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Dipole radiation intensity generated by cylindrical struts Marriner, Duane E. 1979

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DIPOLE RADIATION INTENSITY GENERATED BY 9 \ CYLINDRICAL STRUTS by DUANE E- MARRINER B.Sc. Physics (Honors), University of V i c t o r i a , 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) AT THE UNIVERSITY OF BRITISH COLUMBIA We Accept This Thesis As Conforming To The Required Standard THE UNIVERSITY OF BRITISH COLUMBIA July 1979 (c) Duane E,. Marriner, 1979 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 r e q u i r e m e n t s 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 s t u d y . 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 c o p y i n g 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 g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f 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 not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department n f M e c h a n i c a l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date A u g u s t 1 5 , 1979 DE-6 BP 75-5 I 1 E ABSTRACT The purpose of t h i s work i s to further establish the v i a b i l i t y of the Causality Correlation Technique as a diagnostic t o o l for treatment of noise problems. Acoustical dipole radiation can be generated by obstructing a subsonic flow with a r i g i d s t r u t , i f the s t r u t exerts f l u c t u a t i n g forces on the f l u i d * Such forces would be forces of reaction a r i s i n g from unsteadiness i n the l o c a l flow and would form a d i s t r i b u t i o n of acoustical dipole sources over the surface of the s t r u t . For the experiments reported herein, the subsonic flow issues from a c i r c u l a r nozzle which i s 3^8 x 10- 2 m i n diameters The •quiet' a i r jet operates at an e x i t Mach number of .217,. The exit v elocity i s 72 m/s and i s approximately uniform over the exit plane. In the f i r s t experiment the cylinder model i s stationed at the potential core of the jet ; the Reynolds number i s 6.3x10* (based on cylinder diameter and exit v e l o c i t y ) . Later the cylinder model i s stationed at the turbulent t r a n s i t i o n region of the jet* The "Dipole Radiation Intensity (DRI) * (see Sec.3.1.2) i s a uniquely defined and measurable quantity that i s intimately related to the c l a s s i c a l dipole strength. The ' s p a t i a l d i s t r i b u t i o n ' of the DRI can be constructed on a surface using the Causality Correlation Technique (see Siddon 1). The 'DRI d i s t r i b u t i o n ' i s constructed on the surface of the r i g i d c y l i n d r i c a l strut; A diagnosis i s made of the aerodynamic noise generation mechanism using the said d i s t r i b u t i o n * Circumferential p r o f i l e s of DRI reveal that: for i i i both laminar and turbulent incident flow , a f l u c t u a t i n g l i f t force on the cylinder model produces dipole radiation (dumbell shaped r a d i a t i o n d i r e c t i v i t y ) with i t s major lobes i n the cross stream d i r e c t i o n ; f o r turbulent incident flow , an additional f l u c t u a t i n g drag force produces additional dipole radiation with i t s major lobes p a r a l l e l to the flow* The SPL that i s generated by unit length of the cylinder model becomes 10 db more intense when the model i s i n the turbulent mixing region of the.jet compared to that generated when the model i s i n the potential core;* The far f i e l d SPL o r i g i n a t i n g from the surface exclusively i s predicted from the integrated DEI d i s t r i b u t i o n . For laminar incident flow the predicted SPL i s (69.3 ±2.3) db. This may be compared with an o v e r a l l SPL of (70.1 ±,5) db which was d i r e c t l y measured. However the o v e r a l l SPL contains j e t and wake noise.contamination. The l e v e l of t h i s contamination i s estimated and i s then deducted from the o v e r a l l SPL. The r e s u l t i n g corrected SPL i s (68.5) db. For turbulent incident flow the predicted SPL i s (71.33 ±2)db compared with an o v e r a l l SPL of (74.2 ±-5) db from which the corrected SPL was estimated to be (73.6) db..The error i n the predicted SPLs arises from a systematic error which i s incorporated i n the DRI d i s t r i b u t i o n s * The main factors contributing to the error i n the DRI are: an uncertainty i n the sound t r a v e l time and limited resolving power of the c o r r e l a t o r . Nevertheless the SPLs predicted by the Causality Correlation Technique are i n close agreement with the corrected SPLs. Results of the present work support the claim that useful information on noise generation may be obtained iv quickly by performing measurements at c r i t i c a l points only, on a surface. V TABLE OF CONTENTS Abstract pg. i i L i s t of Figures pg, ix Notation pg. x i Foreword pg. xiv Acknowledgements pg, xvi 1 Introduction pg. 1 2 H i s t o r i c a l Background pg. 5 3 Theory 3.0 C l a s s i c a l Theory 3,0.0 The L i g h t h i l l Wave Equation pg. 7 3,0.1 The General Solution of The Wave Equation pg, 9 3.0,2 A Synopsis on Noise Generated by Flows with Internal Rigid Boundaries pg,,11 3.1 The Causality Correlation Technique 3.1.0 Advantages of Using The Causality Correlation Function pg. 14 3.1.1 Forming The Causality Correlation Function pg. 15 3.1.2 The Dipole Radiation Intensity pg, 17 4 Experimental Procedure 4.0 Design and Fabrication of The Cylinder Probe pg* 21 4.1 Measurement of Probe Cha r a c t e r i s t i c s : Frequency and Phase Response pg. 25 v i 4.2 Frequency Spectra for the Source Point and for the Observation Point pg. 26 4.3 C a l i b r a t i o n of Instruments pg,. 28 4.4 The Measurement of Flow Parameters pg. 29 4.5 The Construction of the DEI D i s t r i b u t i o n pg. 29 5 Calculations 5.0 Procedure pg. 33 6 Discussion of Results 6.0 A Summary of C h a r a c t e r i s t i c Frequencies and V e l o c i t i e s 6.0,0 Cylinder Model Stationed at the Potential Core, Origin at 0(x' = 0, x* = 0, X» = 1D) pg, 36 6,0,1 Cylinder Model Stationed at the Mixing Region, Origin at 0(xJ=0, x'=0, x*=5D) pg. 40 6.1 Discussion of the Dipole Source Distribution pg. 42 6.2 The Circumferential P r o f i l e of the DRI for the Potential Core T r i a l , Origin at O(x'=0, x«=0, X*=1D) pg. 46 6.3 The Spanwise P r o f i l e of the DRI for the Potential Core T r i a l , Origin at 0(x» = 0, x^=0, X' = 1D) pg. 49 6.4 Integral Closure; f o r the Potential Core T r i a l , Origin at 0(x« = 0, x»=0, X«=1D) pg, ,50 v i i 6.5 The Circumferential P r o f i l e of the DRI for the Mixing Region T r i a l , Origin at O(xf=0, x^=0, x'=5D) pg. 51 6.6 The Spanwise P r o f i l e of the DEI for the Mixing Region T r i a l , Origin at 0(x'=0, x* = 0, x$ = 5D) pg..53 6.7 Integral Closure; f o r the Mixing Region T r i a l , Origin at 0(x«=0, x«=0, x«=5D) pg,.54 7 Conclusions S Recommendations 7.0 Conclusions pg, 56 7*1 Recommendations pg, 59 8 References pg, 60 9 Figures pg, 63 10 Appendices Appendix A Derivation of The L i g h t h i l l Wave Equation pg, 88 Appendix B Simp l i f i c a t i o n of The General Solution of The L i g h t h i l l Wave Equation f o r Rigid Surfaces i n Quiet Air Flows pg. 89 Appendix C Simp l i f i c a t i o n of Correlation Functions for Ergodic Random Variables pg. 95 v i i i Appendix D Sample Calculations of the DEI and SPL f o r Potential Core T r i a l , Origin at O(xf=0, x»=0, X« = 1D) pg.i 97 Appendix E Resolving Power of the Correlator Time Base pg,. 103 ix LIST OF FIGUHES 1: 4n Arbitrary Acoustical F i e l d Q, Source Volume V and Source Area S pg. 7 2a: Cylinder Model Stationed at Potential Core, Origin at O(x' = 0, x«a=0, x« = 1D) pg, . 64 2b: Cylinder Model Stationed at Mixing Region, Origin at 0(x« = 0, x^=0, x' = 5D) pg. 64 3a; Stationary Cold Jet (reproduced L i g h t h i l l 2 ) pg* 65 3b: The Mixing of 2-D Jets (reprinted Eoshko 1 3) pg, 66 4: Flow Past a Circular Cylinder at Reynolds Number 6,3x10* (reproduced Reynolds 1 2 1973) pg> 67 5a: Cylinder Model and One Quarter Inch Probe pg, 68 5b: Block Diagram of Correlation Apparatus pg, 6*9 6a: Frequency Response and Phase Cha r a c t e r i s t i c s of 1/4" Probe pg, 70 6b: Frequency Response and Phase Cha r a c t e r i s t i c s of 1/8" Probe pg, 71 7a: Spectrum of Surface Pressure for Potential Core, Origin At O(x' = 0, x^=0, x«=1D), yi=(R=d/2, <f>=Q0, h=0) pg, 72 7b: Spectrum of Surface Pressure for Mixing Layer, Origin at 0(x» = 0, x«=0, X« = 1D), Y\ = (R=d/2, #=0<>, h=1, 91x 10"2m) pg, 73 7c: Spectrum of Far F i e l d acoustical Pressure, Origin at 0(x» = 0, x«a=0, x« = 1D), x»=(x«=0 # x'=-3,00 m, x«=0) pg. 74 7d: Spectrum of Surface Pressure for Mixing Region, Origin at 0(x| = 0, x[=0, x« = 5D) , yi = (R=d/2, c6=0<>, h=0) pg. 75 7e: Spectrum of Far F i e l d Acoustical Pressure, Origin at O(xf = 0, x^=0, xJ = 5D) , xj=(x»=0, xi=-3*00 m, x'=0) pg* . 76 8a: Slope of a Sample Cross Correlation Function pg, 77 8b: A Sample Autocorrelation Function of Surface Pressure pg, 78 8c: Autocorrelation Function of Far F i e l d Acoustical Pressure pg, 79 9a: Circumferential P r o f i l e of DRI; Potential Core T r i a l , Origin at O(x' = 0, x« = 0, X» = 1D), = (R=d/2, <t>=<j>, h=0) pg. 81 9b: Circumferential P r o f i l e of Cross Correlation Function; Potential Core T r i a l , Origin at 0(x' = 0, x*=0, XJ = 1D) , y-i = (R=d/2, <}>=<}>, h=0) pg, 82 9c: Circumferential Profile.Of DRI; Potential Core T r i a l , Origin at O(x«=0, x*=0, X£=1D), yj=(R=d/2, c4=ci, h=1,91x10-2 m) pg. 83 9d: Spanwise P r o f i l e of DRI; Potential Core T r i a l , Origin at O(x» = 0, x'=0, X« = 1D), y^  = (R=d/2, c4=0°, h=a) pg, 84 10a: Circumferential P r o f i l e of DRI; Mixing Region T r i a l , Origin at O(xf=0, x»=0, x«=5D) , yn = (R=d/2, 4=<j>, h=0) pg. 85 10b: Spanwise P r o f i l e of DRI; Mixing Region T r i a l , Origin at O(xf = 0, xj,=0, x» = 5D) , yi = (R=d/2, <f>=QQ, h=h) pg.86 11: Time Scales and Eddy Frequency (reprinted Nayar, Siddon, Chui*) pg. 87 x i NOTATION U0 - j e t exit velocity ti - e x i t Mach number of jet D - j e t nozzle diameter d -diameter of c y l i n d r i c a l strut Xi -components of a fixed point i n the acoustical f i e l d r e l a t i v e to an o r i g i n 0 yi -components of a variable point within the source region r e l a t i v e to an o r i g i n 0 p - t o t a l density of gas u, - t o t a l f l u i d v e l o c i t y vector t -present time dV(y,) -element of source volume Pij -compressive stress tensor Coo - c e l e r i t y of sound T\] - L i g h t h i l l , s guadrupole stress tensor p -hydrostatic pressure <5i] -Kronecker delta T . -viscous stress tensor (p-p„) -density of f l u i d i n acoustical f i e l d n, (y^  ) -unit normal at surface point yn dS (Yi ) -element of source area p s (yi ) -hydrostatic pressure at surface point y-j x i i £-••3 -symbols to indicate that contents are to be evaluated at a particular time ( given by the subscript of 3 ) £ -Kirchhoff retarded time r -distance of sound t r a v e l p' (Xi ,t) -instantaneous pressure fluctuation at f i e l d point Xj P' (Yi #t) -instantaneous pressure fluctuation at surface point y, TTT -symbol to indicate that a time average i s to be performed <•••> -symbols to indicate that an ensemble average i s to be performed on contents * -symbol to indicate d i f f e r e n t i a t i o n of ps with respect to time x -delay time x -Kirchhoff delay time C ( x i / Y i ) -dimensionless cross c o r r e l a t i o n c o e f f i c i e n t pR.w»s (y- j -BMS pressure at surface point y, pd^s (X- j -BMS pressure at f i e l d point xi p R E F -20X10-6 N/m2 SPL (y, ) -sound pressure l e v e l at surface point y\ SPL (xi ) -sound pressure l e v e l at f i e l d point xj x - U i I hcuiE.1 -the h coordinate of a point outside the active flow region 1 E F F - e f f e c t i v e c a p i l l a r y tube length e-ft-eB -phase lag of Helmholtz resonator (B,0,h) - c y l i n d r i c a l coordinates 0 ^-origin for (R,$,h) x i i i (xf,x',xj) -cartesian coordinate system 0' - o r i g i n f o r (x*, xl, x\) y! -components of the surface point r e l a t i v e to the or i g i n 0' x\ -components of the far f i e l d point r e l a t i v e to the o r i g i n 0' U -mean velocity u! - t o t a l v e l o c i t y fluctuation u**4 -RMS ve l o c i t y fluctuation 0C -eddy convection v e l o c i t y -epsilon w -angular frequency r n -(2,867x102 j Kg-* oK) d t - r i s e of cross correlation function (cms) dg -run of cross c o r r e l a t i o n function (cms) dfl -amplitude of autocorrelation function of surface pressure evaluated at zero time delay (cms) d f t -amplitude of autocorrelation function of f a r f i e l d acoustical pressure evaluated at zero time delay (cms) e T -percent uncertainty associated with T xiv FOREWORD The basic p r i n c i p l e s of sound generation, propagation and control were re-examined during the Symposium on Aerodynamic Noise (1969) and also during the AGARD Conference (1973). These Proceedings revealed a c t i v i t i e s in the various aspects of a i r c r a f t noise. The turbofan engine was one topic of concern. The basic design of the turbofan engine incorporates a high bypass r a t i o for increased thrust and quieter jet exhaust* However, additional noise i s produced by the fan and compressor. During the t h r o t t l e d back landing approach the compressor whine from the i n l e t dominates the o v e r a l l a i r c r a f t noise l e v e l . A d i s t r i b u t i o n of acoustical dipole sources are the known main contributors to the i n l e t noise. The abatement of aerodynamic noise i s enabled through the unravelling and understanding of the mechanisms of noise generation and propagation.. This thesis demonstrates a technique used f o r diagnosing mechanisms of dipole noise generation by r i g i d surfaces. The Causality Correlation Technique i s applicable to the problem: 'What f r a c t i o n of the t o t a l far f i e l d noise originates from a patch of area on a noise generating surface ?',. For example t h i s problem arises during fan noise transmission studies; where noise components generated by rotor and stator blades are transmitted through the blade rows and along fan ducting before X V propagating to the observer; where i t i s d i f f i c u l t to simply relate the fluctuating pressure f i e l d on a noisy surface to the eventual acoustical field,. Theoretical results obtained in fan noise transmission studies could be tested using the Causality Correlation Technique,. The said technique yields the f r a c t i o n a l far f i e l d pressure a r i s i n g from unit surface area; transmission effects are accounted f o r automatically i n most cases,. xvi ACKNOWLEDGEMENTS The author i s grateful to Dr, T,E, Siddon for providing an insight into the subject and guidance during the project* My thanks i s offered to Mr, J. Hoar & s t a f f at the Mechanical Engineering Workshop for f a b r i c a t i n g the cylinder models used in the experiment, F i n a n c i a l