 ° . The surface  surface  pressure  and c r o s s  p  s  c o r r e l a t e d with  the s i g n a l from  on  the  the f a r f i e l d .  23 The  signal  p r o c e s s i n g e q u i p m e n t and t h e a n e c h o i c  same a s w i l l  shows  an example o f a c o r r e l a t i o n  Eq. (3.11) r e q u i r e s t a k i n g t h e d e r i v a t i v e time  curve  point  flow  flow  s  "precursor"  the ,  microphone  this  precursor  from  "blip"  the  flow  axis. theory  and, i n p r i n c i p l e ,  are  of  a  lengths,  (Lighthill  subsonic  the  coherent  to separate  function  we  are  travel  i s  times,  on t h e t i m e  delay  (See a l s o A p p e n d i x B) In o r d e r t o  independent of the p a r t i c u l a r  measured  from  c a n be i g n o r e d .  workers  established  i n order  delay  the f u r t h e r the surface  with  characteristic  times  on  o  c h o s e n and t o make them c o m p a r a b l e the  point  o f the p r e c u r s o r does not v i o l a t e the  3.4 N o n d i m e n s i o n a l i z a t i o n . make t h e r e s u l t s  i s  to the  o f t h e time  a r e somewhat  i s moved t o t h e l e f t  However, t h e p r e s e n c e  which  including  t h e s m a l l e r t h e d i f f e r e n c e i n sound  t h e more t h e p r e c u r s o r  of  T = | x - y | / c ) one must n o t p u t  s u r f a c e S, t o o c l o s e t o t h e f l o w ;  from  travel  results,  around  the  c o m p u t e r . The  i s l o c a t e d and t o t h e  the part of the c o r r e l a t i o n  i n (the v i c i n i t y  function at  .  two p o i n t s t h e s i g n a l s  correlation  s  existence  i s measured; i n t h e v i c i n i t y  t o these  a secondary  interested  the  function p p (T).  was done on a d i g i t a l  t o t h e d i f f e r e n c e i n sound  and  and  which  of t h i s  noise r a d i a t e s i n a l l d i r e c t i o n s ,  w a l l where p  corresponding  the  a  where t h e f i e l d  source  0  as f o l l o w s :  The  the  T = |x-yJ/c  also exhibits  explained  were  be d e s c r i b e d i n s e c t i o n 4.4.  Fig.3-4  delay  room  quantities density,  and  experimental  setup  results  of  other  nondimensionalized  using  the  velocities.  1952) t h a t t h e f a r f i e l d  a i r •• j e t  issuing  into  It  acoustic  i s  well  pressure  a medium o f i d e n t i c a l g a s  24  p r o p e r t i e s obeys  the f o l l o w i n g d i m e n s i o n a l  p CC yoU^Dc-ZxTherefore  the  a c o u s t i c pressure  p was  similarity:  (3. 12)  1  nondimensionalized  by  2 (0. 5pU£) M D/x  (3.13)  2  M i s the j e t e x i t Mach number, x i s the distance to the f i e l d p o i n t . P  i s a near  s  variation  with  U  field was  0  diameters  downstream  the  s u r f a c e was  rigid  the  dynamic  proportional varied given by  as  H  to U .  At  o r U*,  i n Eq. (3. 12)  as  t h e same q u a n t i t y  distance  of  the  nondimensionalized Appendix 3.5 strength plotted  the  the  plane  by e x p e r i m e n t .  intersection  of  surface;  the  nozzle exit  the chosen  the e x i t  i n meters of  location  i . e . i t obeyed  plane  (3.13) e x c e p t  p  from  the  by  More  details  o  s  was  t h a t x was  surface D/0 .  i t turned  jet  with H  is  water. H i s out  that  t h e same s i m i l a r i t y  does p. T h e r e f o r e  its  A point s i x jet  chosen as r e p r e s e n t a t i v e . I n F i g . 3 - 5  at  2  at  determined  from  head 2  pressure  p  s  relation  nondimensionalized r e p l a c e d by  center are  h,  the  Time  was  found  in  line.  to  be  B. Results^  F i g . 3-6  distribution  on  shows  the  rigid  the  nondimensional  s u r f a c e . The  actual  source quantity  i s : (for r a t i o n a l e see Appendix B)  -cos  S  U A  =  45°  [  a7  _  P P] -T S  S  T  T  T  { (0.5,uW}^f f  (3.14)  2  The  strongest sources  appear  t o be  concentrated  i n an  area  4  25  to  14  diameters  convenient acoustic back  downstream  t o be ray  where  speaking,  able  to  theory,  the the  of  use  the  as i n d i c a t e d  sound  has  reflection  is  truly  far  field  plane).  character The  present the  rigid,  first  two  experimental  between  near  pressure  fluctuations  far  is  field.)  pressure, scatter  as  angles  in  p  s  and  far  vary  with  i n an a r e a  6  and F i g . 3 - 8 (far f i e l d  was t a k e n the  p  jet. Strictly  wavelength,  are  i f  approximately  are well f u l f i l l e d  by t h e  i s n o t so w e l l f u l f i l l e d a s in  field.  the  transition  (Fig.4-18  distance  before  to trace  a c o u s t i c d i s t u r b a n c e has  from  behaves q u i t e  s  from  be a p p l i e d i f t h e  (Fig.3-5) ,  shows the  where t h e n e a r f i e l d  early  electronics way  were  region how t h e  jet  center  changes i n t o the  like but  a  far  there  field  i s some  strongest  diameters  t o 12 d i a m e t e r s be e x p e c t e d  strength contours  before  the  I t i s however  ray  sources  certain  acoustic  a s 8 goes from i f ray theory  when  afforded  by  discussing the  image  the  problems  method.  refinements useful i n a  argument:  apparently  moves  The  from  9 0 ° t o 6 0 ° . Such a  2  shift  were a p p l i c a b l e .  a p p e a r s t h e r e f o r e t h a t one c a n " b o r r o w "  argument  for  t o j e t a x i s ) o f 9 0 ° and 6 0 ° . T h i s  introduced.  supporting  the  source  i n the research  of  what would  show e q u a l  direction  location  It  The t h i r d  a r e measured  mentioned  the  a typical  requirements  setup.  principle i n order  can o n l y  than  I t would be  i n the data.  qualitative  is  from  i f t h e wave f r o n t s  At 6 d i a m e t e r s ,  Fig.3-7  data  field  h~5D  i n Fig.3-9,  principle  (i.e.,  exit.  reflection  and i f t h e i n c i d e n t  surface pressures  line;  nozzle  emanated  r e f l e c t i n g , s u r f a c e i s much l a r g e r it  the  of  the ray a c o u s t i c s  spatial  Fig.3-9  resolution  illustrates  the  26 hypothetical  "zone of i n f l u e n c e " : the  predominantly whare  p  c o n t r i b u t e to the  i s measured. The  s  angle  of a p e r t u r e  angle  will  be  of  the  r e g i o n of  the  strength  on  source  j e t that the  smaller f o r higher  f r e q u e n c i e s s i n c e the  more v a l i d  as  This  p e r t a i n s both  reflection  mechanism i t s e l f  far  field  c o n d i t i o n . . The  This corresponds  larger  than  expect  a  generally  narrow  produce  argument s h o u l d  Using strictly will  be  the  angle  reflection  to  this  be  of  or  not,  2 to  8  the  1971,  T h i s statement  described study  above,  of  the  flow  about  refractive  effects  the  of  study still  the  noise  more  In of  therefore  not  flows  reflection  due  Powell  1959,  from  much new  region  the  exit  1958,  et  insight The  into  method  t o show i t s f e a s i b i l i t y as  image s y s t e m an  already does not  implicit  to temperature or v e l o c i t y method c o u l d  generation  from  supersonic  flows  not  impossible  to  in  mentioned contain  any  disregard  of  gradients  The  if  is  researchers  Lilley  sound.  difficult  of  turbulent jets.  particular,  except  i t  producing  f i n d i n g s of o t h e r  i n order  the  Mach number  the  700 is  the  downstream  however, add  production  noise.  derivation  the  1959,  i s presented  assumption  path  Dyer  does not,  noise  here  the  which  whether  a c t u a l sound  diameters  1958,  c h a r a c t e r of  to  Supersonic  r e g a r d l e s s of  the  {Ribner  the  decreases. and  0.5m  could  aperture.  argument  T h i s i s c o n s i s t e n t with  al.).  one  reflection  peaks a t about  f r e q u e n c i e s . There,  plane.  Lee  frequency  The  much more a p p l i c a b l e .  applicable found  j e t noise spectrum  much h i g h e r  be  wavelength  to a wavelength of about  h=0.197m. F o r  very  the  the  to s p e c u l a t i o n .  argument becomes more and  Hertz.  surface  zone o f i n f l u e n c e i s c o n e s h a p e d ;  c o n e i s however l e f t  to the  will  t h e r e f o r e be  along  used  to  i t  is  where  l o c a t e the  noise  27  sources  by  probing  Maestrello technigue  the  & HcDaid  for  analysis pressure  on  two  theory  is  quite  process  the  data.  enable  a  paper  reads  on  the  on  the  authors source  work. The  here: time  "...  this  undergoes c o n v e c t i o n  direction  i m p i n g e on  An that  the  implicit  significant quite  IV  from  time  the  could  flow  image t e c h n i q u e  where t h e  of  insight, In t h e  sound  to  sources.  The  detailed later  t o the ones i n  the  Maestrello  technigue  &  developed  between  turbulence  farther  to  enough  j e t but  scattering  be  The  the  of  the  upstream  ..."  as  Chu  (1966),  Lee  p a p e r ] can  shed  more l i g h t  on  can  determine at  waves  are  coming  the  most i s  when  they  surface.  Maestrello  information the  similar  [such  present  not  i n the  from  and  surface  jet.  obtain  i t l e a v e s the  generation  reflecting  in  large.  the  of the  A l l an  advantage  spectral  and  probing  section  problem.  guote  the  the  sound  to  e q u a l l y a p p l i e s to the  p o i n t of  a technigue  (1971),  is  sound  Thus t h e  Only  the  (1970)  i t i s generated  jet.  following  a the  the computations  of  distribution resolution  of  but  expected  There,  a complex F o u r i e r  of  are  image  determining  by  presented  first  the  paper  so  distribution  problems of s p a t i a l  McDaid  by  vicinity  and are  encountered present  different  space-time-correlations  Some r e s u l t s  sound  sources  coming from  complicated,  i f the  a  noise c h a r a c t e r i s t i c s .  sound  surface in  conclusion as  jet  waves a r e  point  a rigid  information  (1970) have a d v a n c e d  locating  where sound of  itself.  determining  means i s shown o f directions  jet  i s not  method.  (1970) a p p r o a c h i s  only obtained However,  t h e amount o f present  & McDaid's  in  very  easily,  but  order  to gain  some  experimental  approach  one  would  work r e q u i r e d think i t i s  is easy  28 to  obtain  spectral information  of  Eg.(3.9)  spectral  resulting  density  surface  on  the  difficult image  derivative  of  on  p p(T)  time  precursor.  that  the  Attempts  transform  occurs  in  3.3.  section  quantitative image  they  by  the  precursor  not  axis.  Fourier  affect  The  f a r enough t o  for  some  of  p p (T),  freguency  this  as  and  were n o t  ways o f  further  the  showed ranges.  destructive  already  ambiguity  the  including  g  constructive  the  Fourier  Fourier operation  the  of  of  this  of  itself  significantly  operation  the  transform  to perform  some  qualitative  i n s p e c t i n g the  seen t h a t  the  nozzle  and  the  the  where  discussed obtaining  explored  in  technique.  be  the  exit  zero  characteristic  between z e r o downstream  the  can  c o r r e l a t i o n s : F i g . 3 - 1 0 shows i n the  frequencies high  crossings  (lower  are  is  small)  distribution  and  of  It  a t the and  be how can  closer  larger  i s of course  downstream  jet.  higher  curvatures  curvature  behaviour  s c a l e s grow i n the  shows  s p e c t r a l information  moves downstream  crossings). This  turbulence  Fig.3-11  one  cross  (the c o r r e l a t i o n has  distance further  between  power  point  record  view  c h a n g e i n s h a p e as  lower  overall  c o n t r i b u t i o n from  because i t l i e s  0  spectral information  Nevertheless, obtained  the  at c e r t a i n f r e q u e n c i e s  In  transform  complete  because effects  a t the  makes the  gives  became n e g a t i v e  interference  the  Unfortunately,  does  delay  i n v o l v e s the  Fourier  shown i n F i g . 3 - 4 ) s u p e r p o s e s  at r = r / c  s  the  the  and  precursor  the  transforming  This  measured.  t o i n t e r p r e t : Eg. ( 3 . 1 1 ) The  the  c o n t r i b u t i o n to  c o r r e l a t i o n f u n c t i o n and  only.  left  is  i n s e c t i o n 3.3  cross  computing  from a u n i t s u r f a c e area  pressure  (discussed  i n the  by  to  peaks  becomes distance  expected  as  direction.  the  root  mean  squared  29 near  field  area  as F i g . 3 - 6 ,  in  the  pressure  of  to identify  successful.  t h e s u r f a c e . I t shows a peak  but the t a i l s  direction  attempt  over  The  the  sound  zone  j e t flow  sources  of  of  correlation  the  transverse  source  "strips"  perpendicular results.  way  would  on  p  cone shaped sort  be  alone  s  region  both  much  less  i s t h e whole  of  the  image  of " p o i n t s " t o the source  tracing  of the  the  from  results 3.6  sources  the  evaluated, direct  (e.g.Ribner  the  field  the  agreement  with  1959).  