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The inlet vortex as a source of tonal rotor noise Leggat, Lennox John 1975

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THE INLET VORTEX AS A SOURCE OF TONAL ROTOR NOISE by LENNOX JOHN LEGGAT M.A. S c . , Univers i ty of B r i t i s h Columbia B. Eng. , Royal M i l i t a r y Col lege of Canada A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mechanical Engineering We accept th is thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1975 In presenting th i s thes is in pa r t i a l f u l f i lmen t of the requirements fo r an advanced degree at the Univers i ty of B r i t i s h Columbia, I agr that the L ibrary sha l l make i t f ree ly ava i lab le f o r reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head o f my Department or by h is representat ives. I t i s understood that pub l i ca t i on , in part or in whole, or the copying of t h i s thes is fo r f i nanc ia l gain s h a l l not be allowed without my wr i t t en permission. Department of Mechanical Engineering The Univers i ty of B r i t i s h Columbia Vancouver V6T 1W5 Date ^ W L - / * m-< i i ABSTRACT The thesis describes experimental and mathematical analyses of the noise resu l t i ng from the in te rac t ion of an ax ia l f low fan with a concentrated i n l e t vortex. A comparison of resu l ts from the two methods reveals the physical phenomena on which the d isc re te tone noise depends. Vort ices were generated with ha l f de l ta wings mounted in the i n l e t be l l mouth of an ax ia l flow fan. Their propert ies of vortex st rength, core rad ius , and core ax ia l ve loc i ty , d e f i c i t were measured in a wind tunnel . The fa r f i e l d spectrum of the fan noise was measured fo r various combinations of these three parameters, and for d i f fe ren t rad ia l pos i t ions of vortex entry. An inves t iga t ion of the o r i g i n and e f fec t on the sound leve ls of na tura l l y occurr ing i n l e t vor t ices was car r ied out using real time blade pressure measurements and a synchronous d isp lay system. The mathematical ana lys is comprised a l i nea r two dimensional aerodynamic model of a vortex passing through a fan. The character-i s t i c s of the blade loading were ca lcu la ted as a funct ion o f the span pos i t ion and c i rcumferent ia l angle. Levels of overa l l noise and of the ind iv idua l d iscrete tones were ca lcu la ted . The theore t ica l dependence of the noise' spectrum on the vortex st rength, core rad ius , and core ax ia l ve loc i t y d e f i c i t were determined. The va r ia t i on of sound pressure level with the rad ia l pos i t ion of vortex entry was also ca l cu la ted . 111 Comparison of the experimental and theoret ica l resu l t s shows that an a r t i f i c i a l l y generated concentrated i n l e t vortex increases the overa l l sound leve l by as much as eight decibels and some harmonics to a maximum of fourteen dec ibe ls , at a t yp ica l set of operat ing condi t ions. The vortex c i r c u l a t i o n has the greatest e f fec t on the tone leve l at the blade pass frequency while the ax ia l ve loc i t y d e f i c i t exerts more inf luence on the higher harmonics o f the spectrum. Local blade s t a l l caused by large values of core ax ia l ve loc i t y d e f i c i t i s a suspected contr ibutor to the observed pure tone noise at the higher harmonics. A na tura l l y occurr ing i n l e t vortex was tracked using the synchronous d isplay system. It was observed to be unsteady in i t s structure and to move about a preferred pos i t ion of entry in to the fan. I ts presence increased the overa l l f a r f i e l d fan noise by a minimum of three decibels and at the blade pass frequency by 4.5 dec ibe l . No change in sound leve l occurred at the four s i g n i f i c a n t harmonics of the blade pass frequency. i v TABLE OF CONTENTS' Sect ion Page / . . . 1. INTRODUCTION . . . . 1 1.1 Fan Noise Generation 2 1.2 The In le t D is to r t ion 4 1.3 Scope of Work . . . . . 5 2. BACKGROUND . . 7 2.1 In le t Turbulence . 7 2.2 In le t Dis tor t ions . . . . . . . . . . . . . . 8 2.3 Stretched Eddies and Vor t ices . . . . . . . . . . 11 2.4 The In le t and Tip Vortex . . . . . . . - . ' . . 16 2.5 Role of Present Work . 18 3. EXPERIMENTAL ANALYSIS . . . 20 3.1 Apparatus 20 3.1.1 Aeroacoustic F a c i l i t y 20 3.1.2 Fan Rig . . . . . . . . . . . . . . . 21 3.1.3 Vortex Generators . . . . . . . . . . 22 3.1.4 Blade Pressure Fluctuat ions 23 3.1.5 Acoust ic Analys is 24 3.1.6 Real Time Polar Pressure Display . . . 25 Section v Page 3.2 Determination of T r a i l i n g Vortex 26 Charac te r i s t i cs . i 3.2.1 Measurement of Core Radius and „ Vortex Strength . . 3.2.2 Vortex Ax ia l Flow D e f i c i t 29 3.2.3 Discussion . . . ." 3 ^ 3.3 Vortex/Rotor Interact ion Experiments . . . . 31 3.3.1 Dependence of Noise on Vortex Parameters . . . . . . . . . . . . . . 33 3.3.2 Dependence of Noise on Pos i t ion of Entry 3 7 3.3.3 Fluctuat ing Blade Pressures 3 8 3.4 Remarks on Experimental Analys is 42 4. MATHEMATICAL ANALYSIS • • • ' 4 3 4.1 General Method . . . . . . . . . . . . . . . 4 3 4.2 Mathematical Formalism 4 4 4.2.1 Fan Blade Geometry 4 4 4.2.2 Ve loc i ty F ie lds . . . . 4 5 4.2 .3 The Quasi Steady L i f t Equation . . . . 4 7 4.2.4 The Acoust ic Transfer Function . . . . 4 ^ 4.2.5 Inclusion of Sears ' Function 5 ^ 4.3 Results of the Ana ly t i c Study . . . . . . . . 52 4.3.1 Empirical Constants and Numerical Sol ution 53 4.3.2 V e r i f i c a t i o n of Model . . . . . . . . 5 4 Sect ion v i ' " Page 4.3.3 Var ia t ion of Sound Level with Vortex Entry Pos i t ion . . . . 57 4.3.4 Var ia t ion of Sound Level with Vortex Parameters . . 58 4.3.5 Magni f icat ion of Input Errors . . . . . 62 4.4 Remarks on Mathematical Ana lys is . . . . . . . 62 5. COMPARISON OF THEORY AND EXPERIMENT 6 4 5.1 Blade Loading 6 4 5.2 Far F i e l d Noise Spectra 66 5.3 Radius of Vortex Entry 69 5.4 Blade S t a l l . . . . . . . 70 6. CONCLUSIONS AND RECOMMENDATIONS • 7 3 6.1 Theoret ical Extensions . . . . 7 6 6.2 Experimental Extensions 7 7 REFERENCES 7 8 APPENDIX A - RADIATION SOLUTION AND SOURCE ANALYSIS BY CAUSALITY CORRELATION . . . . . . . . . . . . 82 Al - Relat ionship Between Surface Pressure and . Far F ie ld Sound . . . . . . . . . . . . . . 82 A2 - Causal i ty Formalism . 84 A3 - App l ica t ion to Per iod ic Corre la t ion Functions . 8 6 v i i Sect ion Page APPENDIX B - LOCATION OF THE VORTEX CORE 89 APPENDIX C - HALF DELTA WING VORTEX CHARACTERISTICS 91 APPENDIX D - DATA FROM VORTEX PARAMETER TESTS 92 APPENDIX E - PARAMETRIC ANALYSIS OF LIFT FLUCTUATION ON A BLADE . . 95 APPENDIX F - TABULATION OF HARMONIC AMPLITUDE DEPENDENCE . . . . . . . 98 v i i i LIST OF FIGURES Figure Page i 1. Lowson's Aero-acoust ic Transfer Function . . . . . . 100 2. Blade Relat ive Ve loc i ty Frequency Spectrum 100 3. Sears' Aerodynamic L i f t Transfer Function for fo r an A i r f o i l Encountering a Sinusoidal Gust . . . 101 4. Fan Blade Pressure Spectra 102 5. Pulse Input Proposed by Hanson 103 6. Comparison of Theory and Experiment (Hanson) . . . . 104 7. Spanwise Source Strength of 420 Hz Tone at 15% Chord 105 8. Le.verton and Tay lor 's Experiment . 106 9. The UBC Fan Noise F a c i l i t y . , 107 10. The Experimental Fan Apparatus . 108 11. The Experimental Half Delta Wings . . . . . . . . ' . " 1 0 9 12. The Method of Determining Vortex Strength . . . . . 109 13. The Instrumented Fan Blade . 110 14. The Frequency Response of the Two Telemetry Systems . H I 15. The Polar P lo t t e r Conf igurat ion . 112 16. Comparison of Experimental and Theoret ical Values of Vortex Tangential Ve loc i ty . . . 113 17. Comparison of Experimental and Theoret ical Values of Vortex Tangential Ve loc i ty 114 18. Vortex Strength Produced by Hal f Delta Wings at Various Angles of Attack . 115 Figure Pages 19. Var ia t ion of Core Radius of Vort ices Produced by Half Delta Wings . 116 20. Ax ia l Ve loc i ty D e f i c i t in Vortex Core . . . . . . . 117 21. Hal f Delta Wing Mounted in Fan 118 ' 22. E f fec t of Imposed Vortex on the Far F i e l d Noise Spectrum . 119 23. Col lapse of Overal l SPL Data onto Parameter Groupings 120 24. Experimental Dependence of Overal l SPL on Vortex Strength . . . . . 1?1 25. Experimental Dependence of 1680 Hz Harmonic. Level on Vortex Strength 122 26. Experimental Dependence of 420 Hz (Fundamental) Level on Vortex Strength 123 27. The Delta Wing Mounted on a F la t P late in the Be l l Mouth 124 28. Var ia t ion of SPL with Radial Pos i t ion of Vortex Entry (Experimental, Pos i t i ve C i r cu la t i on Sense). . 125 29. Var ia t ion of SPL with Radial Pos i t i on of Vortex Entry (Experimental, Negative C i r cu la t i on Sense). . 126 30. Blade Pressure Polar P lo ts 127 31. Sequential Blade Pressure Polar P lo ts 128 32. Sequential Blade Pressure Polar P lo ts (Clean Running F a n ) . . . . . . . 129 33A. Fan and Vortex Pos i t iona l Var iables 130 B.. Superposit ion of Ve loc i ty F i e l d . . . . . . . . . . 130 34. Var ia t ion of the Maximum Instantaneous Rate of Change of L i f t Over the Fan Blade 131 35. Blade Loading from the Ax ia l Ve loc i ty D e f i c i t and C i rcu la t i on of In le t Vortex . . . . . . . . . . 132 Figure 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. x Page Circumferent ial L i f t Var ia t ion at Two Blade Sections and Comparison with Experiment . . . . . 133 Var ia t ion of the Rate of Change of Blade Load Around the Fan Circumference ,134 Far F i e l d Pressure Signature Caused by Vortex Interact ion .135 Comparison of Theoret ical and Experimental Far F ie l d Fan Noise Spectra (Pos i t i ve C i r cu la t i on 126 Sense) Comparison of Theoret ical and Experimental Far F ie l d Fan Noise Spectra TNegative C i r cu la t i on Sense) .137 Var ia t ion of SPL with Radial Pos i t ion of Vortex Entry (Theore t ica l , Negative C i r cu la t i on Sense). . 138 Var ia t ion o f SPL with Radial Pos i t i on of Vortex Entry (Theore t i ca l , Pos i t i ve C i r cu la t i on Sense). . ,139 Change in 4^ Curves as Vortex i s Moved Rad ia l l y Outwards . a ? ; 140 Parametric Dependence of Vortex Interact ion Noise (Pos i t i ve C i r cu la t i on Sense, Overal l Level) ...,.141 Parametric Dependence of Vortex Interact ion Noise (Negative C i r cu la t i on Sense, Overal l Level) y!42 Parametric Dependence of Vortex Interact ion Noise (Negative C i r cu la t i on Sense, 420 Hz) . . . . ,.143 Parametric Dependence of Vortex Interact ion Noise (Pos i t i ve C i r cu la t i on Sense, 420 Hz) . . . . .144 Parametric Dependence of Vortex Interact ion Noise (Pos i t i ve C i r cu la t i on Sense, 1680 Hz) . . . .145 Parametric Dependence of Vortex Interact ion Noise (Negative C i r cu la t i on Sense, 1680 Hz) . . . 146 Circumferent ia l Var ia t ion of Sect ional L i f t and Pressure Fan Blade Sections ,147 NOTATION Four ier coe f f i c i en t s f number of fan blades Four ier coe f f i c i en ts coe f f i c ien ts generated by Sears ' funct ion speed of sound in a i r fan blade chord slope of blade chord var ia t ion funct ion period of revolut ion of fan (1/60 sec) loca l stress vector slope of blade twis t va r ia t i on funct ion f l a r e constant for ax ia l ve loc i t y d e f i c i t funct ion constant con t ro l l i ng radius of polar p l o t t e r reduced wave number (C^ • co/2U) 1 i f t force slope of ax ia l inf low ve loc i t y va r ia t ion funct ion harmonic number acoust ic pressure amplitude surface pressure amplitude acoust ic pressure surface pressure fan rad ia l var iab le fan hub radius fan t i p radius rad ia l pos i t ion of vortex entry distance from rotor to fa r f i e l d microphone vortex rad ia l var iab le vortex core radius surface element of area time since b i r th of vortex time ax ia l ve loc i t y at radius R ax ia l ve loc i t y at t i p radius ax ia l ve loc i t y d e f i c i t at vortex radius r maximum ax ia l ve loc i t y d e f i c i t normal surface ve loc i t y component of ve loc i t y vector ve loc i t y re la t i ve to fan blade vortex tangential ve loc i t y space co-ordinate ind ica t ing point of sound detect ion in the fa r f i e l d space co-ordinate used in source region steady angle of attack wing or blade steady angle of attack f luc tua t ing angle of attack e f fec t i ve blade twist angle, r e l a t i ve to plane rotat ion e f fec t i ve blade twist at rotor t i p x i i i r vortex s t reng th / c i r c u l a t i o n $ angle between tangential ve loc i t y of vortex and component normal to fan blade, (Figure 33A) (J) phase angle of f i l t e r e d cor re la t ion funct ion angle between re la t i ve and rotat ional v e l o c i t i e s p density of a i r E fan c i rcumferent ia l angle 6 angle between surface normal and fa r f i e l d d i rec t ion x time delay between two rea l i za t i ons of a f l uc tua t ing var iab le v kinematic v i s cos i t y w ro ta t ional f requency, (radians/sec) dC^/da aerodynamic l i f t t ransfer funct ion db sound pressure level (SPL) in decibels 7 „ ( S P L >ove ra l l 1 0 l o 9 ° V e r a 1 T 2 ^ db o v e r a " ( .0002 ubarr ~2~ T ^overal l x\ D = pressure of nth harmonic 2 P„ (SPL) n 10 log (.0002 yba r ) ^ ACKNOWLEDGEMENT The work described in the thesis was conducted under the supervis ion of Dr. T .E . Siddon. The author wishes to thank him for his advice and guidance in the research and fo r his cont inual optimism. Thanks go a lso to Dr. I .S. Gartshore fo r his comments on the structure of vor t ices and to Dr. H.G. Davies fo r ass is tance with the mathematical development. The able assistance of summer student, Mr. G. Watt was most appreciated. .. Without the exper t ise of the mechanical and e l e c t r i c a l technicians of the Department of Mechanical Engineering many aspects of the experiments could not have been car r ied out. The major port ion of th is research was funded under the Defence Research Board of Canada Grant No. 9611-03. Some support was received from the National Research Council of Canada under Grant No. A 7106. 1 1. INTRODUCTION Perhaps the most annoying, i f not the most un ive rsa l l y recognized sources of noise to one a n d a l l are the gas turbine engines which * power most of today's commercial a i r l i n e r s . The noise i s generated by unsteady aerodynamic phenomena in the i n l e t , combustor and j e t exhaust of the engine. In f i r s t and second generation turbo-je ts the main noise producing area was the j e t exhaust. Here intense f luc tua t ing shears developed in the mixing region between the exhaust j e t and the ambient a i r . The associated turbulence i s responsib le fo r the broad band rumble that character izes j e t no ise . With the in t roduct ion of the new high bypass ra t io turbofan engines, e s p e c i a l l y to be found on the "jumbo" j e t s , the exhaust v e l o c i t i e s have been reduced subs tan t i a l l y resu l t ing in less shear s t ress f l uc tua t i on in the mixing region and less noise. This reduction in exhaust noise has had the e f fec t of emphasizing sound rad ia t ion from another a rea ; the i n l e t fan . Under cer ta in operating condi t ions such as approach and tax i fan noise dominates the overa l l noise produced by the j e t engine. Fan noise i s not only of major concern in a i r c r a f t pro-puls ion technology, but i t also has been a top ic of much research per ta in ing to noise generation in ven t i l a t i on systems and coo l ing systems for in ternal combustion engines. Although the condi t ions under which fan rotors operate vary subs tan t i a l l y , t he i r noise producing mechanisms may be s i m i l a r ; fo r example, the e f fec t of a ser ies of s t ru ts downstream of a ven t i l a t i ng fan may 2 be s im i la r to the e f fec t of a downstream sta tor sect ion in a turbo-je t engine. 1.1 Fan Noise Generation r The noise generated aerodynamically by fans can be separated into d isc re te tone and broad band rad ia t i on . Often the tonal rad ia t ion tends to stand out over the broad band noise,dominating the annoyance fac tor associated with fan noise. These two types of noise are a resu l t of force f luc tuat ions on the fan rotor and s tator elements. The blade forces can be resolved into time varying l i f t and drag components, rad ia t ing noise normal and tangent ial to the blade elements respec t i ve ly . Under cer ta in condit ions such as high ax ia l f low Mach number or high rotor t i p Mach number, unsteady Reynolds' stresses associated with the combination of the potent ia l f low f i e l d of the rotor and a solenoidal ve loc i t y f i e l d produced by an i n l e t d i s t o r t i on may a lso be s i g n i f i c a n t ^ . Noise problems that plagued ear ly fan design.such as the in te rac t ion of s tator or i n l e t guide vane wakes with a downstream rotor , have been i den t i f i ed and analysed both mathematically and experimental ly. As a r e s u l t , modern designs minimize these e f fec ts through a set of permissible and non-permissible design c r i t e r i a . Yet despite e f fo r ts which have produced sweeping changes in the design of fans, intense pure tones pe rs i s t espec ia l l y in the on-ax is d i rec t ion where,in theory,there should be none. I t seems apparent that other phenomena are contr ibut ing to the d isc re te frequency peaks in the spectra of the fan noise, overr id ing the 2 c l a s s i c a l ro ta t iona l noise f i r s t described by ' «Gutm .. A comprehensive review of the state of the ar t re la t ing to 3 fan noise was published by B.D. Mugridge and C L . Morfey in 1971 . The concept of intake flow d i s to r t i ons was discussed in the 'sec t ion e n t i t l e d "Flow Dis tor t ions Responsible for Tone Generation'.1 The process i s explained as fo l lows : "Although adjacent blade rows are an obvious source of f low d i s t o r t i o n , they may not be the main cause of in te rac t ion tones. Potent ia l f low d i s to r t i ons can a r i se from asymmetry in the duct surrounding the f an ; f o r example, there may be a c i rcumferent ia l va r ia t ion in rotor t i p clearance and hence a per iod ic f l uc tua t ion in blade loading. Other possib le disturbances are cross flow,wakes from upstream bends or obstruct ions in the i n l e t duct, and streamwise vor t ices sucked into the fan from nearby so l i d sur faces. In a l l cases, e f f i c i e n t acoust ic rad ia t ion w i l l resu l t at blade passing frequency i f the c i rcumferent ia l wavelengths of the f low d is to r t i ons are s im i l a r to the rotor blade spacing. Low frequency modulation of the flow d i s to r t i on causes band spreading of the radiated tones, and in fac t most of the energy in what appears to be a d isc re te tone may be associated with unsteady ve loc i t y d i s to r t i ons which would not be detected by a mean ve loc i t y t raverse . " 4 1.2 The In let D is to r t ion During the past four years much in te res t has been di rected towards the subject of various types of i n l e t d i s to r t i ons in te rac t ing with a fan r o t o r V ' ^ ' ^ ' 6 ' ' ' ' ^ ' 9 , ^ 0 . Analyses have in general been i n i t i a t e d on the notion of randomly occurr ing turbulence being drawn past the ro to r , or a s p a t i a l l y f i xed ve loc i t y f i e l d anomaly being encountered by the rotor . These approaches have a common basis in that i n l e t d is to r t ions are represented in terms of a spat ia l sum of very many d isc re te frequency components, each with i t s own cha rac te r i s t i c spat ia l s t ruc ture . Experimental and ana ly t i c approaches to the problem have shown that , where ax ia l length scales of an i n l e t d i s to r t i on are s u f f i c i e n t l y long, an increase w i l l occur in d iscre te tone rad ia t ion at the blade pass frequency and i t s harmonics. In condit ions where s ing le rotors are operating in f ree a i r , turbulence 5 g with ax ia l length scales of up to 100 feet have been measured ' . Some have suggested that these stretched inf low d is to r t i ons may in 7 8 fac t be vor t ices ' ; e i ther produced by the elongation of i n l e t turbulence in the potent ia l sink flow of the fan i n l e t , or o r ig ina t ing at a s o l i d surface near the fan i n l e t , forming the well known 11 12 ground vortex ' . In the l a t t e r case, as pointed out by Siddon 7 8 and Leggat ' , the d i s to r t i on i s perhaps best modelled as a d isc re te potent ia l anomaly enter ing the i n l e t plane at a f i xed p o s i t i o n , ra ther than as a randomly occurr ing f i e l d of i n l e t v o r t i c i t y . L i t t l e i s known about the true nature of the ve loc i t y f i e l d associated with the elongated i n l e t eddy and even less about i t s e f fec t on fan rotor loading, and the harmonic leve ls of the resu l t i ng f a r f i e l d sound. This uncertainty i s complicated by the fact that vor t i ces are d i f f i c u l t to detect even when they are r e l a t i v e l y s ta t ionary . 