support for t h i s project was provided by the National Research Council Awards O f f i c e , and National Research Council Operating Grant No, 67-7106, 1 Sec, 1 INTRODUCTION Surface noise has been i d e n t i f i e d i n many types of turbo-machinery such as low cost v e n t i l a t i o n fans, propellers, turbines, and ducted flow systems,. In general, i t may be observed where there e x i s t conditions of: 1) mechanical vibration; 2) persistent d i s t o r t i o n s i n flow; 3) turbulence in flow; 4) boundary layer turbulence; 5) flow separation; 6) vortex shedding, These conditions may occur either i n combination or i n d i v i d u a l l y . P h y s i c a l l y , there exist three mechanisms capable of causing noise to be radiated from the neighborhood of a s o l i d boundary. These are: mechanism 1) the boundary deforms/vibrates and induces stress fluctuations i n the neighbouring f l u i d which form monopole and dipole sources; mechanism 2) the boundary obstructs an aerodynamic flow and induces stress fluctuations i n the neighboring f l u i d which form dipole sources; mechanism 3) the boundary r e f l e c t s / d i f f r a c t s ambient sound and induces stress fluctuations i n the neighboring f l u i d which form dipole sources* 2 Mechanical vibration develops surface noise by mechanism 1. The remaining flow conditions l i s t e d above may lead to the development of surface noise by a l l three mechanisms. I f , however, the it-boundary i s r i g i d and simple and i s isolated from a l l external sources of sound, surface noise i s generated by mechanism 2 exclusively,. Mechanism 2 i s the main subject of investigation i n these experiments. Following the c l a s s i c a l theory of L i g h t h i l l and Curie, a noise generating surface may be modelled with a s p a t i a l d i s t r i b u t i o n of point dipole.sources concentrated within a thin layer over the surface. The instantaneous dipole radiation i s given by the surface i n t e g r a l of the time rate of change of the dipole strengths. The theory requires that each time derivative be evaluated at the appropriate Kirchhoff retarded time, There i s no instrument for d i r e c t l y measurinq the strength of a dipole source. However, the •Dipole Radiation Intensity (DRI)• provides a useful measure, of the ' e f f e c t i v e ! c l a s s i c a l dipole strength. The DRI i s defined uniquely by the Causality Correlation Function 1 (see Sec, 3.1.1), The primary objectives of the present experimental work are: 1) to quantify the Dipole Radiation Intensity for circumferential points on the surface of a r i q i d c y l i n d r i c a l rod that i s immersed i n the laminar potential core of a free turbulent jet; 2) to quantify the DRI f o r spanwise points on the surface of *The surface i s simple.if the outgoing spherical wavelets diverge without r e f l e c t i o n / d i f f r a c t i o n from other areas of the surface. 3 the rod stationed at the potential core; 3) to quantify the DEI for circumferential points on the surface of a r i g i d c y l i n d r i c a l rod that i s immersed in the turbulent mixing region of a free turbulent j e t ; 4) to quantify the DEI f o r spanwise points on the surface of the rod stationed at the mixing region; 5) to integrate the 'DEI d i s t r i b u t i o n * over the entire surface of the rod to obtain a prediction of the t o t a l far f i e l d sound pressure l e v e l associated with the surface exclusively, f o r each of the two flow cases; 6) to compare predicted noise l e v e l s with d i r e c t l y measured noise l e v e l s ; agreement indicates 'integral closure'; implying that a l l 'noise sources' are adequately represented in the d i s t r i b u t i o n s . During these experiments, an aluminum rod of 1.27 cm diameter and 17,0 cm length was oriented so as to span a free turbulent a i r j e t . The ends of the rod extended out of the flow, so that 'end e f f e c t s ' were not present. The midspan section of the rod was situated on the jet axis and could be moved to the two dif f e r e n t streamwise stations along the jet axis (see Fig,2a&b). The jet issued from a round nozzle 3,8 cm i n diameter* The jet exit v e l o c i t y was set at, U= 72 m/s corresponding to an ex i t Mach number of, M= .217 The Eeynolds number based on the jet o r i f i c e diameter D was, Ee= 1.9x105 4 The Reynolds number based on the cylinder diameter d was, Be= 6 i 3 x 1 0 * The overriding purpose of t h i s work i s to establish further the v i a b i l i t y of the Causality Correlation Technique as a diagnostic tool f o r treatment of noise problems. Results of the present work support the claim that useful information on noise generation may be obtained quickly by performing measurements at c r i t i c a l points only, on the surface. The i n t e g r a l closure check i s used to confirm that a l l •important* points of the d i s t r i b u t i o n have been measured, The secondary purpose of t h i s work i s to v e r i f y the L i g h t h i l l theory on Sound Generated Aerodynamically 2, for the present flow configuration, Agreement between the predicted SPL and the d i r e c t l y measured SPL v e r i f i e s the L i g h t h i l l wave equation at these Reynolds numbers and flow Mach number, The thesis includes: a complete derivation of the Causality Correlation Function from fundamental laws of f l u i d mechanics, the d e f i n i t i o n of the DRI, an outline of the experimental procedure used to construct the DRI p r o f i l e s , a presentation of the relevant ca l c u l a t i o n s and graphics and f i n a l l y a diagnosis of noise generation for the said flow configurations. 5 Sec,2 HISTORICAL BACKGROUND The advent of the turbojet engine was met with a strong concern surrounding the accompanying noise l e v e l s , In response to t h i s concern M, J. L i g h t h i l l (1952) presented a c l a s s i c a l theory on Sound Generated Aerodynamically. L i g h t h i l l described the radiation of sound from within turbulent flows by means of quadrupole sources, an approach which i s v a l i d for i s o t r o p i c turbulence t r a v e l l i n g at a subsonic convection speed, The theory was founded upon a nonhomogeneous scalar wave equation governing the propagation of sound from the region of unsteady flow, C u r i e 8 (1955) applied the standard Kirchhoff solution to the nonhomogeneous wave equation*.The r e s u l t i n g solution i n t e g r a l was generalized to include the e f f e c t s of i n t e r n a l s o l i d boundaries (see Sec, 3,0,. 1), The solution i n t e g r a l predicted the t o t a l instantaneous pressure at a given point; external to the source region, i n terms of measureable source quantities, The potential existed to i d e n t i f y dynamic quantities that are important i n the noise generation process i n a given flow configuration* However, the Curie solution was not suited for d i r e c t application due to complications outlined in Sec.3.1,0 , An alternative time averaged form of the solution i n t e g r a l was proposed by Siddon* (1971). I t provided an experimental technique for obtaining information on s p a t i a l d i s t r i b u t i o n s of monopole, dipole and quadrupole sources. Concurrent studies of aeronoise were made using the new technique, The DRI d i s t r i b u t i o n in the active volume of a free 6 turbulent jet was investigated by E.K. L e e l s (1972) and by E a c k l 3 (1973). S i m i l a r l y , DEI d i s t r i b u t i o n s were measured over the surface of: 1) a round disc embedded i n a free turbulent j e t , see Siddon* (1972) ; 2) a blade i n an axi a l flow fan, see Leggat 5 (1973) ; 3) a strut i n the wake of an a x i a l flow fan, see Leggats (1973) ; 4) a nozzle l i p section of a round coannular j e t , see Siddon & Hoglund* (1977) . The present work i n an extension of the foregoing work and may help to resolve speculations made by Leggat on the magnitude of strut noise in an a x i a l flow fan, 7 Sec. 3 THEORY Sec. 3* 0 C l a s s i c a l Theory Sec.3.0*0 The L i g h t h i l l Wave Equation Consider an acoustical f i e l d consisting of a homogeneous i s o t r o p i c fluid,* Let there be a f i n i t e source region embedded in th i s f i e l d which contains noise generating f l u i d with isentropic fluctuations and s o l i d .boundaries (see Fig, 1) . Let be the coordinates of a fixed observation point i n the acoustical f i e l d and l e t y^  be the coordinates of a variable point within the source region. FIGURE 1: An Arbitrary Acoustical F i e l d Q, Source Volume V and Source Area S The physical continuity of mass i s expressed by, Q x 3P + a(ouj) = 0 at ay; I 8 where p i s the t o t a l density of the f l u i d inside dV(y,) and u\ i s the t o t a l f l u i d v e l o c i t y vector at the volume element dV(y, ) at time t . The Reynolds form of momentum equation requires that, 9(PUj ) + 3 (puj Uj + Pij ) = 0 XT dt ay; where j i s the compressive stress tensor acting on the element dV(yi ) . The L i g h t h i l l wave equation of acoustics i s formed by combining these two fundamental equations and introducing the c e l e r i t y of sound denoted by ca (see Appendix A). We have, a 2 P - c M z a 2 ? = a 2 T i i HI a a t 2 a y 2 ay idy j The L i g h t h i l l wave equation i s characterized by a scalar quadrupole source term on the right hand side, The quantity T^ i s the L i g h t h i l l quadrupole stress tensor, given by, Tij = pu^Uj + pSij - Ti; - c„ 2 XEt> where p, and are the hydrostatic pressure and viscous stress tensor acting on dV(y,) , 9 Sec. 3.0,. ii The General Solution of The Wave Equation An i n t e g r a l form of L i g h t h i l l ' s wave equation follows from the Kirchhoff theory on integration of nonhomogeneous scalar wave equations* Stratton 7 extended the solution i n t e g r a l to include the effects of i n t e r n a l boundaries. C u r i e 8 s i m p l i f i e d the surface and volume i n t e g r a l s using the divergence theorem and the Reynolds equation to obtain the following i n t e g r a l equation, c«,2(p-pco) (xi ,t) = f n i [ l (pui)l dS(ys ) Eg. 3.1 J s l a t J 4Hr monopole radiation - JL_ [ n> [ <Ps Sii ~ I i i ) + P u i u i 3 dS(yj ) d X i J s 4tfr y dipole radiation • ^ 2 ( t Tij 3 dV(y4 ) dxiaxj J u 4itr quadrupole radiation t Here the constant c„ i s the ambient phase velocity of sound, ( p - p e o ) i s the density f l u c t u a t i o n at the point of reception (x, ) , In general the surfaces may be r i g i d or f l e x i b l e , and, they may translate and vibrate. If sound waves undergo r e f r a c t i o n or i f heat transfer occurs, c M would not be constant. 10 fli (Yi) i s the unit normal at the surface point (y; ) and i s directed outward from dS (y; ) i n the right handed sense and p5 (y-;,) i s the hydrostatic pressure at the surface. The variables in the integrand are functions of y, & t. The square brackets [ J indicate that the quantities within are to be evaluated at the Kirchhoff retarded time given by, t = t - |x, -yi | = t - x Eg. 3.2 C oo C oo where t denotes the present time and r/c,» i s the time required for sound to t r a v e l a distance r, Monopole, dipole and quadrupole i n t e q r a l terms appear in the solution i n t e g r a l (Eq.3.1), By comparison, L i g h t h i l l ' s o r i g i n a l wave equation contained only the quadrupole source term. The additi o n a l monopole and dipole integrals arise with the inc l u s i o n of the boundary condition f o r the source region. Contributions to the monopole i n t e g r a l and, i n part, the dipole i n t e g r a l are made by mechanism 1 (see Sec,1). additional contributions to the dipole i n t e g r a l are made by mechanisms 2S3, The quadrupole i n t e g r a l a rises from turbulent unsteadiness i n the flow surrounding the noise generating surfaces, Evidently, noise i s generated by stress fluctuations which are of mechanical and aerodynamic origins* The next section considers i n more d e t a i l the aerodynamic origins of noise for a less general flow configuration. 11 Sec.3.0.2 A Synopsis on Noise Generated by Flows with Rigid Internal Boundaries Consider a flow with an i n t e r n a l boundary that i s r i g i d ; the surface does not deform or vibrate. The boundary condition i s , n, u, = 0 which requires that the normal component of f l u i d velocity at the surface be zero, The Curie solution with the inclusion of t h i s condition reduces to one surface (dipole) i n t e g r a l and one volume (quadrupole) i n t e g r a l , c o e 2(p-p») ( X i ,t) = - a_ r nj T P$^J " Zii 3 as(yj) Eg.3.3 axi J s 4ir r + a* ( t Tij j av(yi) ax-idxj J v 4trr The respective source d i s t r i b u t i o n s represented under the i n t e g r a l signs are by d e f i n i t i o n , a) Dipole sources distributed over the surface S, each of strength equal to the l o c a l force exerted on unit area of the f l u i d by the surface in the direction of the unit normal (includes both normal pressures and shearing stresses). B) Quadrupole sources d i s t r i b u t e d throughout the source ^Of primary importance i n t h i s paper i s the generation of noise by mechanism 2. However, quadrupole noise i s of secondary importance i n the experiments and warrants some discussion here, 12 volume V, each of strength equal to the l o c a l stress T,j acting on unit volume (less the quantity Cg/pS^) . The dipole strength consists of the reaction force that the surface generates against the aerodynamic flow and a small scattering force. The l a t t e r accounts for r e f l e c t i o n and d i f f r a c t i o n of sound waves impinging upon the surface. The dipole source strength must possess nonvanishing s p a t i a l gradients (or else be time varying) f o r the dipole source to radiate. Large force gradients can exist near the surface of a r i g i d c y l i n d r i c a l rod immersed i n an a i r flow (steady or turbulent). Here large gradients would arise from incident turbulence, turbulence in the boundary layer and vortex shedding from the surface of the r i g i d c y l i n d r i c a l rod. The quadrupole source strength T,j (see Sec,3,0,1) consists of several types of stress. F i r s t , the Reynolds stress term ( p U i U j ) describes transport of momentum density (pu,-) by turbulent convection* Second, the viscous stress r-jj describes momentum di f f u s i o n on the molecular scale. The remainder of Tij forms the term (pSij -c,x,2PSij ). This quantity i s i d e n t i c a l l y zero i f the gas i s a perfect gas undergoing adiabatic linear compression. From t h i s , i t i s apparent that Tjj i s exactly the t o t a l l o c a l stress acting in excess of the t o t a l hydrostatic pressure that would exist i f the f l u i d was "acoustically i d e a l , " The quantity T,] must Refraction and convection e f f e c t s are described: completely by the wave equation with the gradients of pu,Uj and s p a t i a l variations i n c^. These e f f e c t s are d i f f i c u l t to account f o r i n practice, 13 possess a nonvanishing double s p a t i a l gradient (or else have a f i n i t e second derivative in t) for the quadrupole source to radiate. Large stress fluctuations of t h i s type can exist in the mixing layer and mixing region of a free turbulent j e t , If a s o l i d object i s inserted into such a j e t , . a d d i t i o n a l stress fluctuations would exist i n the wake of the object. In many p r a c t i c a l s i t u a t i o n s , the dipole radiation ( f i r s t term of Eq,3.3) w i l l be very much stronger than the associated quadrupole radiation, Therefore, the Curie solution for a r i g i d surface can often be approximated by the following equation, This i s a v a l i d approximation provided the surrounding flow may be considered to be 'quiet* and i f the observation point i s taken in the geometric and acoustic far f i e l d (see Appendix B for complete derivation). Here, prime i s used to indicate the f l u c t u a t i n g part of the variable. £=t-r/c Eg. 3* 4 00 14 Sec;3.1 The Causality Correlation Technique Sec,3 .1 . 