the  Fig.3-12  integral  hand  side  jet  noise  the  hand  of is  real  and  integrated  E g . (3.9)  can  calculated  t h e image  distance the  to  image out  done a s i n d i c a t e d measured v a l u e  from  be a  the r e f l e c t i n g  from  s i d e was c a r r i e d  B, t h e d i r e c t l y  is  without  the radiation  nondimensionalization  i n Appendix  becomes  strength of  Jco r e p r e s e n t  on t h e r i g h t  the  between 4 and 5  This i s i n close  surface  between  argument,  diameters,  located  was made f o r t h e d i f f e r e n c e i n  microphone  integration  of  Fig.3-12  c a n be u s e d a s a c h e c k o f t h e method:  f o r r = 0 . The l e f t  In o r d e r  correction  With  axis,  be  10  over  direction  location.  of  1958, 1962; Dyer  Eq.{3.9)  measurement  surface.  would  the  the r e f l e c t i o n  location  the nozzle e x i t .  Check.  i s integrated  i.e., in  peak. U s i n g  peak  When t h e " s t r i p w i s e " s o u r c e along  surface,  area  j e t a x i s a t each a x i a l  back from t h e acoustic  diameters other  to  strength per unit  I t has a pronounced  strongest  and  that  (which  rapidly,  sound).  When  by  technique  much l e s s  and p e r p e n d i c u l a r t o i t . An  influence  h e m i s p h e r e and n o t t h e narrow cross  decrease  i n t h e same  jet a  the f a r j e t . The  graphically.  i n s e c t i o n 3.4  (left  hand  side)  30  {2 and  the  integrated  corresponds errors  (0.5^U  o  surface  c o n s i s t s of  value  i t s imperfect  internal  0.177  x I0~  (3.15)  5  )M D/x}  sources of  and  =  2  computation  surface  c h e c k on  2  (right  to a d i f f e r e n c e of  i n the  derivatives,  -y  a 0.5  of  only  the  0.8  rigidity inch  db.  i s 0.147*10 .  are  the  (recalling  t h i c k sheet lends  of c a u s a l i t y c o r r e l a t i o n techniques  of  Apart  from  finite  numerical  that  extent most  a l s o Siddon  spatially 1973a).  their of of  the the  p l e x i g l a s s ) . This  further confidence to  This  _ s  c o r r e l a t i o n f u n c t i o n s and  uncertainty  consistency  s o u r c e d i s t r i b u t i o n s (see  hand s i d e )  i n the  resolve  use  complex  31  IV.  4.1  technique  such  a  way  a recently  1966,  1970,  Ribner  1972;  section  Siddon  I the By  pressures  and  the  the  elementary causality  & Rackl  Rackl  cross  radiated  i n f o r m a t i o n on  Use  1969;  taken  Lee  As  Lee  mentioned  correlation  the a c o u s t i c source  (Siddon  1971;  as t h e  acoustic pressure  and  i s made  technique  1972b).  pressures are  j e t flow  distribution  acoustic sources.  correlation  1971;  the  t o probe_ t h e  spatial  1973a; C l a r k e & R i b n e r  fluctuating  the  i n order  obtain  computing  S in  basic noise  between i n the  these  far  field  distribution  can  obtained. Section  concept and  developed  to  developed  emitters.  be  of  1971b,  quantitative  was  as  characteristics of  TECHNIQUE  Introduction-  This in  CAUSALITY CORRELATION  4.2  of c a u s a l i t y  correlation  The section  airfoil  about  setup  was  circular  (1.5  0.32)  pressures type  The  spectrum  were  from  a  treats  unit  field  pressure  The  computed  a  cross hybrid  in  the  volume,  Fig.4-1:  into  anechoic  specially  (described  in detail  was  by  an  monitored of  correlator.  the  As i n  subsonic  a  correlations signal  shown  inch diameter),  measured  sensor  far  as  discharged  pressure  microphone. on  u n d e r l y i n g t h e o r y and  volume.  III, a cold,  turbulent  the  correlations,  experimental  (Hachnumber  4.3).  summarizes  by two  Typical  room.  jet The  developed in a  section 1/2-inch  signals  were  correlation  32 functions section  are  remaining  reduced  and  sections after  discuss  of o t h e r  achieved.  the  Difficulties  once the  It  could,  of  various  4.2  self  will  be  found  in  viability  jet quietening  show  how  Comparisons  least  the  data  was  are  made  with  qualitative  encountered  in  agreement  trying  t o make  consistent. Possible explanations of  the  method has  f o r the  f o r i n s t a n c e , be  used  study  the are  been d e m o n s t r a t e d i t  investigation  i n the  is  of  of  other  flows.  the e f f e c t i v e n e s s  devices.  Theory. 4.2.1  Fluid  (1962) i d e a s acoustic  (1.1)  can  on  be  right the  distribution  For  the  ap 2  generated  by  subsonic  -  hand s i d e o f wave e q u a t i o n  of  theory  i s based  flow  as  on  an  Ribner's array  a quasi-incompressible  flows,  V p 2  The  turbulent  r e - w r i t t e n i n the  o  of  viewing  flow.  i C  The  Dilatations^  monopoles  isentropic  term  At  are  becomes a d i a g n o s t i c t o o l  4.4  results.  researchers.  method q u a n t i t a t i v e l y given.  More d e t a i l s  4.4.  The  results  shown i n F i g , 4 - 2 2 a .  Lighthill's  approximate  =  d  wave  form:  (4-1)  dy-, dy.  Eq. (4.1) f o r p.  a c o u s t i c quadrupoles  may  be  and equation  pu^  2  of  regarded  Acoustically, embedded  as a  forcing  i t represents  in,  a  medium  a at  33  rest.  It  can  be  ("dilatations")  converted  into  by s p l i t t i n g  a corresponding  the pressure  monopole  disturbance  field  into  two  parts:  p  -  P =  p  o  +  C 0 3  p  (4.2)  C 1 D  where: p P  = instantaneous o  = local  time  static  pressure  average s t a t i c  pressure  p c o ) i s d e f i n e d by  _ 2  CCO  V  I t has been shown t h a t p<o) within for  f i i  d  U  =  x  (  the a c t u a l  s  2  p<i>  (see K r a i c h n a n  1956 and R i b n e r  is  of  the  propagation  (not  relationship equation"  part  o f p<o>  which  the  the  .results  is  after  3  )  approximation  1962) .  pressure  generation)  and p<i>  U  perturbation pressure  an e x a c t l y i n c o m p r e s s i b l e f l o w and i s a good  M <<1  .  U  of  associated acoustic  expressed  in  the  combining  Eg. ( 4 . 1 ) ,  with  the  waves.  The  "dilatation Eq. ( 4 . 2 ) ,  Eq. (4.3) :  1  c  6 9  2  o  The  right  hand  at  2  —v  2  CD Pw  = - -V ^-V c at  <^>  2  o  s i d e o f Eq. (4.4) c a n be r e g a r d e d  as  the  forcing  34  term  of  the  solution  CO  (  X  equation f o r the a c o u s t i c  f o r the r a d i a t e d  potentials  p  wave  i  t  )  The  l  i n terms of K i r c h h o f f ' s  retarded  becomes:  >  =  d  47rc  where,  sound  p r e s s u r e p< >.  again,  0  2  the  J y  lx-yl  at  L  brackets  3  y  t  2  "  J  indicate  evaluation  at the  earlier  time t=t-|x-yj/c  0  (See F i g . 4 - 2 f o r g e o m e t r y ) . In or  the geometric f a r f i e l d ,  close  to  approximated  p  In  CD  the  as  (x,t) =  1/x  4TTC  from  1/1x|«  0  x  the  total  region.  i f the o r i g i n the  d\  J  c  (many t y p i c a l pressure  variation  of  the  the j e t . S t a r t i n g  diameters  the f a r f i e l d doubling  wavelengths  at a p o s i t i o n  line,  per  m a  Y  be  t  pressure  the  pressure  1/|x-.Yj  in  condition of  (the c r o s s c o r r e l a t i o n  (4.6)  fluctuation  downstream o f t h e n o z z l e and  rms  chosen  Then:  diameters center  term  is  ——r—J. d y  v  far field  shows t h e t y p i c a l distance  =  - - — r J  the a c o u s t i c  approximates  source  and  away),  P(x,t).  o f a 6 db  distance  with  the c e n t e r l i n e  moving a t r i g h t drop  ( 1 >  Fig.4-18  fluctuation on  p  4  a n g l e s to in  i s reached at about  measurements were made a b o u t  the 10 50  35  diameters  away).  The  straight  distance  line  represents the v a r i a t i o n  extended  backward  into  magnitude  of  an  radiation  of  the  field jet is  would  composed  the  equivalent whole  be a b o u t  pressure,  course  indicating  which  on t h e f l o w  arises  purely  from  i s  r e g i o n one c a n e s t i m a t e t h e  point  source  the s i n g l e  we do n o t e x p e c t  line  source  representing  each  the a c t u a l  of  to  or to disturb  have  the j e t will  source.  i s v e r y much s m a l l e r t h a n t h e sound  measured  which  equivalent point  the  i t s pressure  o f p<°>. In f a c t ,  o f many i n c o h e r e n t s o u r c e s  reaction  alone. I f the  20 t o 40 db weaker t h a n  than  per doubling of  j e t . As shown on F i g . 4 - 1 8  s o u r c e r e g i o n t h e r e f o r e p<*>  back  o f p<*>  i . e . , the approximation  be weaker  means t h a t  t h e 6 db d r o p  any  of  In the  p<°>. T h i s significant  t h e measurement o f p<o>  incompressible  turbulent  momentum  exchanges.  In  order  proceeds time  t o o b t a i n t h e mean s q u a r e d  t o square  correlation  derivatives.  and t i m e  o f p<o>  average  radiated  pressure  Eg. ( 4 . 6 ) , i n v o l v i n g  under a d o u b l e  This operation i s similar  integral  with  Ribner  a space-  four  t o t h e one employed  time by Chu  (1966). 4.2.2  Causality  obtain  t h e mean s q u a r e d  sides  o f Eq. (4.6)  t+T.  After  time  Correlation^ radiated  by i t s l e f t  averaging:  In the p r e s e n t formalism,  pressure hand  side,  by  multiplying  e v a l u a t e d a t a new  we  both time  36  p ( x , t ) p(x,t + T)  - „  =  1  /fp  2  4TTC/X  p(x,t+r)  can  be  independent  the  j.  Also,  in  over  assumption  of  function  r only;  of  PP(*,T>'  where pz  V  are  ~  ^  7 1  C  integral this  '2 c  X  on  / V  the  establishes  the  fluctuation  and  According (Crandall ways: ,  by  S  to  Hark  setting  radiated  a 1963)  the  left  hand  3  it  is  averaging  hand  and  Under  the  side  is a  side:  (4.8)  3  T  T = O.  causative  the  time  [ P ° (y) P(x)]. d y C  (4.7)  because  operations.  right  D  obtained  sign  case  f = t + r - t = t + r - t + | x-y | /C =T+ | x-y  is  3  ?  stationarity  similarly  A  J  d y  p(x,t+r)  interchangeable  statistical  =  -  L  under  integration  (y,t)l  w  •(/  taken  of  c  |/c  The  G  integrand  relationship  in  between  Eg. ( 4 . 8 )  the  source  sound.  property  of  stationary  t h i s integrand  can  be  random written  in  variables different  37  The  second  the  form  experiments  correlations Appendix  turned  p<o)p.  agreement  of  an  shape  instead  classical  evaluated;  4.2.3 in  extension  the  Note t h a t o n l y  more  T h e i r shape  i s achieved  (Fig.4-1)  only  the  two  of  two  time  the  of  4TTC  only  i  2  X  s  p<o^-detection of  theory  a  single  cross  antisymmetric. allows  into  °  the  from  J  the  to  be  unit  mean s q u a r e d  to  spectrum by  volume:  T=T  i s obtained  Eq. ( 4 . 1 0 ) :  in  S p e c t r u m ^ W r i t i n g Eg. (4.8)  T  i s obtained  is  flow  are i n v o l v e d .  and  total  the  employed  integral  detected  the  of  f u n c t i o n s ; good  inserted  radiation  L  typical  that  correlation  t o be  Strength  0  c l o s e to  h a v e been n o r m a l l y  t o the  contribution  F o u r i e r Transform  as  shows  i n most  measured f u n c t i o n s .  has  g i v e s the  the c o n t r i b u t i o n  The  the  derivatives  Source  form  the  actual  probe  p o i n t where p<o)  point  with  one  dV  Again,  most c o n v e n i e n t  i s very  of  of  approach;  Local  differential  to be.the  d e s c r i b e d h e r e i n . Fig.4-22a  C treats  prediction  out  by  <&{v)  pressure  setting  of p(x,t)  t a k i n g the  real  from  r=0.  from  the  part  of  38  03  C °  —.  .  00  PP^ ^ T  «i27ri/T  00  n 7 r  3 <£(»/)  °o  where  means F o u r i e r  more d e t a i l e d  derivation.)  The  ~CC071  Q  i27Ti/T  .  -00  Sine  Transform.  (See  Appendix  A for a  C o r r e l a t i o n Volume^  correlation  length  *  5  ^{[Aw].}  av  4.2.4  ff  1  .  volume  is  scale in turbulence.  defined The  by  latter  analogy  to  i s defined  by  the  integral  CO  L(x)  =  u  J ° u ( x ) u(x+£) d£  (4.12)  0  where u i s a v e l o c i t y component  and  vector.  Fig.4-3  integrand  integral  only  L  gives  turbulent  volumes or  the  over a c e r t a i n d i s t a n c e ,  an  indication  energy  correlation  shows t h a t  the  is  the  typical  e x c h a n g e . S i m i l a r l y , we  volume eddies  follow,  of  £  end  we  and  time averaging  g i v i n g an  associated  with  the  space  will  separation to  the  c o r r e l a t i o n length  L.  contribute  lengths would  i n d i c a t i o n of  f o r a moment, the Eg. ( 4 . 6 ) :  a  involved  like  the  to  typical  r a d i a t i o n of  more common method  in  define  a  s i z e s of  s o u n d . To of  the  this  squaring  39  t« = t - | x-y' I / c  where  contribute  c  -co  where  y_"=y'+£  mean s q u a r e d  C O >  \>  2  differentiating  and  J  the  Eg. (U. 13)  pressure  from  5V By  a  consideration  down t h e a l t e r n a t i v e  ;CCO  and  '  use  n  volume V . T h u s V  -=!=-/ ['P  =  c  t«' = t - | x-j | / c  s i g n i f i c a n t l y t o the  correlation  v  and  c  inner  L  J  with unit  only  over  will  the  D 3  y  «  «»•«>  T i ss e t t o zero.  