1.3 Scope of Work This thes is comprises a theoret ica l and experimental ana lys is o f the i n l e t vor tex / ro tor in te rac t ion phenomenon. I t i s hoped that such a study w i l l increase our understanding of the physical processes invo lved, provide a means to predict noise resu l t i ng from vortex blade i n te rac t i on , and o f fe r ins ight into some of the unanswered questions put fo r th by other researchers in the f i e l d . Experimental ly, the problem has been explored by introducing known d is to r t i ons into an ax ia l f low fan . Concentrated vor t ices of known c i r c u l a t i o n , core rad ius , ax ia l ve loc i t y d e f i c i t and pos i t ion of entry are generated by a hal f de l ta wing. The resu l t i ng fa r f i e l d , on-axis sound pressure level i s measured for d i f fe ren t combinations of vortex parameters. A l so , real time polar p lots of blade pressure measurements (sychronized to the ro ta t iona l frequency of the fan) have provided information about the spa t ia l extent and length of na tu ra l l y occurr ing and a r t i f i c i a l l y imposed vo r t i ces . A l i nea r i zed aerodynamic theory has been developed to descr ibe the in te rac t ion of a Rankine-type vortex with a fan ro to r . Empirical parameters descr ib ing the geometry of the fan blade were included and fa r f i e l d on-axis noise spectra ca lcu la ted in terms of vortex parameters. This model has a lso provided information regarding the span-wise d i s t r i bu t i on of unsteady loading over each blade and around the circumference of the fan , as the vortex passes through the rotor d i s c . Both experimental and theore t ica l approaches w i l l be described in d e t a i l . Parametric resu l t s are discussed and , compared. L imi tat ions imposed by the idea l i zed model and the i r inf luence on the accuracy of predict ions are a lso considered. In sequence then, the thesis begins with a d iscuss ion of pert inent background mate r ia l , reviews the extent of previous research, and gives a descr ip t ion of the apparatus and experimental method employed. Then the mathematical model of ro tor /vor tex in te rac t ion is ou t l i ned . A presentation of resu l ts lends evidence to the i n l e t vortex theory and contr ibutes to the physical understanding of the problem. F i n a l l y there is a d iscuss ion of the extent of agreement between the theory and experiments with some comments about the l im i ta t i ons of the mathematical model. 7 2. BACKGROUND Before descr ib ing the analys is undertaken i t i s necessary to discuss the resu l ts of others who have conducted studies into the problem of inf low d i s to r t i on / ro to r i n te rac t i on . Major works in t h i s area w i l l be reviewed, from the i n i t i a l ana ly t i c studies to recent experimental invest igat ions which support the notion of stretched eddies or vor t ices in the inf low of ax ia l flow fans. 2.1 In le t Turbulence Mani was one of the f i r s t to attempt to model the in te rac t ion 4 of a turbulent ve loc i t y f i e l d with a fan rotor . His work was motivated by experimental f indings of So f r in and McCann 1 2 , and F i l l e u l ^ . Both reported on increase in sound level in fans with an increase in turbulence i n tens i t y . Mani postulated that the unsteady ve loc i t y f luc tuat ions produced by the turbulence being sucked into a fan would produce "non-stat ionary f luc tuat ions in angle of attack on the blade row resu l t i ng in unsteady forces [on the b lades ] , and noise r a d i a t i o n . " 4 He developed a two dimensional model of a fan ro tor in the presence of a turbulent in f low. For convenience he chose to model the inf low turbulence as homogeneous and i so t rop i c . His rotor model consisted of a row of f l a t plate a i r f o i l s at incidence. The ana lys is was car r ied out in the frequency domain in that the turbulence was t reated as a superposi t ion of harmonic shear waves. Sears' l i f t response 8 funct ion for an a i r f o i l encountering a s inusoidal gust was used to determine the unsteady blade forces resu l t ing from the inc ident shear waves. This funct ion predicts a diminishing l i f t coe f f i c i en t with increasing frequency or wave number. Among several s i g n i f i c a n t r e s u l t s , Mani found that peaks in the resu l t i ng fa r f i e l d noise spectra appeared at the blade passage frequency and i t s f i r s t harmonic mul t ip le whenever the ra t i o of the length scale of the turbulence to the transverse spacing between the blades exceeded 0:5. For values of th i s r a t i o less than 0.5 the spectra were broad band in nature; the d iscre te tone phenomena did not radiate cons t ruc t i ve ly . 2.2 In le t D is tor t ions Lowson was concerned with the est imation of noise produced when a rotor operated in the presence of a d iscre te inf low d i s t o r t i o n . His pa r t i cu la r in te res t was in the predic t ion of pure tone noise caused by the in te rac t ion of a compressor rotor with the wake of a s ta tor s t a g e 1 6 . Lowson star ted with a previously derived resu l t f o r sound radiated by a f luc tua t ing point source in a rb i t ra ry mot ion 1 ^. He then allowed th is point force funct ion to repeat pe r i od i ca l l y as would the force on a fan blade when ro ta t ing cont inua l ly through a steady pattern of inf low ve loc i t y d i s t o r t i o n . In th i s way Lowson was able to predic t the harmonics of the noise radiated by the f luc tua t ing forces on the ro tor . He deduced a t ransfer funct ion 9 between the i n l e t ve loc i t y d i s to r t i on harmonic order and the no ise , a lso by using Sears ' s inusoidal gust l i f t response funct ion . However, at that t ime, the necessary input information perta in ing to the cha rac te r i s t i cs of s tator wakes was lack ing . Therefore the approach was not pursued fur ther . Lowson's method i s useful fo r pred ic t ing the noise given the unsteady blade fo rces , but i t d id not t rea t the complete problem from the input , the ve loc i t y d i s t o r t i o n ; to the output, the noise. To overcome th i s de f i c iency , Lowson subsequently devised an experiment wherein inf low d is to r t ions could be measured in a frame of reference ro ta t ing with the ro to r . These data were used in the ca l cu la t i on of an aero-acoust ic t ransfer func t ion . He attached a hot wire anemometer to one blade o f an unducted rotor mounted in an anechoic chamber. This enabled him to measure the harmonic leve ls present in the f luc tua t ing ve loc i t y r e l a t i v e to the ro tor blade. By subtract ing these harmonic leve ls from the acous t i c harmonic amp!itudes measured in the far f i e l d , he was able to determine the magnitude of the t ransfer funct ion between input ve loc i t y d i s to r t i on and radiated no ise. He compared these data with the resu l ts obtained from his mathematical model described e a r l i e r which employed Sears ' funct ion. The comparison i s shown in Figure 1. The d i s t r i bu t i on of points taken as one group co l lapse to the v i c i n i t y of the s t ra igh t l i ne described by the theory. The agreement i s best at condit ions of high rotor s o l i d i t y and rpm ( i . e . a seven bladed rotor turning at 1200 and 1600 rpm). 10 At th is point i t i s useful to consider the employment of the Sears ' aerodynamic response funct ion. The funct ion describes the var ia t ion of the magnitude and phase of the aerodynamic l i f t t rans fer funct ion (dC^/dct) with reduced wave number k. I t descr ibes the sect ional l i f t expected of a two dimensional ( i n f i n i t e span) a i r f o i l when inserted into a ve loc i t y f i e l d with: s inusoida l gusts of a pa r t i cu la r wave number act ing perpendicular to the chord of the a i r -f o i l sect ion (Figure 3) As wave number increases the up wash and down wash e f fec ts tend to produce a pa r t i a l cance l l a t i on of inc idence-induced l i f t changes,due to c y c l i c shedding of countersign c i r c u l a t i o n ( that i s e s s e n t i a l l y a consequence of inadequate compliance with the Kutta condi t ion at the t r a i l i n g edge). Turbulence inc ident on an a i r f o i l may be modeled using th is Sears ' funct ion to descr ibe the var ia t ion of force response on each blade sect ion as a funct ion of streamwise wavelength fo r each element of the spa t ia l (wave number) spectrum. The model assumes the gust f i e l d to be l o c a l l y two dimensional; that i s spanwise scale e f fec ts are neglected. Perhaps the most in te res t ing resu l t o f Lowson's work was the shape of the i n l e t r e l a t i ve ve loc i t y spectrum (Figure 2 ) . There are as many as fo r ty higher harmonics of the ro ta t iona l frequency. Strong energy content at high harmonic numbers i s i nd i ca t i ve of an inf low d i s to r t i on of small l a t e r a l extent and giv ing a pulse l i k e shape in the time domain (A t r a i n of de l ta functions has a r e l a t i v e l y " f l a t " d i s t r i bu t i on o f harmonic l e v e l s ) . Such a spectrum would not be expected of an i so t rop i c turbulent 11 ve loc i t y f i e l d of large sca le , but i s more cha rac te r i s t i c of long skinny eddies crossing the rotor plane. The work of both Mani and Lowson presents pos i t i ve evidence that na tura l l y occurr ing inf low d i s t o r t i o n s , whether they be due to random atmospheric turbulence or to a s p a t i a l l y d isc re te inf low d i s to r t i on may be major sources of rotor noise. The purpose of Mani 's model was to "present a systematic approach to determine the e f fec t of turbulence as a noise generator due to i t s impingement on a blade row." His resu l ts show that pure tone peaks at mul t ip les of the blade pass frequency increase in magnitude with an increasing eddy length to transverse spacing between the blades r a t i o . Lowson's experimental resu l ts were unique and in te res t i ng in that they prompted questions as to whether homogeneous i so t rop i c turbulence would be capable of producing an inf low spectrum possess-ing so many higher harmonics. 2.3 Stretched Eddies and Vor t ices fi 8 Hanson and Siddon and Leggat had d i f fe ren t ideas about the nature of the in f low. Hanson used an a u x i l i a r y fan mounted fa r downstream of the intake be l l to draw a i r through an experimental fan i n l e t . In th i s way he was able to measure the cha rac te r i s t i c s of the in f low at the i n l e t plane in the absence of a ro tor . He measured both streamwise and transverse v e l o c i t i e s with a crossed-wire hot wire anemometer. By using a v isual means of cross co r re la t ion between two hot wire s ignals he determined that the inf low d is to r t i ons were cor re la ted ." 12 c i rcumferen t ia l l y over a maximum angle of 30 degrees around the duct a x i s . This corresponded to a disturbance three inches in width ( l a te ra l sca le ) . Auto-corre lat ions of the streamwise ve loc i t y f luc tuat ions were conducted. These produced resu l t s ind ica t ing an in tegra l length sca le of 100 feet in the streamwise d i r e c t i o n . Hanson considered th i s anisotropy in the inf low turbulence to be " c r i t i c a l " , and " leading to short blade loading pulses (due to the narrowness of the eddies) , and high coherence (due to the length of the edd ies ) . " He thought that these long, narrow eddies or ig inated as natura l ly occurr ing atmospheric turbulence. As the eddy i s sucked into the s t a t i c fan i n l e t , i t i s elongated a x i a l l y and contracted t ransversely by the s i n k - l i k e potent ia l f i e l d at the fan mouth. Accompanying th is contract ion i s an increase in i t s " t ransverse" ve loc i t y to conserve angular momentum, g Hodder undertook a ser ies of experiments where the length scale of the turbulence enter ing a fan was var ied . He measured a 10 decibel decrease in the noise at the blade pass frequency and 15 decibels decrease at the f i r s t harmonic when the eddy length was shortened subs tan t i a l l y . In the f i r s t tes t the ax ia l length scale of the i n l e t d i s to r t i on was measured to be very long in tha t i t was cut some f i f t y times by the ro ta t ing rotor blades over i t s co r re la t ion length. The second tes t was conducted with a s u f f i c i e n t l y short eddy length that i t was cu t , over i t s co r re la t i on length, by only one blade. The eddy length was shortened by inser t ing honeycomb in to the fan i n l e t . The s i g n i f i c a n t change in leve l of the two harmonics supports the theoret ica l f indings of Mani. 13 Rao and C h u ^ compared Hodder's resu l ts with t h e i r theory of i n l e t turbulence in te rac t ion no ise. L ike Mani they used i so t rop i c turbulence as an input , and found that by knowing the turbulence and fan parameters of Hodder they could predic t the fundamental and f i r s t harmonic noise l eve ls to wi th in 1 dec ibe l . However, t h e i r ca lcu la ted harmonic spectra exhib i ted less band spreading than Man i ' s . Rao and Chu a t t r ibu ted th is peakiness to di f ferences in the assumptions of loading on neighbouring blades. Mani considered the problem in terms of a Fou r i e r -S t i e l t j e s in tegra l of shear waves producing ve loc i t y f luc tua t ions on the blade. He used Sears' funct ion to obtain the resu l t ing l i f t . Rao and Chu described the turbulence in terms of co r re la t ion lengths of the ve loc i t y f i e l d and then determined the spectral nature of unsteady blade forces resu l t i ng from subsequent passes of blades through the same eddy. The subt le d i f ference of these two approaches may account for the more d iscre te spectrum obtained by Rao and Chu. Measurements of fan blade unsteady pressure were ca r r ied 7 fi out by both Leggat and Hanson on d i f fe ren t fans . Both observed spectra with a dominant fundamental (the fan ro ta t iona l f requency), and many higher harmonics (Figure 4 ) . The presence of i n l e t vor t ices would exp la in these pressure spectra and Lowson's hot wire spect ra . Using a s t a t i s t i c a l d i s t r i bu t i on in p o s i t i o n , i n t e n s i t y , and length of long narrow eddies entering a fan , Hanson proposed a model to predic t the spectrum of noise produced by the fan/eddy i n te rac t i on . He formulated the problem in terms of blade forces produced by the eddies rather than in terms of the eddy ve loc i t y f i e l d . He used measured blade pressure data to deduce the spa t ia l and temporal character of the f l uc tua t ing blade loads. The blade pulses were allowed to vary in i n tens i t y , length, and width over t ime; and in pos i t ion (Figure 5) . The pulse spacing in time was equal to the period of ro ta t ion of the fan. He chose parameters descr ib ing the shape and duration of the pulse t ra in so that the computed l i f t spectrum for the blade matched the re la t i ve harmonic weighting of the measured blade pressure spectrum. Working with th is s t a t i s t i c a l input Hanson ca lcu la ted an expected energy spectrum for any fa r f i e l d point . In the on-axis d i rec t ion h is resu l ts predicted d iscre te tone leve ls at the blade pass frequency and the f i r s t harmonic r i s i n g above the ca lcu la ted broad-band spectral l eve ls (Figure 6 ) . However, his method f a i l e d to forecast information on the second, t h i r d , fourth or f i f t h harmonics which were observed experimental ly in the fa r f i e l d sound. Concurrent with Hanson's studies came the experimental work 7 8 of Siddon and Leggat ' . The aim of t he i r work was to prove the existence of .expected, but as yet unmeasured sources of fan rotor noise- the e f fec t of duct boundary layer separation in the inf low reg ion, t i p clearance modulation and i n l e t ve loc i t y d i s to r t i ons were measured. A method of c ross -co r re la t i on between the blade hydrodynamic pressure and the far f i e l d acoust ic pressure (causa l i t y cor re la t ion) y ie lded spat ia l source strength evaluat ions over the blade s u r f a c e s ^ ' ^ ' ^ (Appendix A ) . A typ ica l source strength curve fo r spanwise points located at 15 percent chord behind the leading edge i s shown in Figure 7. The ordinate gives the cont r ibut ion to the mean squared fa r f i e l d pressure (dp ) coming from each element of blade sur face, dS. The c ross -co r re la t i on measurements ind icate a region of zero source strength at about the 3/4 span pos i t i on . On the outboard side the source strength is negat ive, whi le on the inboard s ide i t i s pos i t i ve . This countersign source strength behaviour indicates that the blade i s experiencing a disturbance which on the time average causes force f luc tuat ions on the blade which are out of phase with each other, in a span-wise sense. The s ign i f i cance of negative source strength i s explained mathematically and phys ica l l y by Leggat^, but b a s i c a l l y i t ind icates . that the port ion of the surface with negative source strength i s rad ia t ing more than ±90° out of phase with the dominant part of the f a r f i e l d sound (coming from the pos i t i ve source reg ion) . Leggat and Siddon explained the existence o f the counter-phase source strength as a consequence of a concentrated i n l e t vortex being phys ica l l y "chopped" by subsequent blade passages. The nu l l at the 3/4 span point would correspond to the pos i t i on of the vortex core. The pos i t i ve and negative source strengths on e i ther s*ide of the n u l l ^ would be caused by decreases and increases in blade angle of attack r e l a t i ve to the mean f low, as the blade passed through the vortex ve loc i t y f i e l d . Such a phenomenon would not be in disagreement with the physical explanation proposed by Hodder and Hanson. Hanson sa id that as the eddy i s sucked into the fan the transverse v e l o c i t i e s would be expected to increase to conserve angular momentum. As Hanson 16 measured only one component of the unsteady transverse v e l o c i t y , and as the hot wire i s incapable of d is t ingu ish ing d i r e c t i o n , i t i s very possib le that the eddies measured were wel l organized v o r t i c e s ; e i ther concentrated ones such as ground f vort ices or na tu ra l l y occurr ing vor t ices being stretched and al igned in the fan inf low f i e l d . 