0 Advantages of Using The Causality Correlation Function Consider now the p o s s i b i l i t y of veri f y i n g Eg.3.4 experimentally* Simultaneous measurement of the instantaneous pressure f l u c t u a t i o n at each point y, on the surface S would be necessary i n order to predict the t o t a l radiated acoustic pressure P" (» i , t ) . Furthermore, the time mean square of Eq.3,4 i s not suited for applications involving large surfaces such as those found in turbomachinery, The time mean square gives r i s e to a more complicated, s p a t i a l c o r r e l a t i o n of pairs of points on the surface and involves a second order time derivative of that c o r r e l a t i o n function* Moreover, the number of s p a t i a l correlations required, for v e r i f i c a t i o n purposes, becomes too large to be considered as p r a c t i c a l . In the next section i t i s shown that when the instantaneous pressure fluctuation i s a random variable possessing the properties of s t a t i s t i c a l s t a t i o n a r i t y , and ergodicity then the solution i n t e g r a l may be reexpressed as, P'P' (x,) a -xi f n, _S_ p| (y, )P'(x,) (t) 4-rrx2Ca, ) L d-& dS(y- ) Eg, 3,6b t =r/Coc For time averaging periods of s u f f i c i e n t length, the integrand i s independent of the present time (t)* Hence simultaneous measurement of the pressure at each point on the surface i s no 15 longer needed to verif y Eg,3.6b . This form of the solution i n t e g r a l i s referred to as the Causality Correlation Function for r i g i d surfaces (see Siddon 1), Sec.3*1.1 Forming The Causality Correlation Function Let the f l u c t u a t i n g pressure at the far f i e l d point x-j be given by the approximate form, p« (x, ,t) - x, f n; j a_ p« (ys ,$) ) dS (y,- ) Eg.3.4 i=t-r/c 0 O Consider now what happens i f Eg.3.4 i s multiplied through by the pressure variation at X i at a new time t' and an ensemble average i s taken, <P' (t) P* (t')> (x;) - x, fni <p«(yi ,t)p» (x,,t')> dS(yi ) 4iTx2c«,Js Eg. 3. 5a Here < > denotes an ensemble average and, P|(Yi^) = ^P|(Yw^)J Eg. 3.5b t=t-r/Co„ Let p| and p 1 be ergodic random functions of time, following the assumption of Sec.3,1,0, Then by d e f i n i t i o n of ergodicity, the ensemble averages i n the above are equivalent to time averages, the l a t e r of which may be s i m p l i f i e d by the following theorem (see Appendix C f o r proof), 16 Theorem 1: P' (Xi ,t)p« (X, , f ) = P'p« (X, ,T) P| (Yi ,t)p« ( X i , f ) = - f a_ p» ( Y l ) p» ( X l ) (t)j EqwC.1 Eq.C.2 T =I+r/C0 The time variables introduced above are as follows: t i s the present time, t* i s the new time, t-=t-r/c» i s the Kirchhoff retarded time, r = t * - t i s the delayed time (assuming t'>t) and Z=t• -t=r+r/c«<, i s the delayed Kirchhoff retarded time,. Applying Theorem 1 to Eg,. 3.5 a produces the Causality Correlation Function, P'P' (x, ,r) * -x, [ n; f _a_ p'( Y l- ) p« (x,) (f) 1 dS (y; ) Eg.3,6a ^x^cm J s L bz J r=r+r/c„ We f i n d that the autocorrelation of the far f i e l d pressure i s related to the surface i n t e q r a l of the time rate of change of the cross correlation between surface pressure and far f i e l d pressure. When the Causality Correlation Function i s evaluated with the delay time (r) set equal to zero the l e f t hand side of Eq,3.6a becomes the t o t a l radiated acoustic pressure - (mean squared value) at the far f i e l d point (x^). We have. p«p« ( X l) - -x, ( n,- j _s_ p'(Yi )P' ( X i ) (r) 1 dS(y,) Eq.3.6b 4-nx2coo J s l a t J t=r/c Q O 17 Sec. 3, il. 2 The Dipole Radiation Intensity The Causality Correlation Function furnishes a unique technique f o r diagnosing noise conditions on r i g i d boundaries. The i n d e f i n i t e i n t e g r a l i n Eg,3.6a may be expressed i n d i f f e r e n t i a l form. Taking the d i f f e r e n t i a l and setting the time delay (r) to zero and di v i d i n g both sides by the d i f f e r e n t i a l area element dS (y; ) we obtain, dp'P1 ( x w y j ) - - X i n; dS UTTXC-d _ P | ( y i ) P ' ( x i ) (r) at Eg, 3. 6c x=r/c a o The l e f t hand side of the foregoing equation gives the net radiated a c o u s t i c a l pressure (mean sguared value) at the f a r f i e l d point ( X i ) that was caused by the fluctuating pressure at the surface point (y; ) . The ri g h t hand side i s proportional to the time rate of change of the cross c o r r e l a t i o n function (p|p') (f) evaluated with the Kirchhoff time delay set egual to r = r/c 0 (see Fig* 8a f o r a plot of a t y p i c a l c o r r e l a t i o n function). The 'Dipole Radiation Intensity (DRI)' at Xi of the surface point yi i s defined to be the guantity dp'p' (x; , y- )/dS* . The DRI i s uniquely defined since the surface S appearing i n Eg,3,6b i s unrestricted. The features of the DRI include a dependence upon the magnitude of the c l a s s i c a l dipole strength and a dependence upon the frequency; the dipole strength (p|) i s contained within the time derivative i n Eg,3,6c. The 1DRI d i s t r i b u t i o n 1 may be defined to be the mapping of 18 the source point to the DEI. The evaluation of t h i s d i s t r i b u t i o n amounts to a decomposition of the t o t a l f a r f i e l d pressure (mean squared value) into component pressures; each component pressure being the net radiat i o n coming from the respective source point. In the forthcoming experiments some confusion can arise in the int e r p r e t a t i o n of 'DEI d i s t r i b u t i o n s * . A circumferential p r o f i l e of the DEI (see Fig,9 a ) bears an obvious resemblance to the f a m i l i a r dipole radiation d i r e c t i v i t y (dumbell shaped) . These two polar plots are i n f a c t d i f f e r e n t . The former i s constructed for a given surface using the d i f f e r e n t i a l equation Eg,3 , 6 c ; the DEI i s evaluated while the source point i s suitably varied and the f i e l d point i s held f i x e d , giving the 'decomposition' of the mean square pressure at the f i e l d point. The radiation d i r e c t i v i t y on the other hand i s customarily obtained by measuring p'p' (x-j ) while the f i e l d point i s varied irresp e c t i v e of source point, giving the 'mapping' of the acoustical f i e l d pressure. A l t e r n a t i v e l y , within the context of The Theory of Measures 1 6, the i n t e g r a l on the r i g h t hand side. of Eg; 3.. 6b can be viewed as being an i n d e f i n i t e i n t e g r a l whose domain i s the set of a l l measureable surfaces S. In t h i s case, i t can be shown that the l e f t hand side of Eg;3.'6b i s i n a sense a 'Lebesgue measure function' which also just happens to equal the far f i e l d acoustical pressure p'p' (x^). I t can be subsequently shown that the set of quantities { dp* p * (x,,y ' ) / d s | y, e S J exists and i s c a l l e d the 'dis-integration' of p'p* (x^) for the surface S. Moreover, the integrand of Eq,3 , 6 b , which we are denoting by dp'p* (xi,y;)/dS, may be shown to be unique. Incorporating the term dis- i n t e g r a t i o n into the d e f i n i t i o n f o r the DEI would y i e l d a 19 more concise d e f i n i t i o n ; the quantity p'p'(xi) would be c a l l e d the DEI, and the set of quantities { dp 1?' (xj,yi)/dS } would be called the Pis-integrated Dipole Radiation Intensity (DDRI). However t h i s change in nomenclature w i l l not be made herein. The cross c o r r e l a t i o n function P5 (Yi) P1 (*i) (£) i s more conveniently expressed as a dimensionless c o e f f i c i e n t . D e f i n i t i o n : We define the cross correlation c o e f f i c i e n t to be, C(x 1,y i) ( f ) PMYOPMX,) (?) -RMS i R M S Eq, 3, 7a r =r/c c The source and far f i e l d RHS pressures are equal to the square roots of t h e i r respective autocorrelation functions evaluated with the time difference (r) set equal to zero, Using Eq,C,1 we obtain. \?T" (Yi ) = J PS (t)p' ( f ) (Yi ) I = J P|P? (Y, ,r) I t «=t r=0 (x P* (t) p' ( f ) (x,) I t • =t -[ P'P' (X,,T) | T=0 Eq, 3, 7 b Eq,3.7c Each EMS pressure i s also related to a sound pressure l e v e l by. ,RMS _ p R E F antilog.o SPL/20 Eq. 3,8 where p R e F = 20 x 10-6 N/m 2« Introducinq these d e f i n i t i o n s i n t o Eq.3,6c yields the following recipe for the DRI at a point, 20 clp'p 1 ( x 1 # Y i ) -XJ n, (y,) _ a _ C ( x 1 / Y i ) ( t ) ds 4irxc» I a t Eg. 3 . 9 r=r/c< p?„ ant R E F ilog.o |SPL(y,) ^+ SPL(xQj Thus the DEI i s porportional to the slope of the normalized cross c o r r e l a t i o n function C(x^ ,y- ) ( r ) * and the antilog of the source & f a r f i e l d sound pressure l e v e l s . The slope i s evaluated with the Kirchhoff time delay set equal ; to r/c M;. Forthcoming sections outline the experimental procedure used to construct the d i s t r i b u t i o n of the DRI over the surface of the emersed cylinder model. 21 Sec,4 EXPERIMENT AL PROCEDURE Sec,4,0 Design and Fabrication of the Cylinder Probe An exploded view of the cylinder model i s shown in Fig.5a. The probe was used to detect the flow induced pressure fluctuation at a point on the surface of the cylinder model. The basic probe design consisted of a miniature microphone set i n a cavity within the midsection of the rod. The cavity was coupled to the surface by a c a p i l l a r y tube which was terminated with a pinhole. The pinhole at the surface could be moved into d i f f e r e n t positions in the flow by rotating or t r a n s l a t i n g the rod (see Fig,2a), The rod measured 1.27 cm in diameter and 17 cm i n length and was d r i l l e d out to accommodate a one guarter inch Bruel and Kjaer Condenser Microphone (Type 4136) connected to a B & K Preamplifier (Type 2618), The microphone cartridge.was sealed in the small cavity by an O-ring, The O-ring was positioned between the microphone membrane and the r e l i e f vent (see Fig.5a). The a i r - f i l l e d cavity i n front of the membrane was coupled to the surface of the cylinder model by the tiny c a p i l l a r y tube* The c a p i l l a r y tube measured .091 cm (,036 inches) i n diameter at the surface of the cylinder and was ,318 cm (,125 inches) i n lengths An a i r f i l l e d cavity that i s coupled to the atmosphere by a narrow c a p i l l a r y constitutes a Helmholtz resonator, The probe acted as a Helmholtz resonator and conseguently introduced an exaggerated response and a substantial phase lag near the Helmholtz resonant freguency. 22 The resonant frequency i s approximately given by the Helmholtz formula, Here S i s the cross sectional area of the c a p i l l a r y , V i s the -jr volume of the cavity and 1 E F F i s the e f f e c t i v e tube length. Near the resonant frequency, the pressure response of the Helmholtz device rapidly increases and the phase angle abruptly increases. The response i s maximum at resonance (after which i t f a l l s off) and pressure lags f l u i d v e l o c i t y by 90°. The Helmholtz resonant freguency of the probe was found by a procedure outlined i n the next section to be about 2 k hz. By comparison, the the far f i e l d pressure was monitored with a one half inch BSK Condenser Microphone (Type 4133) which had f l a t response and introduced n e g l i g i b l e phase s h i f t up to 20 k hz, The probe was subsequently redesigned to achieve a higher resonant freguency. The aim was to make the resonant frequency of the probe higher than freguencies that were known to dominate the spectrum of the far f i e l d pressure f l u c t u a t i o n (see Sec.4,2). With t h i s reguirement s a t i s f i e d , the probe should have a f l a t response ( < ±1.5 ) db and the phase s h i f t between the probe and the far f i e l d microphone should be minimal ( < 15°), over the desired range of frequencies (the range that i s involved i n p' (x^ ,t) or P'P' ( r ) ) . These conditions ensure that the cross co r r e l a t i o n The e f f e c t i v e tube length i s s l i g h t l y greater than the geometric tube length due to i n e r t i a of the a i r beyond the ends of the tube. f Eq,4. 