Therefore,  t o volume r e s u l t s  i n the  volume:  ,6,2 4 2 X  P  '"• ' 15  t o t h e one i n Eg. (4.9)  we c a n w r i t e  forms  C  whichever  ]»  respect  P °V<"}  a  £ 0 >  time d e l a y  similar  integral  The i n t e g r a n d  i s d e f i n e d by  c  ].[P  y  +f.  form  =  i s the  Eg. (4.15) a n d E q . (4.10) a n d u s i n g  { f tP °V° } „ C  most  5  convenient.  Eg. (4.16) we g e t  C.16)  Combining  40  V  The slightly of  x,  «  r  4TTC X  — —  2  0  usefulness  1  location  indicates  the  of  this  correlation  of d e t e c t i o n  average  (4-17)  r /  •CO)*  i n t h a t i t i s dependent  the  " °°  size  volume  V.  upon t h e  c h o i c e of the  of  radiated  of  the an  eddy  suffers direction  sound  radiating  p.  i n the  It x-  direction.  The  j e t i s o f t e n thought  incoherent  sources,  dilating  they  point  as  such  producing of  eddies  eddies  n=Vj /V , et  c  in  into  on  other  results:  et  each o t h e r .  j e t , then  whole i s the  hand, f r o m  the  If V  and  the  be  is  obtained by  a  number  compressing evaluated  size  and at  a  o f a l l sound  of  from  the  of  the the  jet  number quotient  turbulence  n o i s e . S u b s t i t u t i n g Eq. (4.14)  V =const, c  Eq. ( 4 . 8 ) ,  c  average  volume o c c u p i e d to  by  a rough e s t i m a t e  j e t can  significantly w i t h T=0  composed  turbulent eddies,  represents  i n the the  Eq.(4.13),  the  i t  where V j  contributing  i n essence  bump i n t o  that  of b e i n g  the  gives:  following  approximation  41  p (x) * 2  1 A  2  4TTC X  [/- p L  2  o  Therefore,  c 0 5  <3T  combining  pl  J  V f=r/c  (4.19)  j Jet  / o  Eg. (4.18)  and  Eg. (4.19) g i v e s ,  as  a  rough  estimate:  n  Here, in  Vie. —^  =  n i s a coarse  the  jet.  p  _ -  W  -  [&  P  —  measure o f  I f the  spectra  (see Fig.4-27)]dif f e r e n t i a t i o n r  can  be  approximated  p (2™) {p 2  4  [(27T,) p 2  C ( D  C C  P. number o f  incoherent  sources  p e a k y [as they are f o r j e t pressure respect  to time t or  multiplying  by  a  time  delay  characteristic  square  of  .ceo. 1--2 r LP"'Pi  V° } 5  p]  Eq. (4.21) shows t h a t inverse  are  (U.20)  2irv:  frequency  n -  C03  the  with by  —  the  L  n can  be  PrmsPrms  roughly  estimated  correlation coefficient.  by  calculatingthe  42  4.3  D e v e l o p m e n t Of The P r e s s u r e  4.3.1 In  order  L i m i t a t i o n s on C o n v e n t i o n a l  actual  total  present  experiments  of  t h e Mach number  of  measured true  Therefore  fluctuations  instantaneous  amount w h i c h flow  empirical  to  4.2.1 p<o) a p p r o x i m a t e s t h e  i n the flow  f o r M <<1. I n t h e  at theexit  was  eddies  will  2  about  0.32.  be s m a l l e r ; o f t h e  2  inherent i n the accurate  t o be w e l l measurement  i n turbulence are well recognized. For a  t h e c l a s s i c a l c y l i n d r i c a l c o n f i g u r a t i o n (Fig.4-4) the  instantaneous  cross  Probes^  t h e c o n d i t i o n H <<1 a p p e a r s  The d i f f i c u l t i e s  pressure  probe  an  perturbation pressure  0.2.  satisfied. of  i n section  Mach numbers o f t h e c o n v e c t i n g  order  Cylindrical  t o e v a l u a t e e x p e r i m e n t a l l y Eg. (4.10) i t i s n e c e s s a r y  measure p < ° > . As m e n t i o n e d  The  Sensor..  pressure  pressure  p  m  i s generally  than  the  p^ ( i . e . f o r no p r o b e i n t h e f l o w ) by  d e p e n d s on t h e f l u c t u a t i n g  normal  lower  components v a n d w o f  t o t h e a x i s o f t h e probe  (Siddon  1969).  The  equation, p -p a:-Bp(v2 + w -v2-w2) m t  (4.22)  2  where B — 0.5 gives  a  good  approximation  (1971a) e s t i m a t e d can  be  quite  intensities instantaneous  instantaneous  0.04. large,  Although  In  a  typically  error  nondimensionalized the  t h a t t h e rms v a l u e o f t h i s  large.  are  to the instantaneous  turbulent  of  by t h e d y n a m i c error  will  t h e rms v a l u e  the difference  Siddon  instantaneous  error  j e t (where  the order  i s approximately  error.  of  0.025  pressure be l a r g e r ,  of  0.2) or  this  rms  higher  when  the flow  0.5yo(J ; 2  maybe o f t h e o r d e r o f  of the instantaneous  between  turbulence  error  c a n be  t h e rms v a l u e s o f t h e t r u e and t h e  43  measured  pressures  present  application  employed will  necessary  pressure  fluctuation  to  compensating and  the  velocity  evaluating arriving  found  generate  a  Interacting force  microphone. interacts  patch with is  v  w,  could  later  correlation  building  part of  interacting n o i s e as  sending later  value  these  the true s i g n a l ,  of  p , m  p r e s s u r e and  convecting a  a  the  same  leave  acoustic travel  time  another i t  was  the turbulence in  past  Fig.4-5.  the  probe.  fluctuating  pulse  patch  with  the result  i n a rather  with  localized  dipole  pulses  way  research  indicated  o f t h e probe sending  t o be b r o a d e n e d  error  instantaneously  i n this  the  and t h e r e f o r e make an e x t r a n e o u s  than  an  the pressure  and  be i m p r o v e d  t h e nose o f t h e probe  p<°>p. Because  itself.  t h e measured  These d i p o l e p u l s e s a r e c o h e r e n t  correlation  a  t  turbulence  t h e stem  using  p .  probes  induced,  of  and  correcting  extraneous of  values of  e r r o r c o u l d be  fluctuation  simultaneously  the early  A s h o r t time with  fluctuation  correct  components  cylindrical  significant  Imagine  pulse.  by m e a s u r i n g  During  maximum  2  way  It  i t a s t h e measured  o  a  the are  with.  by 0.5^U . T h e r e f o r e ,  showed  the accuracy  appears. that  side  also  In  techniques  probe t h e i n s t a n t a n e o u s  a t the t r u e pressure i f  reduce  i n a turbulent j e t attains  Eq.(4.22) t h e r e b y  Even problem  probe  smaller.  e r r o r we a r e c o n c e r n e d  substantially  cylindrical  (1969)  much  correlation  t h e same m a g n i t u d e a s t h e p r e s s u r e  Siddon  and  cross  0.05 when n o n d i m e n s i o n a l i z e d  conventional of  where  i t i s the instantaneous  be  about  o f t e n t u r n s o u t t o be  to the f i e l d  of  turbulence  o f f another  dipole  the  source  basic  c o n t r i b u t i o n to the the  probe  earlier  i s a tendency  peculiar  way  f o r the  near  the  | x - y j / c . Shortening the 0  44  nose  makes  the  three  contributing  d i s t i n g u i s h a b l e a s shown by t h e d a s h e d It noise  must be s t r e s s e d in  the f a r f i e l d  inserting  the  correlation of  the  noise  i s very  a very  appropriately noise  regions,  called  the  source  without  region,  in  'correlation  the  even  less  Fig.4-5.  although  the o v e r a l l  one d e c i b e l o r s o when effect  on  the  cross  because o f the d i s c r i m i n a t o r y  region  extraneous to  that,  the  flow  (separately  dipole  apparent  the  volume'.  many i n d e p e n d e n t  substantially  on  s i n g l e s o u t the, c o n t r i b u t i o n  localized  a r i s e s from  line  by o n l y  probe,  prominent  It  point  increases  cylindrical  method:  from  at this  effects  t o the o v e r a l l which  is  more  I f the o v e r a l l j e t correlated)  effect  can such a  altering  overall  the  source  contribute  r a d i a t i o n from  noticeably  power  localized decibel  value.  The the  4.3.2  Airfoil  best  prospect  possibility  Probe f o r suppressing  of reducing  the  the d i p o l e  surface  sensor i n the d i r e c t i o n of the f i e l d smaller  than  airfoil-like into  the  direction  correlation  probe s a t i s f i e s  flow  such  that  o f t h e mean f l o w  microphone dipole  the  as  this the  radiation rests in  area  microphone  area  of  the  to something  It  of the f o i l  i s  in  the  A  and o f t h e r a d i a t i o n t o t h e f a r  plane  of  the  foil  much flat  inserted  contains  shown i n F i g . 4 - 6 . The t h i n s t r u c t u r e  radiation  pressure  the turbulence.  requirement. plane  of  the field  ensures'that  is  virtually  eliminated.  In basis  order  t o p u t t h e a b o v e s t a t e m e n t s on a more  i t i s h e l p f u l to introduce  quantitative  the "Probe C o n t a m i n a t i o n  Ratio"  45  C.  C  gives the r a t i o  due  to dipole  squared eddy  is  radiation  acoustic  like  from  the  pressure  or c o r r e l a t i o n  would  between t h e mean s q u a r e d probe  surface  due t o q u a d r u p o l e  volume s u r r o u n d i n g  mathematically,  how in  appear  estimates that  probe b u i l t  these  the  insensitive For  ratio  giving  locating  the  to  dipole  foil  a  the  for  almost  compared  a  thin  cross  using  a  via  case  the f o i l  It  type  would  pressure  dipole  plots  strategically  probe  suggest  the  between  p<o> sound  BSK  airfoil  on F i g . 4 - 9  radiation  cylindrical  can  be  the  pressure  t h a t by l o c a t i n g t h e edge,  t h e probe  can  of attack f l u c t u a t i o n s  both  airfoil.  This  of  The m i c r o p h o n e type  was mounted  course  i s large  dimensions.  show t h a t  the  effects  be e l i m i n a t e d . C u r v e  (as  measured  by  p, w h i l e c u r v e  1/8-inch  made  (Fig.4-7).  can  a l l other c o n d i t i o n s being  an a d a p t o r  the  and  (Fig.4-8),  to angle  probe  n o t be a p r o b l e m .  to the t r a i l i n g  of  we  D shows how C  the s c a l e of the turbulence s t r u c t u r e  and t h e r a d i a t e d  nosecone, "a".  that  correlation  probe)  course  o f aerodynamic o r i g i n  airfoil  insensitive  correlation  extraneous  from t h e  problem.  shape  holes  subsonic  to the c h a r a c t e r i s t i c  The  present  certain  n o r m a l t o and a l o n g t h e s p a n presupposes  Of  Appendix  noise should  sensing  sensing holes close  made  the  pressure errors  in  distributions  be  that  pressure  example  pressure  radiation  mean  d u r i n g t h e p r e s e n t r e s e a r c h h a s a s m a l l enough  contamination  By  rough  the  i t c a n be e s t i m a t e d , and g i v e s  some e s t i m a t e s o f i n t e r e s t from  and  the probe.  t o keep C a s s m a l l a s p o s s i b l e .  defined  acoustic pressure  foil  shaped  with  a  short  t h e same a s f o r c u r v e  on a BSK t y p e  UA0160 m o d i f i e d i n t o  "a" i s a  "b" was o b t a i n e d by  microphone kept  a  of  2618  a right  preamplifier  angle  connector.  46  The  irregularities  radiation and  from  from  d o e s not  probe  of  represents  the  going  cross-spectral  situation  a  the  may  separated  a shape as  of source to  i n the  generated  radiate  at  convection  noise,  and  times,  f o r f u t u r e work.  sensor.  An  early  was  devoted  type  with  the f o i l  and  for  of a random  positive.  from It i s ,  localized  constructive  Such  a  3 . 3 ) . For  dipole  source are  situation the case  source  and  somewhat c o h e r e n t .  the d i f f e r e n c e  being  of  this  the f o i l  d i d not  They the  probe  to  phenomenon i s  type  pressure  trailing  edge o f  exhibit  satisfactorily  t o a n g l e o f a t t a c k f l u c t u a t i o n s i n the p l a n e  to  i n the chordwise  of the  g i v e n by  t h e nose o f t h e  investigation  1971)  negative  density  from  It  curve  possible  where  a s e n s i n g h o l e a t the  to f l u c t u a t i o n s  sensitivity  (section the  "b".  freguencies i f spatially  to d e v e l o p i n g  wing t i p ( S i d d o n S R a c k l sensitivity  seem  arising  Sine "d"  from  consists  coherent.  t h e d i s t a n c e from  left  care  curve  power s p e c t r a l  valleys  image e x p e r i m e n t dipole  does not  at c r i t i c a l  sensing holes. A further  Great  from  situations  a r e t o some d e g r e e  different speed  nosecone,  Curve  calculated  distribution  p e a k s and  the  low  t h e one  hypothesize  c o r r e s p o n d i n g t u r b u l e n t (eddy)  low  calculated  dipole  the F o u r i e r  f l o w s h o u l d a l w a y s be  interference  sources  seen  probe  the  as  the source  exhibit  destructive  is  the  cross-spectrum.  density function  wherein  possible  spectra  "c"  to  becomes n e g a t i v e f o r some f r e q u e n c i e s . A  volume  however,  was  or  attributed  particularly  Curve  o f u n c o r r e l a t e d phenomena: The  unit  and  are  probe,  'cross-spectrum  and  array  "a"  have a s " n i c e " even  "b"  support.  curve  "a",  a  curve  the c y l i n d r i c a l  the  transform  in  direction.  a n g l e s o f a t t a c k i n the d i r e c t i o n  To  of  have  normal  to  47 the plane of the present  a p p l i c a t i o n as  Remember plane  airfoil  that  i s desirable  the f o l l o w i n g  the  field  t h a t c o n t a i n s the  foil  p r e s s u r e i n Proudman's  p(x,t  4TTC  (conf. This  /  x  if  0  2  clearly  shows  x of the  measurement  probe.  