2.4 The In le t and Tip Vortex The presence of ground-generated i n l e t vor t i ces in fans and 1112 a i r c r a f t j e t engines i s well documented ' , and has been explained 12 using potent ia l flow methods . During f l i g h t concentrated vor t i ces may or ig ina te from stagnation points in the boundary layer on the fuselage and enter rear mounted engines, whi le during tax i and run up vor t ices are l i k e l y to grow from both the fuselage and the ground. The ro tor /vor tex in te rac t ion problem i s most acute when a he l icopter rotor in teracts with i t s own t i p vo r t i ces . Here the vortex generated at the t i p of one rotor blade may pass close: to the underside of the. fo l lowing blade. The associated ve loc i t y f i e l d in te rac t ion may cause e f f i c i e n t noise rad ia t i on . I f the he l i cop te r i s hovering,the t i p generated vor t i ces w i l l remain r e l a t i v e l y s t a t i o n -ary with respect to the rotor blade. The resu l t i s a f i xed region on the blade over which there occurs a vortex induced s t a l l . Broad-band noise i s produced. Per iod ic slapping occurs when the he l i cop te r i s in forward f l i g h t . The forward ve loc i t y of the he l i cop te r tends to skew the path of the shed vortex. The vortex then i s no longer stat ionary with respect to the blade and a per iod ic modulation of the blade load r e s u l t s . Actual chopping of t i p vor t ices takes place when vor t ices shed by a rotor in tersect a tandem ro to r ' s plane of ro ta t ion . These problems are somewhat d i f f e ren t from the fan ro tor / vortex in te rac t ion problem as the vortex in f luence i s f e l t over only a small percentage of the rotor blade span; general ly near the t i p . Regardless, resu l ts of work concerned with th i s problem deserve cons iderat ion. 20 Leverton and Taylor analyzed the vortex "chopping" phenomenon using a model he l icopter rotor and two je ts blowing against one another to simulate the vortex (Figure 8). They found that the harmonic l eve ls var ied as the fourth power of ro tor ve loc i t y and the square o f equivalent vortex strength. The simulated vortex used in the experiments would create more coherent wing loads than would be produced by a true vor tex. For th is reason true vortex resu l ts may prove to be somewhat d i f f e ren t . 21 ' Pat terson, Amiet, and Munch c ross -cor re la ted blade pressure and fa r f i e l d acoust ic pressure to determine the noise generating mechanisms involved when a vortex passed beneath a s ta t ionary a i r -f o i l . They found that when the vortex passed very c lose to the encountering blade, loca l blade s t a l l developed. This process i s not modulatory in t ime. The noise generating mechanism i s associated with the passage of s t a l l ed wake eddies over the a i r f o i l t r a i l i n g edge. ; Complete theoret ica l studies of the noise resu l t i ng from 22 he l icopter blade/vortex in te rac t ion were ca r r ied out by Widnall f o r a vortex passing under a blade of a he l icopter in forward f l i g h t . 18 Using l i n e a r and unsteady aerodynamic theory, she was able to predict features of the radiated noise inc luding to ta l power and d i r e c t i v i t y as a funct ion of the vortex c h a r a c t e r i s t i c s . Widnall showed good agreement between theory and experiments at low t i p Mach numbers. 2.5 Role of Present Work I t i s evident that there i s s t i l l : much to be resolved about the problem of a fan rotor i n te rac t ing with an i n l e t d i s t o r t i o n . I t i s general ly agreed among researchers that long coherent eddies are responsible fo r much of the tonal rad ia t ion from ro to rs . The physical nature of these eddies; whether they should be character ized as steady or in termi t tent vor t ices or whether indeed they possess net c i r c u l a t i o n at a l l i s not c l ea r at t h i s t ime. Theoret ical approaches to the problem predic t the leve ls of sound at the blade pass frequency and i t s f i r s t harmonic, but do not agree with experiments at higher harmonics. Two dimensional methods using l i n e a r aerodynamic assumptions and employing Sears ' s inusoidal gust l i f t response funct ion have given acceptable p red ic t -ions of the noise due to turbulence i n te rac t i on . However, th is approach has not been appl ied previously to the problem of a d isc re te vortex penetrating the rotor d i s c . The dependence of the fan noise on the vortex parameters and on the pos i t ion o f vortex entry into the fan are unknown. The fo l lowing Sect ions, Three and Four, describe the experimental and theoret ica l approaches respect ive ly . The author has 19 presented the experimental work f i r s t as i t i s f e l t that the develop-ment of the theory f a l l s into place more l o g i c a l l y i f the l im i t a t i ons of the experiments are r e a l i z e d . Sections Three and Four are essen t i a l l y se l f -con ta ined . The reader* who i s pr imar i l y in teres ted in the theory may read sect ion Four f i r s t without loss o f c l a r i t y . 20 3. EXPERIMENTAL ANALYSIS ft The experiments were designed to enable a de ta i led ana lys is of the noise produced in the fa r f i e l d when a vortex of known s t rength , ax ia l ve loc i t y d e f i c i t and core radius was introduced into a fan. Small ha l f de l ta wings at incidence were used to generate vor t ices of varying strength and core rad ius. Their propert ies were measured in a low-speed wind tunnel using a unique v o r t i c i t y detect ion device. Subsequently, the wings were mounted in the fan be l l mouth, and the vortex in te rac t ion e f fec ts measured fo r various operating cond i t ions . On-blade f luc tua t ing pressures were measured with and without the vortex generator i n s t a l l e d . The resu l t ing pressure s ignal was plot ted in real time on a c i r c u l a r polar p lo t synchronized to the rotat ion of the fan . This procedure allowed the study of the in te rac t ion of na tura l l y occurr ing vor t ices with the fan ro to r , and a confirmation of the predicted blade load f luc tuat ions due to in te rac t ion with the contr ived vo r t i ces . The Univers i ty of B r i t i s h Columbia aeroacoustic f a c i l i t y and other apparatus w i l l be described p r io r to discussions of the experimental techniques. 3.1 Apparatus 3.1.1 Aeroacoustic F a c i l i t y A l l acoust ic measurements were car r ied out in an anechoic chamber of ins ide dimensions 16 feet by 14 feet by 8 feet high. The chamber i s constructed of s t e e l - f i b r e g l a s s sandwich panels 4 inches th ick mounted on a spring i so la ted concrete pad. The wedges i n s t a l l e d in the chamber are made of semi - r ig id f i b r e g l a s s , and encased in a protect ive g r id of galvanized hardwire c l o t h . The 23 chamber has been ca l ib ra ted and va l ida ted and was found to exh ib i t free f i e l d (anechoic) condi t ions from a lower cu t -o f f frequency o f 150 Hz to a maximum measured frequency of 100 KHz. Removable hatch covers were incorporated into the i n i t i a l design of the chamber to al low experiments on fan and j e t no ise , which involve the passage of a i r through the chamber. 3.1.2 Fan Rig The research fan was located i n one corner of the chamber, i t s centre l i n e 30 inches from the wal l wedges (Figure 9 ) . The fan began l i f e as a Woods of Colchester 19 i nch , seven bladed ax ia l f low fan un i t . The ro tor was c a r e f u l l y r e b u i l t , po l i shed, and balanced. I t operates in a remanufactured casing made from r ings o f cast and machined aluminum (Figure 10). The clearance between a l l . blade t i p s and the casing i s constant and equal to .040 inches. The e a r l i e r sheet metal shroud provided by Woods was subs tan t ia l l y "out of round". The rotor was driven by a 10 hp d i r ec t coupled induct ion motor d r iv ing the ro tor at a constant rate of 3600 rpm. For the vortex in te rac t ion experiments the fan was operated at a constant head of 2.54 inches of water and a flow rate of 7000 CFM with a blade stagger angle of 20 degrees. This operating point was near the optimum and gave a throat i n l e t ve loc i t y equal to that under which 22 the vortex generating wings were tes ted. The importance of t h i s matching of ve l oc i t i e s w i l l be discussed l a t e r . The fan casing was v ibra t ion i so la ted from the chamber using rubber machine mounts and a f l e x i b l e coupl ing to the exhaust duct. The downstream d i f f use r sect ion was acous t i ca l l y treated to minimize extraneous noise. Var ia t ion of the fan head and flow rate was made possib le by the i n s t a l l a t i o n of an a i r f o i l louvre in the wind tunnel . A 1/4 inch ce l l ed hexagonal honeycomb mesh measuring 4 feet square and 6 inches in thickness was made to be i n s t a l l e d over the a i r intake hatch when required. An i n l e t be l l mouth was molded from f i b reg lass to an aerodynamically optimum shape g iv ing uniform and unseparated 24 flow at the throat . I t was bolted to the upstream end of the fan cas ing. 3.1.3 Vortex Generators The vor t ices that were introduced into the fan were produced by ha l f de l ta wings with sharp leading edges. The various sweep back angles and dimensions as shown in Figure 11. For tes t ing in the wind tunnel the wings were mounted in a fa i red support bracket 5 1/2 inches long and 3/8 inches square. A ba l l vortometer was used to measure the vortex tangential ve loc i t y a t any radius (Figure 12). I t consisted of a styrofoam ba l l .10 inches in diameter attached by a th in nylon thread to a support pin mounted to the t r a i l i n g edge of the a i r f o i l . The ro ta t iona l ve loc i t y f i e l d in the wing t r a i l i n g vortex causes the ba l l to ro ta te , 23 accelerat ing to an angular ve loc i t y where the net drag on the ba l l i s c lose to zero. This condi t ion corresponds to the point where the ba l l ve loc i t y equals the vortex tangential ve loc i t y at the pa r t i cu la r vortex radius that the ba l l occupies. Simultaneous measurement of the angular v e l o c i t y , co, and the radius of the c i r c l e described by the b a l l , r, y ie lded the ro ta t iona l ve loc i t y of the vortex v( r ) ( resu l ts are discussed in sect ion 3.2) . The ba l l rpm was measured with a Strobotac and the c i r c l e diameter with a re f lex camera mounted on a ve r t i ca l t raverse screw. 3.1.4 Blade Pressure Fluctuat ions Measurements of the blade pressure f luc tua t ions were made by i n s t a l l i n g a .125 inch Ku l i t e semi-conductor pressure transducer in to a brass casing which was f i t t e d into a m i l l ed s l o t in the bottom of the blade. Ten pin holes .030 inches in diameter were d r i l l e d from the base of the s l o t through to the blade upper surface (Figure 13). A hole in the brass casing was al igned with the pin hole corresponding to the desired measuring point . The resonant frequency corresponding to the p in -ho le /cav i t y resonator created by the volume ins ide the transducer casing was ca lcu la ted to be 10,000 Hz, which i s one o rde r , of magnitude above the highest frequency of i n te res t i n the blade pressure spectrum. Shielded leads from the transducer were d i rected back along the m i l l ed s l o t to the hub of the fan where a two channel EKEG F.M. telemetry system was located. The s lo t was f i l l e d with RTV mold-making rubber. This system enables the t rans fer of two s igna ls 24 simultaneously from the rotat ing fan rotor to the s tat ionary telemetry rece iver . Gain and phase s h i f t charac ter is t i cs for the system are shown in Figure 14.The twotelemetry t ransmit ters were mounted in a p lex ig lass housing mounted on the front end of the fan shaf t . Bat ter ies to power both the transmitters and two pressure transducers were also f ixed in the housing. The rotor and rotor-mounted instrumentation were s t a t i c a l l y balanced. An aerodynamically shaped nose cone covered the hub-mounted telemetry t ransmit ter package (Figure 10). 3.1.5 Acoust ic Analysis The fa r f i e l d acoust ic pressure was measured at a distance of 6 fan diameters"(9.5 f t ) in f ront of the rotor plane on the fan ax i s , using a Bruel and Kjaer 1/2 inch microphone model 2134 ( f ree f i e l d corrected) . Ca l ib ra t ion of the microphone was accomplished using a Bruel and Kjaer piston phone generating a pressure level of 124 db at 250 Hz. The transducer-telemetry system was ca l ib ra ted using the same piston phone f i t t e d with an adaptor designed to seal around the blade surface pin holes. A Bruel and Kjaer narrow band spectrum analyzer with a band width equal to 6 percent of the centre frequency was used to measure the far f i e l d and on-blade pressure spectra and d iscrete tone amplitudes. Due to a slowly varying modulation of the rms pressure under "natura l " i n l e t condi t ions, a Schlumberger time domain analyser had to be used to time average the f i l t e r e d harmonic leve ls over a period of 30 sec. This procedure produced an improvement in the accuracy to a repea tab i l i t y wi th in ±0.1 db. A l im i ted amount of co r re la t ion analys is was car r ied out, using a SAICOR SAI43A cor re la t ion computer. Cross-corre la t ions between the fan blade surface pressure and fa r f i e l d acoust ic pressure y ie lded surface source strength information. The technique i s described in Appendix A. Magnitudes of the source strengths at blade spanwise locat ions were s im i l a r to those measured before the ' modif icat ions to the fan were car r ied out (Figure 7) . This ind icated that the modif icat ions had T i t t l e e f fec t on the i n l e t vortex detected in e a r l i e r work. 3.1.6 Real Time Polar Pressure Display A special analogue c i r c u i t was developed to sychronize the c i rcumferen t ia l l y varying blade pressure f luc tua t ion to the circumfer-ent ia l pos i t ion of the rotor blade in which the transducer was mounted. Thus a real time polar p lot of the blade pressure at any rotor pos i t ion could be displayed on an osc i l l oscope . The method incorporated two analog mu l t i p l i e r s which mu l t ip l i ed the blade pressure signal by sine and cosine waves respect ive ly of frequency equal to the fan rotat ional frequency. The governing equation reads as fo l lows: x y . ep i s the ampl i f ied transducer voltage a r i s i ng from blade pressure f luc tua t ions . The constant accounts for the overa l l voltage to pressure ca l i b ra t i on of the system. x and y are used as inputs to the hor izontal and ve r t i ca l drives of an osc i l loscope t race, oo i s the frequency of fan ro ta t ion . K i s constant which determines the radius of the polar c i r c l e . The c i r c l e d is to r ts with time on each revolut ion to depict the spa t ia l and temporal var ia t ion of blade pressure. The schematic of the system and an example of resu l t ing polar p lo t are shown in Figure 15. 3.2 Determination of T ra i l i ng Vortex Charac te r i s t i cs Measurements of the st rength, core r a d i i , and pos i t ion of t r a i l i n g vor t ices behind del ta wings have been car r ied out by other pc pc 01 OO researchers ' ' ' . However, in these cases the resu l ts obtained, were pa r t i cu la r to the dimensions of the model and the Reynolds' number under which i t was operated. For th i s reason, i t was decided to experimental ly measure the vortex cha rac te r i s t i cs corresponding to the model s izes and flow rates associated with the fan experiments. A del ta wing of 5 1/2 inch root chord was mounted in an open c i r c u i t wind tunnel with tes t sect ion dimensions 6 1/2 inches by 12 1/2 inches. The vortex pos i t ion re la t i ve to the wing t r a i l i n g edge, tangential ve loc i t y , core radius and ax ia l flow d e f i c i t were measured for angles of attack ranging from 10° to 30° in f i ve degrees increments. Detai ls of the method used to locate the vortex core re la t i ve to the wing,using a hot wire anemometer are contained in Appendix B. 3.2.1 Measurement of Core Radius and Vortex Strength To ensure that the ba l l vortometer accurately fol lowed the vortex tangential ve loc i t y , i t was necessary to pos i t ion the pivot point on the axis of the vortex. Using the core pos i t iona l data recorded in Appendix C, the pivot point was placed at the point in space corresponding to the vortex core pos i t ion fo r the wing angle of at tack. With the tunnel at operating ve loc i t y (63.7 f t / s e c ) , the pos i t ion of the pivot point was adjusted un t i l the rpm of the ba l l maximized. This l as t minor adjustment ensured the pivot point to be as c lose to the axis of the vortex as the ba l l vortometer was capable of sensing. The nylon thread was extended to a length of 1.5 inches and then decreased in length by small i n te rva l s . At each in terva l the rpm of the ba l l and the diameter of the c i r c l e i t swept were measured. In some cases, the ba l l t ra jec tory would l i e wi th in the rotat ional ( so l i d body) core. In other cases i t would spin at a radius larger than the core radius where the magnitude of the tangential ve loc i t y f a l l s as 1/r. Here the vortex ve loc i t y f i e l d behaves l i k e that of an i r ro ta t i ona l ( ideal ) vortex. When the ba l l was in th is region the ve loc i t y gradient over the nylon thread caused the ba l l to spin fas te r than the loca l vortex ve loc i t y . On the other hand, when the radius of the vortometer c i r c l e became too small , s t i f f ness in the nylon thread became important, and with decreasing c i r c l e radius the ba l l eventual ly stopped spinning. The former of these two ef fects became important at r ad i i exceeding twice 28 the core radius of the vortex, while the l a t t e r occurred when the c i r c l e radius became less than approximately 25 percent of the core rad ius. The measured data was compared with a mathematical model fo r 29 real vor t ices obtained from vortex tube experiments:, r 2 V(r) = 2 iF (1 - e~4?T ) . . . . (3.1) where r i s the vortex st rength, v i s the kinematic v i s cos i t y of a i r and T i s the time elapsed since the b i r th of the vortex. T determines the vortex core radius . To achieve a comparison we referenced the measured radius of peak tangential ve loc i t y to the theoret ica l value. This i s accomplished by set t ing the f i r s t der iva t ive with respect to radius of the vortex tangential ve loc i t y equal to zero dV(r) dr (3.2) y ie ld ing 2 - r • . 4vT , „ 2 / + 4vT subst i tu t ion of the measured value of core radius (r ) for r gives the value of T for which the theoret ica l and measured core rad i i w i l l a l i g n . The strength of the vortex may then be simply ca lcu la ted as fo l lows: 29 g w V^> . (3.4) 1 - e" 4vT Curves showing the amount of agreement between measured and theoret ica l vortex tangential ve loc i ty f i e l d s are shown in Figures 16 and 17. Here also i s shown the col lapse of the ca lcu la ted values of vortex strength. Curves descr ib ing the var ia t ion of vortex strength and core radius with angle of attack for the 5.5 inch wing were plot ted (Figures 18 and 19). Vortex strength was found to vary l i n e a r l y with angle of attack while core radius was given by the fo l lowing re la t ionsh ip Log r c = .439 a +"1.27 . . . . .(3.5) Having these r e l a t i o n s h i p s , i t was possib le to define vortex strengths and core r ad i i for vor t ices generated at a l l angles of attack from 0 to 30 degrees,by the other three ha l f del ta wings, by measuring these propert ies at only two angles of attack for each wing (Figures 18 and 19). These data along with those perta in ing to the vortex measurements from the 5.5 inch wing are presented in Appendix C 3.2.2 Vortex Axia l Flow D e f i c i t The amount by which the ax ia l (streamwise) ve loc i t y decreased ins ide the vortex core was measured by t ravers ing a hot wire anemometer l a t e r a l l y through the core. The spat ia l extent and magnitude of the d e f i c i t was found to vary with the wing angle of at tack. The core remained very t igh t and the d e f i c i t was d i f f i c u l t to detect un t i l the wing angle of attack increased to the point where t r a i l i n g vortex 30 burst ing was expected . At th is point the spat ia l extent of the d e f i c i t increased rapid ly with angle of attack (Figure 20). The magnitude of the d e f i c i t i s tabulated in Appendix C. Deletions in th is data for wings at low angles of attack ind icate that the magnitude of the d e f i c i t could not be measured. For these cases the vortex had not burst. The term "vortex burst ing" refers to a t rans i t ion that occurs in del ta wing t r a i l i n g vor t ices when the wing i s operated at high angles of. at tack. The vortex core changes from laminar to turbulent . The t rans i t i on i s accompanied by a spreading of the rad ia l extent of the ax ia l ve loc i ty d e f i c i t in the vortex core, and an increase in the growth rate of the vortex core radius with time. The s t a b i l i t y of the vortex deter iorates more quick ly than for the laminar case. The resu l t is a more rapid break up of the vortex f i e l d . The onset of burst ing i s useful because i t allows two types of vor t ices to be tested (laminar and turbulent cores) . Not only may the e f fec t of vortex strength on the noise be examined, but a lso the contr ibut ion of various forms of vortex cores and percentage ve loc i t y d e f i c i t s . 