1 23 function p|p* (r) which i s the es s e n t i a l ingredient used to obtain the DEI, i s not distorted by extraneous resonance peaks or by phase s h i f t s introduced in either channel (see Appendix B for more d e t a i l on the phase s e n s i t i v i t y of the cross c o r r e l a t i o n function). The freguency range of the probe can be extended by optimizing the parameters given i n Eq;.4, 1 , Several changes were made to the basic probe design i n order to achieve the optimum design. Cardboard discs made with a paper punch were c a r e f u l l y inserted on the inside of the microphone grid cap, The s l o t s of the grid cap were f i l l e d in with s i l i c o n e rubber. These two modifications reduced the cavity volume thereby increasing the resonant frequency, Furthermore, the c a p i l l a r y tube was tapered, From the surface, half the length of the tube was bored out to ,091 cm (.036 inches);. The other half was bored out to > 198 cm (,078 inches), Tapering the tube resulted i n a smaller e f f e c t i v e tube length and an increase i n the resonant frequency. The resonant frequency of the re-designed one guarter inch probe was found to be about 5 k hz. Comparing Fig,7c with Fig.6a, shows that the upper frequencies i n the far f i e l d spectrum s l i g h t l y overlap the resonance frequency of the one quarter inch probe. The proximity of these frequencies creates a p o s s i b i l i t y of high frequency d i s t o r t i o n s a f f e c t i n g the cross co r r e l a t i o n function (p^p* (f) )• A second probe was designed i n an attempt to achieve a resonant frequency that was higher than the marginal 5 k hz that was measured f o r the one guarter inch prober The design of the second probe was aimed towards further reducing the cavity volume thereby increasing the Helmholtz 24 resonant freguency,. To achieve t h i s , a one eighth inch BSK Condenser Microphone Cartridge (Type 4138) was mounted in a smaller cavity in a d i f f e r e n t cylinder model, The membrane of the one eighth inch microphone was very close to the grid cap so cardboard discs could not be employed, after several attempts at optimization i t was found that the resonant freguency of the one eighth inch probe was also about 5 k hz, The actual freguency response curves for the.one guarter and one eighth inch probes were recorded along with th e i r phase response by a procedure outlined i n the next section. The response curves are shown i n Figs;,6a & b , For both probes, the freguency response was f l a t (within ±1,5 db) out to about 3 k hz with a resonant boost occurring at the upper freguency range. The phase s h i f t -e-A - -e-o was recorded near probe resonance and i t was observed to increase abruptly near resonance but was less than 3 degrees for frequencies below 4 k hz, f o r both probes, Thus i t was found that the resonant frequency and the phase c h a r a c t e r i s t i c s of the two probes were about the same, However the design of the one quarter inch probe was apparently superior to that of the one eighth inch probe. Although the one eighth inch probe was advantageously the smaller of the two probes , 1EFF. was larger in the one eighth inch probe than i t was in the one quarter inch probe. Furthermore the cavity volume i n the one eighth inch probe was probably larger than i t was i n the one guarter inch probe since the volume i n between the s l o t s of the one eighth inch microphone grid cap was not f i l l e d in with s i l i c o n e rubber, The g r i d cap could not be removed or modified without risking damage to the microphone membrane contained therein* These disadvantages 25 that are found in-the design of the one eighth inch probe l i m i t that probes performance over the one guarter inch probe. Sec; 4; 1 The Measurement of Probe Ch a r a c t e r i s t i c s : Freguency and Phase Response The frequency response of the probe was obtained by exciting the device with a constant amplitude sinusoidal sweep tone generated by a loud speaker, The tests were carried out i n the anechoic chamber at the Department of Mechanical Engineering at UBC. ft one half inch BSK Condenser Microphone (Type 4133) was positioned several excitation wavelengths down the primary lobe of a good guality loudspeaker* The cylinder model was placed beside the microphone. The loudspeaker was driven with B&K Sine Random Generator (Type 1024) connected i n series with a 50 watt power amplifier. The SPL of the excitation was held constant by a feedback loop* The SPL near the probe was fed back into the Compressor Input of the generator via the one half inch microphone. The compressor section of the generator modulated (increased or decreased) the output of the generator i f the SPL started to d r i f t . The probe response was amplified by a BSK Measuring amplifier (Type 2606). This amplified signal was connected to the y axis of a B&K Level Recorder (Type 2305); the x axis was synchronized with the sweep frequency on the generator* The recorder plotted probe response (in db) versus frequency (see Fiqs,6a & b). . 26 The phase s h i f t introduced by the Helmholtz resonator probe was measured using an accurate dual beam oscilloscope. Channel B on the C,E,0. was triggered with the generator signal (used as a reference) and channel A was triggered with the probe s i g n a l . The two traces were observed on the C.B,0. , From t h i s , the phase s h i f t -e-c, - -e-6 was recorded near resonance, for each probe (see Figs, 6a & b). Sec.4* 2 Freguency Spectra f o r the Source Point and for the Observation Point Freguency spectra of surface pressure were obtained with the cylinder model spanning the potential core region of the j e t ; the position of the o r i g i n 0 i s given by the coordinates (xf = 0, x£=0, X'=1D=3. 8x10- 2 m) in t h i s case (see Fig.2a). The pinhole was moved onto the jet axis to the point yi=(E=d/2 # 0=O<>, h=0) and the pot e n t i a l core spectrum was recorded (see Fig,7a), The pinhole was subseguently moved into the mixing layer to the point yi = (E=d/2, 0=0°, h=1,91x10-2 m) , and the mixing layer spectrum was recorded (see Fig.7b). A one half inch BSK Condenser Microphone (Type 4133) was moved to the far f i e l d point x|=(x|=0# x«=-3.00 m, x«=0) so that i t was 90° to the jet axis, and the f a r f i e l d spectrum was charted (see Fig*7c), The amplifier and loud speaker introduced n e g l i g i b l e phase s h i f t between channels A and B for the range of freguencies near probe resonancei. 27 Frequency spectra of pressure were obtained with the cylinder model spanning the mixing region of the j e t ; the.position of the or i g i n 0 i s given by the coordinates (x* = 0, x» = 0, x'=5D=19.0x10-2 m) in t h i s case (see Fig. 2b) „ The mixing region spectrum was recorded with the pinhole at the point yi=(R=d/2, 0=00, h=6) , the same orientation as was used for potential core t r i a l (see Fig;„7d). Once again the f a r f i e l d spectrum was charted with the one half inch microphone at the previously given far f i e l d point (see Fig.7e), The locations of the mixing layer and the mixing region are shown in Fig-3a. Spectral analysis of the pressure signals was performed with a B&K Band Pass F i l t e r Set (Type 1614),. The said f i l t e r set has a constant percentage bandwidth which i s 23% of the center frequency or equivalently one t h i r d of an octave* Consequently, i t was necessary to account for the e f f e c t of increasing bandwidth in the one t h i r d octave spectra i n order to obtain a more meaningful measure of high frequency energy i n the pressure signals. For each one t h i r d octave spectrum, spectrum l e v e l s associated with frequencies above the center frequency of the peak spectrum l e v e l were corrected f o r increasing bandwidth. The correction results i n a 3 db decrease i n the spectrum l e v e l * each time the freguency i s doubled from the center frequency of the peak spectrum l e v e l . These corrected l e v e l s are indicated by arrows i n Figs.7a to 7e i n c l u s i v e , The one t h i r d octave spectra were obtained in the following manner. The f l u c t u a t i n g signal from each microphone was fed into the input of a measuring amplifier. The f i l t e r set was introduced at an intermediate stage of the amplifier; the amplifier was 28 provided with Ext. F i l t e r jacks on the rear panel for t h i s purpose.. The f i l t e r e d output of the amplifier was recorded on the yi axis of the B&K Level Recorder (Type 2305) . The recorder provided a voltage pulse back to the f i l t e r set. This pulse triggered the f i l t e r set each time a t h i r d octave band had been charted. The combined e f f e c t was to plot the f i l t e r e d response in db against the center freguency of the one t h i r d octave band. Sec. 4;3 Ca l i b r a t i o n of Instruments The sound pressure l e v e l i n a i r i s defined i n units of decibels by. SPL = 10 log 1 0 f p'p 1) Eq,4,2 where p R e p= 20x10-* pascals which i s the minimum sound i n t e n s i t y perceptible by the human ear. The measuring amplifier was eguipped with a precision galvanomet er. i The scale, marked in decibels (SPL) required accurate c a l i b r a t i o n * A B&K Pistonphone (Type 4220) was used for t h i s purpose. The sig n a l from the Pistonphone generated a scale reading on the measuring amplifier that corresponded to (124, 0±, 1) db. The Pistonphone generated a high SPL. The excitation freguency was 250 hz, This combination was suited f o r c a l i b r a t i o n of the probe; a high SPL tone dominates ambient (background) level s and produces a steady scale reading; a low freguency tone 29 remains unattenuated as i t t r a v e l s down a c a p i l l a r y tube. However, a t i g h t seal was reguired between the pinhole and the Pistonphone to avoid leakage of low frequency energy* An adaptor was used that f i t t e d snugly into the Pistonphone, The end of the adaptor had a b u i l t in O-ring which formed a seal between the pinhole and the Pistonphone. Sec. 4.4 The Measurement of Plow Parameters The value of the atmospheric pressure was obtained from the airport and the temperature inside the anechoic chamber was measured with a mercury thermometer marked to an accuracy of ±.05°C. The stagnation pressure head at the jet nozzle was measured with a mercury manometer to an accuracy of ±5% on the scale reading. Sec,4,5 The Construction of the DRI Dis t r i b u t i o n The DRI i s related to the cross c o r r e l a t i o n function (PfjP' (r)) by the following recipe (see Sec.3. 1.2), dp'p' ( x 1 # Y l ) n, (yj) dS 4iT xCco ±_ C (x, ,y, ) it) | Eg. 3. 9 a-c J r =r/c«, Pf« ant i l o g l o |SPL( Y i)^+ SPL(x 1)| Here C(r) i s the dimensionless cross correlation c o e f f i c i e n t o defined to be, 30 C(xi,y-) (£) = P K T i l P ' l x i ) (r) Eq.3.7 a t =r/c 0 0 The d i s t r i b u t i o n i s constructed by evaluating Eq.3.9 at an appropriate number of source points (y^ ) with the.far f i e l d point (x^) at a fixed distance 90° to the jet axis. The parameters used to evaluate Eq.3.9 were obtained as follows: the anqle subtended by x} and n, (y, ) i s equivalent to (j> to f i r s t order (see Fiq, 2a), and <jt was indicated by marks scribed into the cylinder support at 1 5 ° i n t e r v a l s ; the distance x=|x1| was accurately obtained using a steel a tape measure; the ambient temperature was recorded at three minute time i n t e r v a l s with the thermometer described above (see Sec.4,4); the sound pressure l e v e l at the pinhole was monitored with the calibrated scale on the measuring amplifier f o r each pinhole position. Three types of correlation functions were used to evaluate the contents of the sguare brackets in Eg,3 . 9 - A SAICOB Honeywell SAI-43A Correlation and Probability Analyser was used to process the pressure signals (see F i g . 5 , b ) ; the various c o r r e l a t i o n functions were given as output. For each pinhole location the following correlation functions were recorded graphically from the given output: 1) the cross c o r r e l a t i o n function; p|(yi)p*(xi) (?) (see F i g . 8a) 2) the source point autocorrelation function; p° P* (y-j ) (t) (see Fig* 8b) 3) the far f i e l d point auto c o r r e l a t i o n function; 31 P'P'(xi) (T) (see F i g . 8c) A c y l i n d r i c a l coordinate system (B, <j>, h) with o r i g i n denoted by 0 was introduced to designate the pinhole position (y^) and where necessary, the position of the far f i e l d microphone ( X i ) (see Fig . 2a) « A cartesian coordinated system ( x j , x{, x^) with o r i g i n denoted by 0' designates the position of the origin 0 and where necessary, the position of the source point (y.j) and the far f i e l d microphone (x?), The DEI d i s t r i b u t i o n was obtained i n two p r o f i l e s : the DEI was evaluated f o r a series of circumferential points around the cylinder model, and for a series of spanwise points along the cylinder model. The circumferential p r o f i l e of the DEI was obtained by incrementing the 0 coordinate of y, from 0° to 360° i n 15° steps at a specified h. The spanwise p r o f i l e of the DEI was obtained by incrementing the h coordinate of y-j from 0 to h a m e r i n 5 cm steps at a s p e c i f i e d <f> coordinate. Circumferential p r o f i l e s of the DEI were constructed for two di f f e r e n t flow cases: Fig.9a i s the r e s u l t for a laminar incident flow and was obtained with the midsection of the cylinder model stationed i n the potential core of the jet (the o r i g i n was at 0 ( x , = 0 , x[=0, X|=1D) for t h i s case); Fig.10a i s the result for a turbulent incident flow and was obtained with the midsection of the cylinder model stationed in the turbulent mixing region (the o r i g i n was at O(x'=0, x«=0, x|=5D) for t h i s case). An additional circumferential p r o f i l e i s given for turbulent flow (see Fig.9c),. h Q u i E T 9 i v e s the h coordinate of a point outside the active flow region; where the SPL has decreased by 10 db or more. 32 Here the o r i g i n was at 0(x' = 0, x£=0, X^=1D) and the incident flow comes from the turbulent mixing layer (see Fig;. 3,a). The spanwise p r o f i l e of the DEI was constructed along the l a t e r a l edge of the cylinder model for the two di f f e r e n t flow cases: Fig.9d i s the r e s u l t for a laminar incident flow and was obtained with the cylinder model spanning the potential core/ mixing layer (the o r i g i n was at 0(x'=0, x'=0, x| = 1D) f o r th i s case); Fig,. 10, b i s the resu l t for turbulent incident flow and was obtained with the cylinder model spanning the mixing region (the o r i g i n was at 0(x'=0, x£=0, x'=5D) fo r t h i s case). Circumferential p r o f i l e s provide information on the mechanisms of noise generation for laminar and turbulent flows over c y l i n d r i c a l struts,; Spanwise p r o f i l e s are. obtained primarily f o r the purpose of demonstrating i n t e g r a l closure, which indicates that a l l the important points of the DEI d i s t r i b u t i o n have been accounted for. "*The l a t e r a l edge i s the set of points on the surface of the cylinder model where 0=0°, or where 0=180°. 33 Sec* 5 CALCULATIONS Sec* 5,0 Procedure The pointwise evaluation of the DEI requires the measurement of parameters and processing of the pressure signals, as was described i n Sec.4.5. In addition a slope computation must be performed before the DEI can be evaluated* The slope of the cross correlation function (p|p' (r)) i s taken at the Kirchhoff delay time given by r =r/Coo (see Fig. 8a) and i s then normalized by p!^ and p R > r^ , which are obtained from their respective autocorrelation functions (see Figs*8b 6 c ) . The value of r/Coo should be obtained as accurately as possible; in Appendix B i t i s shown that the slope changes quite rapidly with T. and f i n a l r e s u l t s could be sensitive to t h i s uncertainty. To ensure minimal slope error, the geometric path length (r) and speed of sound (c„) were recalculated for each of the pinhole locations to obtain an updated Kirchhoff delay time (see Appendix D for example). The co r r e l a t i o n functions were computed with the highest resolving power available on the correlator. Nevertheless, systematic error i n the slope s t i l l remains, and arises largely from the uncertainty i n the speed of sound. In accounting for t h i s slope error, three values of the slope were estimated. They comprise the minimum, best, and maximum slope "The sample slope c a l c u l a t i o n given i n Appendix D was made at an i n f l e c t i o n point and as such i s given f o r just the best and minimum slope estimates,. 