that  field  errors  interacting  (Fig.4-6).  f o H at  L  2  with  Errors  only  show:  i  The  far  same field  J  d y  (4.23)  3  f  are  due u  to  lies  x  w  contribute  t o the c r o s s c o r r e l a t i o n  contribute  t o the r a d i a t i o n  Various  p e r p e n d i c u l a r to t h i s  Pressure  configurations  some t y p i c a l a stainless  upper s u r f a c e of of  of  turbulent  the  steel  plane u,v  The  the  foil  c o r e o f a 1.5  type  velocities  of  the  in this  in  because  w  foil  plane  the  not,  pressure  are  velocity principle, does  Pressure  flow  inch  were  They  not  connects errors  were d e t e r m i n e d  diameter  j e t about  tried  consist  i n b a l s a wood w i t h an  of the tube  foil.  probes  shown i n F i g . 4 - 1 1 .  embedded  end  the onset  pCo)p  will  Pressure  Calibration^  of  tube  plane  p.  i n the  x  p.  examples being  cross section.  potential  in  to  t h e p r o b e d e s i g n . E r r o r s d e p e n d e n t on  Static  inclination  the  to turbulent v e l o c i t i e s  component  4.3.3  component u  microphone c o n t r i b u t e s  the probe. due  the v e l o c i t y  by  type  the  is  minimized  of  will  in  Fig.4-10)  direction  out,  necessary  microphone i s p l a c e d i n the  form  2 ~  not  argument  probe  (1952)  =  but  airfoil  to the lower due  to  and  changing  statically  i n the  0.5 diameter  from  48 the e x i t probe is  plane, using a P i t o t  configuration  t h e wedge  that  shaped  deviations  from  P*,) / (°-5p £) • u  F  o  true  of  static  of the f o i l  (v-component) the  less  a  reference.  in  than  4 . 3 . 1 ) . The o u t o f p l a n e  form  error  p  than  error  the  C = (Pp  .008.  curve  By  way  probe  value of  p  to  i n the plane  for a cylindrical  an i n s t a n t a n e o u s C  gives  ±12° ( e q u i v a l e n t  t h a n 0.2) t h e C  error  the  The  performance  Fig.4-13  pressure,  v a r i e s by l e s s  incidence  turbulence could attain  as  Fig.4-12.  angles of a t t a c k l e s s  r  intensities  (section  tube  gave t h e most s a t i s f a c t o r y  sensor  turbulence  comparison,  static  of  i n 20%  about  0.04  ( w - d i r e c t i o n ) has a  marked a n t i - s y m m e t r y .  T h i s i s most p r o b a b l y due t o one o r b o t h o f  2  airfoil  reasons:  symmetry flow  1)  section  and t h e p r e s s u r e t a p s  due t o an i m p e r f e c t m a n u f a c t u r i n g  development  the c e n t e r l i n e minimize  sensing  was i n f l u e n c e d  different  static  angles  pressure  i n t h e u-component  holes  thickening  itself  at  the  fluctuations  a  the  are  trailing  tendency  for  located  of  attack.  measurement  (direction  of  due  p  the j e t  n o t b e i n g on In  order  to  due  to  error mean  i n a d i p , immediately  C  2)  by t h e p r o b e  edge. The r e s u l t i n g  negative  process;  flow)  forward  pressure recovery  to  the  lack  finite  the  of the offsets  thickness  (Fig.4-14).  The  C  estimated  p  from  error  due  Fig.4-13.  A =  where P P^,  is  m  to We  u-component  can  be  define:  (P -P )/(0.5 oU|) m  ee  i s t h e measured s t a t i c the s t a t i c  fluctuations  pressure that  (4.24)  y  p r e s s u r e f o r 0° angle of would  occur  without  attack,  t h e probe i n  49 the  flow.  At 0°  angle  of attack,  pressure  i s shown t o be  known. By  varying  reference  Pitot  uncertainty In the  of  deviation  tube  where t h e  from  =-0.006. However, P^  p  the c o n f i g u r a t i o n of static  in  the  the  C =0-line  foil  as  not e x a c t l y  found  the  true  order  and  the  that  the  of  0.003.  ,  = -0.009 =  p  d i s c u s s e d i n the  remains v a l i d  the  probe  j e t i t was  l i e s i s of  p  was  worst c a s e :  C  If,  C  the  i n the  next  (P - oo)/(°- /> o) p  5  = A  U  m  paragraph,  t u r b u l e n t flow  we  the can  (4.25)  static  write  calibration  (Siddon  1969):  o  In  20%  turbulence  the  most  -0.0036.  The valid  suggestion  in  the  assumption. small  Such  with  the  mm.  the  -U/4L-2500 Hz  in a  restriction. even  smaller  frequencies Hz  In  up  the  p u  must  upper 100  zone o f  t o about  5 kHz  implies a sensor  the approaching be  small,  foil.  For  This  maximum s h e a r and  although  t h e r e f o r e expect  the  we the  dimensions flow. In L  will  is  the probe  be  about  a  severe  quite  the  mean v e l o c i t y  into  are  other  present  interested  j e t noise  t o run  remain  is  are  at  guasi-steady  where  frequency  flow.  m/sec),  w i l l t h e r e f o r e be  calibrations  i f the  limiting  m/sec  70  -error  flow  holds  l e n g t h of the  the  may  C  incidence  s c a l e s of i/L/0  (about  ( F i g . 3 - 2 ) . We  static  assumption  chord  Thus  The  time-varying  ratio  characteristic L-10  an  = 0.2.  that  real,  compared  words,  u/U  peaks at  is in 700  d i f f i c u l t i e s of  50 resolution. (section  This  4.5.2)  may  d o e s n o t work o u t . As t h i s  p r o b e o f an a i r f o i l It  is  with  hoped  regard  4.3.4 Fig.4-15  the  type  i t should  that further  The C o m p l e t e consists  Pressure  of  the  i s connected  2618 p r e a m p l i f i e r .  Sensor  foil  pressure  upon a s a p r o t o t y p e . follow, i n particular  a s shown i n F i g . 4 - 1 2  p r o b e mounted t o a BSK  v i a an a d a p t o r  type  and  1/8-inch  UA0160  to  a  was f i l l e d  with  p a r a f f i n and  p r e a m p l i f i e r - a d a p t o r combination  was c o a t e d  with  paraffin  mechanical  resonances.  to  t h e microphone combines with  to  form  about  a Helmholtz  resulting flat  The c a v i t y  frequency  connecting  out with  resonant  o f the p r e s s u r e sensor.  at the f r e q u e n c i e s c o n t r i b u t i n g  a c o u s t i c spectrum.  holes  response  a c o t t o n insert.. Fig.4-16  response  to  the f o i l  the small pressure sensing  r e s o n a t o r . The r e s u l t i n g  2500 Hz was damped  essentially the  will  check  The a d a p t o r  eliminate  the  the f i n a l  i s the f i r s t  be l o o k e d  development  why  to m i n i a t u r i z a t i o n .  microphone which type  be one o f t h e r e a s o n s  at  shows Itis  predominantly  to  51 4.4  Experiments The  experiments  dimensions with  were c o n d u c t e d  2. 7m*2.7m*1.8m  a lower  cutoff  i n an a n e c h o i c  (measured  from  f r e q u e n c y o f about  chamber  with  t h e t i p o f t h e wedges)  300 Hz. The j e t n o z z l e was  placed  n e a r one c o r n e r o f t h e room; t h e n o z z l e e x i t  38.1mm  (1.5 i n c h e s ) .  diameter  was  i  4.4.1  A i r Supply^ S e t t l i n g  ( Fig.4-17) It  A reciprocating  fedinto  flexible chamber heat  and sound  settling a  the  insulation.  chamber. I m m e d i a t e l y  cotton  wool  the  filter  mesh s i z e  uniform  velocity  analogy  t o t h e magnetic  1944) .  The  mould  obtain  B&K  located  inside  The F a r F i e l d  (1.9  type  used  for  the  entry  into the placed  d u s t and a s e t o f s c r e e n s w i t h velocity  distribution  was d e s i g n e d s u c h  result  that a  at the nozzle exit current  using  (Smith S Wang  machine,  the  g e l coat as the f i r s t  The  nozzle layer  Microphone.. was  the j e t axis  meters)  milling  by  surface.  microphone  a t 45° from  diameters  the  o f t h e n o z z l e were  through a r i n g  on a n u m e r i c a l  a smooth  1/2-inch  of the s e t t l i n g  n o i s e p r o p a g a t i o n from  a uniform  would  b e i n g made o f f i b r e g l a s s  4.4.2 A  upstream  Host  and  The n o z z l e i t s e l f  flux  as the a i r supply.  o f t h e n o z z l e was g e n e r a t e d on a c o m p u t e r .  was p r o d u c e d  itself to  shape  valve  to retain  distribution  of  eliminated  the  t o ensure  nozzle entry.  chamber.  fibreglass  This  from  was used  tank. The a i r then f l o w e d through a  settling  with loose  especially  diminishing at  to  was p a c k e d  upstream,  compressor  a large recieving  hose  Chamber^ and N o z z l e ^  from  used at  as a  the f i e l d  distance  the nozzle exit.  That  of  microphone about  50  the f a r field  52 condition of  was met i s shown i n F i g . 4 - 1 8 :  t h e rms p r e s s u r e  across  the  near  fluctuation field  to  I t presents the v a r i a t i o n  from  the  inside  far  field  characterized  by t h e 6 db d r o p f o r e v e r y  The  was done p e r p e n d i c u l a r  at  traverse  an a x i a l l o c a t i o n  4.4.3 S i g n a l (Fig.4-19)  The  amplified, and  signal  signal  from  p  the  far  of  functions acoustic  delay  peaks  eliminating correlation spectra  r=r/c .  "a"),  is  indicated computer graphical  The  function.  the  digitizer.  measure  order  correlation The  and  fed  raw  into  to record  correlation  correlator  itself  and  judgement  local  in  derivative  tails  only  in  process  was  weak  used  in  of the  the  cross-  o f the complete  source strength,  line  functions  was  the  extracted to  by  the source-reciever  affects  (vertical quantity  In  B6K  Transformation  The u n f i l t e r e d  by E q . ( 4 . 1 0 ) .  by a 1 / 2 - i n c h  the s i g n a l  contamination  principal  was  The  was u s e d  manifests  subjective  0  sensor  Correlator.  The  time a f t e r  Fourier  at r=r/c  the plotted  p<o>p. from  some  This  from  measured  latter  acoustic  functions.  the derivative  exit.  r e f l e c t i o n s o f f t h e p r o b e s u p p o r t and  Some  0  resulting  correlation to  from  101 S i g n a l  An x-y p l o t t e r  noise  occurring  the  distance).  and s t a r t i n g  pressure  t h e same way,  functions  some  mechanism.  time  filtered  correlation  contamination  secondary  being  270 - 22500 H z ) , d i f f e r e n t i a t e d ,  field,  the c o r r e l a t o r .  contained  traversing  the  A o f a PAR M o d e l  channel  resulting  to the j e t axis  from  (passband  was a m p l i f i e d ,  the  (t)  ( 0 >  microphone, B  latter  of  the n o z z l e  region  Processing^  channel  p(t)  flow  (the  doubling  o f 4 d i a m e t e r s from  filtered  fed into  the  Fig.4-9,  from the  equal curve  the data as data  on  were d i g i t i z e d u s i n g calculated  by  a a  least  53  square  fitting  vicinity  a polynomial  of  r=r/c .  Eg. (4.11) was (Rackl  45° jet  estimated  1972a).  the  to  j e t a x i s the  on  the  and  of  the  Sine  Transform  microphone always a t the  Source  procedure  distribution  The  pressure  sensor  was  microphone,  probe s u p p o r t  Strength  with  i n the  source  strength  of  Nondimensionalization  s  uv ~~  2{0.5 U ) M P  nozzle 0 /c o  Q  nozzle o  package  moved a r o u n d  i n order  t h e sound  to  field  in  of the  minimize (Fig.4-1).  locations.  2 2  Z  previous section  was  source  unit  effected  exit =  ttach exit  distance  volume as  in  the  shown  (4.27) 2  number. diameter, - field  in  shown i n A p p e n d i x  velocity.  nozzle exit  results  strength plotted i s :  D/x (U/D) 0  per  where:  x  utility  to  Distribution.,  described  nondimensionalized  D  according  Reduction  4.5.1  Fig.4-20.  the  same l o c a t i o n  t r a v e r s e s were done a t v a r i o u s downstream  4. 5 D a t a  in  Fig.4-22b).  the s i d e o p p o s i t e to the  Radial  B.  (Fig.4-22a field  Fourier  function  using a Fourier transform  Keeping  interference  The  The  0  to the c o r r e l a t i o n  microphone.  It  i s not  similarity  t h a t the d i s t r i b u t i o n  to the d i s t r i b u t i o n  traditionally flows  surprising  been argued  s h o u l d be  o f mean s h e a r  of  bears  in a jet. It  t h a t sound g e n e r a t i o n  from  s t r o n g e s t i n t h e r e g i o n s o f most  has  turbulent  intense  t u r b u l e n c e o c c u r r i n g where t h e g r a d i e n t s o f mean v e l o c i t y largest. half  a diameter  shifts this not  Thus, over from  toward the  first  the  indicate  a x i s and  the  to the  few  diameters,  j e t axis;  p o i n t i t s h o u l d be  contribution 45°  the  further  decreases  emphasized  t o t h e mean s q u a r e d  i s strongest  rapidly  peak  i n amplitude.  that this  pressure  are  downstream t h e  s o u n d power r a d i a t e d f r o m  some  distribution  At does  u n i t volume, but i n the  far field  the' at  j e t axis.  F o r e a c h downstream integrated a "slice"  resulting of  location  i n the  the  radial  distribution  distribution  of source  strength  was from  the j e t :  00 Ssiice  Fig.  4-21  qualitatively 197 2; Laufer  Ribner & Kao  techniques far of  field about  are based  =  1958,  1962;  (causality location  on  27rr  dr  only with r e s u l t s  1972). The  0.3.  u v  shows t h e p r e s e n t  The  of  and  correlations); to the  H.K.  (Lee see  &  Ribner  also  Chu,  Lee's works use  similar  L e e ' s d a t a were t a k e n  j e t a x i s and  are not  without  I t i s known t h a t h i g h e r  compares i t  J o n e s 1968;  semi-empirical techniques a slice,  and  of other workers  present  4 0°  (4.28)  result  D y e r 1959;  other curves  power r a d i a t e d f r o m ality.  /S r=0  a t a Mach number  experimental  data;  and  the  indicate  c o n s i d e r a t i o n of  frequencies are  for a  they  sound  direction-  r a d i a t e d more  55 p r e f e r e n t i a l l y i n d i r e c t i o n s i n the v i c i n i t y o f 45°, t h i s being due t o source convection  and r e f r a c t i o n  frequencies generated  closer  therefore to  more t h a n  expect  peak e a r l i e r  would  be  the  to  lower  the e x i t  the a x i a l  ones. than  source  for  discussed  the  overall  except  has t o r e a d j u s t  inside  the nozzle to a f r e e l y  point  out  of  that  similarity  i.e.  the  f r e q u e n c i e s : One f o r 45°  very  power  4.3.3  axial  location  where  rapidly  distribution.  The  prediction  from  to the nozzle  exit  from  shear  a boundary  layer.  the  layer  One must  where t h e  of  the  radiation than  very close  downstream  might  must however be b o r n i n  i t i s n o t known a c c u r a t e l y  become v a l i d  are  downstream)  Ribner's  growing  the higher  frequencies  distribution  i n section  considerations  where t h e f l o w  affecting  Higher  lower  mind. The g e n e r a l t r e n d i s t o s u p p o r t similarity  layer  ( i . e . between 2 a n d 3 d i a m e t e r s  expected  restrictions  by t h e s h e a r  also  assumptions  potential  x~ -Law  takes  7  core,  over  is  uncertain.  4.5.2 Further the  p  Cheeky C l o s u r e  integration  2  as  of S  measured  p r o v i d e s a check integration  of  calculations  j  c e  along  directly  a  10 times  However,  mistake  many t i m e s , a l s o  possibilities  considered  and many were d i s c a r d e d . Among  of  the  from  t h e probe  order  microphone.  of  Possibly  explaining  trailing  edge  14 K i l o h e r t z ) , valid  the p  is  Numerous  shedding  axis  should  recover  2  a s o b t a i n e d by  a s l a r g e a s t h e measured one.  numerical  were c h e c k e d  the j e t  a t x, a c c o r d i n g t o Eg. ( 4 . 8 ) . T h i s  of the procedure.  i s nearly  possibility  s !  Difficulty  ruled  out  by d i f f e r e n t  this  (the shedding  explanations are:  were:  were Vortex  frequency  shaking  the  persons.  discrepancy  the l a t t e r  mechanical  as  The  of  is the  56  X)_ The d e r i v a t i v e o f p<o>p i Eg. (4.10) travel  at  r =r / c  time o f t h e  microphone. T h i s  sound  time  from  "geometrical"  delay the  that corresponds  source  time delay  to  the  i s i n fact  to  to the  far  field  not equal t o  side  o f t h e j e t (Fig.4-1)  jet  flow  where i t i s a c c e l e r a t e d and r e f r a c t e d . The " e f f e c t i v e "  time  delay  i s therefore slightly  geometrical at  a  i s  time delay  where t h e s l o p e i s  collected  showing  that  the  would  steepest. the  the  p<o>p shown  According  to  the  have t o be c a l c u l a t e d  o f t h e p o i n t o f symmetry  on  time delay  the f a r  through  The t y p i c a l  t h e d e r i v a t i v e would t o the right  The e f f e c t i v e  center  smaller.  antisymmetrical.  the " c e n t e r " ) , but s t i l l  slope.  i s i n s e r t e d from  and t h e s o u n d must t r a v e l  quite  point s l i g h t l y  called  time as the.probe  according  real  Fig.4-24  travel  t o be e v a l u a t e d  the  in  sound  the  o f  S  portion  move t h i s  with  time  positive  p o i n t toward t h e  Experimental  effective  (here  evidence  delay  was  should  be  shorter: Some e x p e r i m e n t s inserted delay the  from  was v e r y sound  another about  in  not  0.1 i n c h e s from  with  the  travel  this  was p l a c e d  the f a r side) The  cross  and w i t h o u t  correlation  calculated on  side.  the  In t h i s case  through  immediately  shifted  pressure  the geometric  would  with  under  was  the j e t flow. was  time delay.  a  since  diameter  the pressure noise  computed  In the l a t t e r  observed  very  When t h e j e t f l o w  to the l e f t  time  expect  and d r i v e n by random  correlation  sensor  the j e t exhaust stream. In  a point source  function  geometrical peak  with  c l o s e t o t h e " c e n t e r " a s one  does  tones.  manner  near  s e t of experiments,  (inserted pure  the  were p e r f o r m e d  of  probe  and  by  i n the usual case  a  peak  c l o s e to the was  toward a s h o r t e r t i m e  turned delay.  57  This  shift  v a r i e d somewhat  different shift  frequency;  with  performed  the  i.e.  for  simple  setup  a  by  selected  vicinity  centered remained  [...].£}  the  parts,  which d o e s n o t  section effective This  time  It  decision  therefore  not  delay  have  integrated  p , but t h e change observed  not  However, for  a  Appendix  turbulence  plane;  there,  the  would f a l l  with  D deals only one  t h e probe s i z e .  been  of  should  the with  give  This  will  be  dipole noise could  s  or  going in  that the slope.  The s l o p e a t t h e in  a  f a r short of  smaller  accounting  the  that dipole  present gross  more  pressure  indeed  o f the probe i n t o  sensor.  consideration  when t h e y case  noise  turbulence properties  become  The  o f s e c t i o n 4.3.3  the  to  the  exit  dominant.  the flow  The s m a l l e r t h e probe, t h e l e s s  to  are of the order  closer  change t h e f l o w . discussion  assume  a  that  mentioned  steepest  best.  w  before  showed n e g a t i v e  D indicate  scales especially  3)_ The i n s e r t i o n  p<o>p  resulted  already  point  i n Appendix  problem  x/D=4. Haybe  smaller of  be  was v a r i e d i n  when  to  in  discrepancy.  2]_ The c a l c u l a t i o n s should  (indicated  i s s m a l l e r and would r e s u l t  2  the t o t a l  a v a r y i n g T,  centered  decided  by t h e  time  was  a sine transform  as  exact  transform  T-axis  If  the  was  at  expected,  possible  geometrical  for  be  tones of  experiment  Sine  the transform  i s given  may  third  the  frequencies.  seem was  delay  A  Fourier transforming  o f symmetry  p o i n t o f symmetry  4.3.2.  determine  "centered"  on  As may  for a l l  after  p<o>p  before 0  positive  to  used.  shift  of r=f=r/c .  on t h e p o i n t  and f o r pure  o n l y : The F o u r i e r  t h e amount o f o r i g i n  Eq. ( 4 . 1 1 ) the  i t was i m p o s s i b l e  on t h e c o m p u t e r  calculated  f o r random n o i s e  will, this  of  course,  disturbance.  shows t h a t t h e p r o b e may  indeed  58 be  too  l a r g e to adequately  point than  in  the  present  about 2 d i a m e t e r s  frequencies  are  error  no  would  4}_  It  longer  inserted  the  additional  the  p r o b e was  the  axis  behind  velocity  not  significantly  5\_ The  across  the  as  dynamic c a l i b r a t i o n  the  S p e c t r a . The  unit  frequency  i . e . , the  for various  radially  one  f r e q u e n c i e s are  results  small  f o r the  and  pressure  there i s a  the  probe  strong  when i t i s  T h i s may  error.  lead  to  However, when  directly  where t h e r e  would  above  hardly  the c o r r e l a t i o n  over  frequency  the  p<o>p  the  the  response  domain, 9c  some  the  be was  from  9a  source  slices  1972;' Chu,  typical cross  spectrum  according strength  cross  Again of  probe.  obtained cross  locations.  done  sensing  a  the  a g a i n s t downstream  & Ribner  of the  shows  typical  circles)  was  pressure  experimentally  s p e c t r a from  (full  sensor  Fig.4-9  d e n s i t y of  shows  plotted  (Lee  pressure  that flow  downstream  Fig.4-23  available.  the  domain c a l c u l a t e d  spectral  obtains  in  (closer  a diameter  probe,  time  Fig.4-22b  functions  nozzle exit  1/2  of  9a  f u n c t i o n i n the  volume.  that  pressure  upper p a r t of  cross-functions:  i n the  a  different.  alters  Eg. ( 4 . 1 1 ) ,  and  the  substantially  function  assumption  author  i n Fig.4-1)  be  of  where s c a l e s a r e  s u r f a c e of  at a p o i n t  I t may  correlation  peak  to the  only.  4.5.3 pair  out  gradient across  acoustically holes  quasisteady  distortion  inserted  any  exit)  at  f u r t h e r upstream  f a r s i d e a s shown i n F i g . 4 - 1 .  flow  (not  the  pressure  valid.  pointed  gradient  an  from  be  mean v e l o c i t y  turbulent  jet, especially  h i g h . The  was  from  r e s o l v e the  the  per  spectrum  integrating jet. Their  d i s t a n c e from  together Laufer  and  to  the  with  other  Kao  1972).  59 The  empty  argument:  squares  We know t h a t  derivative spectral probe the  of Fig.4-23  of  were a r r i v e d  the radiated  the source f l u c t u a t i o n .  density  o f p<o)  (Fig.4-27)'  was  a  s  measured  account  for  d e p e n d s on  Bearing  with  this  the  Fig.4-23.  directionality  and  simple  the  second  i n mind, t h e  stationary  m u l t i p l i e d . by t h e s q u a r e  peak f r e q u e n c i e s e n t e r e d i n t o  not  sound  a t by a v e r y  o f f r e q u e n c y and  Although i s  foil  not  this  does  amenable  to  *  quantitative  analysis  of source s t r e n g t h the  curves  fall  close  together. It 23  i s interesting  attains  Strouhal  t h e v a l u e o f about number  of  contribution  of that  must  dominate  case  the  part  the  curve  The  low  that  from  at  as  This  r e g i o n s beyond  i.e. in  Fig.4-21  to  the  beyond  omitted  X=7D, which  f a r downstream  can  be  as  10  t o bend  interpreted  in  i s  the  emitted  however,  still  further  a l l available  contribution  data  to the j e t  10 d i a m e t e r s .  sound  nozzle  resonators. I f this  the spectrum  sound  t h e r e i s h a r d l y any s i z a b l e  2)_ The low f r e q u e n c y  acoustic  the j e t axis.  does n o t show much t e n d e n c y  numbers.  peak  sound e m i s s i o n . " I n t h e p r e s e n t  done  frequency  downstream. L o o k i n g  plane,  "... due  the  ways:  21  noise  is  o f the j e t extending  still  to  n o i s e a t 4 5 ° from  this  were  Strouhal  none o f t h e c u r v e s i n F i g . 4 -  0.2 which c o r r e s p o n d s  t h e low f r e q u e n c y  down t o l o w e r  suggests  that  measurements  diameters;  that  the o v e r a l l  Lee (1971) h a s commented  different  t o note  a low f r e q u e n c y  i s generated  and  upstream  the s e t t l i n g  of the e x i t  chamber a c t i n g a s  were t r u e one may e x p e c t  to  see i n  hump above t h e j e t n o i s e which  would  60 vary  markedly  3}_  from  one  j e t to another;  of  the  points discussed i n section  Some  course,  be  reflected  i n the  smaller  t u r b u l e n c e s c a l e s are  Therefore,  if  enhance the  high  frequency  higher  Strouhal  peak  at  dipole  large,  the  produce  probe  measured  may  pressure  this  i s not  a s s o c i a t e d with  contents  were  and  make  numbers. I n a d d i t i o n ,  interaction a  of the  signal  signal  and  flow  which  again  of The  i t would  the  spectra  f a r downstream,  with  is  will,  freguencies.  problem  thus  case.  p o i n t 2) :  higher a  the  4.5.2  s p e c t r a . In p a r t i c u l a r  contamination  where s c a l e s a r e  but  the  stem  of  coherent  with  the  contaminate  the  cross  correlation.  The from  directly  the  jet  measured  axis  and  elementary  s p e c t r a over  the  [ E q . (4.11)  same  squared the  pressure  directly  spectra. of  high  The  spectrum  t h e one  the  j e t volume s h o u l d , over  check does not and  the  r e c o n s t r u c t e d spectrum  by  integrated apparently  45°  integrating a l l  theoretically,  v o l u m e ] . As  work o u t .  at  f o r the  Fig.4-28  be mean  compares  nondimensionalized i n c l u d e s an  excess  energy.  Number o f I n c o h e r e n t  correlation  i n the f a r f i e l d  that results  integrated  measured  frequency  4.5.4 The  this  sound  coefficient  Sources. of  Eg. (4.21)  at  a  location  the  jet axis  3  •  diameters out  downstream and  t o be  approximately  number o f  1/2  0.08.  incoherent sources n ~  It the  would  appear reasonable  order  of  100  to  a diameter  T h e r e f o r e , a rough i n the  to i n f e r One  estimate  turned of  the  jet n i s  (0.08) -2 ^  200.  from  150  from  this  number  t h a t n i s of  must however i n v e s t i g a t e  what  61 external  i n f l u e n c e s may  increase  or  decrease  the  correlation  coefficient: 21  Dipole  magnitude  of  coefficient: the  n o i s e o f f the the  In  correlation  because  p ms  ^  r  p<o)p r e p r e s e n t s  the  probe i s i n s e r t e d . be  given  Scharton  inch to  by  one.  