31 3.2.3 Discussion of Vortex Ve loc i ty F ie lds The wind tunnel measurements of the ha l f del ta wing t r a i l i n g vor t ices provided a range of c i r c u l a t i o n values from a minimum measured 2 2 strength of 7.5 f t /sec (r c ~.625 inches) to a maximum of 40 f t /sec (r = 1.23 inches). The vortex ax ia l ve loc i t y d e f i c i t s varied from the l i m i t of being undetectable with a hot wire (no vortex bursting) to a maximum of 82 percent ve loc i t y d e f i c i t ' The agreement of the measured data with the mathematical vortex model was good in most cases. Therefore, th is model was used as the vortex input to the theoret ica l analys is of the problem with the Values for r and r c being based on the ca l i b ra t i on experiments. The hot wire measurements of ax ia l ve loc i t y d e f i c i t s may be con-sidered accurate only in cases where the d e f i c i t region is wide compared to the length of the wire. However, in cases with t igh t cores such as the 5.5 inch wing at 10° angle of attack (Figure 20), the ax ia l ve loc i t y measurement i s probably somewhat inaccurate because the act ive port ion of the hot wire sensor (-.05 inches) was probably larger than the true core radius. Less confidence can be placed in the magnitude of the d e f i c i t in these cases. The pos i t ion of the vortex core, however, i s thought to be reasonably accurate. 3.3 Vortex/Rotor Interact ion Experiments The experiments were designed to determine the e f fec t of the vortex parameters of vortex st rength, core rad ius, and ax ia l ve loc i t y d e f i c i t on the overa l l on-axis fan noise resu l t ing from a 32 s ing le vortex passing through the rotor d i sc . We also studied the dependence of the far f i e l d noise leve l on the rad ia l pos i t ion of vortex entry into the fan. As the resu l ts of these experiments were to be compared to those derived by the mathematical model, i t was necessary to ensure that condit ions in the fan i n l e t were as s im i l a r as possible to those under which the vortex cha rac te r i s t i cs had been measured in the wind tunnel . Furthermore, i t was necessary to ensure that other forms of na tura l l y occurr ing d is to r t ions be minimize^ to the ult imate degree. The former of these two condit ions was achieved as fo l lows, the ha l f del ta wings were mounted in the throat of the fan i n l e t be l l mouth 2 inches from the fan rotor (Figure 21). This distance corresponded to the pos i t ion where a l l vortex parameters were measured re la t i ve to the wing in the wind tunnel. The operating point of the fan and duct system was a l te red with the con t ro l l i ng louvre un t i l the ax ia l inf low ve loc i ty at the be l l throat matched the ve loc i t y used fo r the wind tunnel tests (63.7 f t / s e c ) . At th is condi t ion the fan was operating with a s t a t i c head of 2.54 inches of water with a blade stagger angle of 20°. The presence of contaminating noise being generated as a resu l t of an inter ference between the upstream potent ia l f i e l d of the rotor and the del ta wing surface was found to be minimal. No increase in sound level was detected at a l l the harmonics with the ha l f de l ta wing at 0° angle of attack compared with the case of no del ta wing inser ted. 33 The e f fec t of natura l ly occurr ing d is to r t ions was minimized by placing the honeycomb over the anechoic chamber hatch. The resu l t was a decrease in the overa l l sound pressure leve l of 3 db and in the blade pass frequency of 4.5 db. No e f fec t was apparent at the higher harmonics. A decrease in the degree of f luc tua t ion in the magnitude of the overa l l sound pressure level with time accompanied the inser t ion of the honeycomb. The resu l t ing base- l ine spectrum i s shown as the s o l i d l i n e in Figure 22. (No imposed vortex) . 3.3.1 Dependence of Noise on Vortex Parameters Tests were car r ied out fo r each of the ha l f del ta wings mounted at angles of attack varying from 10° to 30° (both burst and unburst vo r t i ces ) . A vortex with pos i t i ve c i r cu l a t i on (counter-clockwise rotat ion) was generated by the wing at pos i t i ve angles of at tack, and vice versa for negative angles of at tack. The fan rotor was also turning counter-clockwise as viewed from the i n l e t . A change in the angle of attack of one wing produced a change in a l l three vortex parameters of in te res t : core rad ius , c i r cu l a t i on s t rength, and ax ia l ve loc i ty d e f i c i t . Operating a l l four wings at the same angle of attack resul ted in a change in vortex c i r cu l a t i on and ax ia l ve loc i t y d e f i c i t from wing to wing, but no change in core rad ius. In a few cases, wings operating at d i f fe ren t angles of attack would produce vor t ices of equal vortex s t rength, but varying core radius and ax ia l ve loc i t y d e f i c i t . I t i s apparent, then, that the e f fec t of a l l three vortex parameters may not be studied indepen-dent ly , but must be analyzed in groups of two and three. 34 Overal l fa r f i e l d sound leve ls and spectra were measured fo r wing angles of a t tack , both pos i t i ve and negat ive, i n 5° increments. The far f i e l d microphone was at the on-axis pos i t ion at a distance of s i x fan diameters. Typ ica l l y the overa l l l eve ls increased by anything from 1 to 8 db and some harmonics showed an increase of up to 14 db as the angle of attack was increased from 0 to 30 degrees. Figure 22 contrasts the e f fec t on the far f i e l d spectrum for vor t ices of i den t i ca l cha rac te r i s t i cs except that the i r d i rec t ion of ro ta t ion i s reversed in the two cases. Notice that the vortex of negative (c lock -wise) ro ta t ion creates two new harmonic peaks (2100 and 2520 Hz) . This resu l t i s fo r only two condi t ions, but they are representat ive of the trends observed in a l l the t r i a l s . They serve to give an overa l l impression of the e f fec t of the vo r t i ces . A tabulat ion of the e f fec t of the varying vortex parameters on the overa l l leve l and blade passing harmonic leve ls i s presented in Appendix D. The trends of pos i t i ve and negative vortex sense do not show any common features. Therefore they w i l l be discussed separately. A parametric analys is was car r ied out of the expected l i f t f l uc tua t ion amplitude on the blade (Appendix E) . This produced a way of sca l ing the dimensional dependence of acoust ic noise l eve ls r e l a t i ve to the vortex proper t ies. Two terms invo lv ing the three vortex parameters evolved: ( r ^ c r c a n c * r ^ 0 r c )• The overa l l noise leve ls fo r both pos i t i ve and negative vortex sense were p lo t ted against each of the two terms. The data col lapsed most favourably : against the grouping r. U^r,.. The curves for these data are shown in Figure 23. Vort ices of negative sense ( ro ta t ion opposite to that of the fan) e f f ec t the overa l l noise leve l more strongly than do pos i t i ve vor t ices at the higher values of r u"c r*c whi le the opposite i s true at low values of r u r • Unfortunately, the magnitude of the d iscre te tone l eve l s of the measured blade pass frequency and i t s higher harmonics did not co l lapse well when p lo t ted against e i t he r of the two parametric groupings calcu lated in Appendix £. This i s not su rp r i s i ng , however, as each frequency w i l l possess i t s own funct ional dependence on the three vortex propert ies ( r , U and r ); and therefore, the data may col lapse against one grouping of the parameters, but more l i k e l y some compromise of the two groupings. Most of the d iscre te frequency sound leve ls show an increase in amplitude with an increase in vortex st rength, core rad ius , and ax ia l ve loc i t y d e f i c i t . The e f fec t on the overa l l noise of changing only the vortex c i r c u l a t i o n and ax ia l ve loc i t y d e f i c i t parameters i s shown in Figure 24. Here each set of four points (white c i r c l e s , black t r i -angles, e tc . ) represents the noise level produced when each 1 of four vor t ices of equal core r a d i i , but increasing vortex strength pass through the fan ro to r . The four equal core r a d i i vor t ices were produced by the four hal f del ta wings of equal root semi-span operat-ing at the same angle of at tack. The ax ia l ve loc i t y d e f i c i t in the vortex increases as the root chord of a ha l f de l ta generating wing decreases. Therefore in any set of four po in ts , the ax ia l ve loc i t y d e f i c i t decreases as the vortex strength increases. 36 Take fo r example the set of white squares fo r a vortex strength with pos i t i ve sense. Here as the vortex strength increases, the resu l t ing noise level increases very l i t t l e . I t would seem that the e f fec t on the noise of the increase in the vortex strength- i s being can-ce l l ed by the e f fec t of the decrease in ax ia l ve loc i t y d e f i c i t . This trend i s general ly true for a l l sets of points for the overa l l sound pressure leve l case. Now look at Figure 25 which shows the var ia t ion of amplitude of the 1680 Hz harmonic with changes in vortex strength. There is a large increase in harmonic amplitude with increasing vortex strength for both pos i t i ve and negative vortex sense. For the case of negative vortex sense and the white squares, a doubling a vortex strength causes a 8 db increase in the sound pressure l e v e l . The reason for th is trend i s not immediately obvious. On f i r s t look i t i s c lear that the blade loading at 1680 Hz is very sens i t i ve to changes in the vortex st rength, but there are no obvious physical explanat ions. The resul ts of the mathematical model may help to c l a r i f y t h i s trend in the data. In Figure 26, an en t i r e l y d i f fe ren t trend i s d isplayed. Here the sound pressure level at the blade pass frequency ac tua l l y decreases with increasing vortex strength. I f the ax ia l ve loc i ty d e f i c i t were the dominant mechanism cont r ibut ing to the sound leve l at th is frequency, then th is behaviour would be expected. That i s , as the vortex strength increases, the dominant d i s t o r t i o n , ve loc i t y d e f i c i t , decreases; and so the sound level should decrease. In tu i -t i v e l y , however, we know that because the ax ia l ve loc i t y d e f i c i t tends to be small in spat ia l extent compared to the vortex tangential 37 ve loc i t y f i e l d , i t should contr ibute more favourably to the higher harmon-ics of the spectrum. This resu l t i s indeed a paradox, and i t too must await the resu l ts of the mathematical model for a proper explanat ion. 3.3.2 Dependence of Noise on Pos i t ion of Entry The change in the far f i e l d noise spectrum with vortex pos i t ion of entry was measured by introducing the vortex into the fan at a number of d i f fe ren t rad ia l pos i t ions . The 5 1/2 inch wing was mounted on one of a ser ies of th in f l a t p la tes . The plate was then placed in the throat of the bel l mouth so as to cut the c i r c u l a r i n l e t plane as does a chord of a c i r c l e (Figure 2 / ) . The wing was set at incidences of plus and minus 30°. When the rad ia l pos i t ion of entry was to be changed, the wing was mounted on a longer f l a t plate thus bringing i t c loser to the hub of the fan. At each rad ia l pos i t ion the ef fect of the plate alone on the far f i e l d noise spectrum was measured. The e f fec t was s i gn i f i can t at a l l harmonics, when the plate was 2 inches in f ront of the fan ro tor . When moved upstream to a distance of 4 inches the e f fec t was very much less s i g n i f i c a n t . For the worst case i t s presence changed the overa l l level by .6 db. The worst increase in a s ing le frequency was 2.3 db at 840 Hz. The minimum increase in the sound pressure level was produced when the tes t vortex (r=30 f t / s e c , r c=0.078 f t , U c /U =' 77%) was introduced into the fan at a radius of R/R .^= . 4 5 . The resu l t was a 4 db increase in the overa l l noise l e v e l . The lowest increase for a s ing le frequency was 3 db for the 1680 Hz harmonic. The increase in sound level resu l t ing from the introduct ion of the mounting plates are shown in Appendix D. In Figures 28 and 29 i s p lo t ted the resul tant va r ia t ion in sound leve l produced when the vortex was introduced at rad ia l pos i t ions varying from a minimum fan radius of .36 f t (R/R^. = .45) to a maximum of .64 f t (R/R^. = .81) . Figure 28 shows the e f fec t of pos i t i ve sense vor t ices and 29 of vor t ices of negative sense. In both graphs the trend i s fo r the overa l l sound pressure leve l and that of the fundamental tone to decrease uniformly as the vortex approaches the root of the fan blades. The higher harmonic leve ls tend to cross over each other as the rad ia l pos i t ion of vortex entry i s va r ied . Discussion of th i s phenomenon would best be l e f t un t i l resu l t s of the mathematical model can be presented simultaneously. 3.3.3 Fluctuat ing Blade Pressures A study of the e f fec t of the imposed vortex and na tu ra l l y occurr ing ground or atmospheric vor t ices was made poss ib le using the blade pressure measuring apparatus described e a r l i e r in Sect ion 3 .1 .6 . The pressure transducer was mounted in the fan blade .6 inches from the blade t i p . This point was deemed most su i tab le as i t corresponded to the region on the fan blade where the pressure f luc tua t ions and acoust ic source strength had been found to be st rongest . These resu l ts come from e a r l i e r work^. The source strength curve of Figure 7 shows that the magnitude of the source strength i s greatest between non-dimensional fan r a d i i of .86 and .97. The present measuring point i s " at a non-dimensional fan radius of .93. At t h i s point the e f fec t of na tura l l y occurr ing vor t ices should be most apparent. 39 Tests were run with imposed vor t ices of pos i t i ve and negative 2 sense (r = ±20 f t / s e c , TQ = .65 f t , U c /U = 75%). The pressure measuring hole was ;324 inches outboard of the vortex core rad ius . The vortex gen-erator was then removed from the fan. Measurements were repeated for the clean running fan both with the honeycomb in place over the anechoic chamber hatch and with i t removed. Pictures taken of the osc i l loscope trace under these four condit ions are shown in Figure 30. For the imposed vortex with c i r c u l a t i o n sense opposite to that of the fan ro to r , the pressure on the upper surface decreases on encounter with the vortex (Figures 30A, 31A). The vortex tangential ve loc i t y at the pressure tap H ro tat ing opposite to the rotat ion of the fan. The resu l t ing increase in the coR component f e l t by the blade sect ion causes an increase in the loca l angle of at tack. The resu l t i s an increase in sectional, l i f t which involves a decrease in the pressure on the upper surface of the blade sec t ion . A vortex of opposite c i r c u l a t i o n sense produces the opposite sequence of events and leads to an increase in the pressure in the region of the vortex (Figures 30B, 31B). Figures 30C, 31C and 30D, 31D contrast the e f fec t of p lac ing the honeycomb over the anechoic chamber hatch. Figure 30C shows a subs tan t ia l l y greater degree of blade pressure, f luc tua t ion than i s apparent in Figure 30D. I t i s th is f luc tuat ion that i s responsible for the 3 db increase in far f i e l d noise level produced when the honeycomb is removed from the anechoic chamber hatch. A ser ies of 16 mm exposures of the var ia t ion of the blade surface pressure signal are shown in Figure 32. The f igure shows what appears to be a large natura l ly occurr ing vortex entering the 40 lower port ion of the fan general ly near 5 o 'c lock pos i t i on . I t i s ind icated by a l oca l i zed "hump" on the polar p lo t . I t can be seen a f te r examination of Figure 32 that th is cha rac te r i s t i c hump (marked fo r convenience) is recurrent over time and pers is ts for several revo lu t ions . The camera shutter speed was 1/50 sec and the f i lm speed was 18 frames per sec. Therefore each exposure displays approximately one revolut ion of the fan rotor . Two revolut ions occur between frames. In the ser ies of exposures shown in Figure 32, the charac te r i s t i c hump pe rs i s t s for about four consecutive frame:-, or twelve revolut ions of the ro tor . The double-humped bl^de pressure signature produced by the vortex i s i nd ica t i ve of a vortex of pos i t i ve sense ( i . e . counter-clockwise rotat ion) entering the fan inboard of the pressure tap. ' I t possesses a s i g n i f i c a n t ax ia l ve loc i t y d e f i c i t . The pos i t i ve sense vortex c i r c u l a t i o n causes a decrease in angle of attack at the i n s t r u -mented blade sec t i on , and an increase in pressure on the blade upper sur face. However, as the blade passes through the core of the vortex i t experiences a decreases in ax ia l ve loc i t y and a resul tant increase in angle of a t tack. This explanation accounts fo r the dimple (suction) between the two humps. The spat ia l extent of the natural vortex var ied although i t usual ly encompassed about 60° on the i n l e t plane, with a core radius covering about 30° or 1.48 inches. In cont rast , the largest vortex generated by the ha l f delta' wings possessed a smaller core radius of 1.15 inches. 41 The fac t that the double humped pressure signature i s recurrent over time indicates that the vortex is moving cont inua l ly " over the face of the fan in a preferred area (the lower sector of the i n l e t ) . The group of f i ve marked frames shown in Figure 32 not only show th i s motion of the vortex, but also suggest that the vortex strength i s changing with time. The magnitude of the pressure signature increases and then decreases as successive frames are encountered. These i r regu la r increases and decreases in the rate of vortex ro ta t i on , coupled with the varying pos i t ion of vortex entry into the fan are probably responsible for the low frequency modulations in the fa r f i e l d overal l .sound level which were r e l a t i v e l y sporat ic and a-s large as 6 db over a time in te rva l of one sec. I f the vortex were as strong and pers is tent as the photo-graphs ind ica ted , then i t could only have been generated by the gathering of v o r t i c i t y from the boundary layer on a nearby surface ( i . e . , a f l oo r vor tex) . The c losest and la rgest surface present was the f l oo r of the laboratory outside the hatch of the anechoic chamber. Fine styrofoam beads were spr ink led over the f l oo r while the fan was running at maximum flow ra te . The beads were seen to rotate over the surface of the f l oo r in concentr ic clockwise t r a j ec to r i es , t h e i r rate of ro ta t ion var ied somewhat, in an e r r a t i c fashion. Also the pos i t ion on the f l oo r over which the c i r c u l a t i o n centered, tended to move about, as the vortex wandered to regions of high v o r t i c i t y concentrat ion. 42 3.4 Remarks on Experimental Analys is The experiments described above have provided a good i n f o r -mation base on the problem o f . ro to r /vo r tex i n te rac t i ons , but the resu l ts obtained are only appl icable to the physical condi t ions under which the experiments were conducted. Some of the experimental resu l ts are func t iona l l y dependent upon a group of parameters, rather than j us t one. A mathematical model would help in the extension of the resu l t s beyond the experimental l im i ta t ions and in the determination o f how the vortex physical propert ies a f fec t rotor no ise. 43 4. MATHEMATICAL ANALYSIS A 1inearized,two dimensional model was developed to describe the in te rac t ion of a fan rotor with a s ing le i n l e t vortex. The model leads to predic t ions of the d iscrete tone noise leve ls and helps to expla in some of the resu l ts obtained in the previous sec t ion . Further-more, the model i s general enough to be used as a bu i ld ing block for other more complex combinations of vor t ices and ro to rs . 