34 values (see Fig* 8 a),. The Mach number of the jet was calculated for reference by setting the the pressure head measured d i r e c t l y across the jet nozzle equal to the dynamic pressure,. The predicted SPL for the far f i e l d point was obtained from a surface i n t e g r a l of the DBI distribution,. The integration was performed numerically using Simpson's Rule* The scheme used a scaled up version of the circumferential p r o f i l e at midspan to approximate circumferential p r o f i l e s at other spanwise positions. This approximation should y i e l d an overestimated SPL since Fig.9c indicates that the area enclosed by the circumferential p r o f i l e i n the mixing layer encloses l e s s area than a scaled up potential core prototype, The o v e r a l l SPL was measured d i r e c t l y at the f a r f i e l d point and includes acoustical energy radiated from the jet and wake as well as the cylinder generated surface noise. It was possible to estimate the SPL associated with the surface, exclusive of a l l other sources of acoustical energy. This was done by subtracting the energy associated with jet and the wake from the ov e r a l l SPL. The corrected SPL may now be compared with the SPL predicted by the Causality Correlation Technique. The SPL associated with the basic undisturbed jet was measured d i r e c t l y with the cylinder model removed. . I t was assumed that t h i s jet noise i s present to the same degree a f t e r the introduction of the cylinder model. The SPL associated with the wake was subseguently estimated from L i g h t h i l l * s g u a d r u p o l e source i n t e g r a l (see Eg.3 , 1 ) . In the i n t e g r a l the vel o c i t y f l u c t u a t i o n within some active (acoustic) wake volume was assigned the nominal 35 value of Uo/2; half the wake v e l o c i t y d e f i c i t (on the j e t axis). The active volume was taken to be two cylinder volumes. Time derivatives were simply replaced by the vortex shedding frequency (for the jet c e n t e r l i n e ) . Acoustical conversion e f f i c i e n c y of the wake fluctuations was assumed to be no more than 50%* Sample calc u l a t i o n s are given i n Appendix D f o r the potential core t r i a l (origin at O(x , =0, x*,=0, X|=1D)). Similar calculations were performed for the mixing region t r i a l (origin at 0(xf=0, x^=0, x ^ D ) ) . 36 Sec.6 DISCUSSION OF RESULTS Sec.6.0 A Summary of Ch a r a c t e r i s t i c Frequencies and V e l o c i t i e s Sec.6.0.0 Cylinder Model Stationed at the Potential Core, Origin at 0(xJ=0, x»=0, x|=1D) A nominal frequency for vortex shedding from the portion of the cylinder model i n the pote n t i a l core of the j e t (see Fig, 3a) i s given by the Strouhal frequency. The Reynolds number, based on cylinder diameter i s i n t i e range 50,000 < Re < 200,000. At these Reynolds numbers the wake i s dominantly turbulent, however, periodic vortex structures s t i l l e x i s t i n the wake of a cylinder. The drag c o e f f i c i e n t i s constant i n t h i s range of Reynolds number and the t r a n s i t i o n point which marks the t r a n s i t i o n between the laminar boundary layer and the free turbulent boundary layer, coincides with the point of separation (see Fig.4). The nominal vortex shedding frequency based on a Strouhal number of *2 and the j e t core v e l o c i t y (UG) i s , fsiRouHAL " • 2U - ,2Uo = ,2(72 m/s) = 1,13 k hz Eq. 6, 1 d d (1.27x10-2 m) Vortex shedding at the above frequency i s evident i n Fig. 7a. There the spectrum of surface pressure for the potential core exhibits a peak i n the one t h i r d octave band centered at 12 50 hz, The peak i s probably due to vortex shedding at the above Strouhal frequency. Irregular turbulence i s superimposed on the vortices. Turbulence 37 acts together with spanwise variation of the mean incident velocity to broaden the Strouhal peak. A nominal freguency f o r acoustical radiation a r i s i n g from the portion of the cylinder model i n the potential core i s also given by the Strouhal frequency. The existence of a guasi tone at the Strouhal frequency i s evidenced in the spectrum of far f i e l d pressure (see Fig.7c), There a r e l a t i v e l y narrow peak i s centered at 1250 hz. The acoustical peak appears to be narrower f o r two reasons: the l e f t hand t a i l f a l l s off more rapidly since i t i s weighted by a dipole radiation e f f i c i e n c y of f _ l ; the right hand t a i l f a l l s o f f ' f r e e l y 1 , without probe resonance,. A nominal frequency for vortex shedding; from the part of the cylinder model i n the middle of the mixing layer should be about one half that given for the potential core; assuming that the mean velocity i n the middle of the mixing layer i s about 0 o/2. Thus, the 400 hz peak i n the spectrum of surface pressure for the mixing layer (see Fig.7b) i s probably the Strouhal peak. This would require that the mean velocity at that point at which the spectrum was measured (see Sec. 4*2) would be, 0. = f s™ouH.n. d - 400 hz (1,27x10-2 m) = 25 m/s Eq.6,2 Strouhal* .2 I Considering that the exact position of the mixing layer point and the exact value of the e f f e c t i v e Strouhal number are unknown, the mean velocity inferred by Eg.6,2 seems reasonable. Moreover, i t was observed that the 400 hz peak i s associated with points s l i g h t l y around the backside the cylinder model r e l a t i v e to the approaching flow. This observation also suggests that the 400 hz 38 peak i s a vortex peak. In the wedge-shaped mixing layer , a highly turbulent flow develops. Nominal freguencies can be predicted for eddy impingement upon the cylinder model in the mixing layer from an eddy model. A recent paper by fioshko13 (1976) reported the existence of large organized structures i n a two dimensional mixing layer; the spacing of those eddies was about ,31x1 (see Fig,3b). Assuming that the spacing between three dimensional eddies in the coannular mixing layer i s about , 3 1 x 5 and i f the eddy convection velocity (0 C) , i s about ,5U 0 then the eddy impingement freguency ought to be about (by Taylor's hypothesis), f EDDY ~ - . 5Uo = .5(100 m/s) = 4.3 k hz Eg, 6,3 .31x' ,31D (1.27x10-2 m) The spectrum of surface pressure for the mixing layer (see Fig.7b) exhibits a narrowly defined peak i n the one t h i r d octave band centered at 400 hz and a broadly defined peak i n the one t h i r d octave band centered at 1600 hz. Evidently, the former of these two peaks i s v i s i b l e in the spectrum of f a r f i e l d acoustical pressure (see Fig.7c), This implies that the dipole radiation which i s associated with t h i s peak is,not weak, The 1600 hz peak i s not v i s i b l e i n the aforementioned spectrum due to the masking e f f e c t of the strong peak at 1250 hz which was previously attributed to vortex shedding from the portion of the cylinder model in the potential core. The existence of a peak at 4300 hz i s not apparent. This could mean that the spacing of the alleged three dimensional eddies i s somewhat greater than ;*31x|. Further i n v e s t i g a t i o n revealed that the 400 hz/1600 hz mixing 39 layer peak becomes more pronounced for points s l i g h t l y around the back side /front side of the cylinder model r e l a t i v e to the approaching of flow* A c h a r a c t e r i s t i c freguency for noise that i s emitted from the mixing layer i s given by L i g h t h i l l 9 (1962).~A quadrupole model of an axisymmetric a i r jet was developed by L i g h t h i l l using the measurements of Laurence 1 8 (1956), The model i s depicted i n Fig.3a for a Reynolds number of 6 x 10 s (based on nozzle diameter). The eddies i n the mixing layer are viewed as being eguivalent to moving frame c o r r e l a t i o n volumes. Associated with each c o r r e l a t i o n volume i s a quadrupole (s) , T-jj = pu.,Uj. The c h a r a c t e r i s t i c frequency of the broad band noise emitted from the mixing layer was inferred tc be of the order U0/x' and was believed to originate from the convecting guadrupoles. C u r i e 8 (1955) demonstrated that the c h a r a c t e r i s t i c freguency of surface dipole noise (pu-j ) ari s i n g from guadrupoles 'in contact' with a surface i s less than the c h a r a c t e r i s t i c frequency of the 'free' quadrupole noise (pu;uj) by a factor of two. The c h a r a c t e r i s t i c frequency of surface dipole noise a r i s i n g from mixing layer guadrupoles should be, according to the above model, f EDDY ~ i l 5«> = = .5(100 m/s) = 1.13 k hz Eg. 6.4 x' D (3,8x10-2 m) Hence the quadrupole model when modified for surfaces y i e l d s a predicted frequency which i s i n rough agreement with the 1600 hz peak, In summary, the 400 hz peak which appears in the spectrum of surface pressure for the mixing layer and the associated spectrum 40 of far f i e l d acoustical pressure, i s a phenomena a r i s i n g from the surface's i n t e r a c t i o n with the shear layer, and i s not due to mechanical resonance. The 1600 hz peak i n the previously stated spectrum i s also an unresolved p e c u l i a r i t y of the surface's i n t e r a c t i o n with the shear layer. For l a t e r reference, the velocity components may be assigned to the jet flow at one nozzle diameter downstream of the exit plane and to the wake behind the cylinder model at that station. The velocity f l u c t u a t i o n ( u R M S ) f o r turbulence in the mixing layer was found to be ,15Uo at a r a d i a l position where the mean velocity (U) was ,5U0 (see Nayar, Siddon, and Chu**); the vel o c i t y f l u c t u a t i o n displayed l i t t l e a x i a l v a r i a t i o n . The r a d i a l variation in the mean velocity corresponding to the 1D region i s given i n E e i c h a r d t 1 7 . The a x i a l variation of mean ve l o c i t y i s given i n Nayar, Siddon, and Chu»*, The maximum velocity fluctuation behind the cylinder model should equal the wake velocity d e f i c i t ; the variation of the wake velocity d e f i c i t along the cylinder model i s equivalent to the r a d i a l variation of U i n the j e t . Sec,6,. 0* 1 Cylinder Model Stationed at the Mixing Region, Origin at 0(x'=0, x«[=0, x'=5D) A nominal frequency for vortex shedding when the cylinder model i s immersed i n the mixing region i s given by the Strouhal freguency. The mixing layer has eaten away the potential core after about four nozzle diameters downstream of the exit plane. Thereafter the flow i n the mixing region i s i n t r a n s i t i o n ; becoming f u l l y turbulent after about eight nozzle diameters. At 41 the point where the potential core vanishes the j e t centerline velocity begins a hyperbolic decay. Nayar, Siddon, and Chu** measured a jet centerline velocity of *8Uo at a point f i v e nozzle diameters downstream. The Strouhal freguency corresponding to t h i s velocity i s , • fsTRouHRL " ±20 -.2 (. 8a 0) = ,2(, 8) (72 m/s) = .889 k hz Eg* 6* 5 d d (1,27x10-2 m) The spectrum of surface pressure for the mixing region (see Fig*7d) exhibits a peak i n the one t h i r d octave band centered at 800 hz. There i s a 10% difference between t h i s observation and the value predicted by Eg.6.5, The difference i s probably due to s l i g h t differences between the two j e t s . The st a r t i n g point for the hyperbolic decay of the centerline velocity may not be the same for the two j e t s . Thus the broad 800 hz peak i s apparently the Strouhal peak; broadened by turbulence and the mean vel o c i t y p r o f i l e . A c h a r a c t e r i s t i c freguency for noise a r i s i n g from mixing region guadrupoles i n contact with the cylinder model may be inferred from the i n t e g r a l time scale of the moving frame autocorrelation. Physically, there e x i s t s a known one to one correspondence between the far f i e l d spectrum of the j e t noise and the 'convecting frame* source spectrum; convecting at the eddy convection velocity U c. This would imply that the f a r f i e l d noise originates from changes that occur within the moving frame (i . e , the decay of turbulence). Therefore the c h a r a c t e r i s t i c freguency for f a r f i e l d noise a r i s i n g from the convecting guadrupoles i s eguivalent to the r e c i p r o c a l of the i n t e g r a l time 42 scale of the convecting frame autocorrelation; the i n t e g r a l time scale being a measure of the average l i f e t i m e of an eddy measured within the moving frame. The c h a r a c t e r i s t i c freguency of dipole noise a r i s i n g from quadrupole contact should be one half of the aforementioned frequency, fEDDY ~ 1 = Do = (72 m/s) = .63 k hz Eq.6.6 2T 3D 3(3.8x10-2) The i n t e g r a l time scale was obtained from Fig.11 (reprinted Nayar, Siddon and Chui*) for the point y(= (y« = 1.00x10-2m, y«.=0, y»=4.5D). The spectrum of surface pressure for the mixing region did not exhibit a DISTINCT peak at thi s freguency however. For l a t e r reference, the ve l o c i t y fluctuation f o r turbulence in the early mixing region varies r a d i a l l y from .08Uo on the jet axis to ,14U0 at a point where the mean ve l o c i t y was ,50 0 (see Nayar, Siddon, and Chu**). The mean vel o c i t y p r o f i l e f o r the early t r a n s i t i o n region, also of intere s t here, i s given in Nayar, Siddon, and Chu**, The turbulence i n t e n s i t y i n the wake behind the cylinder i s larger than that i n the incident flow by some factor ranging between 1 and U 2 / ( . 0 8 0 o ) 2 - 150, Sec; 6,1 Discussion of the Dipole Source Dis t r i b u t i o n The o v e r a l l SPL, measured at x,, increases by 20 db or more after the cylinder model i s introduced into the flow* The observed The convecting frame autocorrelation i s the Fourier transform of the convecting frame source spectrum. 