The  probe  2]_  one  and  or the  White  both  of  present to  measured  but  at  two  case,  filtered  a point  will  by  affected  by  while  volume where  the  i n s e c t i o n 4.3.1). low  arrived  j e t with  value at  of  n«3  a  1/8-  using  Mach number  (Appendix  D)  T h i s high  probe  than  not  egual  for  this  value  would  only  plays  a  i t i s dominant.  i n f l u e n c e on  using jet  i n the  band,  be  the  be  the  correlation  whole j e t ,  small  1.8.  s i g n a l s being  uninfluenced  r m s  of  from  the  The  ratio  order  the  will  discussed  i f a significant  amount o f  by  wish  to  octave  band  filter  noise  spectrum.  filtering.  smaller  parts of  than  the  If  p<o>p  On  the  the  the  energy  of out.  increase  spectrum  correlation  other  the  may  j e t where t h e  the  magnitude of  correlated i s filtered  one  an  the  amount o f e n e r g y o u t s i d e  c o n t r i b u t e d from p  from the  diameter  have an  of the  a significant  is  r m s  of  increase  i n c r e a s e much more  (1972) was  for instance,  ratio  same o c t a v e  contains  that  may  the  noise  essentially  which  the  coefficient  peak f r e q u e n c y  the  of  Filtering  signal  in  already  contamination  is  role,  correlation  coming  c o n t r i b u t i o n from  that dipole noise  significant  p  p<o>p w i l l  noise  e  microphone i n a 5/8-inch  indicate  the  n  and  considerably underestimated.  configuration  In  t  ( T h i s was  Thus n may  p<o)p and  However,  s  significantly  function  Eg. (4.21) b o t h  dipole radiation.  Prms  p r o b e can  centred p  < 0 )  (t)  o f p<o> m  ay  other this  the  is  peaks  be  left  hand,  p(t)  octave  band  j e t . Therefore  unfiltered  on  one,  the  again  62  increasing  the  underestimation Scharton value  5  correlation of  n  White  coefficient  [octave  band  (1972) w h i c h  may  to  an  employed  by  have c o n t r i b u t e d t o t h e i r  low  coefficient  unaffected):  In Eg. ( 4 . 2 1 ) ,  other  signals  decrease  the  than  correlation  must t h e r e f o r e be the  present  pass  p(t)  n  pass  Lee the is  points  be  and  the  thereby  r m s  can  contain  j e t . This  superposing  can  frequency  frequency  p  n.  sources.  n o i s e from out  by  of about  the  a  270  4th Hz.  substantially  decrease  the  itself  p  and  and  r m s  on p  be  the  ( 0 )  Electronic  usually  would  overestimate  filtered  also  increasing  r m s  < 0 >  in  Eq.(4.21).  noise  e l i m i n a t e d by  (t)  has a  high  suitable  filter.  S Ribner  (1972, p.  1289)  o f 2500. A l t h o u g h  known i f i t may 3)  be  a cutoff  may  by  to  overestimated.  character  order not  and  p  the  function  to e l i m i n a t e e x t e r n a l noise  a i r had  noise  thereby  would  freguency  r a d i a t e d from  coefficient  with  coefficient  signals  Again,  filter  Electronic  correlation  acoustic pressure  one  decrease  the c o r r e l a t i o n  work f o r i n s t a n c e , low  order  4)_  significantly  leaving  the  taken  supplying the  high  (but  the  compressor  low  was  o f n-3 ].  correlation  In  leading  filtering  3|_ E x t e r n a l a c o u s t i c n o i s e c a n  Care  and  and  4)  above.  this  roughly may  estimated  n  to  come c l o s e t o t h e  have been o v e r e s t i m a t e d  as  be  of  truth i t  described  in  63  V.  SUMMARY  CONCLUSIONS^. RECOMMEND ATI ONS.. ii  Two  experimental  learning  more a b o u t  turbulent measure by  both  jet.  pressure  representative resulting some  acoustics used  diameters  If  the  disappear Eg.(1.2)  sources  principle  in a  i n that  i t to the radiated  the  s u r f a c e i s put c l o s e i s  limited  region  of source  they sound  taken  as a f l u c t u a t i o n  of  s t r e n g t h over  to the  the  jet.  The  the surface allows  noise sources i n the j e t using the r a y  principle.  technique  F o r t h e low  speed  t o be l o c a t e d  be  of  i s employed  advantage  to  t o measure b o t h of  no  [terms would  normal  (Siddon velocity  velocity  containing  in  replace  plane as the presence  development o f t h e flow  condition  t h e aim o f  subsonic j e t between 4 and 6  downstream.  hypothetical  necessary  sound  surface  s t r o n g e s t s o u r c e s appear  this  probably  on  reflection  the  a rigid  certain  distribution  conclusion  and r e l a t e  with  method.  on  of a  of  use the c a u s a l i t y  t h e image t e c h n i q u e  The  were d e v e l o p e d  distribution  fluctuation  a cross correlation  In  the  the  j e t . They a source  techniques  Uj  the the  of the r e a l 1973b).  future rigid  i t would  s u r f a c e by a  surface  I t may,  influences  however,  be  and p r e s s u r e , a s t h e b o u n d a r y on in  the the  surface surface  n o t v a n i s h , b u t may be n e g l i g i b l y  would  then  i n t e g r a l s of  small  in  some  c a s e s ].  In  the  direct  turbulence probing  technique  the f l u c t u a t i n g  64 pressure  i s measured  correlated  with  the  (1962) d i l a t a t i o n acoustic to  the  source  the  attempt  over  the  sensor  shape  was  the  also  required  using  a modified  point of  inclined varying  the  directly  angle  or into  should  of  by  even  flow.  the  be  nozzle attack.  be  Kulite the  of  i s in  However,  failure  shape of  the  over  an  the  optimum  presence  carried  of  the  further  to  However, t h e c o m p u t i n g  sensor  be  very  i s very  using  explored.  airfoil  large.  desirable and  Such c o u l d  surface.  a flow  be  which -  done i n a known u n s t e a d y which g e n e r a t e s  time  of  flow  flow  with  than  of  the  could  The  -  both  transducers other  semiconductors  A  [ f o r instance  to  the  would  are  guessing".  lead  be  similar  result:  theory  i n t o account  of  integrating  distribution  dimensional  pressure  feasibility  type  rotating  researchers.  view o f d i p o l e n o i s e c o n t a m i n a t i o n  The  Ribner's  jet; this  "educated  S m i t h ' s method  microphones should  calibration  by  pressure  takes  varying incident  piezoelectric  at  analysis could  of  the  sensor:  5 H e s s 1967) ] s h o u l d  time  disturbance.  pressure  cross  integration  Reasons f o r t h i s  (Smith  Miniaturization  Radial  of  of other  a  include  embedded  shear.  using  which  is  distribution  radiated pressure  i n the three  and  j e t which l o o k s somewhat  arrived  The  condenser  volume  f o l l o w i n g recommendations  investigation  surface  the  a  jet failed.  probe support.  from  to  results  total  f u r t h e r develop  S m i t h ' s method foil  sensor  r a d i a t e d sound. Using  leads  with  i n the  these  theoretical foil  field  type  mean v e l o c i t y  to r e g a i n the  To  present  of  agreement  From  XI  far  s t r e n g t h i n the  a l l sources  given.  airfoil  a c o u s t i c s t r e n g t h from " s l i c e s "  qualitative an  an  theory  distribution  gives  by  be  dynamic  such  as  an  sinusoidally  65 2)_ The  larger  problems w i l l at  higher  the  arise  j e t and  from  used  ratio  i n proportion to M  i n modern a i r c r a f t  3]_ The "center"  the  transducer  location  of  the  to  impossible  phase r e s p o n s e ,  known and  corrected for  The  thesis  Ribner's  pressure  correlation  exact  or  technique.  i n the  proposed.  broad  band  gives above  Since  i t i s an  a good  the  model o f The  possible.  delay  of  be  the  reduced  to the  far  field  some  way.  For  be  used  the  at  the  i n place. Since i t point source must  flow  noise  for  'at  method. In relative  are of  with  least  a be  can  viable be  its  source  source  the  exploiting causality  i n h e r e n t i n the  implemented the  of  using  others  correlations  the  measurements  feasibility  difficulties  the  recommendations  quantitative  Eg. (D.12).  from  j e t could  a strong  models of  time  in  latter  run  later.  efficient  p i c t u r e of  travel  response  were u n c o v e r e d ; some were s o l v e d , were  the  phase  established source  sound  build  this  D,  fluctuation  without  to  less  contamination  function should  calculated  o f sound  transducer,  probe  the c o r r e c t time  source  measured  the  more r e a l i s t i c  reduce the  correlation  the  a point source  next  uniform  t o be  be  shown i n A p p e n d i x  1 ) ] : The  measuring  m i c r o p h o n e has  is  causality  4.5.2, p o i n t  instance,  as  - 2  but  probe,  disturbances. Also, tests  only  uncertainty regarding  of  s m a l l e r the  extraneous  Mach numbers would n o t  jets  [section  the  solutions  performed  present  in  the  state  i t  distribution. highly  distribution  method  If  the  accurate should  be  66  REFERENCES  (Alphabetically according to preceded by a "*" i n d i c a t e s an 1963.)  the first author. A citation e x p e r i m e n t a l work p u b l i s h e d s i n c e  * J . Atvars, " R e f r a c t i o n of Sound by a M.A.Sc. T h e s i s ( u n p u b l i s h e d ) , UTIAS, D.I.  Jet Velocity 1964.  Field,"  Blokhintsev, " A c o u s t i c s o f a Nonhomogeneous Moving Medium," i n R u s s i a n 1946. T r a n s l a t e d as NACA T e c h . Memor. No. 1399 (1956).  * A . J . Bowen, J.H. Dunmore, and D.C. S t e v e n s o n , "An I n v e s t i g a t i o n of t h e N o i s e F i e l d P r o d u c e d Near a H a l f - I n c h D i a m e t e r Steam Jet," J . Sound V i b . 5, 113 (1967). * P.  Bradshaw, D.H. 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Hess, " C a l c u l a t i o n o f P o t e n t i a l Flow a b o u t A r b i t r a r y B o d i e s , " P r o g . A e r o n . S c . , V o l 8, pp.1-138, 1967. R.H. S m i t h & C.T. Wang, " C o n t r a c t i n g Cones G i v i n g S p e e d s , " J . A e r o n . S c . , O c t . 1944, p. 356.  Uniform  Throat  H. T e n n e k e s 8 J . L. Lumley, "A F i r s t c o u r s e i n T u r b u l e n c e , " The MIT P r e s s , C a m b r i d g e , M a s s a c h u s e t t s , and L o n d o n , England, 1972. *  R.B. W e b s t e r , " J e t N o i s e S i m u l a t i o n on S h a l l o w Mech. 4 0 ( 2 ) , pp. 423-432, 1970.  Water," J . F l u i d  70  APPENDICES  71 Appendix A  Mathematical  Eg. ( 4 . 7 )  where [ , . . ] ^  =  p (t)  D  i s independent  independent,  time  delay  -  can  t h e argument  be t a k e n  o f y. Time a v e r a g i n g and  argument-|x-y|/c sign  integration  stationarity r  only  c  the  The  time  left  over  Consider:  d at  Put:  ->  ~  d ar  are f u n c t i o n s  hand s i d e o f E g  a more c o n v e n i e n t  into  averages  sound  pp ( r ) .  c  0  because V are  i n t e r c h a n g e a b l e o p e r a t i o n s . Dnder t h e a s s u m p t i o n  can  t'=t+r  (t) by  under the i n t e g r a l  of the f a r f i e l d  put  Through E g . J 4 . J . l L i  (A.1)  autocorrelation be  Eg_._J4 .,7]_  4TTC,fx  means: R e p l a c e  (t-|x-y|/c ).  statistical  Concerning  i s repeated:  p(t)p(t+r)  it  Details  of  of the  ( 4 7 ) _ becomes t h e  The  e  right  hand  side  form:  dr dt  [p  C 6 5  (t -r)] p(f) ,  r  (A. 2)  where  [...]£  again  means:  replace  the  argument  by  72 argument* | x-yj / c differential  ( f = T + | x-j£| / c ) .  0  becomes i n  form:  app(r)  I  av  Now  Eg. (4.7) t h e r e f o r e  Q  - V TTC„  take the F o u r i e r  X  L  r a  -  [f dr  P  (  0  3  P ] ^ T  CA.3)  J  Transform  00  /  /.-.. e  dr  =  ,Z7rvT  ?{  ... }  (A.4)  -CO  of  Eq. (A.3).  The r i g h t hand s i d e  becomes:  CO  f\4-  r.h.s. = — V -  4TTC X 2  By p a r t i a l  u  r.h.s =  L  3  P °'PL ? C  T  J  T  (A.5)  dr  f [ COD 1  / | p " " p ] e \2ttvt d. r  g—  ** o  *  C  -co  the r e a l  part  Therefore,  the r e a l  density  T{...  3^ = c o s i n e When i t  £  T  app(r)/av  taking  ( A .,„6.)  A  autocorrelation  method  ^  CO  . — l2m/  By d e f i n i t i o n ,  where  i 2  integration:  4  spectral  e  of is  part  of p per u n i t  C  the  Fourier  is  usually  Fourier  transform  power  spectral  the  on  both  sides  results  of  the  density. in  the  volume:  i-% {• • • }»  } = T{ • • • } • transform,  the  3^=  sine  transform.  transform necessary  i s . estimated to  correct  by a the  numerical result  73  ("smoothing") to  infinity.  functions length made.  because a n u m e r i c a l In  the  present  p<o>p d e c r e a s e  of the record  integration  case,  sufficiently  available  so that  however, rapidly such  cannot the  be  extended  correlation  to zero within the  a correction  was  not  -74 Appendix  Nondimensionalization  There 1)  two  on  the  compare  final  with  reasons  inches  [in],  [msec],  and  data.  results  Lengths are  B.1  two  nondimensionalizing  R e s u l t s are  pressures  have l e s s  data:  degradating  more g e n e r a l and  easier  workers.  