4.1 General Method The sect ional load var ia t ion i s a funct ion of span-wise pos i t ion for a s ing le blade as i t completes one revo lu t ion. This loading i s then appl ied to a l l seven blades taking into considerat ion the appro-pr ia te time lag for vortex encounter by each blade. This rotor loading funct ion i s then used to predic t the far f i e l d overa l l and d iscrete tone sound pressure l e v e l s . A completely closed-form ana ly t ic approach was attempted, but proved to be too complicated in end form to be of general use. Therefore, a hybrid so lut ion was developed. The sect ional loading on a blade was determined a n a l y t i c a l l y by superimpos-ing the inc ident ve loc i t y f i e l d s . The remainder of the problem (Four ier ana lys is and rad ia t ion ca lcu la t ions) was solved numerical ly using the UBC IBM 370 computer. 44 4.2 Mathematical Formalism The geometry of the vortex entry into the fan i s shown in Figure 33A. The vortex i s f i xed in space, enter ing the fan at a rad ia l pos i t i on , R Q , and at a 3 o 'c lock c i rcumferent ia l angle (Z=0). Counter-clockwise c i r c u l a t i o n i s taken as pos i t i ve . The vortex tangen-t i a l ve loc i t y v(r) i s a funct ion of i t s rad ius , r, measured from the vortex centre. The fan i s spinning with counter-clockwise rotat ion and angular ve loc i t y so. I ts t i p radius i s R^ . and hub radius R^  (the fan rad ia l var iab le i s R) . 4.2.1 Fan Blade Geometry In the analys is the sect ional l i f t on each blade was calcu lated as a funct ion of rad ia l pos i t i on . . Radial changes in blade shape resu l t from var ia t ions in chord, tw is t , and zero l i f t l i n e . These were measured and accounted for as fo l lows. The chord var ia t ion was modelled as a l i nea r funct ion. C. h = C h + D(R t - R) . . . . . (4.1) 0 The change in twis t and zero l i f t l i ne were lumped together to give an e f fec t i ve l i nea r tw is t funct ion 3 = 3 n + G(R . - R) •(4.2) 45 The equation describes the measured var ia t ion in blade twist over the span. I t was found to increase l i n e a r l y as the blade root was approached. The sect ional shape of the a i r f o i l was measured and found 31 32 to be closed to a NACA 4412 wing sect ion ' , which has a l i f t curve slope of 5.72 per radian over a range of Reynolds numbers from 83,000 to 3,000,000. The fan blade sect ional Reynolds' numbers var ied from 340,000 to 560,000 for the tes t flow ra te . 4.2.2 Ve loc i ty F ie lds The blade was subjected to two pairs of orthogonal ve loc i t y components re la t i ve to the moving blade element. The steady components comprise the ax ia l ( inf low) ve loc i t y and the loca l rotat ional ve loc i t y . Both of these steady components were modulated by ve loc i t y disturbances imposed by the time varying vortex f i e l d ( re la t i ve to the blade). The ax ia l ve loc i t y was modulated by the ve loc i ty d e f i c i t in the vortex core. The v e l o c i t y , coR, resu l t ing from the fan rotat ion was modified c y c l i c a l l y by the c i r cu la to ry ve loc i t y f i e l d of the vortex. The ax ia l ve loc i t y d e f i c i t of the vortex core was modelled by f i t t i n g a Gaussian funct ion to measured p ro f i l es of the d e f i c i t . The form of equation which su i ted the measurements of the burst vortex cores was U c ( r ) = U c e " H ( r ~ ) 2 . . . . (4.3) 46 Here U (r) i s the loca l ve loc i t y d e f i c i t and U i s the maximum value of the d e f i c i t at the vortex centre. The vortex core radius i s given by r c > and H i s an empi r i ca l l y derived constant which describes the rate of f l a r e of the d e f i c i t with vortex radius. A comparison of th is funct ion with experiment i s shown in Figure 20. The mathematical form of the tangential ve loc i ty f i e l d was discussed e a r l i e r and i s given by v(r) = (1 - e 4 v T ) . . . . . (4.4) In ca lcu la t ing the sect iona l l i f t versus time, we are only in terested in the component of v( r ) which is tangential to the blade element t ra jec tory ( i . e . that which produces an upwash f luc tua t ion on the loca l blade element). This component acts perpendicular to the span of the blade and i s given by v(r) cos $ (see de f i n i t i on of Figure 33A). I t contr ibutes to the instantaneous l i f t on the blade, whereas the span-wise component i s inconsequential to the l i f t , according to our two dimensional model. In terms of the fan parameters, t h i s upwash component of the vortex ve loc i t y may be expressed as fo l lows: _l_ r R • v(r) c o s * (1 - e 4 v f ) ( " 0 ° ° S ) . . . . .(4.5) where: 47 IT - 2R R n cos I + R, '0 2 . (4.6) The angle Z locates the pos i t ion of the quarter chord span l i n e in the rotor plane. The ro ta t iona l ve loc i t y of the fan remained constant, but the ax ia l v e l o c i t y , U( r ,u ) t ) , was found to decrease r e l a t i v e l y l i n e a r l y with decreasing fan radius at the one fan operating condi t ion used. I t was decided to model th i s e f fec t with the equation Here UQ i s the ax ia l ve loc i t y measured at the fan rotor outer extremity. The ve loc i t y drop to the rotor hub amounts to 18 percent. Figure 33B depicts the var ia t ion in blade re l a t i ve ve loc i t y due to the vortex Note that U(R,t) incorporates the c y c l i c va r ia t i on of the ax ia l ve loc i t y 4.2.3 The Quasi-Steady L i f t Equation , . The instantaneous sect ion l i f t on the blade may now be expressed using the l i f t equation of l i nea r i zed aerodynamics: ^ ( R , t ) = 1 p V 2 r e l ( R , t ) Oc )a (R, t )C h (R) . (4.8) U(R,t) = U Q - M(R t - R) - U q ( r ) . . . . (4.7) ve loc i t y components, which w i l l occur upon each vortex encounter. d e f i c i t , U c ( r ) . dC where V^ e l(R , t ) = U 2(R , t ) + (coR - v(r) c o s $ ) 2 48 •(4.9) The instantaneous angle of attack a (R , t ) = [S ( r ) - ij/] - a ' (4.10) a Q - a . (4.11) where a.g i s the steady angle of attack a ' = Tan •1 U(R , t) coR - v(r) cos $ Tan •1 U(R,t) coR = Tan -1 U v(r) cos $ p ? —Y to R - coR v(r) cos a + U (4.12) (4.13) Having expressed the c y c l i c var ia t ions of v e l o c i t y and angle of attack in terms of known geometric propert ies of the f an , the time varying l i f t over the ent i re blade i s wr i t ten as : L( t ) dC 2 P da 0 f t 1 (k) r [U 2 + (coR - v(r) cos $ ) 2 ] ' n [ a 0 - ( Uv( r) cos $ co2R2 - coR v ( r ) cos $ + U 2 -)]dR. . . (4.14) L( t ) is a repe t i t i ve pu l se - l i ke funct ion of time fo r each blade. The amplitude of the c i rcumferent ia l l i f t va r ia t ion for each blade i s i d e n t i c a l . However, because the blades encounter the vortex one a f te r the other , the l i f t f luc tua t ion on one blade leads that of the fo l low-49 ing blade by 1/B of the period of revo lu t ion . Therefore, the method of ana lys is was to sum the contr ibut ions to the fa r f i e l d noise from l i f t f l uc tua t ion on each of the blades. (As shown in the fo l lowing Sect ion) . 4.2.4 The Acoust ic Transfer Function In terms of the f luc tua t ing force on the B blades o f the fan ro to r , the fa r f i e l d acoust ic pressure i s given by where 6 i s the angle between l i f t vector , L ( t ) , and the f a r f i e l d measur-ing point . F i s the period of revolut ion of the fan . This resu l t 33 comes from Curies surface in tegra l so lu t ion . I t s o r i g i n i s b r i e f l y described in Sect ion 1 of Appendix A. Equation 4.15 sums the f l uc tua -t ing l i f t s from the B blades. I t i s v a l i d only where the distances from each blade to the fa r f i e l d measuring point are equal ( i . e . an on-axis evaluat ion po in t ) . I f an o f f - a x i s point i s chosen, we must sum the l i f t f luc tuat ions from the blades accounting fo r the phase d i f ference caused by the va r ia t i on in path length f o r each b lade. This problem w i l l not be considered in the present work. f i e l d , on -ax is , acoust ic pressure in terms of the f l uc tua t ing rate of change of l i f t on a s ing le blade. p(R,t) = cos 9 4TTRC . . . . . (4.15) Using Equation 4.15 i t i s poss ib le to ca lcu la te the fa r 50 The notion of using the to ta l blade force f l uc tua t ion to ca lcu la te acoust ic rad ia t ion i s va l i d i f the compact source condi t ion i s met. That i s , the quarter wave length of the frequencies of i n -teres t must be long compared to the di f ferences in path length to the fa r f i e l d measuring point between the fan t i p and hub. This condft ion holds for the range of frequencies being considered here. The aerodynamic l i f t t rans fer funct ion dC^/da i s a funct ion 15 of reduced wave number k , which i s i t s e l f dependent on frequency and the rad ia l pos i t ion of the blade sec t ion . Idea l ly the wave number spectrum of the c y c l i c sect ional incidence var ia t ion should have been ca lcu la ted at each blade sec t ion . The ca lcu la ted amplitudes would then have been mu l t i p l i ed by the appropriate value of the l i f t t rans fer func t ion . However, th is procedure proved to be lengthy and very cos t l y in computer time. Calcu la t ions of the rate o f change of l i f t over the blade sect ions showed the maximum ef fec ts were at the fan rad ia l pos i t ion where the vortex was being ingested (Figure 34). For th is reason i t was argued that a span-wise var ia t ion i n the magnitude of the' l i f t t rans fer funct ion would have l i t t l e e f fec t on the integrated blade loading. We therefore used a frequency weight-ing o f dC^/da appropriate to the blade chord and re l a t i ve ve loc i t y at the point o f vortex entry. 51 The resu l t ing acoust ic pressure s ignal i s expanded in a Fourier se r i es . The re la t ionsh ip between the rate of change of l i f t on the blades; and the fa r f i e l d acoust ic pressure i s l i nea r . Therefore the l i f t t ransfer funct ion i s car r ied through the analys is to the point where the acoustic spectrum i s evaluated. At th is time i t i s incorporated into the analys is as a funct ion of frequency. An elaborat ion of th is method fo l lows . 4.2.5 Inclusion of Sears ' Function Because the Sears' funct ion (dC^/da) i s not a funct ion of time i t may be taken outside the summation s ign in Equation 4.15 ( re fe r to Equation 4.14 for the dependence of L( t ) on dC^/da). The process for inser t ing the appropriate values of the funct ion into Equation 4.15 i s to expand the per iod ic fa r f i e l d acoustic pressure into a Fourier ser ies using a constant value for dC^/da ( i . e . the l i f t curve slope for zero frequency l i m i t , k 0) oo p(R,t) = f 0 + £ A„ cos n out + B„ s in n cut . . . .(4.16) Sears ' funct ion i s a funct ion of reduced wave number k . . . .(4.17) For each d iscre te frequency, co, there is a appropriate correct ion fac tor 52 The fa r f i e l d d iscre te frequency tone leve ls P n (R,o) ) = C • . 2 + B 2 ... .(4.19) n n and the correct fa r f i e l d acoust ic pressure i s p( r , t ) = Cn ^ + Z c A cos nu t + CB s inntot . . . .(4.20) u c n n n n 4.3 Results of the Ana ly t i c Study The nature of the model i s such that there ex i s t i n f i n i t e p o s s i b i l i t i e s for the var ia t ion of fan and vortex parameters. In order to i l l u s t r a t e the most essent ia l features of the i n t e r a c t i o n , a planned approach to the v e r i f i c a t i o n and app l i ca t ion of the model was devised and fo l lowed. The v e r i f i c a t i o n consisted of examining resu l t s a t various steps of the development of the problem, and eventual ly of comparing the ca lcu la ted overa l l and d iscre te tone leve ls with those measured (as discussed in Sect ion 3). In th i s way the physical elements of the problem were revealed and assessed at each point of the development. The degree of agreement with experimental resu l t s gave a f i n a l tes t of the v a l i d i t y of the model. 53 The model was used to determine the cont r ibut ion of each of the vortex propert ies (ax ia l ve loc i t y d e f i c i t , core rad ius , and vortex strength) to the overa l l and d iscrete tone sound pressure l e v e l s . The e f fec t on these leve ls of varying the rad ia l pos i t ion of vortex entry into the fan was also determined. 4.3.1 Empir ical Constants and Numerical Solut ion In Section 4.2 expressions were developed to describe the blade tw i s t , chord, inf low ve loc i t y p r o f i l e , and ax ia l ve loc i t y d e f i c i t At the fan operating point chosen for the experimental work, the empir ical constants in the expression were determined as fo l lows: Chord (Equation 4.1) C. = .287 f t D = .177 Blade Twist (Equation 4.2) 0 O = .388 radians 6 = .238 rad ians / f t Ax ia l Ve loc i ty De f i c i t s (Equation 4.3) H = 3.65 U„ = i s d i f fe ren t for each vortex Inflow Veloc i ty (Equation 4.7) U Q = 63.7 f t / s e c M = 20.2/sec . The numerical so lu t ion to the problem uses s i x t y s tat ions over the span o f the fan blade and four hundred twenty around the circumference of the fan. Simpson's ru le i s employed to integrate l i f t over the span of the blade. A f i r s t centra l d i f ference method determines the f i r s t der iva t ive of the integrated l i f t with respect to t ime. ' • - \ / • ' • ' • . > — 7 • •; The UBC computer l i b r a r y programme en t i t l ed FCF car r ies out the Four ier ana lys is of the fa r f i e l d acoust ic pressure. I t i s a curve f i t t i n g rout ine which ca lcu la tes a Four ier ser ies f i t to a spec i f ied funct ion using the fas t Fourier transform alogr i thm. The ca lcu la ted Four ier ser ies coe f f i c i en t s give the real and imaginary parts of the spectrum. Their modulus renders the d isc re te tone acoust ic pressure amplitudes. 4 .3 .2 V e r i f i c a t i o n of the Model In t h i s sec t i on , there w i l l be frequent reference to the loading on the blade sec t ions . The reader i t reminded that the sect ional l i f t was ca lcu la ted using a constant value f o r the l i f t t ransfer funct ion (5 .76/ rad ian) . The e f fec t on the so lu t ion of the wave number dependence of the l i f t t ransfer funct ion was not incorporated into the model un t i l the f a r f i e l d acoust ic pressure was expanded in a Four ier s e r i e s . The magnitude of the loading curves therefore,represent the l i f t exc lus ive of the e f fec ts of the unsteady aerodynamics. Figure 35 shows the time dependent loading over the span of the blade at the c i rcumferent ia l angle Z = 0. In the case shown, the vortex sense is negative (opposite rotat ion to the fan) and the curves are p lot ted so that the contr ibut ions of the vortex c i r c u l a t i o n and ax ia l ve loc i t y d e f i c i t may be viewed separately. The vortex and fan condi t ions were se lected to match the experimental test employing the 5.5 inch ha l f de l ta wing at an angle of attack of -20° ( T = -20 f t 2 / s e c , rQ = .065 f t , U"c = 47.8 f t / s e c ) . The vortex c i r c u l a t i o n e f f ec t i ve l y decreases the steady loading inboard of the core and increases i t on the outboard side. The ax ia l ve loc i t y d e f i c i t causes a s i g n i f i c a n t increase ir. loading in the region of the vortex core (the resul tant increase in angle of attack over-r ides the decrease in the re la t i ve v e l o c i t y ) . However, i t has no strong e f fec t outside the vortex core l i m i t . The c i rcumferent ia l va r ia t ion in sect ional loading for the same fan condit ions is shown in Figure 36. The s o l i d l i nes show the loading on a blade sect ion (R/R^ = -94) outside the vortex core fo r pos i t i ve and negative values of vortex c i r c u l a t i o n sense. The ef fec t of the ax ia l ve loc i t y i s not f e l t at th is pos i t ion of fan radius. However, the loading on a blade sect ion that passes through the vortex core region i s quite d i f fe ren t (R/R t=.83 , dashed curve). The ax ia l ve loc i t y d e f i c i t creates a sharp spiky increase in l i f t . The contr ibut ion from the vortex ve loc i t y i s i n s i g n i f i c a n t in th is region. According to Equation 4.15, the acoust ic rad ia t ion from each blade depends on the rate of change of force on that blade. The der iva t ive of the c i rcumferent ia l to ta l blade l i f t with respect to time i s shown in Figure 37, fo r both pos i t i ve and negative vortex sense. Both curves are odd funct ions symmetric about the vortex core c i rcumferent ia l pos i t ion (E = 0) the two antisymmetric spikes in the region of the core resu l t from the ax ia l ve loc i t y d e f i c i t . The pos i t i ve and negative magnitude shoulders outboard of the spikes are caused by the pos i t i ve ( s o l i d l i ne ) and negative (dashed l ine) sense vo r t i ces . By evaluat ion of Equation 4 .15, the resul tant fa r f i e l d pressure s ignal i s obtained due to the l i f t f luc tuat ions on the seven bladed ro tor . The waveform for one complete rotor cyc le i s shown in Figure 38. The corresponding pure tone spectra are given in Figures 39 and 40 for pos i t i ve and negative c i r c u l a t i o n sense respect ive ly . For comparison with the predicted l e v e l s , experimental values are shown as a s o l i d l i n e . The dotted l ines on top of the s o l i d l i nes at 2100 and 2520 Hz in Figure 39 ind icate that th is measured harmonic level dropped below the broad band noise l e v e l . The s o l i d l i nes rep-resent the leve l a f te r cor rect ion was made for the s i x percent band pass f i l t e r . The true d iscre te tone leve ls at these f r e -quencies may l i e anywhere in the dotted l i ne regime. The agreement between spectra i s good; that i s , general ly w i th in two to three db. Having followed the progression of the mathematical develop-ment through the i n i t i a l stages of the problem i t became apparent that the resu l ts made sense phys i ca l l y . The d i rec t i ons , r e l a t i ve magnitudes and spat ia l extents of the time varying l i f t forces due to the vortex encounter agree with our i n t u i t i v e expectat ions. The 57 agreement between experiment and theory fo r t h i s one case i s very respectable. The di f ference between predicted and measured overa l l leve ls in a set of twenty-f ive d i f fe rent comparisons ra re ly exceeded 5 db. 4 .3 .3 Var ia t ion of Sound Level with Vortex Entry Pos i t i on The rad ia l pos i t ion of vortex entry was var ied ( in the theor-e t i ca l model) from a minimum radius of .35 f t ( R 0 / R t - . 