43 increase i s caused i>y dipole sources that are formed by mechanism 2, The p o s s i b i l i t y that dipole sources are formed by mechanism 1 i s ruled out by the following considerations. The resonant freguency for mechanical vibration of the cylinder model i s 250 hz assuming i t i s uniform i n density and clamped at one end; the actual model i s bored out along half i t s length and, as such, i t should resonate at about 5 00 hz, However, the spectrum of surface pressure for the potential core i s without a resonant peak i n the v i c i n i t y of 500 hz, In addition, the second term i n the second i n t e g r a l of Eg, 3,1 shows that the model would need to o s c i l l a t e with an amplitude of 20 cm at 500 hz i n order to produce a SPL of 69,3 db; actual amplitudes should be at least four orders of magnitude smaller than 20 cm, The spectrum of surface pressure for the f a r f i e l d does exhib i t a low freguency peak centered at 400 hz* However, t h i s peak moves away from the mechanical resonant freguency as the exit velocity i s increased. Further inves t i g a t i o n reveals that the low frequency peak i s common to both the aforementioned spectrum and the spectrum of surface pressure for the mixing layer. Moreover, the mixing layer peak becomes more pronounced for points that are s l i g h t l y around the backside of the cylinder model r e l a t i v e to the oncoming flow. In Sec.6.0,0 i t was argued that t h i s low frequency peak i s associated with vortex shedding at a reduced mean ve l o c i t y rather than a mechanical vibration. Having determined the mechanism by which dipole sources are formed, we consider the importance of interference i n these experiments, Sound that i s radiated by distributed sources can i n t e r f e r e destructively i n some directions and constructively in 44 others. To account for t h i s the dipole radiation term of Eg.3,4 requires that the component of each force, when taken in the dir e c t i o n of the f a r f i e l d point be evaluated at i t s Kirchhoff retarded time; the o v e r a l l dipole radiation i s proportional to the in t e g r a l of the time rate of change of these components. Accordingly, the radiations from dipole sources i n t e r f e r e at the far f i e l d point at the present time ( t ) . In acoustics the following i s often the case: the differences i n t r a v e l time from various part of the surface are generally regarded as unimportant i f the surface dimensions are small r e l a t i v e to the c h a r a c t e r i s t i c radiation wavelengths* When t h i s i s the case the source region i s termed •compact1 and the force components are evaluated at a common Kirchhoff retarded time (i.e. differences i n retarded time are overlooked), Nevertheless the p o s s i b i l i t y s t i l l e x ists for interference or indeed, f o r incoherence between adjacent 'patches' of the surface which w i l l a ffect the o v e r a l l radiation that i s scattered into a given d i r e c t i o n from the cylinder surface. It i s the unigue feature of the Causality Correlation Technigue which enables one to measure the ' l o c a l e f f i c i e n c y of radiation' while giving due regard to these e f f e c t s . This feature i s i l l u s t r a t e d by the following example. Consider the case where two dipoles less than a wavelength apart are i n counter-phase. Their DRIs for a given far f i e l d point are found to be opposite i n sign, Destructive interference occurs when the two DBIs are integrated; the sum of the DEIs i s less than or egual to the sum of the absolute values of the DRIs. By contrast, the c l a s s i c a l dipole strength i s always a positive quantity belonging to the source region..The DRI, on the other hand, f u l l y 45 accounts for f a r f i e l d interference a r i s i n g from incoherent di s t r i b u t e d sources. The shape of the sound f i e l d that i s generated by a compact d i s t r i b u t i o n of dipoles i s not necessarily that of a point dipole giving a true 'dumbell' d i r e c t i v i t y pattern. The actual shape depends upon the aforementioned e f f e c t s of source incoherency and indeed upon the shape of the surface i t s e l f * A. true •point d i p o l e 1 radiation f i e l d w i l l only r e s u l t i f the f l u c t u a t i n g surface pressure f i e l d i s i n phase, coherent and uniformly distributed i n magnitude over a compact guasiplane surface (180° out of phase, counter-coherent and uniformly distributed i n magnitude on the opposite side of the surface) . In t h i s case the DRI w i l l be uniform i n magnitude and also of the same sign at a l l surface points and the radiation d i r e c t i v i t y w i l l exhibit a true 'dumbell' d i r e c t i v i t y pattern* C h a r a c t e r i s t i c radiation wavelengths were estimated from far f i e l d freguency spectra of pressure and found to be in excess of 17 cm ; the main acoustical energy was contained i n bands below 2000 hz . Comparing the shortest wavelength (17 cm) with surface dimensions (1.27 cm x 17 cm) implies that the dipole source region on the surface of the cylinder model i s s p a t i a l l y compact i n cross section but i s not compact in length, The dipole source d i s t r i b u t i o n c e r t a i n l y may not be modelled by a point dipole source. Furthermore i t may not be modelled with a l i n e dipole source since the flow i s three dimensional; there being considerable variation i n phase, amplitude and source coherence along the cylinder. We have seen that the dipole r a d i a t i o n term from the 46 c l a s s i c a l theory (see Eg* 3.1) gives some basis for the following simple description: the dipole source d i s t r i b u t i o n on the emersed cylinder model i s compact i n cross section;. Retarded time difference between d i f f e r e n t spanwise points may not be neglected i n general (depending on the location of the far f i e l d point). The DRI p r o f i l e s determined by the correlation technique y i e l d additional and useful time averaged information. Sec. 6. 2 The Circumferential P r o f i l e of the DRI for the Pot e n t i a l Core T r i a l , Origin at 0(x'=0, x*=0, X'=1D) The circumferential variation of the DEI around the cylinder model i s plotted i n Fig;. 9a for the cross section of the model i n the p o t e n t i a l core, with the streamwise coordinate of 0 set at xf^lD. The 'swept back' cos 2 <f> shape of t h i s p r o f i l e arises from several f a c t o r s : surface geometry introduces a cos <p factor; a fluct u a t i n g l i f t force i n d i r e c t l y introduces an additional cos <j> factor; an anomaly i n the <j> variation of the surface pressure causes t h i s p r o f i l e to deviate from a true cos 2 <f> shape. The source region geometry introduces a cos <j> factor into the abovementioned p r o f i l e . This factor appears as Xj n^  (y, ) / x i n the recipe for the DEI and originates from the Curie solution. Eeferring to Fig. 2a, the factor x i n , ( y ^ ) / x becomes cos <j> when the points x^  and y-j l i e within the plane swept out by E and containing 0. There the DEI i s related to the component of the t o t a l force fluctuation in the d i r e c t i o n of the f a r f i e l d point, The t o t a l force f l u c t u a t i o n i s i n the di r e c t i o n of n-j {y^ ) . 47 The existence of an additional cos <f> factor i s revealed in Fig;.9b. The geometric cos 0 factor has been eliminated from t h i s p r o f i l e , yet a basic but very skewed cos o" shape i s s t i l l present. Consequently, the circumferential p r o f i l e of the DEI has a basic cos 2 0 shape* A true cos 2 ^ shape results when an external force i s applied i n a constant d i r e c t i o n to the cylinder model causing i t t o o s c i l l a t e back and f o r t h in a motionless perfect f l u i d . The circumferential variation of pressure around the cylinder model in t h i s case would have a true cos 0 variation. Hence, t h i s factor would be introduced into the circumferential p r o f i l e of the DEI giving i t a true cos 2 ^ shape. Mechanical o s c i l l a t i o n s of t h i s kind were shown to be unimportant i n the generation of the far f i e l d SPL (see Sec,6.1). Furthermore, i t was shown that the dipole sources are formed by reaction forces; exerted on the f l u i d by the r i g i d surface in reaction to the l o c a l aerodynamic flow. The net force must then resemble a f l u c t u a t i n g l i f t force giving r i s e to a basic cos 2 0 shape to the circumferential p r o f i l e of the DEI,. Such a force would indeed arise from the guasi periodic shedding of vortices from a point near the l a t e r a l edge of the cylinder model, The DEI reaches i t s maximum value at a point s l i g h t l y around the backside of the cylinder model where (ft - 15°/165°, giving the p r o f i l e a swept back appearance. The EMS value of p| also reaches a peak value at t h i s point (1.38 x 10 - 2 Pa). Evidently, there i s a suction peak in the v i c i n i t y of 0 - 15°/165° near the separation point, probably where the vortices are f i r s t formed. Since the nominal frequency for vortex sheddinq i s 1.13 khz, t h i s region 48 would o f f e r e f f i c i e n t radiators of dipole sound and a maximum DEI. The possible existence of dipole sources at the t r a i l i n g extremity of the cylinder model (where 0=90°) was investigated* For t h i s , the far f i e l d point was positioned at 45° to the jet axis; x1 = (R=3.00 m, <^=45°, h=0) . The DEI associated with the point on the surface of the cylinder model where $=90° was found to be negl i g i b l y small. Consequently, the existence of drag dipoles at the t r a i l i n g extremity was ruled out, Although the RMS value of p| (the c l a s s i c a l dipole strength) reaches a peak value at the t r a i l i n g extremity (1.44x10 - 2 Pa), most of thi s energy appears to be at low freguencies and, as such, would not offer e f f i c i e n t dipole sources. Nevertheless, l i f t / d r a g dipoles contribute to the radiated acoustical pressure. This i s apparent from the deviation i n the circumferential p r o f i l e of the DRI from a true cos 2 <f> shape. Since the suction peak i s not concentrated at 0=0°, the force f l u c t u a t i o n has both l i f t and drag components; 96% of a force f l u c t u a t i o n exerted i n the 0=15° d i r e c t i o n i s i n the cross stream di r e c t i o n ( l i f t ) , the remaining 4% i s i n the stream direction (drag). . The l i f t component of the force fluctuation contributes d i r e c t l y to the SPL at the far f i e l d point; i t i s the component of the t o t a l surface force i n the di r e c t i o n of the f a r f i e l d point that contributes d i r e c t l y to the SPL there. The nominal frequency of t h i s f l u c t u a t i n g l i f t component i s 1.13 khz;. The drag component does not make a contribution to the SPL at the f a r f i e l d point which i s x|=(x«=0, x«=-3.00 m, x'=0). 49 Sec,6.3 The Spanwise P r o f i l e of the DEI for the Potential Core T r i a l , Origin at O(x*=0, x«=0, x^=1D) The spanwise variation of the DRI along t h e . l a t e r a l edge of the cylinder model i s plotted i n Fig,9c, The spanwise p r o f i l e i s weighted by the geometric factor x ^ n ^ y , ) / x which i s the cosine of the angle between X , and n^ ( Y l ) (see Fig*2a), This factor i s very nearly egual to unity for points yi along the l a t e r a l edge of the cylinder model, Along the l a t e r a l edge the DRI reaches a maximum value at a point about 2.4 cm o f f the je t axis. This corresponds to a point in the mixing layer of the j e t . There the DRI has the value 5.19x10-* N*/m2 which i s a factor of six greater than the value corresponding to h=0, The i n t e g r a l of the DRI d i s t r i b u t i o n was separated into the contribution from the potential core and the contribution from the mixing layer. The contribution from the mixing layer was found to be 6i3 times that from the potential core, The r a t i o of former to l a t e r was 8 db, indicating that the SPL developed by the portion of the model i n the mixing layer i s 8 db more intense than that developed by the portion i n the poten t i a l core.. However, t h i s r e s u l t i s dependent upon the Reynolds number and the streamwise location of the cylinder model; about two third s of the model i s in the mixing layer assuming the conical spreading angle of the l a t t e r i s 25°. Conseguently, the 8 db figure i s not universal. 50 Sec.6.4 Integral Closure; for the Potential Core T r i a l , Origin at O(xJ=0, x!,=0, x^ilD) The SPL generated by the surface exclusively was predicted from the integrated DBI d i s t r i b u t i o n using Simpsons rule (see Sec,5), The results were, SPL = (71.7) db SPL P R E D = _[69.31 db SPL = (67.0) db The three SPL values correspond to the maximum, best, and minimum SPL; most of t h i s spread arises from graphical error associated with the slope of the cross c o r r e l a t i o n functions (see Sec.5). Without t h i s e r r c r the predicted SPL would be, SPL = (6 9,3±.8) db. The o v e r a l l SPL generated by the surface, j e t and wake was measured d i r e c t l y . The resu l t was, SPL = (70. 1±.5) db This value l i e s within the range of SPL P P i t D. The SPL generated by the surface exclusively was also estimated from the above o v e r a l l SPL by correcting for the residual j e t and wake noise (see Sec.5) giving a value of, 51 SPL = (68.5) db In the estimate the j e t noise correction was ,5 db (see Appendix D) . Turbulence i n t e n s i t i e s i n the wake were not available for the estimate, so some reasonable values were assumed (u R m s= 0"o/2). The acoustical conversion e f f i c i e n c y f o r the wake was taken to be the order of 5058, The resulting wake noise correction was approximately 1 db. Agreement between the predicted.and corrected SPLs implies that i n t e g r a l closure i s s a t i s f i e d within an acceptable degree of uncertainty. The IEC have published standards for precision sound l e v e l meters which w i l l measure continuous sounds. * A precision sound l e v e l meter must measure the f r e e - f i e l d sound pressure l e v e l with an accuracy of ±1 db under s p e c i f i e d reference conditions simulating normal use'. Hence, the difference between the predicted SPL and the estimated SPL i s l e s s than the uncertainty that i s associated with a precision SPL. Sec; 6.5 The Circumferential P r o f i l e of the DEI f o r the Mixing Region T r i a l , Origin at O(x» = 0, x«=0, x'=5D) The circumferential variation of the DRI around the cylinder model i s plotted in Fig. 10a f o r the cross section of the model i n the mixing region of the j e t , with the streamwise coordinate of 0 set at x|=5D. The shape of t h i s p r o f i l e exhibits a s i g n i f i c a n t deviation from a true cos 2 <f> shape. .The DEI reaches a peak value at a point s l i g h t l y around the front side of the cylinder model, 52 and also at a point s l i g h t l y around the backside of the model r e l a t i v e to the approaching flow. Moreover, the t o t a l area enclosed by t h i s p r o f i l e i s twelve times that enclosed by the circumferential p r o f i l e for the potential core t r i a l * The existence of the forward peak at <f>- -25° indicates that turbulence in the incident flow i s important i n the generation of e f f i c i e n t dipole radiators. The force fluctuations are influenced by the factors of turbulence i n t e n s i t y i n the incident flow, freguency, surface geometry, and mean velocity gradient, Apparently, the force fluctuations o f f e r the most e f f i c i e n t dipole radiators at the portion of the surface where <f>~ -25°, The rearward peak at 0-25° i s apparently due to the guasi periodic shedding of vortex structures i n t h i s region* The marked deviation of the circumferential p r o f i l e , for the mixing region t r i a l , from the cos 2 <f> shape implies that f l u c t u a t i n g l i f t / d r a g forces on the cylinder model are probably responsible f o r a s i g n i f i c a n t contribution to the f a r f i e l d SPL at x1 = (E=3,00 m, <j>=0°f h=0) bearing i n mind that the drag component on the model does net make a contribution to the far f i e l d radiation in the d i r e c t i o n of x-i, The said p r o f i l e also exhibits a small degree of asymmetry* The lack of symmetry i s believed to be due to a s l i g h t off-centering of