measured i n g e n e r a l i n N/m  i n m e t e r s [m],  (N=Newton), t i m e  2  sometimes  in  in milliseconds  voltages in Volts [V].  signal  input  C (T)  flow  channels;  of the  c  c  2)  of o t h e r  is  computes e^eg ( T ) , where e  S  for  C a u s a l i t y C o r r e l a t i o n Technique  The  are  of C o r r e l a t i o n F u n c t i o n s  F l u c t u a t i o n s i n ambient c o n d i t i o n s . w i l l  effect to  are  B  of the  electronics  are we  e (t)  = RC  eg(t)  = G g  A  C  eg  in are  dimension  c o n s i s t s of  identical  correlator  What we  and  A  the  correlator  numerically  shown  to the  Fig.4-19. the  The  v o l t a g e s a p p l i e d to  of e e g A  i s [ V ] . The 2  v o l t a g e s the  v a l u e s o f e^eg.  values of The  G g M  G  in  is  p<°>p.  Tracing  M A  Sg  S  A  p<o>( ) t  p(t)  where: RC  = time  output which  sensitivity  c  interested  C A  the  i s t h e r e f o r e S = 1[1/V],  get: G  correlator  constant  of the  differentiator  [msec],  through  the  75 GQ  = input gain  G|Y| = g a i n S  is  c  [V/in]), x y  o f measuring  amplifier,  = microphone s e n s i t i v i t y  C (T)  S  of c o r r e l a t o r ,  recorded  the plotted  C(T) = e e A  S  Q  on  an  x-y-plotter i s measured  (sensitivity i n inches.  S  x y  Then:  ,  c  C ( T )  = <  A  S  6 M  A  R  C  G  Nondimensionalization acoustic  2  f u n c t i o n C (T)  S.„  p™p(r)  [V/N/m ].  C A " S  G  B  M  B  i s effected  G  C  B  ) S  C  by d i v i d i n g  the f a r  field  p r e s s u r e by 2 (0.5^0-2) M D / x , 2  Q  the  hydrodynamic  pressure  f l u c t u a t i o n by 0.5/3U , 2  time  and t i m e  Using  delay  the  by D/U . Q  second  form  of  Eg. ( 4 . 9 ) ,  Eg. (4.8)  i s  nondimensionalized:  pp(r)  {2(0.5^U )M D/x } 2  2  2  -  o  P > r r a ospu? P a ( r ^ ) a(tuy ) { 2 ( 0 . 5 / > U ) M C0  (Uo/D) 4TTC  The  2 O  X 2M D/X 2  o  nondimensional  measured  -I  a  2  J O  L  y  2  D  integrand  cross correlation  data.  [...]»  i s  2  D/x }-  extracted  o  from  3  y  the  76 B.2 Image T e c h n i g u e The  signal  substitute and  omit  a quarter  is  inch  S  A MA CA G  G  i t was d e t e r m i n e d  nondimensionalized replaced  Equation  same  a s i n F i g . 4 - 1 9 ; we  x y  sensor  CM  1  S  pressure  Then:  B MB CB G  that  G  p  s  S  C  s c a l e s much t h e same way a s p, p  by t h e same q u a n t i t y  by h, t h e d i s t a n c e  (3.10)  the  microphone f o r the f o i l  S  T  S  is  i s nearly  the d i f f e r e n t i a t o r .  , , p p( ) = Since  flow  (3.13) e x c e p t  of the surface  from  that  x  s  G  the j e t axis.  becomes:  7 {2(0.5 U ) M P  o  2  2  D/x } c  cos 8 ( l l / D ) 4 T T C X h/x O  H = 0.325 The measured  Q  4  f \-A P§ J L.dfru/D) ?(0.5oU )M D/h D/h J L.d( u/D) 2(0.5 U )M s  /3  T  (typical), 8 -  integration  22  P _ _ l 2(0.5 2(0.5 MU)M )M D/x D/x k o/ 0  2 2  d  S  22  5 5 ° , h/D = 5.17. (See F i g . B - 1 . )  nondimensional integrand cross  o  22  correlation  was done by hand.  data  [ ... ] ^ was e x t r a c t e d by  a  computer  from  the  program. The  77  Appendix  Of The  The (1970,  procedure  1973a).  P "(x,t')  Eg. (4.7) p(x,t).  E g . (4.6)  was Here,  function  P  is similar  - ~ i r -  =  c  Shajse Of The  ( 0 )  time  average.  under  the  is  to  the  one  pCn,  so  by  we  the r i g h t  i n t e g r a l sign  outlined  by  Siddon  rewritten:-  d y' 3  multiplying  are i n t e r e s t e d  On  Cross Correla tion Function^  '/^(y'.t'-r^)  obtained we  C  (  both  sides  in predicting  multiply  Eq. (C. 1) by  hand s i d e because  the  the  c.i,  o f Eg. (4.6) shape  of  the  pt o > (j, t - r / c )  and  Q  factor  can  the i n t e g r a t i o n  be  by  taken  i s with  respect  to j ' :  p (y,t-r/c )p(x,t')  =  c w  o  (C2)  /  4TTC X 2  O  Retarded speed  time  V  C W  (y,t-r/cJ p  differences  flows,  compared  .  p  since  due  C 0  V,t'-r'/c )  to r#r' can  the a c o u s t i c  with c o r r e l a t i o n s c a l e s  d / 3  o  be  wavelengths (Lighthill  neglected  in  are generally  1962),  i.e.,  we  low large set  r=r«.  Assuming and  with T=t'-t  that we  a l l variables get:  are s t a t i o n a r y  random  functions  78  p (y,t-r/c ) p(x,f)  CO). p""p (x, r+r/c )  =  c0)  o  p^(y,t-r/c ) p (y' t'-r>c ) =  p > p '(C,y,r)  C05  0  t  where  c0  o  _  (C.3)  0  yi  dr  , u ,  C0?  £ = y ' ~ Y.  Hence  p  < 0 ,  p U + r/c.) =  -  T  -  1  4 "C  !  We s e l e c t which  x  pcoyo).  (the  =  the  p  v  P  C 0 5  p  C O 5 ,  (C,y,T)  d £  ( c . 5 )  3  0T form f o r t h e  property  and t i m e d e l a y  of  correlation  convecting  decay  with  p* ^* * 0  as  0  1  space  are increased:  co>2 f l ^ - u ^ J / L ^ / L ^ ^ / L ^ r / T }  1-direction  structure  0  a functional  exhibits  separation  f-jj-  -  77  i s the d i r e c t i o n convection  o f motion  velocity  (c.6)  f o r the  U ) . The f u n c t i o n c  turbulence f has the  properties:  (1)  f (|=0, T = 0 ) = 1  (2)  f#T^  o arg  all  > 0 f o r a r g —> 00 , s u f f i c i e n t l y  integrals  to converge.  (3)  | f (arg=0) | > | f (| a r g | >0) |  (4)  f (arg)  = f (-arg)  rapidly  for  79  (5)  f and  are continuous,  d org where  arg  stands  Xj = C / j > ^ T / T ,  real  p « V  The  here.  the  *t i s a n o n d i m e n s i o n a l  2  time  scale  of the  turbulence  i s parallel  is  for  to the j e t axis.  There  i n t h e x - and X j - d i r e c t i o n s 2  effect  i s very  in  due t o s p r e a d i n g  s m a l l and n e g l e c t e d  Put  of  X| some the  here.  consider:  dr  p  the  i s also  dt  d dr  no  d e l a y , not the  allowed  convection  -g-  of  (C.7)  which  Now  time  3  direction  this  a r g u m e n t s o f f . Now p u t  f ( x , - a t , x , x , t)  convection  jet;  4  E g . (C.'6) becomes:  =  w  of  C  concern  time.  any  a = U T / L | . T i s some s u i t a b l e  L  further  for  co) co}' t  u  V  dt  =  x, =x, - a t . 1  I  p  In  (C.8)  dr  co} W J  T  d  2  r •>> Y\ f ( x , - a t , x , x , t)  -jzz  what  o t h e r s being  with  (C.9)  f  2  follows  I I  differentiation  at  — —  3  d org  means  partial  r e s p e c t t o one o f t h e 4 a r g u m e n t s o f f ,  held constant.  Then:  the  80  a f  _  df_  dx]_  ar  "  ^  dT  =  - a ——•= + tfx,  df a  d dtf  .  =  tfx  dT  2  dj_ </x  +  3  dj_  dx^ dT  +  dT  (c. 10)  d\  di ,df dx] d\  (  —TI.  a  _dx^  —773-  1  d  2  +  dj_  -  ,  ,  2a  d d\  +  df ^di , dx] d\  ,  z  ( C 11)  I t i s mathematically convenient to assume that f i s separable:  f  and  =  f,(x}) f ( x ) f ( X ) f ( T ) 2  2  3  3  4  (C.12)  define:  CO  /  -CO  Then :  f:dX;  =  F;  (C.13)  81  p  c w  p ( r + r/0  x { «  2  1  The  =  'Vf, ^ / ^ -co .  property  =  -  properties  (1)-(5).  function  which  turbulence.  X  5  t  d  x  1  d  property  integral  P  L  '  l  L L  of It  0 9  T  (2) and  v a n i s h e s . With  3  F  2  T  F  For  a  dt*  convection  for  very  no sound  acoustic  output  nondecaying would  be  ( 5 ) . Because o f  dx =dx^  any  f|  strongly  radiated. that  For  satisfying  on  T  and  the  of  the  p a t t e r n £4  is a  properties  f r o z e n convected  i t i s necessary  finally  (  (CIS)  4-TTC^XT*  depends  (CU) J  |  t o g e t h e r d e s c r i b e the decay  constant;  then  0 2  2  i s z e r o by  independent  £4  F F, ' ' 4TT C„ x T  H  d  (4) t h e s e c o n d  is  L, U U ; ?  2  integral  7 V r + r / 0  p<°>p  ,2  ( 0 )  . . d f . °R d f, . d f, dx, - 2 a ^ f / ^ j - dx,+ ^ / f t f j d x , } ' -i) . ' -co' ' '  d x  first  -p  a  flow  to  produce  p a t t e r n s are generated  and  decay. If  Gaussian  we  make t h e r a t h e r  special  assumption  of  a  convected  for f:  f = exp{-(x,-aT) CO /f,dx, -CO  2  -xz  CO =  (C.16)  2  CO  /f dx 2  -CO  - x f -T }  2  =  /f dx 3  -CO  3  =  J¥  (C.17)  82  The is  total  area  no dc-component  acoustic  under e ~ ^ ( 1 - 2 T ) 2  i s zero.  2  i n p<o>p a s one might e x p e c t  pressure.  This  This  means  since  i s t r u e f o r any f ^ s a t i s f y i n g  p  there i s an  properties  (D-(5). The s h a p e o f t h e c r o s s c o r r e l a t i o n p<o>p c a n now  easily  be  predicted:  p -  p  _ A  =  -  p  TjT  p  co)  p  L, L  2  2 Fig.4-24 fitted  c  shows  a  c  2  2  p  L3  co>  2  e  - T  (  cross  the convected  fitted  function  _  2  T  3  }  {  to numerical  C  correlation p  -  i  g  )  f C )  p and  (C.20)  Gaussian  one  the could  f o r f i s a good appropriate  could  transform  assumption  coefficients  e m p i r i c a l l y estimate the  u n i t volume. The F o u r i e r t r a n s f o r m  s p e c t r a l density)  recourse  T  3  t h e agreement i s good. Knowing  from  3  e T> { 3 ( f ) - 2 ( f ) }  as  radiation  (  s q u a r e s a f u n c t i o n c f t h e type  appears that  the  2  measured c r o s s  It  from  £  p  xT  typical  t o i t by l e a s t  A  _ £  =  be o b t a i n e d techniques.  (equal  i n closed  form  to the without  83 Appendix C  Probe C o n t a m i n a t i o n R a t i o  When  a  interaction dipole  probe  of the flow  nature.  radiated  from  measurements this  dipoles  are  acoustical properties  V  c  (no  turbulent  radiation.  For  The  probe  acoustic  C t o be d e f i n e d  can  be  in  the  noise  the guadrupole  the  cross  principle  emitters  small  on  flow  creates  of a sound  correlation  i t i s necessary to  reader  may  recall  that  than g u a d r u p o l e s o f s i m i l a r surface  may  be  measurements.  able  The  to  probe  h e r e i s a number by which t h e  performance of a probe i n a t u r b u l e n t  probe  D. 2  causality  distort ratio  itself  the  a  a  the probe s u r f a c e  itself.  more e f f i c i e n t  Radiation  Eg. (4.23)  with  into  flow  dipole  i . e . , even  contamination  inserted  superposes  the  considerably  D.1  It  using  minimize  strength,  is  flow  with  given  correlation  volume  evaluated.  f r o m One C o h e r e n t Eddy the  flow).  It  with  follows  from  Eg. (4.15)  and  t h e probe  does  that:  Probe  Interaction  Noise.. A s s u m i n g  that  84 not  move and t h a t s h e a r  section  D.5),  integral  o f Eq. (1.2)  p.  =  and  stresses radiate  using  Eg.(3.2)  is left.  negligible  and E g . ( 3 . 5 ) ,  and  time  and p  i s t h e s u r f a c e normal  g  averaging, becomes:  If  process  s m a l l compared indicating  r a d i a t i o n , /? i s t h e  t h e n o r m a l t o t h e s u r f a c e S and  Eq. (D.2)  the  the second  - i f c V H p s ] ^  between  radiation,  only  (see  I t becomes i n t h e f a r f i e l d :  where t h e s u b s c r i p t " d " i n d i c a t e s d i p o l e t y p e angle  sound  and  is  with  setting  a  typical  evaluation  at  pressure.  cos/3 = 1  statistically  the  for  direction After  the  of  squaring  worst  case,  s t a t i o n a r y and t h e s u r f a c e i s  wavelength,  retarded  time  the  brackets  c a n be d r o p p e d  [..-] i n this  case.  Assuming probe the  (i.e. i t  integrals  squared probe. if  that p  i s well correlated at least  derivative  ever  contained  i n Eg.(D.3) r e p r e s e n t  nothing  of the f l u c t u a t i n g  and  i f  The  requirements  velocities  else  lift  but  the  force exerted  to estimate  a r e s m a l l compared  of  estimate  dipole  designed  "poisoning".  to gain C  L  is  the l i f t  L flow  shape.  contamination  some f e e l i n g the  mean  on t h e  to t h e mean  a r e n o t t o o s t r i n g e n t as t h e p r o b e  the  volume),  t h e p r o b e s h a p e comes c l o s e t o a s l e n d e r body  i s o n l y a rough amount  the surface of  in a correlation  We c a n u s e u n s t e a d y a i r f o i l t h e o r y  U,  the  is  the disturbance  ratio  s  unsteady  for lift  85 coefficient, a  i s the instantaneous  0.5yOU2SpC  L(t)  angle  of attack:  0.5/)U2S -^-a(t)  L  (D.4)  p  dC It  i s w e l l known t h a t t h e l i f t  curve  slope  L-  depends  on  the  da wavenumber see  of  the  incident  disturbance [Sears  a l s o Liepmann' 1952, p. 796 ; and- s e e  can  be  which  generally  the  moderate,  k value  of  as i n d i c a t e d  with  L  typically  o f 2 i s t h e r e f o r e used  subseguent  calculations.  i s t h e component  U i n the plane becomes a f t e r time  the  wave  Most  finite number  by t h e dashed  line  curve  probes  wings f o r is  only  i n Fig.4-25.  slope  in  the  With ^  v(t)/U  of the v e l o c i t y  (C.5)  disturbance  that contains the d i r e c t i o n differentiating  with  p e r p e n d i c u l a r to  o f r a d i a t i o n ) Eg.(D.4)  r e s p e c t t o time,  s g u a r i n g , and  averaging:  which  i s an e s t i m a t e  Eg. (D.3).  of the  ~  d  D.3  Definition  ,  acoustic  probe s u r f a c e t o guadrupole  type  ^  of  the  2  2  2  x  2  two  integrals  2  (  o f Probe Contamination ratio  pressure the  2  16 7T c  probe contamination  squared  product  in  Therefore:  pZ  the  ratio  f o r the l i f t  a(t) (v  Fig.4-25].  