4 4 ) to a maximum of .70 f t (R Q /R t=.88) . The ca lcu la ted var ia t ions in overa l l SPL are shown in Figures 41 and 42 for negative and pos i t i ve vortex sense respect ive ly . Both sets of curves show an increase in level with an increasing rad ia l point of entry. For the case of pos i t i ve vortex sense (Figure 42) the rate of increase of overa l l SPL r i ses as the fan t i p i s approached ( i . e . , the curve i s concave upwards). However, the overa l l leve l fo r negative vortex sense (Figure 41) droops down as the maximum radius i s reached. The decrease i s caused by a drop out in the level of the fundamental blade pass tone occurr ing at a radia l pos i t ion of .70 f t ( R Q / R t = .88). An unexpected resu l t such as th i s one ( i . e . , fo r a vortex of negative sense) deserves some a t ten t ion . The far f i e l d acoust ic pressure i s d i r e c t l y proport ional to the varying blade l i f t (Equation 4.15) . By studying a sequence of the time h is tory of dL/dt curves for condit ions of increasing radius of vortex entry (approach-ing the drop-out rad ius ) , we obtain some ins igh t in to the physical nature of the anomaly. 58 For the negative sense vortex in accordance with Figure 35, i t i s evident that the increase in l i f t resu l t ing from core ve loc i t y d e f i c i t and outboard "upwash" e f fec t i s p a r t i a l l y cancel led by a reduction of l i f t due to the downwash e f fec t experienced inboard of the vortex core. For pa r t i cu l a r spectral components of the d i s to r t i on ve loc i t y f i e l d , i t i s possib le to imagine a perfect span-wise cance l l a t i on , such that dL/dt integrated over the blade span goes to zero, when the vortex penetrates the rotor d isc at a unique value of R. (Figure 43). The consequence of th is cance l la t ion i s evident in Figure 41 for the fundamental blade pass tone. On the other hand, for a vortex of pos i t i ve sense,.such a dropout cannot occur because the upwash f i e l d i s now inboard of the vortex core. Moving the vortex towards the t i p increases the amount of upwash induced l i f t , and produces an increase in the rate of change in SPL. This e f fec t produces the concave up curve observed fo r the overa l l SPL, blade pass frequency, and i t s f i r s t harmonic (Figure 42). 4.3.4 Var ia t ion of Sound Level with Vortex Parameters A set of computer ca lcu la t ions were ca r r ied out to determine the e f fec t of vortex parameters on the overa l l sound pressure leve l and i t s pure tone spectrum. An a rb i t ra ry vortex of core radius 0.6 i n , ( r c / R T = . 0 6 3 ) ax ia l ve loc i t y d e f i c i t of 25 f t / sec ( U C / U Q = 39%) 2 and vortex strength of 20 f t / sec was chosen as a datum. The fan operating condit ions remained as described in Section 4 . 3 . 1 . One of the vortex parameters was var ied in magnitude whi le the other two were held constant. The resu l t ing sound leve ls were ca lcu la ted . 59 The parameters were var ied as fo l lows: core radius from .12 inches to 1.2 inches, ax ia l ve loc i t y d e f i c i t from 5 f t / s e c (7.8%) to 50 f t / s e c (48%), and vortex strength from 4 f t 2 / s e c to 40 f t 2 / s e c , fo r both pos i t i ve and negative vortex sense. The upper l i m i t s of these ranges corresponded to the maximum measured experimental values. The vortex entered the rotor at a rad ia l distance o f . 6 5 f t ( RQ /R^ = . 8 2 ; with in the previously noted dropout region) . This pos i t ion corresponded to the distance used in most of the experimental analys is of the vortex in terac t ion noise. The predicted overa l l and d iscrete tone sound pressure leve ls are p lo t ted against each varying parameter in turn. The overa l l l eve ls for pos i t i ve and negative vortex sense are shown in Figures 44 and 45 respec t i ve ly . They are plot ted against a combined parameter, r U r , fo r convenience only and not to j u s t i f y c c a l i nea r interdependence on the parametric grouping. As discussed already in Section 3, such a s i m p l i s t i c dependence i s probably not a r e a l i s t i c approximation fo r the general inf luence of the vortex propert ies on the noise l e v e l s . I t i s in te res t ing to note that in Figure 44 (r pos i t i ve) 4 ? for values of r u"c r, .greater than 25 f t /sec , the r £ and U c curves have approximately the same shape. Thus the overa l l SPL appears to be equal ly sens i t i ve to changes in U c and r . However, the sound level appears to exh ib i t much less var ia t ion with vortex c i r c u l a t i o n , 4 2 r . Below 25 f t /sec , the r c curve a lso f l a t tens out. In th is region the overa l l SPL seems to be only sens i t i ve to changes i s U c . 60 Figure 45 shows that for the case of negative vortex sense, the over-a l l SPL i s almost t o t a l l y independent of the vortex st rength, and equal ly sens i t i ve to changes in U c and r c . As previously discussed the vortex-induced up and down wash inboard and outboard of the vortex core causes resul tant increases and decreases in the span-wise loading on the fan blade (Figure 35), as the vortex i s encountered. In cases where the di f ference between pos i t i ve and negative loading i s la rge , changes in vortex strength w i l l contr ibute s i g n i f i c a n t l y to changes in the overa l l SPL. ( i . e . fo r R^/R^values approaching the hub or t i p ) . However, i f the c i r c u l a t i o n inf luence cancels almost per fec t ly along the span, then the e f fec t of changing the vortex strength w i l l be minimal, and the core d e f i c i t e f fec t w i l l dominate the noise rad ia t i on . The blade shape determines the sect ional loading. Therefore the cha rac te r i s t i cs of the design ( i . e . var ia t ion of blade chord from root to t i p , blade twist funct ion) and the rad ia l pos i t ion of vortex entry do have a profound inf luence on the degree of cance l la t ion that i s ac tua l l y r e a l i z e d . With these points in mind, and re fe r r ing to Figure 45 i t i s apparent that for the case of negative vortex strength (with core radius of 0.6 inches and entering at R Q / R ^ . 8 2 ) , the span-wise acoust ic rad ia t ion resu l t ing from the vortex c i r c u l a t i o n must be mutually cance l l i ng . A change in vortex strength produces very l i t t l e change in the overa l l sound pressure l e v e l . Figure 46 shows resu l ts for the fundamental blade pass frequency leve ls with negative vortex sense. The resu l ts here 61 confirm the foregoing explanat ion. As the core radius i s increased the sound level passes through a sharp minimum at a value of 4 2 r U c r £ equal to 25 f t / sec , the value fo r the datum vortex. Var ia t ion of the ax ia l ve loc i t y d e f i c i t or the vortex strength appears to change the value of the grouping r U c r c at which the drop out occurs. Computation of sound level versus r U c r c at other harmonic frequencies indicates that the fundamental tone i s the frequency most af fected by the cance l l a t i on , as was the case with the drop out,dependent on the pos i t ion of vortex entry (Figure 41). However, the curves in Figure 46 show that th is nul l may occur for a var ie ty of parameter combinations at one pos i t ion of vortex entry ( RQ/R^. = . 8 2 ) . The var ia t ion of the fundamental tone level fo r pos i t i ve vortex sense i s shown in Figure 47. For values of r U c r c less than 4 2 20 f t / sec , a change in the vortex strength causes the greatest 4 2 change in the tone l e v e l . For values greater than 20 f t / sec , changes i n r, r and U have roughly equivalent e f f ec t s , with r showing the strongest in f luence. Data s im i la r to that given in Figures 44, 45, 46 and 47 were ca lcu la ted and plot ted for the higher harmonic leve ls (840 Hz, 1260 Hz, 1680 Hz, 2100 Hz, and 2520 Hz). At the highest frequencies (2100 Hz and 2520 Hz), the sound leve ls were dominantly inf luenced by changes in ax ia l ve loc i t y d e f i c i t . The curves for 1680 Hz are shown in Figures 48 and 49. The harmonic level i s seen to be equal ly sens i t i ve to changes in both ax ia l ve loc i t y d e f i c i t and core radius. The lower harmonic amplitudes (840 Hz and 1260 Hz) are most sens i t i ve to changes in the vortex core rad ius. These curves are .tabulated in Appendix F. 4.3.5 Magni f icat ion of Input Errors The curves of SPL vs r U r a lso serve to ind icate the c c s e n s i t i v i t y of the model to errors in the spec i f i ca t i on of the input vortex parameters. The degree of confidence in the a b i l i t y of the model to predic t accurate noise leve ls depends upon the amount by which input errors in the vortex parameters are magnif ied. The overa l l sound leve ls for both pos i t i ve and negative vortex strength seem to be r e l a t i v e l y insens i t i ve to errors in the input c i r c u l a t i o n , but more susceptable to errors in the core radius and ve loc i t y d e f i c i t . Over estimation of the vortex c i r c u l a t i o n (using 20 f t /sec instead of 16 f t /sec) produces an error in the predicted overa l l level of only +1 db for pos i t i ve vortex sense and +2 db fo r negative sense. The frequency component most sens i t i ve to input er ror appears to be the fundamental, fo r negative vortex sense. Here a twenty percent er ror in the vortex c i r cu l a t i on strength would cause a 14 db error in the predicted l e v e l . This high magnif icat ion of error i s caused by the drop out discussed e a r l i e r where small changes in the d i s t r i -bution of blade loading cause large changes in output. 4.4 Remarks on the Mathematical Analys is The value of th is analys is of the vor tex/ ro tor in te rac t ion problem i s dependent upon the degree of real ism in the mathematical s imulat ion. There are numerous l im i ta t i ons on the simple l i nea r i zed approach taken here. However, the model i s viewed as an essent ia l f i r s t step to understanding the problem. A t . t h i s t ime, a more rigorous extension of analys is does not seem feas i b l e . The fo l lowing sect ion discusses the degree of agreement between theoret ica l and experimental r e s u l t s , and underscores some of the areas in which the mathematical model might be viewed as inadequate. 64 •5. DISCUSSION OF RESULTS Direct comparison of experimental and theoret ica l resu l ts can be made in the areas of c y c l i c blade loading, far f i e l d noise spect ra, and the dependence of the noise leve l on the radia l pos i t ion of vortex entry. These features of the study w i l l be discussed, and an attempt w i l l be made to explain the discrepencies between theory and experiment. The c a p a b i l i t i e s and l im i ta t i ons of the model w i l l emerge as a by product of the d iscuss ion . 5.1 Blade Loading I t was not possible to measure the chord-wise loading at a sect ion experimental ly. However, the time h is tory of the pressure f luc tua t ion of the imposed i n l e t vortex may be determined from the polar diagrams, as in Figure 30. Recall that for these experimentsj 2 the vortex parameters were r = 20 f t / s e c , U C / U Q = 75 % , r c = 0.65 f t and pos i t ion of entry, RQ/R^ = .82. The angular extent of the pressure increase on the upper surface of the a i r f o i l was about 60 to 70 degrees of ro ta t ion (measured from zero crossing to zero c ross ing) . The measured extent of the corresponding decrease in l i f t predicted by theory i s 72 degrees (Figure 35) . I t i s d i f f i c u l t to compare the magnitudes of the loading d i r e c t l y , as the chord-wise pressure d i s t r i bu t i on over the a i r f o i l at the instant of peak suction i s 65 unknown. However, i t can be approximated by using known t h i n - a i r f o i l loading funct ions. The measured decrease in suct ion on the a i r f o i l upper surface at 15 percent chord was .015 psi (an increase in the pressure c o e f f i c i e n t of .09 on a steady value of - 1 . 2 ) . This would correspond to an angle of attack change of about -0 .4 degrees on a two-dimensional th in a i r f o i l (dC^/da = 5.72). The corresponding de-crease in sect ional l i f t i s approximately 2.1 l b / f t . On our fan blade at the appropriate span-wise po in t , the theoret ica l minimum sect ional l i f t (corrected fo r the Sears' funct ion e f fec t ) was about 3.1 l b / f t fo r s im i l a r operating cond i t ions. The c i rcumferent ia l sect ional l i f t ( theore t ica l ) and pressure (experimental) va r ia t ions are p lo t ted fo r comparison in Figure 36. The experimental pressure p lo t i s r e l a t i v e l y symmetric about Z = 0 as i s the sect ion l i f t graph. There i s ac tua l l y a decrease in the pressure in the region of I = ir /4 . The negative f luc tua t ion in the surface pressure i s thought to be a resu l t o f the decrease in ve loc i t y i n the de l ta wing wake below the vortex centre (Figure 20). This anomaly in the a x i a l ; v e l o c i t y d e f i c i t p r o f i l e i s probably caused by the shear layer being shed from the t r a i l i n g edge of the del ta wing vortex generator. The double humped blade pressure signature observed on the polar p lots fo r na tura l l y occurr ing i n l e t condi t ions ( ca l l ed "natural vortex",, Figure 32) was simulated su rp r i s ing l y wel l with the ana ly t i c model. The resu l t i s shown in Figure 50. The rad ia l blade pos i t ion was selected in the mathematical model so that i t passed through the core of a pos i t i ve sense vortex, but f a r enough away (on the outboard side) from the vortex axis to ensure that c o n t r i -butions to the f luc tua t ing l i f t from both the ax ia l ve loc i t y d e f i c i t and vortex c i r c u l a t i o n were approximately equal. (U/U c =39.2%, 2 r = 20 f t / s e c , r = .065 f t , R /R =.82). The experimental pressure c 0 t signature was measured at a radius R Q / R ^ = .94. The spiky increase in l i f t in the core (predicted theore t ic -a l l y ) i s consistent with the experimental ly observed decrease in pressure. However, the "humps" of the experimental vortex are broader than those of the theoret ica l vortex. This resu l t suggests that the natural vortex i s larger than the theoret ica l vortex, but the actual extent of the natural vortex may not be determined using only one blade-mounted pressure transducer. 5.2 Far F ie ld Noise Spectra The theoret ica l model predicted the measured overa l l noise leve ls to wi th in 5 db for twenty-f ive comparisons with experimental data where the vortex propert ies and the pos i t ion of entry were var ied . The average discrepency was 3.5 db. In a l l cases, the measured leve ls were higher than the ca lcu lated ones. As the theory considers the in teract ion of the rotor and one vortex on ly , and the experiment involves the generation of noise in excess of the clean running fan noise leve l (80.7 db) , the base noise leve l of the fan adds to the noise produced by the rotor /vor tex i n te rac t i on . This may in part account for the discrepancy between theory and experiment. The comparison of leve ls at the d iscre te tone frequencies f e l l wi th in the 5 db l i m i t with two exceptions; at the blade pass frequency and the th i rd harmonic frequency (1680 Hz), for negative source strength. 67 At the fundamental frequency, fo r r = -20 f t / s e c , the theoret ica l predic t ion f e l l 10 db short of the measured value (Figure 4 0 ) . Here the vortex was being introduced in the t i p region of the blade ( RQ /R^ . 8 2 ) . AS previously discussed in Section 4, a drop-out occurs in the theoret ica l predic t ion of the fundamental leve l in the region. As the pos i t ion of entry of the vortex i s moved toward the hub of the ro to r , the agreement between theory and exper i -ment improves, to wi th in 3 db (compare Figures 29 and 41) . Experimental ly, a drop-out in leve l i s not seen to occur at large rad i i for negative vortex sense (Figure 2 9 ) . To understand th is d i f ference we reca l l the explanation of the drcp-out phenomenon given in Section 4. As the negative sense vortex is moved outward r a d i a l l y , the outboard, upwash-induced l i f t f luc tua t ion i s developed over a diminishing f rac t ion of the to ta l blade span, while the inboard downwash-induced negative l i f t f luc tua t ion grows increas ing ly dominant. Thus the l o c a l i z e d , co re -de f i c i t induced l i f t f luc tua t ion can be t o t a l l y cancel led at some c i rcumferent ia l posi t ions by the res idua l , c i r c u l a t i o n induced load f l uc tua t i on , (of opposite s ign ) , for a spec i f i c rad ia l pos i t ion of vortex entry. The foregoing p ic ture i s predicted by our idea l i zed two dimensional blade element theory. In actual f a c t , however, the outboard upwash e f fec t of the vortex does not merely vanish as the outboard segment of the blade diminishes in s i z e . As the vortex c i r cu la t i on f i e l d i s crowded into a narrowing space between the core f i lament and the bell-mouth w a l l , the loca l vortex v e l o c i t -ies are ampl i f ied , so that an almost to ta l cance l la t ion of vortex 68 c i r cu l a t i on e f fec ts remains operat ive. The residual core-induced l i f t f luc tua t ion ensures that a drop-out canno't occur. This explanation of the re la t i ve ro les of the c i r c u l a t i o n induced upwash e f f ec t s , and core d e f i c i t induced load f luctuat ions helps to explain the somewhat paradoxical resu l ts shown in Figure 26 and discussed in Section 3 . 3 . 1 . At small negative values of r , the increasing c i r c u l a t i o n seems to induce an increase in tone l e v e l . However, the fundamental tone sound pressure level was seen to eventual ly decrease as vor t ices of equal core rad ius , but increasing negative strength and decreasing ax ia l ve loc i t y d e f i c i t were introduced into the fan at a radius of -.65 f t ( RQ/R^= . 8 2 ) . As the ax ia l ve loc i ty d e f i c i t decreased, the contr ibut ion to the upwash produced l i f t on the blade also decreased. The e f fec t of the accompanying r i se in the magnitude of the vortex c i r c u l a t i o n tended to cancel over the span of the blade. The net change in l i f t over the span due to the c i r c u l a t i o n , i f any, was smal l . The resu l t ing decrease in ax ia l ve loc i ty d e f i c i t produced up-wash has the same ef fec t on the fa r f i e l d noise as does the decrease in upwash when a theoret ica l vortex i s moved r a d i a l l y outwards. That i s , there i s a decrease in the amplitude of the fundamental frequency. Discrepancies between theory and experiment at the th i rd harmonic ( in some cases a di f ference of 7 db) are not immediately expla inable. Figure 25 ind icates a rapid increase of sound level with negative sense vortex strength, whereas Figures 28 and 29 both show that as the vortex i s moved r a d i a l l y outwards, the th i rd harmonic 69 frequency becomes dominant over a l l tone l e v e l s , except f o r that of the fundamental (blade pass) frequency. 5.3 Radius of Vortex Entry . The agreement between theory and experiment fo r pos i t i ve (Figures 42 and 28 and negative (Figures 41 and29) vortex sense i s best near the blade t i p , with the exception of the fundamental tone f o r negative vortex sense. Inboard of the rad ia l pos i t i on R Q / R ^ - 7 , however, the t heo re t i ca l l y predicted sound leve ls tend to be lower than those measured. For example, at Rg/R^ = . 5 , the four th harmonic frequency (2100 Hz) experimental ly measures 69 db (negative sense vortex) whi le i t s t heo re t i ca l l y predicted leve l i s 48.5 db, f a r below the level produced fo r natural i n l e t condi t ion (no vortex) which was 66.4 db. Therefore, some other mechanism i s dominating the noise leve ls fo r small R, even when the a r t i f i c i a l vortex i s imposed. The experimental curves of harmonic sound leve ls tend to cross over and general ly show no simple trend as the vortex i s moved toward the t i p . In cont ras t , the predicted l eve l s increase smoothly as the t i p i s approached, and except for the fundamental frequency f o r negative vortex sense, do not cross over one another. This d isc rep-ency i s most surely a consequence of the combination of a vortex crowding e f fec t at la rger R, as discussed e a r l i e r , and a poss ib le non- l inear ( s t a l l i n g ) e f f ec t , to be discussed in the next sub-sec t ion . 