the cylinder model with respect to the j e t axis. Proper alignment of the probe was d i f f i c u l t to achieve at the xJ=5D station. The area enclosed by a circumferential p r o f i l e can be used as a prediction of the SPL that would be generated by unit length of the cylinder model. The circumferential p r o f i l e depicted in Fig,9a was integrated along with that depicted i n Fig,10a, The r a t i o of 53 the i n t e g r a l s (expressed i n decibels) was 10 db. We may conclude i n the case of t h i s j e t nozzle and t h i s cylinder model that a turbulent flow generates 10 db more dipole noise than a laminar flow when these flows are obstructed with a c y l i n d r i c a l s trut and when the Reynolds numbers (based on nozzle diameter and cylinder diameter) and flow Mach number are those given i n Sec.1. Sec.6,6 The Spanwise P r o f i l e of the OBI for the Mixing Begion T r i a l , Origin at O(xf=0, x»=0, x'=5D) The spanwise variation of the DBI along the l a t e r a l edge of the cylinder model i s plotted i n Fig.10b. The spanwise p r o f i l e i s weighted by the geometric factor x\n- i y . ) / x - 1 (see Fig*2b). Along the l a t e r a l edge of the cylinder model the DBI reaches a maximum value at the point about 1.1 cm off the jet axis. There the DBI i s 9.54 N*/m2 which i s greater than the DBI for the mixing layer by a factor of aicut two, apparently (see Sec.6*3). On the other hand, the maximum value of the velocity fluctuation quoted fo r the mixing region (see Fig.3a) i s s l i g h t l y l e s s than that which was quoted f o r the mixing layer (see Sec.6.0.0 & 6.0.1), I t i s possible that the peak value of the DEI for the mixing layer i s actually somewhat greater than that given i n Sec,6.3. The peak value was d i f f i c u l t to ascertain since i t occurs within a small range of the mixing layer* I t i s also possible that since the eddies i n the mixing region have a larger c o r r e l a t i o n volume, they form a more coherent d i s t r i b u t i o n of dipole sources over the surface of the cylinder model than that formed by the eddies i n the mixing region* The more coherent d i s t r i b u t i o n could radiate 54 more e f f i c i e n t l y which would boost the peak i n the spanwise p r o f i l e of the DEI for the mixing region r e l a t i v e to the mixing layer peak found i n the spanwise p r o f i l e of the DEI for the potential core, Sec* 6.7 Integral Closure; for the Mixing Eegion T r i a l , Origin at 0(x»=0, x«=0, x^=5D) The SPL generated by the surface exclusively was predicted from the integrated DEI d i s t r i b u t i o n using Simpsons rule (see Sec,5 for an explaination of how t h i s integration was performed). The r e s u l t s were, SPL = (71.7) db SPL P R t 0 = (71,33) db SPL = (67.0) db The three SPL values correspond to the maximum, best, and minimum SPL; most of t h i s spread arises from graphical error associated with the slope of the cross c o r r e l a t i o n functions (see Sec,5). Without t h i s error the predicted SPL would be, SPL = (7-1. 33**05) db, The o v e r a l l SPL generated by the surface, jet and wake was measured d i r e c t l y . The r e s u l t was. SPL = (74. 2±, 5) db 55 This value l i e s within the range of SPL P R E D* The SPL generated by the surface exclusively was also estimated from the above o v e r a l l SPL by correcting for the jet and wake noise (see Sec.5) giving a value of, SPL = (73.6) db In t h i s estimate the j e t noise correction was .5 db* Turbulence i n t e n s i t i e s i n the wake were not available for the estimate, so some reasonable values were assumed (u R M S= U c/2). The acoustical conversion e f f i c i e n c y f o r the wake was taken to be the order of 50%. The r e s u l t i n g wake noise correction was about 1 db. Agreement between the predicted and corrected SPLs i s implies that i n t e g r a l closure i s s a t i s f i e d with an acceptable degree of uncertainty. In Sec.6.5 i t was stated that there i s a 10 db difference between the SPL for unit length of the strut i n the potential core and that for unit length of the strut i n the mixing region. This figure may appear to be i n contradiction with the difference i n the o v e r a l l SPLs for the two t r i a l s which i s 4.1 db. In fa c t , t h i s i s not the case because the o v e r a l l SPL for the potential core t r i a l includes the contribution from the mixing layer dipoles (see Sec.6.3). When t h i s contribution i s properly deducted from the o v e r a l l SPL for the potential core t r i a l , the difference between the o v e r a l l SPLs becomes 10 db. 56 CONCLUSIONS & RECOMMENDATIONS Conclusions Integral closure was obtained for each of the two t r i a l s a f t e r making the wake noise correction of 1 db. After t h i s correction was made, the difference between the predicted SPL and the corrected SPL was .8 db for the potential c o r e . t r i a l and 2.3 db for the mixing region t r i a l . Integral closure i s found in both cases since these differences are within the combined uncertainty of tJie predicted and estimated SPL's, which was 2.8 db. Integral closure was obtained i n t h i s way with an •incomplete' DRI d i s t r i b u t i o n ; the integration scheme used the circumferential p r o f i l e f o r h=0 and and the spanwise p r o f i l e f o r 0=0 0,. More comprehensive representations do not lead to any s i g n i f i c a n t change i n the predicted SPL, at least f o r the present flow configuration. In f a c t the predicted SPL for the potential core t r i a l becomes 69 db instead of 69,3 db when the circumferential p r o f i l e f o r the mixing layer i s introduced into the integration scheme. Diagnostic information on the mechanisms of noise generation can be obtained from the p r o f i l e s . The following diagnosis was given for the potential core t r i a l : 1) The portion of the cylinder model inside the potential core of the j e t exerts f l u c t u a t i n g forces on the f l u i d i n reaction to a fl u c t u a t i n g l i f t force, The l a t e r arises from the quasi-periodic shedding of vortices near the l a t e r a l edge of the cylinder model. 2) The dipole sources that are formed by these reaction forces Sec. 7 Sec.7, 1 57 become the most e f f i c i e n t radiators of sound at a point near 0-15°, probably where the vortices are f i r s t formed. The portion of the cylinder model i n the mixing layer also exerts reaction forces on the f l u i d . These forces arise from turbulence i n the incident flow and vortex shedding, The rearward peak i n the DEI (at 0-15°) should gradually diminish along with the mean ve l o c i t y (U) and the Strouhal frequency as the cylinder model i s traversed i n the h d i r e c t i o n . The value of the DEI along the l a t e r a l edge of the cylinder model reaches a maximum at a point s l i g h t l y beyond the mid-point of the mixing layer (h/d=1.9), pin-pointing the loc a t i o n of the most e f f i c i e n t dipole radiators for the potential core t r i a l . This i s also where the turbulence intensity and c h a r a c t e r i s t i c freguencies thereof probably reach t h e i r respective peak values, The portion of the cylinder model i n the mixing layer develops a far f i e l d SPL which i s at least 8 db more intense than that developed by the portion of the cylinder model inside the potential core (at these Eeynolds numbers and and flow Mach number and when the cylinder model i s stationed one nozzle diameter downstream of the exit plane)*. In f a c t the far f i e l d SPL developed by the mixing layer portion i s evidently 68.4 db* I t combines with the far f i e l d SPL developed by the potential core portion which i s 60*4 db to y i e l d the o v e r a l l SPL of 69.3 db. The following diagnosis was given for the mixing region 58 t r i a l : 1) Once again the cylinder model exerts fluctuating forces on the f l u i d i n reaction to turbulence i n the incident flow and vortex shedding from the model. 2) The rearward peak i n the DRI (at 0-25°) should gradually diminish along with the mean vel o c i t y (0) and the Strouhal freguency as the cylinder model i s traversed i n the r a d i a l (h) d i r e c t i o n . 3) The combined e f f e c t s of turbulence i n t e n s i t y , turbulence scale, surface geometry and mean velocity gradient are such as to y i e l d the most e f f i c i e n t dipole sources in the v i c i n i t y of <t>- -25°, 4) The value of the DRI along the l a t e r a l edge of the cylinder model reaches a maximum value at the point (h/d=, 87), where the turbulence in t e n s i t y i n the incident flow should also be a maximum. 5) The far f i e l d SPL developed by unit length of cylinder model in the mixing region (x^SD) i s 10 db more intense than that developed by unit length i n the potential core at these Reynolds numbers and flow Mach number. In conclusion, the Causality Correlation Technigue provided an e f f i c i e n t means i n terms of man-hours f o r obtaining diagnostic information on the mechanisms of noise generation. For the present flow configuration satisfactory i n t e g r a l closure was obtained from measurements made at a series of 24 circumferential points around the models midsection and 10 spanwise points along the l a t e r a l edge of the model* I t was not necessary to obtain a series of 59 circumferential p r o f i l e s at d i f f e r e n t spanwise locations to achieve i n t e g r a l closure. The L i g h t h i l l wave equation i s v e r i f i e d f o r the present Eeynolds numbers and flow Mach number* V e r i f i c a t i o n follows from i n t e g r a l closure and the uniqueness of the DBI. Sec* 7,. 1 Becommendations The alleqed wake noise should depend on the e x i t Mach number. L i g h t h i l l 2 has shown that the surface (dipole) noise (p'p') should be proportional to the sixth power of M whereas the volume generated (guadrupole) noise should depend on the eighth power of M, Conseguently, the difference between the predicted SPL and the o v e r a l l SPL (less the. j e t noise) should increase with the sguare of the e x i t Mach number as the jet and wake noise become more dominant. An experiment should be devised to check for t h i s Mach number e f f e c t . Additional experiments could be devised where the Causality Correlation Technigue i s used to measure the SPL f o r unit length of strut as a function of Eeynolds number. The said technigue could also be employed i n fan noise transmission studies as was outlined i n the Foreword. 60 Sec.8 REFERENCES 1, . I,E, Siddon, 'New Correlation Method for Study of Flow Noise', Presented to The 7 International Congress on Acoustics. Budapest (Aug. 1971) . 2, M>J. L i g h t h i l l , 'On Sound Generated Aerodynamically•, Proc. Roy. Soc, A, 211, 564 (part 1 ) 1952. 3, Robert Rackl, 'Two Causality Correlation Techniques Applied to Jet Noise', PhD Thesis for The Department of Mechanical Engineering 0\.B,C. (Apr- 1973). 4, T.E. Siddon, 'Surface Dipole Strength* by Cross Correlation Method', J, of Acoustical Soc. of America vol 53 No. 2 (1973). 5, Lennox John Leggat, 'Experimental Investigations of On-axis Discrete Frequency Fan Noise', M-A.Sc. Thesis for The Department of Mechanical Engineering U.B.C. (oct> 1973). 6, T.E,> Siddon, L. Hoglund, 'Investigations of Nozzle Lip Noise Generation by a Round Coannular Jet', Department of Mechanical Engineering O.B.C. (1977). 7, J>A. . Stratton, 'Electromagnetic Theory', New York: McGraw-H i l l (1941). 61 8. N, Curie, 'The Influence of Solid Boundaries Upon Aerodynamic Sound 1, Proc. Roy. Soc. London Ser. A vol 231 (1955). 9. M,J, L i g h t h i l l , 'The Bakerian Lecture 1961 Sound Generated Aerodynamically', Proc. Boy. Soc. London Ser. A vol* 267 (1962). 10, H.S. Bibner, 'The Noise Of A i r c r a f t ' , Proc. ICAS, 4 Annual Congress, Paris (1964) (Spartan Books, MacMillan Ltd.), pp. 13-71 11. A.A. Townsend, 'The Structure of Turbulent Shear Flow', (Cambridge 0. P., New York, 1956)* 12. A.J, Beynolds, 'Turbulent Flows in Engineering', (John Wiley & Sons Ltd., 1974, pp,-9). 13, A* Eoshko, 'Structure of Turbulent Shear Flows : A New Look', AIAA Journal vol 14, No. 10 (1976). 14. B.M. Nayar, T,E. Siddon, W.T. Chu, 'Properties of Turbulence in the Transition Begion of a Bound Jet', UTIAS Tech Note No. 131 (Jan. 1964). 15. H.K, Lee , 'Correlation Of Noise And Flow Of A Jet', OTIAS Bep, 168 (1971). 62 16. M.E. . Munroe, 'Measure and Integration', (addison -Wesley Publishing Company, 1971). 17. H, Beichardt, 'On a New Theory Of Free Turbulence', JiE , a.E,S. 47, Pg, 167, 1943, TEANS Z.a.M.M, 21_i No, 5 1941. 18. J.C, Laurence, 1956 'Intensity, Scale and Spectra Of Turbulence In Mixing Eegion Of Free Subsonic Jet', N, a . C.A Sep, No. 1292. Shown in FIGURE 2.a & 2.b: 0 i s the o r i g i n of the c y l i n d r i c coordinates (E, f, h) and 0' i s the o r i g i n of the cartesian coordinates (xj, x\ ,xj) . 64 FIGURE 2.b: C y l i n d e r Model S t a t i o n e d a t M i x i n g Region, O r i g i n a t 0(xi=0,.x_=0, X3=5D) U Q p o t e n t i a l c o r e ^ ^ ^ T H » m i x i n g r e g i o n B B g ^ d * X T ' " > I * ' n fl 1 0 V e r d e v e l o p e i x'=0 x> 4D~~"~ _____ x|=8D ~~ FIGURE 3.a: S t a t i o n a r y C o l d J e t ( r e p r o d u c e d L i g h t h i l l 2 ) Plane mixing layer between (wo streams with velocities V • and (.',, densities p, and p,. Mixing layer between helium and nitrogen V'2I'Vt = 0.38; p2lpi = 7;p,LlL/iii = I.2, 0.6 and 0.3x10s, respectively, from top to bottom. (L is the width of the picture.) 4 0 0 K ^ Thn-rt-M O O . I 0.2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 Distribution of normalized eddy spacings. (Total number of measurements = 3622; total number of eddy pairs ± 100.) FIGURE 3.b: The M i x i n g o f 2-D J e t s ( r e p r i n t e d R o s h k o 1 3 ) Re = 6. 3 x 10 4 FIGURE 4: Flow P a s t a C i r c u l a r C y l i n d e r a t Reynolds Number 6.3 x 1/4 i n c h p r o b e A s s e m b l y B S K M i c r o p h o n e C a r t r i d g e T y p e 4 1 3 5 69 1/2 BSK C o n d e n s e r M i c r o p h o n e (Type 4 1 3 3 ) BSK M e a s u r i n g A m p l i f i e r ( T y p e 2 6 0 6 ) C y l i n d e r M o d e l BSK Measuring Amplif ier SAICOR Honeywell S A I - 4 3 A Corre la t ion 8 Probabi l i ty Ana lyser C o r r e l a t i o n F u n c t i o n .FIGURE 5.b: B l o c k Diagram o f C o r r e l a t i o n A p p a r a t u s 70 FIGURE 6.a: Frequency Response and Phase C h a r a c t e r i s t i c s o f 1/4" Probe FIGURE 6.b: Frequency Response and Phase C h a r a c t e r i s t i c s o f 1/8" Probe S P L db u n f i l t e r e d . I it 8 H I I IT |c h • A I 3 0 1 2 0 I I 0 _ : - H -too ULiz ± t b a c k g r o u n d c h • A 50 100 200 500 1000 2000 5000 10000 20000 f r e q u e n c y h z E X I T V E L O C I T Y U . = ( 7 2 ) m B - ' S T R O U H A L F R E Q U E N C Y f 8 = (I • I 3 ) k h i FIGURE 7.a: Spectrum o f S u r f a c e P r e s s u r e f o r P o t e n t i a l Core, O r i g i n a t 0(x_=0, x_=0, x.=lD), y.