r e g a r d e d a s low a s p e c t d Ci  variation  (1941) f u n c t i o n ,  mean  radiation  as  the  R a t i o . . We now  ratio  due t o d i p o l e t y p e squared  from  acoustic  the adjacent  ' C = pj/p^ '  between radiation  7  )  define  the  mean  from t h e  pressure  eddy o f  *  D  due  to  fluid: (D.8)  86  Combining  Eq.(D.1),  E q . (D.6) , and Eq. ( B . 7 ) :  .2 / p c 2c S x-  ,_U_f C  2  c  0  speed  (you/)  For  low  then  c a n c e l s o u t i n Eg. (D.9). D.4  downstream and h a l f  representative.  where  U„ °  i s  by  correlation diameters  major  C  Also  from  is  imagined V — L£.  -  /0.6U \2 (-7- )  c  with  2irv  (0.6M)  we d e s i r e  S  2  5_ D  U  is that to  27r0.2U/L , c  Laurence  an e l l i p s o i d  1969):  0.13x 1957).  with  2  v* ^  about  0.6U , o  v — u __. — rms rms r m ( ?  r  time where L  can  c  \  (x i s measured  I f the c o r r e l a t i o n  minor a x e s  1/3 o f t h e  2  (D.10)  (D.11)  v~* = 3v* )  4  =  T  (277-0.2 0 . 6 U / L ) 2 ( 0 . l 5 U ) c  2  in  becomes:  ^ o  be  i s the  c  2v"2  f o r a Gaussian:  4  i s chosen  (277-i/) V — = -  2  L°  2  as  4 <V  that  S  2  (from  point  the a x i s  respect  m  T h e r e f o r e Eq.(D.9)  o  0  14  If  by  from  I t i s assumed  1  the exit)  then  M  velocity.  c o n s t a n t and  J e t T h e  velocity  l e n g t h , and i s q i v e n bv L_ —  (from S i d d o n  *  t h e mean  multiplying  (note  C  Turbulent  a diameter  differentiation  q  replaced  There,  the e x i t  0 . 1 5 U , and t h a t  volume  p c a n be r e g a r d e d  flows the density  E s t i m a t e s f o r a Low Speed  diameters as  -2  P  o  "  2  / Sp  (-tV) * L c  that  C be l e s s  than  0.1 f o r good  accuracy,  we  must  87  require  For  that  x=4D, L. —D/2.  Then  we  need S  < D M/48 2  p  e In  the  experiments described S  For  cylindrical  probes S  because the  pressure  velocities  are  in  Fig.4-26.  diameter  of  the  small  size  bend  downstream  noise  by  The has  a  of  and  amount o f  D.5  Is the  P r o b e ? The obtained Eq. (D.3) :  the  However,  the  jet  exit  present  probe  an  "effective"  the  turbulent  nose o f  the  to 2 d  (where  elements  vortex  by  to  sensor.  of  2  2.2 In  mm  shown  d  is  the  less,  a  very  the  90°  addition,  would  cross  p r o b e as  or  area  create additional  shedding.  i n s e c t i o n IV about  10  of mm , 2  this  thesis  thus  barely  c a l c u l a t i o n s here are the may  requirements be  done f o r  become  radiating a  more  significant  noise.  Skin  Friction  r a d i a t e d sound by  sensing  M=0.32: (D.13)  have t o be  area  the  the  i s egual  p  and  due  near  probe d e s c r i b e d  the  dipole  S  D=38.1 mm,  2  replaced  pressure  sectional  to  be  d would  separation  Eq. (D. 13).  x=4D. C l o s e r stringent  that  probe)  pressure  satisfying  can  thesis:  mm  fluctuations  f o r a dynamic  cross  < 9.7  p  concentrated  Assuming  flow  p  in this  due  considerations  Indeed  Unimportant  to s k i n f r i c t i o n similar  to  the  or  for  the  Foil  shear s t r e s s i s  ones  arriving  at  88  P.r  =  ,,  2 '2  16 7r [from the  the t h i r d same  result  c  Q  2  x  term  R e y n o l d s number  viscosity  steady  drag  to  the  a flat  we  becomes  6  f o r c e W.  length  the  of  The  probe  2  (D. 15)  for  rough  use B l a s i u s '  w  m /sec:  fr  number  For a  =  w  drag 2  velocity  the 2 i n t e g r a l s  transition,  estimate Law  of  i.e.  the  quasi  f o r the laminar  i t is strictly  the  applicable  skin only  plate:  W  p  Tinder  o  W=C S 0, 5/o (D + u) =  S^  Eg, (1.2) ]  U=0.6U = 66 m/sec, and t h e k i n e m a t i c  c , although  the instantaneous  where  of  of t h e drag  chord  < 5*10^ = Re  c  then,  on t h e  i s laminar.  coefficient  ' " " " I  f o r c e case  derivative  Reynolds  fluctuation  '  S  integral  as f o r t h e l i f t  Re i s b a s e d  is  t r  d  s  t h e mean v e l o c i t y  layer  friction  ] '  T  o f t h e a i r a t 20°C z>=14.9*10~  Re  boundary  / , [  S  i n the second  Re = bU/i/ = 4.36*10*  where  d  t h e mean s q u a r e  2  b=9.85 mm,  [+]  s  assumptions  i n W,  I  is  now  the  1  3  upper in  (D. 16)  f o r c e becomes:  f  fluctuation after  1.328//Re~  the  differentiating,  2  8  Sp 0.5/o(U2 + 20u+~0)  v Re f  s u r f a c e area mean  flow  of the f o i l ; direction.  s q u a r i n g , and t i m e  (D. 17)  u i s the E g . (D.17)  averaging:  -89  (1.328 S oU)^  —  p  W  We  -  2  now  U  f  Re  define  fluctuations  the  probe  by a n a l o g y  c  fr  present  IcJ  -  u  —  2  fr  = H  friction  (  D  '  1  9  )  (P^f  I  v  i  c  ;  u  JL76_  2  —  R  and  2  2 Q  e  putting  ( D  S  — IOS f o r  pf  p  '  2 0 )  the  probe:  Eg. (D.21) friction  shows i s very  to f o r c e s  normal  unimportant  in  present  type  foil  -  C  that  the  (D.21,  probe  contamination  much s m a l l e r t h a n to the s u r f a c e . the  dipole  Shear  stresses  calculations  p r e s s u r e s e n s o r . S i n c e C^  h a r d l y ever  ratio  t h e one f o r sound  noise  r  m a g n i t u d e s m a l l e r t h a n C, d i p o l e  flows.  f o r skin  /Pvc"  v" , u s i n g Eg. (D.9) ,  C„  probably  ratio  Eg. (D.19) and Eg. ( D . 1 ) :  _  Assuming  contamination  t o Eg. (D.8) : C  Combining  (D.18)  2  o f any i m p o r t a n c e  radiated  are  i n low s p e e d  due  therefore  affecting  i s about  n o i s e due t o s h e a r  for skin  the  2 orders of stresses  is  turbulent a i r  90  Fig.3-1  Image geometry  91  Fig.3-2  Nondimensional spectrum o f j e t n o i s e a t 45 a x i s , (a) i s o l a t e d j e t , (b) w i t h s u r f a c e S the j e t . (computer p l o t )  1  to j e t behind  92  Fig.3-3  M i c r o p h o n e m o u n t i n g . 1: % - i n c h B r u e l & K j a e r m i c r o phone, 2: b r a s s s l e e v e , 3: % - i n c h t h i c k p l e x i g l a s s panel.  93  F i g . 3-4  T y p i c a l c r o s s c o r r e l a t i o n f u n c t i o n between s u r f a c e p r e s s u r e p and f a r f i e l d p r e s s u r e p. D e r i v a t i v e t o be e v a l u a t i d a t T = r / c . N o t i c e p r e c u r s o r to the l e f t . 0  Fig.3-5  R e l a t i o n s h i p between s u r f a c e p r e s s u r e p  $  and e x i t v e l o c i t y  H = s e t t l i n g chamber p r e s s u r e i n m e t e r s o f w a t e r .  U. Q  U1  20  Fig.3-8  IS  12  8  4  Equal s o u r c e s t r e n g t h c o n t o u r s f o r 6=60°  VO,  98  far field  Fig.3-9  Zone o f I n f l u e n c e . A n g l e o f a p e r t u r e o f t h e d a s h e d cone n o t known.  99  Fig.3-10  3 t y p i c a l p ^ p ( x ) a t d i f f e r e n t downstream p o s i t i o n s  x.  Distance j e t axis to surface  i s 5.17D. p  d i r e c t l y opposite of j e t axis,  p measured i n f a r f i e l d  a t 45° t o j e t a x i s , ( c o m p u t e r p l o t )  s  measured  Fig.3-12  " S l i c e w i s e " i n t e g r a t e d s o u r c e s t r e n g t h f o r 9=45°  102  Fig.4-1  Exnerimental setuo of c a u s a l i t y c o r r e l a t i o n technique. P r e s s u r e s e n s o r measures p(°)(t), m i c r o p h o n e i n t h e f a r f i e l d measures D U ) ( t ) . . I) = e x i t v e l o c i t y .  origin  Fig.4-2  Geometry  uu(i) 4  space separation £  Fig.4-3  D e f i n i t i o n o f correlation length L  104  Fig.4-4  C l a s s i c a l c y l i n d r i c a l probe c o n f i g u r a t i o n . Upper p a r t : p r e s s u r e d i s t r i b u t i o n due t o c r o s s f l o w , f u l l l i n e : 40 < R e y n o l d s number < 40000, d a s h e d l i n e : p o t e n t i a l f l e w . Lower p a r t : V i n s t a n t a n e o u s v e l o c i t y v e c t o r , V a x i a l component, V c r o s s f l o w component, a n  105  Fig.4-5  T y p i c a l c r o s s c o r r e l a t i o n between f l u c t u a t i n g p r e s s u r e i n t h e j e t p(°) and a c o u s t i c p r e s s u r e p, c o n t a m i n a t e d by d i p o l e n o i s e : T u r b u l e n c e i n d u c e s a f l u c t u a t i n g l i f t on t h e nose o f t h e c y l i n d r i c a l p r o b e s e n d i n g o f f a d i p o l e p u l s e . Dashed l i n e : e f f e c t o f s h o r t e n i n g nose piece.  P r i n c i p l e o f a r r a n g i n g f o i l p r o b e i n t h e f l o w ( d i s t a n c e s and s i z e s n o t p r o p o r t i o n a t e ) : P l a n e o f f o i l c o n t a i n s mean f l o w d i r e c t i o n and d i r e c t i o n o f r a d i a t i o n toward f a r f i e l d microphone ( r i g h t ) .  L o c a t i o n s o f p r e s s u r e s e n s i n g h o l e s on a i r f o i l : (a) s u p e r s o n i c , (b) s u b s o n i c . The measured p r e s s u r e i s t h e a v e r a g e between t h e p r e s s u r e s on t h e u p p e r and t h e l o w e r s u r f a c e .  108  Fig.4-8  S u b s o n i c a i r f o i l . Upper p a r t : f o i l c r o s s s e c t i o n and d i s t r i b u t i o n o f p r e s s u r e c o e f f i c i e n t due t o t h i c k n e s s . Lower p a r t : c due t o t h i c k n e s s and a n g l e of a t t a c k , c : upper s u r f a c e , c_,: lower s u r f a c e . p  V  C o n t a m i n a t e d and u n c o n t a m i n a t e d c o r r e l a t i o n s and t h e i r F o u r i e r t r a n s f o r m s . p(°) d e t e c t e d a t x=3D, r=0.4D; p m e a s u r e d a t 45 t o  o  j e t a x i s . C u r v e a: fi(°)p (p(°) m e a s u r e d by f o i l t y p e s e n s o r ) , c u r v e b ( ° ) D (p(°) m e a s u r e d by 1 / 8 - i n c h m i c r o p h o n e w i t h s h o r t n o s e p i e c e . C u r v e c ? o u r ? e r t r a n s f o r m o f a , c u r v e d: F.T. o f b. T t i m e d e l a y , v f r e q u e n c y . D  Fig.4-10  V e l o c i t y v e c t o r s on a i r f o i l  probe  112  Fig.4-12  D e t a i l e d drawing o f f o i l type pressure sensor. 1 b a l s a l e a d i n g edge s e c t i o n , 2 s e n s i n g h o l e , 3 s t a i n l e s s steel tubing, 4 cotton wool, 5 t e f l o n c o n n e c t o r , 6 1/8-inch Bruel & K j a e r microphone.  0.020" 0.010 -w- component ./  -./  -.2  -0.010 0.005  -0.010  Fig.4-13  ^  .2  tana  v- component  -  J  S t a t i c pressure c a l i b r a t i o n s o f f o i l type pressure s e n s o r shown i n F i g . 4 - 1 2 . c = {? - P )/(0.5pU£). q  m  t  v-component: a n g l e o f a t t a c k a i n t h e p l a n e o f t h e f o i l , w-component: a p e r p e n d i c u l a r t o p l a n e o f f o i l .  114  location of pressure tap  Fig.4-14  Estimated pressure d i s t r i b u t i o n over f o i l with d i p n e a r t h i c k e n i n g t r a i l i n g edge.  Fig.4-16  Frequency response of f o i l type p r e s s u r e sensor.  117  compressor  anechoic room Schematic o f a u x i l i a r y equipment  Fig.4-18  V a r i a t i o n o f pressure f l u c t u a t i o n w i t h d i s t a n c e from j e t , s t a r t i n g a t 4 diameters downstream i n t h e j e t and p r o c e e d i n g a t r i g h t a n g l e s t o t h e j e t a x i s .  Channel  Channel  A  B  p >(t)[N/m ] (o:  119  p(t)[N/m ]  2  2  ^ 2  preamplifieF  1/8 "microphone S  [v/N/ 2]  A  i  m  amplif.  measuring gain  S  [^N/m J 2  B  MA  6  measuring gain  amplif. G^ B  i HP  filter  differentiator, time constant RC CmsecJ  'CA  input gains  y (»[v]  * (t)[v]  B  A  C (z)[v] r  'CB  CORRELATOR  S [l/V] r  e7e^(v)LV J A^B 2  graphical digitizer  Fig.4-19  Signal flow  deck of punched cards  l  122  Fig.4-22a  Some t y p i c a l n o n d i m e n s i o n a l i z e d c a u s a l i t y c o r r e l a t i o n f u n c t i o n s p ^ p a t h d i a m e t e r f r o m t h e j e t a x i s , downs t r e a m p o s i t i o n x v a r y i n g . P l o t t e d h i n c h above i s f i t t e d 7th order polynomial, (computerplot)  123  x = ID I  O  Fig.4-22b  Some t y p i c a l c r o s s s p e c t r a ( S i n e t r a n s f o r m o f f u n c t i o n s shown i n F i g . 4 - 2 2 a , m u l t i p l i e d by f r e q u e n c y ) ( c o m p u t e r plot)  124  Fig.4-23  Peak f r e n u e n c i e s v e r s u s downstream p o s i t i o n .  Fig.4-24  Measured c a u s a l i t y c o r r e l a t i o n p^ 'p ( c r o s s e s ) and f i t t e d t o i t the p r e d i c t e d f u n c t i o n .  126  k&(4 chordlengthsF'  waven. k  steady case Fig.4-25  L i f t c u r v e s l o p e as a f u n c t i o n o f wave number (Sears f u n c t i o n )  '^^pressure  distribution  estimated effective chordlength ~2d  Fig.4-26  E f f e c t i v e c h o r d l e n g t h on c y l i n d r i c a l  probe.  127  -1  0  Fig.4-27  1  0.5  1  1.0  —t  1.5  1  vD/U  g*—  0  Nondimensional s p e c t r a l d e n s i t y o f t h e "pseudosound" p(°^ a t 4D d o w n s t r e a m , %D f r o m j e t a x i s , as measured with f o i l type sensor.  128  Fig.4-28  D i r e c t l y measured and i n t e g r a t e d s p e c t r u m a t 45 i n t h e f a r f i e l d . C o m p a r i s o n by shape o n l y .  129  Fig.B-1  Image g e o m e t r y t o s c a l e  

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