70 5.4 Local B lade .S ta l l The cross-overs in the curves of harmonic leve l versus rad ia l pos i t ion (Figures 28 and 2 9 ) , together with the unusual increase in the th i rd harmonic sound leve l may be explained by a loca l s t a l l on the blade. This s t a l l would be induced by the sudden incidence change a r i s i ng from the ve loc i t y d e f i c i t in the vortex core. The vortex used to generate the curves of Figures 28 and 29 had a maximum ax ia l ve loc i t y d e f i c i t of 49.1 f t / s e c . When coupled with the steady ve loc i t y f i e l d s at each rad ia l pos i t i on , th is core d e f i c i t could produce a momentary increase in angle of attack which might exceed the c r i t i c a l s t a l l i n g angle fo r the blade sec t ion . The ca lcu lated momentary peak incidences range from 17 degrees at Rg/R t = . 4 2 to 16 degrees at RQ/R^ =.82. According to a i r f o i l measurements on blade 31 p ro f i l es s im i l a r to ours (NACA 4412) , s t a l l should begin at an angle of attack of about 12 degrees for our range of Reynolds' numbers (340,000 to 560,000). As mentioned in the Background Sec t ion , loca l blade s t a l l can provide a strong sound generating mechanism in he l i cop ter , ro tor / vortex in te rac t ions . However, the deta i led nature of such a phenomenon i s not eas i l y predicted theo re t i ca l l y because of uncertainty about the time dependent behaviour of three dimensional separated f lows. Since our vor t ices pass r ight through the rotor d i s c , there appears to be a high l i ke l i hood for such non- l inear l oca l i zed e f fec ts . If th i s conjecture were cor rec t , then i t would appear that f luc tuat ing blade s t a l l may. a f fect the harmonic leve ls in d i f fe ren t ways at 71 varying rad i i of vortex entry. For example, near the t i p ( R Q / R ^ . 8 2 ) the s t a l l could cause the observed emergence of the th i rd harmonic (1680 Hz) as the second stronaest tone in the noise spectrum (Figures 28 and 2 9 ) . Presumingthis statement to be t rue, l e t us t ry to explain the resu l ts of Figure 25 where at RQ/RJ. = . 8 2 , increases in the negative sense c i r cu l a t i on resu l ts in a more rapid increase in the th i rd harmonic leve ls than for the other pure tones. For the four ha l f del ta wings operating at angles of attack of 30, 25, and 20 degrees, the generated vor t ices a l l had "burst" cores and exhibi ted large ax ia l ve loc i t y d e f i c i t s at the rotor plane d is tance. I t i s at these large angles of hal f del ta wing incidence that we observe the rapid increase in harmonic level with increasing vortex strength. Perhaps the core d e f i c i t i s large enough to i n i t i a t e the c y c l i c s t a l l . The e f fec t of the tangent ial ve loc i t y of the negative sense vortex i s to increase the loca l angle of attack outboard of the vortex centre. Here the re l a t i ve v e l o c i t i e s are higher than on the inboard port ions of the wing. This s i tua t ion may be exaggerated by the boundary-induced crowding of the vortex rotat ional ve l oc i t y . The e f fec t of increasing r i s to increase the tangential v e l o c i t i e s , and so, there w i l l be a spreading of the extent of s t a l l in the t i p region. Since the re la t i ve ve loc i t y i s higher as we approach the t i p , the s t a l l becomes more intense with increasing R ^ . For a pos i t i ve sense vortex, the c i r cu la t i on tends to reduce the angle of attack outboard of the vortex core, and to increase i t on the inboard s ide. However, the increased angle of attack i s 72 in a region of lower re la t i ve v e l o c i t y , and so the noise radiated by the s t a l l , even i f i t occurred, would be expected to be less intense compared to the case for negative vortex sense. This physical explanation may account for the rapid increase in harmonic leve l with increasing vortex strength for negative sense vo r t i ces . A somewhat weaker increase i s observed for pos i t i ve sense vo r t i ces . Such a non- l i near i t y i n the blade l i f t funct ion would appear to have i t s greatest e f fec t at the th i rd harmonic at R Q / R T = -82. Perhaps th i s occurs because the c i rcumferent ia l spa t ia l extent of the separation i s such that i t contr ibutes energy p re fe ren t i a l l y to the tn i rd harmonic. Stated very s imply, at RQ/R^ - -82 the re la t i ve v e l -oc i t y seen by a blade sect ion i s 253 f t / s e c . For th is v e l o c i t y , the c i rcumferent ia l wave length that contr ibutes most e f f i c i e n t l y to the 1680 Hz harmonic i s 1.8 inches. The extent of the s t a l l producing ax ia l ve loc i t y d e f i c i t (core diameter), for the vor t ices generated, ranges from 1.3 inches to 2.3 inches. 73 6. CONCLUSIONS AND RECOMMENDATIONS The phenomenon of rotor /vor tex in terac t ion in an ax ia l f low fan was studied both experimental ly and t heo re t i ca l l y . The e f fec t of na tura l l y occurr ing and a r t i f i c i a l l y generated vor t ices on the unsteady blade loading and on the fa r f i e l d sound was examined. The quant i ta t ive agreement between predicted and measured overa l l and d iscre te tone sound l eve l s establ ished the value of using, a:- a f i r s t model, the case of a steady s p a t i a l l y f ixed vortex passing through a fan ro tor . The chosen model i s en t i r e l y l i nea r in i t s formulat ion, and does not account fo r boundary or blade to blade confinement e f fec ts . It i s incapable of pred ic t ing the non- l inear consequences of c y c l i c blade s t a l l , should these occur in r e a l i t y . There i s no evidence to confirm that such s t a l l ac tua l l y occurs when a large scale natural vortex i s drawn through a fan. However, where the point of vortex o r i g i n i s c lose to the fan (as i s the case when vor t ices or ig inate on a runway or on the fuselage of an a i r c r a f t and are drawn through the intake fan) the magnitude of the vortex ax ia l ve loc i t y d e f i c i t in the plane of the fan rotor may be great enough to cause a loca l s t a l l , and s i gn i f i can t increases in the tonal sound rad ia t i on . Measurements of the natura l l y occurr ing " f l oo r vortex" f i e l d showed that the i n l e t disturbance f luctuated slowly and 74 randomly about cer ta in mean posi t ions of entry into the fan. There was a lso some low frequency modulation of the vortex strength, apparently being inf luenced by the large scale external disturbances in the aerodynamics laboratory. These time varying charac te r i s t i cs account for some of the observed i r r e g u l a r i t y in the overa l l and d iscrete tone l e v e l s . The mathematical model does not account fo r temporal modu-la t ions of vortex strength and pos i t i on . The theoret ica l acoust ic wave-form is steady with time. When the honeycomb was placed oyer the anechoic chamber i n l e t hatch, a 3 db decrease in the overa l l sound level was rea l i zed . Thus the ex terna l ly generated " f l oo r vortex" was at leas t p a r t i a l l y suppressed. However, i t i s probable that addi t ional i n l e t v o r t i c i t y was s t i l l being generated from corners and obstruct ions wi th in the chamber i t s e l f . Therefore, a more r e a l i s t i c model would be one incorporat ing more than one i n l e t vortex. Each would possess time varying vortex propert ies and pos i t ion of entry into the fan. This model would then exh ib i t the band spreading of the d iscrete tones observed experimental ly, and would also produce a broad band contr ibut ion to the noise spectrum (as in Hanson ). A r t i f i c i a l l y generated vor t ices were employed to check the v a l i d i t y of the model. In r e a l i t y , however, the model should be useful for descr ib ing the in teract ion of natural v o r t i c i t y with any rotor . A knowledge of the structure of natura l ly occurr ing vor t ices i s there-fore required. Vortex cha rac te r i s t i cs may be determined by the simultaneous measurement and polar d isplay of blade surface pressure s ignals from more than one su r face .pos i t i on ; or a l t e rna t i ve l y by using modern techniques of laser velocimetry (non- intrusive detect ion of ve loc i t y f i e l d s ) . Comparison of the two or more polar p lots at the same instants of time over a period of many fan revolut ions would enable the t racking of the magnitude of the vortex strength and i t s pos i t ion of entry. Evaluation of the spat ia l extent of the blade/vortex i n te r -act ion i s possible using c ross -cor re la t ion between two surface pressure s igna ls . Measurement of the pressures near the blade t r a i l -ing edge might a lso ind icate i f the natura l l y occurring vortex i s producing a l o c a l , time modulated s t a l l . The vortex c i r c u l a t i o n and ax ia l f low d e f i c i t s combine to produce a counter-phase unsteady loading over the blade span. The descr ip t ion of the instantaneous d i s t r i bu t i on of blade load serves to confirm the v a l i d i t y of the source strength curve (Figure 7) obtained by Leggat and Siddon. The postu lat ion that a f l oo r vortex was entering the fan at a preferred pos i t ion ( R Q / R t = . 7 5 ) now appears to be confirmed. The resu l ts of the experiment and theory indicated that the changes in the fa r f i e l d sound level resu l t ing from changes in vortex strength were confined to the blade passage frequency (neglect ing the e f fec t o f s t a l l ) . The magnitude and spat ia l extent of the ax ia l ve loc i t y d e f i c i t determined the sound leve ls at the higher harmonic f requencies. Previous models of inf low d is tor t ions i n te r -act ing with r o t o r s ^ ' 6 ' 1 0 have been successful in predict ing the blade pass frequency and i t s f i r s t harmonic leve ls but have been 76 unable to ca lcu la te the observed peaks at the higher harmonic f r e -quencies. The present model, however, does predic t s i gn i f i can t sound leve ls with reasonable accuracy up to the f i f t h harmonic of the blade pass frequency for the cases considered. In view of the above d iscuss ion , the fo l lowing recommendations are proposed. 6.1 Theoret ical Extensions 1. The s ing le vortex model should be ref ined to include the e f fec ts of the be l l mouth boundary on the vortex tangential ve loc i t y . This addi t ion to the model may bemade by including an image vortex of opposite sense the same distance outboard of the blade t i p , as the real vortex i s inboard. 2. The rotor /vor tex in terac t ion model should be extended to include the e f fec ts of more than one vortex,and also to account for the time varying vortex propert ies and pos i t ion of entry.This modi f icat ion may be implemented by using random funct ions of vortex propert ies or a l t e rna t i ve l y by employing a s t a t i s t i c a l approach s im i la r to that car r ied out by Hanson 6. 6.2 Experimental Extensions 3. The present two channel instrumented blade should be redesigned to enable a more rapid means of re locat ing the pressure transducers in the blade. In the present system, the brass casing which protects the transducers must be placed in the mi l led blade s l o t so that the hole penetrating the side of the casing i s al igned with the hole communicating to the upper surface of the blade. A proper'alignment of the two holes proved to be d i f f i c u l t to achieve and therefore was time con-suming. Having completed the redesign, pressure measure-ments could be made conveniently at many blade surface loca t ions . Experiments to determine the st ructure of the observed f l oo r vortex, and to test for loca l blade s t a l l (as described above) could then be car r ied out. 4. Further invest igat ion into the o r i g in of the on-axis d iscre te tone no i se , i s required. E l iminat ion of the " f l o o r vortex" by inser t ing the honeycomb did not cause the blade passage frequency and i t s harmonics to disappear. The i n s t a l l a t i o n of a f ine wire mesh cone over the fan i n l e t may help to reduce v o r t i c i t y o r ig ina t ing ins ide the chamber. I f the mathematical model i s modi f ied, and i f the blade i n s t r u -mentation i s improved, a powerful means w i l l ex i s t for the continuation of research on the sources of d iscrete tone fan noise. 78 REFERENCES MANI, R. " Isolated Rotor Noise due to In let Dis tor t ion/Turbulence," Final Progress Report Prepared for the National Aeronautics and  Space Admin is t ra t ion, Lewis Research Centre, June 1974. GUTIN, L. "On the Sound of a Rotating P rope l l e r , " Translat ion of "Uber da Scha l l f e l d einer Rotierenden Luftschraube," Physika l ishe Zei tscher i f t der Sonjetunion, Band 9> Heft 1 * 1936; NASA Tech. Memo. No. 1195, 1945. MUGRIDGE, B.D. and MORFEY, C L . "Sources of Noise in Axia l Flow Fans," Journal of the Acoust ica l Society of America, Vo l . 51, No. 5, 1972, pp. 1411-1426. MANI, R. "Noise Due to the Interact ion of In let Turbulence with Isolated Stators and Rotors, " Journal of Sound and V i rba t ion , Vo l . 17, No. 2, July-August 1971, pp. 251-260. LOWSON, M.V. "Studies of Noise Radiation by Rotating B lad ing , " Proceedings of the Interagency Symposium on Univers i ty  Research in Transportat ion Noise, March 28-30, 1973, pp. 211-224. HANSON, D.B. "Spectrum of Rotor Noise Caused by Atmospheric Turbulence," Journal of the Acoust ical Society of America, Vo l . 56, No. 1, 1974, pp. 110-126. LEGGAT, L . J . Experimental Invest igat ions of On-Axis Discrete Frequency Fan Noise, M.A. Sc. Thesis , Univers i ty of B r i t i s h Columbia, October, 1973. SIDDON, T .E. ' and LEGGAT, L . J . "Blade Load Modulation as a Source of Discrete Frequency Fan Noise, " Proceedings of Internoise  '73, August, 1973, pp. 176-185. HODDER, B.K. " Invest igat ions of the Ef fect of In let Turbulence Length Scale on Fan Discrete Tone Noise, " NASA Tech. Memo. X-62,300, September, 1973. 79 10. RAO, G.V .R. , CHU, W.T., and HODDER, B.K. "Rotor Noise Due to Inflow Turbulence," AIAA Paper No. 73-632, J u l y , 1973. 11. PLUCINSKY, J . C . "Quiet Aspects of the Prat t and Whitney A i r c r a f t JT15D Turbofan," Presented at the Society of Automotive Engineers Business A i r c r a f t Meeting, Wich i ta , Kansas, A p r i l , 1973. 12. COLEHOUR, J . L . and FARQUHAR, B.W. " In le t Vortex," Journal of  A i r c r a f t , Vo l . 8, No. 1, January, 1971, pp. 39-43. 13. SOFRIN, T.G. and McCANN, J . C . "Pra t t and Whitney Experience in Compressor Noise Reduct ion," Presented at the 72nd Meeting of the Acoust ica l Society of America, 1966. 14. FILLEULs N. l e . S. "An Invest igat ion of Ax ia l Flow Fan Noise, " Journal of Sound and V ib ra t ion , 3(2) , 19.66, pp. 147-165. 15. SEARS, W.R. "Some Aspects of Non-Stationary A i r f o i l Theory and Its P rac t i ca l A p p l i c a t i o n , " Journal of the Aeronaut ical . Sciences, 8(2) , February, 1941, pp. 104-108. 16. LOWSON, M.V. "Theoret ical Analys is of Compressor No ise , " Journal  of the Acoust ical Society of America, 47(2), 1970, pp. 371-385. 17. LOWSON, M.V. "The Sound F ie l d fo r S ingu la r i t i es in Mot ion," Proc. Roy. S o c , (London), A286, 1965, pp. 559-572. 18. SIDDON, T .E . "Surface Dipole Strength by Cross Corre la t ion Method," Journal of the Acoust ica l Society , Vo l . 53, No. 2, February, 1973, pp. 619-633. 19. RACKLjR. Two Causal i ty Cor re la t ion Techniques Appl ied to Jet  Noise, Ph.D. Thesis , Univers i ty of. B r i t i s h Columbia, A p r i l , 1973. 20. LEVERT0N, J.W. and TAYLOR, F.W. "Hel icopter Blade S l a p , " Journal of Sound and Vibrat ion 4(3) , 1966, pp. 345-357. 80 21. PATTERSON, W.R. , AMIET, R.K. , and MUNCH, C L . " Iso lated A i r f o i l Tip Vortex Interact ion No ise , " Presented at the AIAA 12th Aerospace Sciences Meeting, January, 1974. 22. WIDNALL, S. "Hel icopter Noise Due to Blade-Vortex In te rac t ion , " Journal of the Acoust ica l Society of America, Vo l . 50, No. 1, 1971, pp. 354-365. 23. SIDDON, T . E . , HOGLUND, L . , and DAVIS B. "Va l ida t ion Report; UBC Anechoic Chamber and Jet Noise F a c i l i t y , " Submitted to General E l e c t r i c Company L imi ted, A i r c r a f t Engine Group, November, 1974. 24. SMITH, R.H. and WANG .C. "Contract ing Cones Giving Uniform Throat Speeds," Journal of the Aeronautical Sciences, October, 1944. 25. ELLE, B . J . "An Invest igat ion at Low Speed of the Flow Near the Apex of Thin Delta Wings with Sharp Leading Edges," Performance Sub-Committee, Aeronautical Research Council ( U . K . ) , Perf . 1621, F.M. 2629 , January, 1958. 26. MARSDEN, D.J.. , SIMPSON, R.W., and RAINBIRD, W.J. "An Invest igat ion into the Flow over Delta Wings at Low Speed with Leading Edge Separat ion, " Report No. 114, The College of Aeronaut ics, C ran f i e l d , U . K . , February, 1958. 27. FINK, P.T. and TAYLOR, J . "Some Low Speed Experiments with 20 Degree Delta Wings," Performance Sub-Committee, Aeronautical Research Counc i l , Per f . 1382, F.M. 2339, September 1955. 28. LAMBOURNE, N .C and BRYER,. D.W. "The Burst ing of Leading Edge Vort ices - Some Observations and Discussion of the Phenomenon," F lu id Motion Sub Committee, Aeronautical Research Counc i l , F.M. 3085, ARC 22, 775, A p r i l , 1961. 29. HALL, M.G. "The Structure of Concentrated Vortex Cores," Royal A i r c r a f t Establishment, Tech. Memo. Aero. 869, January, 1965. 81 30. SUN, Y .C. Experimental Invest igat ion of the F ie ld About Sharp- Edged and Rectangular Wings, M.A.Sc. Thesis, Univers i ty of B r i t i s h Columbia* December, 1961. 31. ABBOTT, I.H. and VON DOENHOFF, A . E . Theory of Wing Sect ions, Dover Publ ica t ions Inc . , New York, 1959. 32. JACOBS AND SHERMAN, NACA Report, No. 586, 1937. 33. CURLE, N. "The Influence of So l i d Boundaries Upon Aerodynamic Sound," Proc. Roy. S o c , A 231 , 1955, pp. 505-514. 34. LIGHTHILL, M.J . "On the Sound Generated Aerodynamically, I I , Turbulence as a Source of Sound," Proc. Roy. Soc. A 222, 1954, pp. 1-32. 35. KRAICHNAN, R.H. "Noise Transmission from Boundary Layer Pressure F luc tua t ions , " Journal of the Acoust ical Society of America, Vo l . 25, 1957, pp. 65-80. 82 APPENDIX A RADIATION SOLUTION AND SOURCE ANALYSIS BY CAUSALITY CORRELATION* In the diagnosis of mult istage systems such as turbomachines one i s frequent ly faced with determining which components of a system contr ibute most to the overa l l generate sound. Spectral analys is of the noise and systematic modi f icat ion of machine parts have in the past provided l im i ted information. A proven means of source l o c a l i z -at ion i s the causa l i t y co r re la t ion t e c h n i q u e ' ' ' ^ ' ^ . It establ ishes a causative re la t ionsh ip between ind iv idual noise source phenomena and the overa l l sound radiated in a given d i r e c t i o n ; thus y ie ld ing quant i ta t ive information on acoust ic source d i s t r i b u t i o n s , t he i r loca l spectra and scales of coherence. The theoret ica l development of the causa l i t y approach was car r ied out by Siddon. A complete der ivat ion of the technique,appl ied to surface generated aerodynamic noise>may be found in reference 18. A condensed version fo l lows. A l . Relat ionship Between Surface Pressure and Far F ie ld Sound • 33 Star t ing with Cur ies general ized so lu t ion to the L i g h t h i l l 34 equation , i t fo l lows that the sound resu l t ing from the in teract ion between a flow and surface S i s given by the fo l lowing far f i e l d _ This Appendix i s included to provide the reader with an abbreviated presentation of the der ivat ion of Equation 4.15 and an ex-planat ion of the technique used to obtain the curve in Figure 7. 