=(R=d/2, 0 = 0 ° , h=0) 73 E X I T V E L O C I T Y U . = (I 0 0 ) m 8-FIGURE 7.b: Spectrum o f S u r f a c e P r e s s u r e f o r M i x i n g L a y e r , O r i g i n a t 0 ( x l = 0 , x 2=0, x £ = l D ) , y.=(R=d/2, 0 = 0 ° , h=1.91xl0-' 2m) 74 EXIT VELOCITY U 0 =(72)ms-FIGURE 7.c: Spectrum o f Far F i e l d A c o u s t i c a l P r e s s u r e , O r i g i n a t 0(x_=0, x_=0, x^=lD), x!=(xi=0, xi=-3.00 m, x.=0) 75 FIGURE 7.d: Spectrum o f S u r f a c e P r e s s u r e f o r M i x i n g Region, O r i g i n a t 0(x_=0,x_=0, x^=5D), y;=(R=d/2, 0=0°, h=0) 76 EXIT VELOCITY Uo = ( 7 2 ) m s - ' » FIGURE 7.e: Spectrum o f Far F i e l d A c o u s t i c a l P r e s s u r e , O r i g i n a t 0(xi=0,. xi=0, x 3=5D), x! = (xl=0, x^=-3.00 m, x^=0) 77 b) FIGURE 8 .a: S l o p e o f a Sample 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 c a l e : I c m = I cm FIGURE 8.b: A Sample A u t o c o r r e l a t i o n F u n c t i o n o f S u r f a c e P r e s s u r e p'p' ( x i f T) s c a l e - I cm = I c m FIGURE 8.c: A u t o c o r r e l a t i o n F u n c t i o n o f Far F i e l d A c o u s t i c a l P r e s s u r e The far f i e l d microphone i s held at the point x«=(x»=0, x[=-3.00 m, x«=0) on Figs.9a to 9d i n c l u s i v e and on Figs.10a and 10b. 81 FIGURE 9.b: C i r c u m f e r e n t i a l P r o f i l e o f Cross C o r r e l a t i o n F u n c t i o n ; P o t e n t i a l Core T r i a l , O r i g i n a t 0(xi=0, x_=0, x.=lD),. .y. = (R=d/2, 0=0, h=o) FIGURE 9.c: C i r c u m f e r e n t i a l p r o f i l e o f DRI; P o t e n t i a l Core T r i a l O r i g i n a t 0(x_=0, x_=0, x . = l D ) , y f =(R=d/2, 0=0, h = 1 . 9 1 x l 0 - 2 m) M ( X j . y , ) cole- I cm = • 5 M m 2 10 2 0 3 0 h s c a l e : I c m = 2 mm FIGURE 9.d: Spanwise P r o f i l e o f DRI; P o t e n t i a l Core T r i a l , O r i g i n a t 0(x' = 0, x i = 0, x i y. = (R = d/2, 0 = 0 ° , h = h) 1 6 c m FIGURE 10.a: C i r c u m f e r e n t i a l P r o f i l e o f DRI; M i x i n g Region T r i a l , O r i g i n a t 0(xj = 0 , x 2=0, Xa=5D), y i=(R=d/2, 0=0, h=0) 86 d p p ( x j f y,) dS sca le . I cm = 5 N 8 J 6 J 2 J - „ s c a l e : I cm FIGURE 10.b: Spanwise P r o f i l e o f DRI; M i x i n g Region T r i a l , O r i g i n a t 0 ( x i = 0 , x 2=0, x 3 = 5 D ) , y . = (R=d/2, (4=0°, h=h) 87 2-5 r 2 0 -D 15 -10 0-5 0 1-4 r _ f 10 -0 6 a) RADIAL VARIATION OF INTEGRAL TIME SCALE 8 10 12 y/R G x /D • 4-5/ai , * 301 rad./sec. • o • x/D > 6 0 / O J < • 228 rad./sec. • x/D B 7-S/6j f * 169 rad./sec. b) TYPICAL EDDY FREQUENCY I i I •8 10 _ i 12 y/R FIGURE 1 1 : Time S c a l e s and Eddy F r e q u e n c i e s ( r e p r i n t e d Nayar, Siddon & Chulh) 88 Sec.10 APPENDICES Appendix A± Derivation of The L i g h t h i l l Wave Equation The continuity equation i s , a£ + a(puj ) =0 J_ at dy. The Eeynolds form of the momentum equation may be derived from the better known form by the addition of a multiple of the continuity equation. The re s u l t i s , a ( p u ; ) + a f p u j u , + p,j ) = o j£ S t a y . A nonhomogeneous wave equation i s obtained by forming the p a r t i a l of X with respect to variable t and t h e . p a r t i a l of with respect to variable y, and combining the two r e s u l t s . Introducing the c e l e r i t y of sound denoted by c ^ f i n a l l y y i e l d s a wave equation. S t 2 ay? e y i 3 Y j Here T\t = pu^ uj + p,j - c„ 2 p£jj . Nonlinear source terms are retained i n the nonhomogeneous wave equation. 89 Appendix B__ Simp l i f i c a t i o n of The General Solution of The L i g h t h i l l Wave Eguation f o r E i g i d Surfaces i n Quiet Air Flows S i m p l i f i c a t i o n of Eg,3,1 i s possible for the case of an i n f l e x i b l e surface that i s fixed i n a •guiet 1 j e t , Provided that the Mach number of the jet i s s u f f i c i e n t l y small, the f r a c t i o n of noise at the observation point x, that originated from a surface located within an a i r j e t i s known to dominate the f r a c t i o n of noise that originated from the acoust i c a l l y active volume of the j e t . . L i g h t h i l l 9 investigated t h i s Mach number ef f e c t and demonstrated that the surface (dipole) noise should be proportional to the sixth power of the jet Mach number whereas the volume generated (guadrupole) noise should be proportional to the eighth power of M. Hence, for small Mach number (M < . 3 ) the solution may be approximated by the dipole i n t e g r a l term alone, p» (x, ,t) *____( _ p« Sij " T,-J ] dS( Y l) Eq.B.la dx,- J s 4irr The isentropic r e l a t i o n , p« (x, ,t) = (9-p»)c„2 Eg.B.lb was introduced into Eg,B.1a ( l e f t hand side), This introduction i s permissible because the acoustic f i e l d i s usually i s e n t r o p i c for subsonic flows. The hydrostatic pressure (p s) and shear stress (t,j) at the surface were replaced by t h e i r f luctuating parts (p£ S 90 TTij respectively) ; the D.C. Components do not contribute to the radiation. For subsonic flows the surface pressure fluctuations (p') are greater than the shear stress f l u c t u a t i o n ( ) at the surface by a factor of 6. The root mean square pressure on a f l a t surface under a f u l l y developed turbulent boundary layer has been measured to be about 0*6% of the free stream dynamic pressure (1/2pUf) for subsonic flows over a wide range of Reynolds numbers1o. By comparison, the skin f r i c t i o n factor a r i s i n g from shear stress fluctuations at the surface was found to be less than 0.1% of the free stream dynamic p r e s s u r e 1 1 . Therefore the removal of from Eg,B.1a i s j u s t i f i e d since the error introduced i s , at most, one part i n s i x . accordingly Eg.B.la becomes, p« (x, ,t) - -_b [ nj f p« £ij ,] dS (yj ) Eq.B.2 2>x, J s 4irr The solution i n t e g r a l , may be further s i m p l i f i e d after imposing the far f i e l d approximations. The path length of sound t r a v e l (r) i s replaced by x i n the denominator provided the point X j i s many c h a r a c t e r i s t i c source dimensions away from S, Moreover, the s p a t i a l divergence i n Eq.B.2 i s equivalent to a p a r t i a l time derivative multiplied by the r e c i p r o c a l of the speed of sound i f the point x, i s also many c h a r a c t e r i s t i c wavelengths away from S, Applying these far f i e l d approximations to Eg,B,2..yields, p» (x n,t) * xi [ a,' j d_pj 1 dSCy,) Eq.3,*4 4 T T X 2 C O O J s [ d f [ £=t-r/Co<> Even though the far f i e l d conditions are imposed, the exact 91 value of the Kirchhoff retarded time must be used i n Eg.3.4 in order to account for interference at x v. Consequently, the exact path length (r) of sound t r a v e l i s used i n the expression for the Kirchhoff retarded time rather than the approximate path length (x). The v a l i d i t y of the far f i e l d approximations as well as the s e n s i t i v i t y of the integrand to changes i n Kirchhoff retarded time are now examined* The path length r may be approximated by x in the denominator of Eq,B.2 provided the observation point i s i n the geometric far f i e l d * In addition, the s p a t i a l gradient i n Eq, B.2 i s equivalent to a time derivative multiplied by the r e c i p r o c a l of the speed of sound provided that the observation point i s also i n the acoustical far f i e l d . Proof: Letting r = I x j - y j l and i f f T = f ^ y - j t ) i s a vector we claim that, ____ [ f; ( Y l ,£) i] dS (yj ) ax-i r Beginning with the l e f t hand side of Eg, B, 3,3/axi i s passed under the i n t e g r a l sign, Applying the product rule the integrand becomes, __L £ f i (Yi .1) 3 = ____ J L £ f i J + _! J S L U f i J) Eg.B,4a axi r sx-j r r ax; X2C 0 at dS(y-) Eq,B.3 t=t-r/c 0 O 92 Applying the chain rule of d i f f e r e n t i a t i o n , the second term on the r i g h t hand side becomes, J_ _3_ [ f i ,] = J_ a r ax-j r (t - r / c M ) £f ,• (y, ,t-r/ C o o)} ( t - r a: <Lft-r | I Xj l Co, ) Eq.B.4b or more concisely, r 3x, r I a t ax. \ 1 c„ 1 Eq, B.5 The gradient of r i s required to simplify Eg.B,5 . It i s easy to show that a/ax-; (r) = (x,-y^)/r by using the chain rule* However, in the geometric far f i e l d where |y, | < < | x i l , we have (x^y,) -Xi and r - x. In t h i s case the gradient of r i s simply, _d_ (r) = jo, dX, X Eq. B.6 Combining Eqs.B*5S6 and Eq.B*4a y i e l d s . ax,- r X 3 x 2 c „ I a £ xi Eq,B.7 If the observation point i s also in the acoustical far f i e l d where A < < r , i t i s evident that the second term i n Eq,B.7 dominates the f i r s t term. 93 In f a c t , the second term i s larger than the f i r s t by the factor 2irc»x/X since, 3f; ] - wf, = 2TTCq> f{ Eq, B.8 Here w i s the angular freguency, Neglecting the f i r s t term which describes the near f i e l d , the integrand i n Eg, B. 3 becomes. B £ f i (Yi A) j. = -*i [ a f i Q, E,D Eg. B.8 t=t-r/c_> The path length r was not approximated by x i n the Kirchhoff retarded time (see Eg,3,4) due to the integrand's s e n s i t i v i t y towards changes i n the Kirchhoff retarded time. Within the integrand, the time dependent part of p| i s of the form Re {w Exp(iwt)) . A Taylor expansion of the time dependent part of p' i n a neighborhood €(epsilon) of x with t fixed y i e l d s , w Exp|iw|t -x+€ j| = w Exp (i»t) • w |-iw. j Exp (iwt) Eg, B.9 + w 1  ||w2f  j Exp (iwt) + ,,. Considering only the r e a l part of Eg.B,9 yi e l d s , EeJw Expfiw/t - __+_£]))= » cos(wt) • w /we\sin(wt) + .... I ( » c_ I)» I cj\ Eg. B. 10 Hence, the f i r s t correction term in Eg,B.10 may be omitted only i f 94 2lf€ « A Eq.B. 11 The above condition i s equivalent to sayinq that the source reqion i s 'compact'. 95 Appendix C; S i m p l i f i c a t i o n of Correlation Functions f o r Ergodic Random Variables Theorem 1: P' ( X i ,t) p« (x, ;,t«) = p«p« ( x , , T ) Eg.C.1 Pi (YT ( X i , f ) = p's ( Y l ) p» (x, ) (x) j Eq.C,2 Proof: The f i r s t formula i s t r i v i a l . F i r s t , the variables are stationary since they are ergodic* Hence, t h e i r s t a t i s t i c a l properties are invariant with respect to a translation i n time. In pa r t i c u l a r the autocorrelation function may be s h i f t e d by t* p f ( t )p«( t«) = p'( t+t*)p'( t«+t*) Eg.C*3 Let t = - t and define r = t ' - t i n the above, p 1 ( t ) p« ( t« ) = P' ( 0 ) p« ( r ) Q.E.D. Eq.C*4 Hence the autocorrelation function depends only on the elapsed time difference, To prove the second formula l e t r = t ' - t . We have, Pl( t )p'( f ) = _a_ p« ( t ) p« ( f ) Eq.C.5 a t = < S )p' ( t» ) at 96 = *S A. P?P' ( £ ) at at P| ( £ )P* ( f ) = - J ^ . P ' P ' ( £ ) J Q.E.D,. r =r+r/c„ 97 Appendix D: Sample Calculations of the DRI and SPL for the Potential Core T r i a l , Origin at 0(x' = 0, x_=0, X«=1D) Acoustical Parameters a) Speed of Sound Ceo = J pr-i T ± Coo/2 (e T ) = ,/ (1, 40) (2,867x102 j Kg-- o K) (299°K) ± c M/2 (1/299) c«» = (346.4 ± .61 ms-__ ii) Kirchhoff Delay Time (r) corresponding to r=|x;-y. |; x\= (R= 3.003 m, <^=0Of h=0) and y-j = (R=1.27cm, <ji = 0°, h=0),r=0 TZ = r/Cco ± (e r+e C o o) = (2.997m /346, 4 ms-i) ± (.005/2.997 + .6/346,5) __. 5 (8.65 ± ,02) ms 98 Sample Calculation of DEI a) Minimum Value of Slope of Cross Correlation C o e f f i c i e n t for the Aforementioned Values of x., & Y l [ a / d r C ( - c ) j] ( x i r Y l ) = d c / {Jd Ad b d.j x=r/c_ = -20. 8/{/(49, 2) (40,4) (3.52x10-* s) j L a Zs£ C J r l J Jx. ____, 1 = (-1.33x103 s z H b) Best Value of Slope of Cross Correlation Coefficient for the Aforementioned Values of x-j & y^  [ a/ar C(r) >] (x7 ,y, ) = d./ {Jdftd6 d.j r=r/c e o , = -22.7/{ J (49,2) (40.4) (3, 27x10-* s) } L& Z s i t C i U J _ _ _ . i l = j-1.56x103 s z i l c) Maximum value of Slope of Cross Correlation C o e f f i c i e n t for the Aforementioned Values of x; & y^  LI Z f l i C j r l J i x ^ l = i-1.__6xlQ_! s r i L d) Minimum Value of DRI f o r the Aforementioned Values of x} & Yi dp'p'/dS (x, ,y_ ) = -cos(^) /{4TTXCCO} _ a / d r C ( r ) J p 2 £ F x=r/c 0 0 antilogy { SPL (x^ ) +SPL (y,) }/20 The best value and the maximum value are the same since the point (r, £ C(t) ]) i s an i n f l e c t i o n point (see Sec. 5 & Fig,8a), 99 = -cos(0<>)/{4TT(3.002 m) (347,0 ms-i)_ (-1.33x103 s-i) (20.0x10-6 Nm-2) 2 antilog,. {135.3 + 69,6}/20 dp'p'/dS ix 1 J L__ 11 = (7. 14x10~* N 2 m - 6 ) e) Best Value of DRI f o r the Aforementioned Values of x, & y. dp'p'/dS ( x , ^ ) = -cos(0) / £ 4 T r x C o o } [ a/_r C(t) fl . p« E F r=r/Coo antilog.o {SPL ( x , ) +SPL (y; ) } / 2 0 = -cos ( 0 ° ) / { 4 T T ( 3 . 0 0 3 m ) ( 3 4 6 . 4 ms"Mj ( - 1 , 56x103 s-i) ( 2 0 . 0 x 1 0 - 6 Nm-2) z antilog,o {135,8 + 7 0 . 1 ] / 2 0 dp'p'/dS i__i_t_.il = (9,41x10-* N__m____l f) Maximum Value of DRI f o r the Aforementioned Values of x, & y; dp'p'/dS (x1 ,y 11 = (9. 41x10-1 N 2 M - * ) 100 SPL Prediction (see See.5) a) Minimum SPL SPL ( X l) = 10 l o g l o [ p'p' ( X l) / p| e f } = 10 l o g l o { 2.006x10-3/(20x10-6)2 } SPL (x,) = (67.0) db b) Best SPL SPL (x,) =10 l o g t o { p'p« (x,) / p2R£f j = 10 log 1 0 { 3,410x10-3/(20x10-«)2 } SPL (x,) = (69.3) db c) Maximum SPL SPL ( x O = 10 l o g 1 0 { p'p' (x,) / P2 f t t F j = 10 log, 0 { 5.970x10-3/(20x10-6)2 } SPL J X i l = (71.7) db SPL Estimation (see Sec*5) a) Overall SPL (measured directly) SPL J X T I = (7 0, 1 ± ,51 db b) Overall Hean Square Pressure (from above) P!.MU(*I) = P!EP antilog, 0 { SPL (x,) /10 } = (20x10-6 Pa) 2 antilogy { 70.1 /10 } £&*MVJX;L = ^4^09x10^3 Pa£l c) Radiated Acoustic Pressure from the Basic Undisturbed Jet (measured directly) Pier ( X i ) = Pf e F a n t i l o g l o { SPL (x,) /10 } = (20x10-6 Pa)2 antilog 1 0 { 51.1 /10 } MET (X;) = (.05x10-3 p d 2 ) d) Radiated Acoustic Pressure from the Model's Wake (estimated) PlLe <XT) " * X; f | e ui (Y, ) U , (Yj l (t) ]• dV( Y i) 4TTX*C 2 t=t-r/c = 1/ {4TT(3, 00 m) 2 (346,4 ms-i) 23 (1-13 khz) 2 (1.17 kgm-3) (36 ms-*)2{2 (6,35x10-3 m) 2 ( 6 A 66x10-2m)} e) corrected SPL SPL = 10 log'° j-fe^  103 appendix E: Resolving Power of the Correlator Time Base The evaluation of the slope of the cross c o r r e l a t i o n c o e f f i c i e n t i s almost independent of the amplifier gain s e t t i n g because C ( r ) i s dimensionless. Complete independence i s possible i f the SAICOB SAI 43A input attenuators i s adjusted in the following manner: Step 1: Set the SAICOB SAI 43A in the PDF mode and measure the probability density function for channels A and B. Step 2: Adjust the attenuators so that one or two standard deviations (6) of the PDF occupies the f u l l 400 bins of the analyser. The overload l i g h t of the channel should be flashing red i n d i c a t i n g that input peak voltages (i»e* i n excess of one or two 6 ) are being clipped. This ensures best use of the signal. Step 3: (optional) I f the input s i g n a l increases by x db on the measuring amplifier scale then, to compensate, increase the attenuator setting of the SAICOB SAI 43A by x db. This w i l l maintain a constant l e v e l of 104 signal clipping™ This procedure w i l l y i e l d a maximum resolution of the correlation functions. 

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