83 P ( x . t ) . ^ I [ p U n c + ^ ^ ( f . + p u i U n ) ] dS (Al) S z~ c (Here we have discarded the volume-distr ibuted quadrupole sources). In cases where the surfaces are non-vibrat ing and have r i g i d steady motion, the terms invo lv ing surface v e l o c i t i e s u n and u^  vanish. I f the surface s t ress i s dominated by i t s normal component (and i t general ly 35 i s ) a simple re la t ionsh ip resu l ts p(x, t ) = ^ | cos e [ ( t - £ ) ] dS . . . . .(A2) S I f the dimensions of S are small compared to the wavelength of the highest frequency o f in te res t in the radiated sound, the var ia t ion in retarded time over the surface may be neglected, and R - x. The integral in Equation A2 now describes the resul tant force exerted by the f l u i d on the sur face. I f the surface i s an a i r f o i l in a f low, the force on the a i r f o i l may be resolved into instantaneous l i f t and drag force components. When the drag force i s neg l i g i b l e , Equation A2 may be wr i t ten as 84 P<*' = W [ f t ^ R * (A3) t " c A2. Causal i ty Formalism I f both sides of Equation A2 are mu l t i p l i ed by the acoustic pressure at a new time t ' , time averaging y i e l d s : p ( t )p ( t ' ) = ^ c o s e ( y ) [ ( y , t - f ) p ( x , t ' ) ] dS(y) . S <St — . . . .(A4) Then assuming p and p s are s t a t i s t i c a l l y s tat ionary random var iables P P ( X ) = " 4 H C " { C O S 0 [ p s p ( T - | ) ] dS; x = t - V . (A5) S We evaluate the mean square acoust ic pressure by se t t ing T = 0 85 P 2 (x) = 4TTXC cos 9 [ ^ p s p ] x dS . . . . .(A6) S t = " c Thus the contr ibut ion to the mean square sound pressure at a far f i e l d point x_, a r r i v ing from the surface element ds(y) , when p s i s being measured, i s given by the integrand of A5. This quanti ty may be viewed as the strength of the acoust ic source at that po int , and fo r the case where the dipole rad ia t ion from the surface i s predominant, (as is the case in app l ica t ions such as rotor no ise) , i t i s ca l l ed the surface dipole source strength 2 dp _ _ cos 8 [ £ PcP 3 x • • • -(A7) dS 4TTXC L <5T H S ~ c The d i s t r i bu t i on of source strength resu l ts in an acoustic model f i xed to the geometry of the surface. The surface may then be considered as an array of acoust ic sources of various strengths. I n i t i a l experiments showing the v a l i d i t y of th is approach 18 were car r ied out by Siddon in the inves t iga t ion of broad band noise radiated from a f l a t c i r c u l a r p late embedded in an a i r j e t . The success of these experiments pointed the way towards the analys is of more complicated s i tua t ions involv ing surface in te rac t ion noise Q as were car r ied out by Siddon and Leggat at a l a t e r date . Typ ica l l y the surface re la ted cor re la t ion function P S P ( T ) w i l l have a cha rac te r i s t i c antisymmetric shape for source f luc tuat ions which are broad band in spectral nature.This trend resu l ts because the acoust ic rad ia t ion i s proport ional to the time rate of change of the surface pressure. One merely evaluates the slope of many such functions at points on the surface, at appropriate time delays, in order to generate the contours of source strength over the surface. In cer ta in cases invo lv ing anticoherent sources, some elements of area may exh ib i t an apparent negative source strength. When source strengths are integrated over the sur face, these negative strengths w i l l cancel a port ion of the pos i t i ve st rength, but the resu l t ing integral w i l l always be pos i t i ve . In the specia l case where two counter phase sources exact ly cancel one another, there w i l l be no far f i e l d sound, and hence no co r re l a t i on . Where there i s a strong harmonic coupling between source and fa r f i e l d pressure spect ra , the resu l t i ng c ross -co r re la t ion funct ion w i l l not decay qu ick ly with time delay, but w i l l be per iodic in nature. A3. Per iod ic Corre la t ion Functions It i s useful to f i l t e r both the surface pressure and the far f i e l d s igna ls with frequency and phase matched f i l t e r s i f the fa r f i e l d spectrum is dominated by pa r t i cu l a r f requencies, or i f the d i s t r i bu t i on of a component frequency of broad band noise i s to be analysed. This approach allows for the examination of the d i s t r i -bution of sources producing the f i l t e r e d frequency. For f i l t e r e d source ana lys is , the re la t i ve pu lsat ion phases of sources at various posi t ions are important. 87 A4. Accounting for Phase Var ia t ion in the phase of f l u i d d i l i t a t i o n s on the surface at a pa r t i cu la r frequency can be most important when re la t ing the source d i s t r i bu t i on to the net overa l l sound (at that frequency) by in tegrat ion over the surface in quest ion. I f the source and far f i e l d s ignals are f i l t e r e d , then the c ross -co r re la t i on funct ion w i l l be s i nuso ida l , having a frequency iden t i ca l to the centre frequency of the band pass f i l t e r . The cor re la t ion funct ion can be represented mathematically by the expression. Here c|> i s the phase angle between the measured cor re la t ion funct ion and a cor re la t ion funct ion of the same period but with a zero crossing of negative slope at the correct time delay — . P S P ( T ) - | P s P j s in [u> (T - f) + <j>] . . . . .(A8) c D i f fe ren t ia t i on with respect to time gives ^ - P S P ( T ) = • - | P S P | (tocos M T - |)]cos<(> . . .(A9) evaluat ion at the correct time delay gives . . . .(A.10) 88 subst i tu t ion into A6 gives 2 dp _ cos 6 cos i p p . , f l r j , "dT ~ 4lF*c" w l p s p l • • • • - ( A l l ; Thus for a f i l t e r e d c o r r e l a t i o n , i t i s possib le to predic t the source strength at a pa r t i cu la r source loca t ion by knowing the amplitude of the c ross -co r re la t i on func t ion , the frequency, the angle between the surface normal and the fa r f i e l d po in t , the phase angle <j>, the speed of sound, and the distance to the f i e l d point x_. 89 APPENDIX B LOCATION OF THE VORTEX CORE In order to pos i t ion the pivot point of the ba l l vort imeter co r rec t l y , i t was necessary to know the locat ion of the vortex core centre re la t i ve to the t r a i l i n g edge of the ha l f del ta wing. A hot wire was traversed v e r t i c a l l y and hor i zon ta l l y through the vortex produced when the 5 1/2 inch wing was mounted in the wind tunnel . Two sets of orthogonal traverses were conducted. One with the wire oriented perpendicular to the ax ia l f low, and the other with i t pa ra l l e l to the ax ia l f low. The wire served to ind icate the point of minimum ve loc i t y : in the f i r s t case, fo r the resul tant of the ax ia l and vortex tangential ve l oc i t i es and in the second fo r the resul tant of the ve r t i ca l and hor izontal components of the vortex tangential ve loc i t y . The resu l ts of the two methods agreed wel l and are presented below. 90 a h a X Y 5° 5 ,1 127 26 3 32 '8 '32 32 16 10° 10 ,1 ,27 27 3 32 '4 '32 32 8 15° 15 32 l 5  116 , 3 *4 30 32 9 16 20° 17 32 i 5  116 111 116 1 21 32 25° 20 32 i 3  116 , 21 '32 1 13 16 30° 23 32 13 '8 ,17 '32 31 32 35° 24 32 , 7 16 , 7 16 32 4 APPENDIX G HALF DELTA WING VORTEX CHARACTERISTICS Angle of Attack r ( f t 2 / s e c ) r c ( f t ) U c ( f t / sec ) 4.25 inch wing 10 7.5 .053 42.7 15 11.25 .059 51.0 20 15.0 .065 46.5 25 18.75 .073 49.7 30 22.5 .081 56.6 5.5 inch wing 10 10.0 .057 28.1 15 15.0 .059 39.2 20 10.0 .065 47.8 25 25.0 .070 49.6 30 30,0 .078 49.1 6.875 inch wing 10 11.9 .056 -15 17.5 .063 38.2 20 23.75 .070 47.8 25 30.0 .079 51.0 30 36.25 ' .089 52.9 8.25 inch wing 10 13.5 .057 • -15 . 20.3 .065 -20 26.8 .074 38.3 25 33.75 .084 40.8 30. 43.7 .096 49.7 92 APPENDEX D DI. Data from Vortex Parameter Tests ( a l l l eve ls are SPL in decibels) Clean Fan Frequency Hz Overal l 420 840 1260 1680 2100 2520 80.9 72.2 66.1 68.9 69.6 • 66.4 63.8 Wing angle of attack +10° Wing Overal l 420 840 1260 1680 2100 2520 4 .25" 82.4 75.2 70.6 72.1 73.2 68.9 65.9 5 .5 " 82.3 75.2 69.9 71.1 73.3 67.7 65.5 6.875" 83.0 75.9 71.2 72.9 74.0 68.8 66.7 8 .25" 83.2 76.1 71.1 72.5 74.8 68.8 67.0 Wing angle of attack -10° Wing Overal1 420 840 1260 1680 2100 2520 4 .25" 81.7 74.0 70.3 70.5 71.8 67.0 65.0 5 .5 " 81.7 73.9 70.1 70.3 72.0 66.7 64.7 6.875" 82.3 74.6 70.3 70.6 72.2 67.3 65.9 8 .25" 82.0 74.2 69.7 70.3 72.7 67.0 65.8 Wing angle of attack +15° Wing Overal1 420 840 1260 1680 2100 2520 4 .25" 83.5 77.2 71.9 72.7 74.4 70.6 66.9 5 .5 " 83.0 76.9 70.1 71.6 74.4 70.0 67.7 6.875" 84.2 77.9 71.4 72.8 76.0 71.1 67.7 8 .25" 84.4 78.0 71.6 73.0 77.7 70.8 69.0 93 Wing angle of attack -15° Wing Overal l 420 840 1260 1680 2100 2520 4 .25" 82.0 75.3 70.6 70.2 71.3 67.6 65.6 5 .5 " 81.9 74.8 68.9 69.3 71.4 68.8 66.7 6.875" 82.5 74.6 68.2 69.8 73.5 69.3 68.5 8 .25" 82.7 73.4 67.2 70.9 76.0 70.7 67.7 Wing angl e of attack +20° Wing Overal1 420 840 . • 1260 1680 2100 2520 4 .25" 83.2 78.6 71.1 73.2 73.0 67.5 66.3 5 . 5 " 83.3 78.0 69.9 72.4 75.1 67.7 66.9 6.875" 84.2 78.3 71.4 72.7 77.6 69.1 68.0 8 .25" 84.6 78.7 11A 72.6 78.3 • 68.1 67.7 Wing angle of attack -20° Wing Overal1 420 840 1260 1680 2100 2520 4 .25" 83.2 77.2 72.0 71.1 73.6 70.1 69.4 5 . 5 " 83.4 76.6 70.0 70.2 76.5 71.8 68.8 6.875" 84.1 75.7 70.3 72.1 78.4 72.0 69.6 8 .25" 84.1 73.3 71.5 74.5 78.9 72.2 68.1 Wing angle of attack +25° Wing Overal l 420 840 1260 1680 2100 2520 4 .25" 84.1 80.2 70.8 73.4 73.4 67.4 65.7 5 .5 " 83.6 78.9 69.1 72.2 76.4 66.6 65.4 6.875" 84.9 79.6 72.7 72.7 78.0 67.7 67.0 8 .25" 85.0 79.5 75.0 73.9 78.6 67.9 67.1 94 Wing angle of attack -25° Wing Overal l 420 840 1260 1680 2100 2520 4 .25" 85.1 80.5 74.7 . 74.0 75.9 70.8 69.3 5 .5 " 84.8 79.9 73.6 70.5 79.1 70.5 70.2 6.875" 85.9 78.7 75.5 73.4 80.5 70.7 72.1 8 .25" 86.3 77.1 76.7 76.4 81.8 70.4 71.7 Wing angle of attack +30° Wing Overal l 420 840 1260 1680 2100 2520 4 .25" 85.6 82.7 70.4 74.0 75.2 69.2 66.2 5 .5 " 85.5 82.4 68.8 72.9 76.4 69.9 68.1 6.875" 85.6 82.1 72.6 72.4 76.1 68.3 67.1 8.'25" 86.2 81.3 77.6 75.2 77.5 68.8 67.0 Wing angle of attack -30° • Wing Overal l 420 840 1260 1680 2100 2520 4 .25" 87.1 83.5 77.6 76.7 76.7 69.7 71.2 5 .5 " 87.7 84.3 77.5 71.5 80.1 71.6 74.4 6.875" 88.2 83.0 78.4 73.7 82.8 73.3 74.1 8 .25" 88.9 82.0 78.6 78.8 84.0 76.5 75.0 D.2. Far F ie ld Noise Levels with Mounting Plates Inserted Plate , R R a d i u s ^ ) Overal1 420 840 1260 1680 2100 .2520 Clean fan 81.6 72.7 67.6 70.0 71.2 67.1 65.1 .94 81.7 72.4 68.0 69.9 71.7 67.2 65.2 .89 81.8 72.4 68.1 71.0 72.1 67.7 65.6 .84 81.8 72.7 68.1 69.9 72.5 67.9 66.0 . 78 81.7 73.0 68.8 70.3 72.7 68.1 65.9 .73 81.8 72.4 68.7 70.1 72.1 68.3 65.8 .68 82.2 73.4 69.5 70.1 72.9 68.3 66.1 .63 82.4 74.5 69.9 70.9 71.9 68.3 66.5 95 APPENDIX E PARAMETRIC ANALYSIS OF LIFT FLUCTUATION ON BLADE — The pressure in the fa r f i e l d produced by a force f l uc tua t i on on a surface i s proport ional to the time rate of change of the to ta l force act ing on the sur face. P - o T • . - - ( E D From Equation 4.14, L( t ) = 1 d C £ 2 p "do- Ck) U2 + wR V(r) cos <J>)2 a . U V(r) cos 4) w 2 R 2 + o)R V(r) cos 4. + (U dR (E2) The resu l ts of the theory indicated that the noise was pr imar i l y a funct ion of the angle of attack f luc tuat ions over the blade. Therefore i f U c « U and V(r) cos § « wR, 96 then, ex. (U Q - U c) V(r) cos <f> • A (E 3 ) i e , p = dt ^ P -A^ AN v « - I p -A A —°- , 2 K da o 2 K da ,,2 (E4) 1 d C £ d 2 p ~da~ BT [ U o V ( r ) C 0 S * + U c V ( r ) c o s * ] A • • '•' Now the area over which the vortex acts i s proport ional to r c and V(r) cos <j> i s proport ional to the vortex strength d iv ided by the core radius A r (E6) V(r) cos <f> a c .(E7) P - i p - d ^ d t [ V rc + uc r vc] (E8) Thus, we see that where (dU /d t ) i s l a rge , then U Tr w i l l be the c c c parameter grouping having the strongest inf luence on the fa r f i e l d no ise. However, i f (dU /d ) i s sma l l , then because U > U , the C L 0 c U Q r r c grouping w i l l dominate. 98 . APPENDIX F TABULATION OF HARMONIC AMPLITUDE DEPENDENCE ON VORTEX PARAMETERS (Theoret ical) Frequency 840 Hz Pos i t i ve Vortex Sense Negative Vortex Sense r Varies U„ Varies r Varies r Varies U„ Varies r Varies Ac c c c c c f t /sec?) (db) (db) (db) (db) (db) (db) 5 63.4 51.0 57.9 63.8 48.8 25.4 10 64.0 56.8 56.0 63.6 55.2 35.8 15 64.0 60.0 57.9 63.5 59.2 53.2 20 64.2 62.2 61.0 63.5 61.6 59.6 25 64.2 64.2 64.2 63.5 63.5 63.5. 30 64.2 65.0 62.8 63.5 65.2 66.8 • 35 64.2 67.0 69.2 63.5 66.4 69.0 40 64.2 68.0 71.0 63.4 67.6 71.2 45 64.2 69.0 73.0 63.4 68.8 72.8 50 64.3 70.0 74.0 63.3 69.6 74.0 Frequency 1260 Hz 5 64.0 50.8 47.8 64.6 50.6 40.4 10 64.8 56.8 50.8 64.6 56.6 50.0 15 64.7 60.0 -56.8 64.6 60.2 56.6 20 64.7 62.0 63.2 64.6 62.6 61 .2 25 64.7 64.7 64.7 64.6 64.6 64.6 30 64.7 66.4 67.2 64.6 66.0 67.2 35 64.7 67.6 69.2 64.6 67.6 69.2 40 64.7 68.4 70.4 64.6 68.6 70.8 45 64.7 69.6 70.8 64.6 69.8 71.6 50 64.7 70.8 72.4 64.6 70.4 72.4 99 Appendix F(continued) Frequency 2100 Hz Pos i t i ve Vortex Sense Negative Vortex Sense ru r 4 C °? ( f t 4 /seO r Varies (db) U c Varies . (db) e c Varies (db) TVaries (db) U c Varies (db) r c Varies (db) 5 10 15 20 25 30 35 40 45 50 62.4 64.0 64.0 64.0 64.0 64.0 64.0 64.0 64.0 64.1 50.0 56.0 59.2 62.0 64.0 65.0 67.0 68.1 69.0 70.0 40.0 51.2 57.6 61.6 64.0 , 64.9 65.9 65.2 64.3 62.8 64.0 64.0 64.0 64.0 64.0 64.0 64.0 64.0 64.0 64.0 i 49.6 54.6 59.2 62.0 64.0 65.2 66.8 68.0 68.8 70.0 38.4 51.2 56.5 61.4 64.0 65.2 66.0 65.6 64.5 62.8 Frequency 2520 Hz 5 62.3 48.3 40.0 62.3 48.4 38.4 10 62.3 54.6 51..8 62.3 54.6 51.4 15 62.3 58.2 57.6 62.3 58.0 59.2 20 62.3 60.9 60.4 62.3 60.4 61 .0 25 62.3 62.3 62.3 62.3 62.3 62.3 30 62.3 64.1 62.3 62.3 64.0 65.1 35 62.3 65.6 62.3 62.3 65.6 62.3 40 62.3 66.8 60.8 62.3 66.8 61 .0 45 62.3 67.5 58.2 62.3 67.6 58.1 50 62.3 68.7 55.2 62.3 68.6 55.2 mB (rpm) m= modal number B= number of blades Figure 1 Lowson's Aero-acoust ic Transfer Function Frequency, Hz Figure 2 Blade Relat ive Ve loc i ty Frequency Spectrum (Lowson) 101 Figure 3 Sears' Aerodynamic L i f t Transfer Function for an A i r f o i l Encountering a Sinusoidal Gust D 200 4 0 0 6 0 0 8 0 0 1000 Frequency, Hz (3Hz bandwidth) Figure 4 Fan Blade Pressure Spectra. Upper, from Leggat and Siddon; Lower, from Hanson C E N T E R O F E N V E L O P E 2 V ( y ) U T 7 T U T y S T A N D A R D P U L S E E N V E L O P E Figure 5 Pulse Input Proposed by Hanson 104 Figure 6 Comparison of Theory and Experiment (Hanson) Figure 7 Spanwise Source Strength of 420 Hz Tene at 15 Per Cent Chord o Figure 8 Leverton and Tayl h e l i c o p t e r r o t o r wing a i r jet o anechoic wind tunnel ID w a i n fan in et beil anechoic chamber honeycomb 18' I^ IO" Figure 9 The UBC Fan Noise F a c i l i t y Figure 11 The Experimental Hal f Delta Wings Figure 12 The Method of Determining Vortex Strength Figure 13 The Instrumented Fan Blade 20 Figure 14 The Frequency Response of the Two Telemetry: Systems pressure transducer input 2 0 K 2 0 K "WWW-residual fan imbalance gives occelerometer input. 60 Hz 5 0 0 K mtmt— 2 K 9 0 ° phase shift at 60 Hz Scope Display fan blade pressure signal synchronized to fan rotational frequency Figure 15 The Polar P lo t te r Configurat ion 113 # • measured v(r) . theoretical v(r) o vortex strength calculated from data 1 0.4 0.6 0.8 1.0 L2 1.4 1.6 1.8 Vortex Radius, r, inches Comparison of Experimental and Theoret ical Values of Vortex Tangential Ve loc i ty 114 r e so-78 inches r=20- Ft/sec -• measured v(r) theoretical v(r) o vortex strength calculated from data I • l; I i | 0 2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Vortex Radius, r, inches 17 Comparison of Experimental and Theoret ical Values of Vortex Tangential Ve loc i ty 115 116 Figure 19 Var ia t ion of Core Radius of Vort ices Produced by Half Delta Wings \ — S •\ ^ • -.\ i effect of delta wing shear layer 5 - WING Ur63 .7Pt/seC -r=30 Ft/sec »a=30 a T= 20F#sec ,a*20* F= 10 Ft5sec » a = i o ° Gaussian fit to a = 2 0 ° axial velocity deficit l I vortex core radii I •\ 0 I Distance from Vortex Centre, inches j Figure 20 Axia l Veloc i ty De f i c i t in Vortex Core Figure 21 Half Delta Wing Mounted in Fan overall levels 80 70 - 60 T= ± 2 0 Ft/sec rc= 0.065 Ft/sec Uc=47.2 Ft/sec R Q /R T = 0.82 •clean fan -negative circulation •positive circulation 100 I I 1 1 1 I 1 1 1 1000 Figure 22 Frequency, Hz Effect of Imposed Vortex on the Far F ie ld Noise Spectrum 8 0 75 70 o vortices wif h negative circulation • vortices with positive circulation ±25 ±50 ±75 ±100 rucrc ±125 F t%ec 2 1 ±150 ±175 ±200 Figure 23 Collapse of Overall SPL Data On.to Parameter Groupings O < o x t 4* O i O J O) OJ ro ro oo rb i ro o o> j_ ro i 00 I 4v 2 o • B D O • • Sound Pressure Level,db oo "71 — -•^ Vt <V — o 0> ro O ro 4> ro co, OJ ro OJ o> 4* o 00 o 4~ 00 ro ^ 1-—| 1 00 4* 00 oo oo • c s s • o • 9 a i+ i+ — — ro ro OJ O cn o o O 0 O 0 o o p o p O 8 O O O o -r- 0)->l->| m 4* ooo) o5 3 3223 3 LZL Q O B jQ •o : 78 0> a 76 v> CD G L •o 74. c o (f) Delta Wing Angle of Attack r- 0.088 R £ 0.076 Ft ifc- 0.070Ft Ttf 0.060 Ft r<f 0.054 Ft O B • • o o A A * A 7 2 . a H 1 I i I i i i 1 0 4 8 12 16 20 24 28 32 36 40 Vortex Strength, I\ Ft5sec Figure 25 Experimental Dependence of 1680 Hz Harmonic on Vortex Strength • a A A • J . 84' CC§3» 82, J Q -80 0> > ft) _J £ 7 8 3 V) (0 0) 76 T 3 C O to 74 I I I t i I I Delta Wing Angle of Attack • ± • ± 0 ± A ± 30° 25° 2 0 ° 15° 10° r£Q088 Ft r*0.076 Ft r*0.070Ft r*0.060 ° r«0.054 Ft • A A 40 -36 -32 -28 -24 -20 -16 -12 - 8 - 4 0 Vortex Strength 1 1 > » 1 ' j f f 4 8 II ,r, Ft7sec j j 12 16 20 24 28 32 36 40 CO Figure 26 Experimental Dependence of 420 Ht (i Fundamental) "On Vortex Strength Ooverall r = 3 Q n % e c ® 420 Hz rc= 0.078 Ft O 840 Hz A 1260 Hz A 1680 Hz • 2100 Hz B 2520 Hz Uc=49.i Ft /sec - J 1 L i i - i 0.4 0.5 0 6 0.7 0.8 0 Radius of Vortex Entry, R / R f Rf=0.792 Ft Var iat ion of SPL With Radial Pos i t ion of Vortex Entry (Experimental, Pos i t i ve C i rcu la t ion Sense) 6 0 5 0 2 Ooverall T=-30 Ft/sec © 4 2 0 Hz rc= 0.078 Ft O 840 Hz IjJ5 49.1 Ft/seC fT* A 1260 Hz A 1680 Hz O 2100 Hz 401— H 2520 Hz • i J — - J L 0.4 0.5 0.6 0.7 0.8 0.9 Radius of Vortex Entry, R / R t R t 50.792FT Figure 29 Var iat ion of SPL With Radial Posi t ion of Vortex Entry (Experimental, Negative-" C i rcu la t ion Sense) ro CTl B ure 30. Blade Pressure Polar Plots A. Negative Sense Vortex B. Pos i t i ve Sense Vortex C. Clean Running Fan (no honeycomb) D. Clean Running Fan (honeycomb in hatch). (Exposure: 1/50 sec) c D 31. Sequential Blade Pressure Polar P lo ts A. Negative Sense Vortex B. Pos i t i ve Sense Vortex C. Clean Running Fan (no honeycomb) D. Clean Running Fan (honeycomb 1n hatch) . (Exposure: 1/50 sec, 18 frames/sec) 12S Figure 32. Sequential Blade Pressure Polar P l o t s . Fan (no honeycomb). (Exposure: 1/50 sec, 1 8 frames/sec) (White markers show vortex s ignature. ) Clean Running Figure 33A Fan and Vortex Pos i t iona l Var iables Figure 33B Superposit ion of Ve loc i ty F ie lds Figure 34 Var iat ion of the Maximum instantaneous Rate of Change of L i f t Over the Span of the Fan Blade Uc= 47.2 Ft/sec r = - 2 0 . Ft/sec R 0 / R = 0 . 8 2 •- 0 0 6 5 Ft steady loading loading,vortex velocity deficit loading.vortex circulation — total loading 0.4 0.5 (R-R,) (RfR.) 0.6 0.7 0.8 0.9 1.0 Rj = hub radius, .25Ft Rt= tip radius, .792Ft co ro Figure 35 Blade Loading from the Ax ia l Ve loc i ty De f i c i t and C i r cu la t i on of In le t Vortex f = - 2 0 Ft /sec , R/R = 0.83 / r = - 2 0 Ft/sec \ theory experiment gives -2.1 Lb/Ft 2 Circumferential Angle,i /Radians w 2 r= 2 0 Ft/sec R/R = 0.94 rc= 0 . 065 Ft Ro/Rf0 .82 U c = 47.2 R/sec 0.0.1 Blade Surface Pressure l-o PS I —0.01 Figure 36 Circumferential L i f t Var ia t ion at Two S e c ^ Comparison with oo 3 0 0 0 ^ 2 0 0 0 -a 1000 0 > a •n - 1 0 0 0 a H - 2 0 0 0 - 3 0 0 0 — TT 1 1 T = - 2 0 R / s e c r= 2 0 F?%ec — R0/R=0.82 rc= 0.065 Ft IT 2 C ircumfcrentjal Angle,$ 9 Radians TT 2 Figure 37 Var iat ion of the Rate of Change of Blade Load Around the Fan Circumference co 4* Far Field Acoustic Pressure Level , //.bars Sound Pressure Level ,SPL,dbRe 0 .0002 fibars o -j o O o 4* O O CD O 1^ O O 00 o C O _ ^ ro t - O ) J o 00 o o < —t a < CD (0 0> OD O ro o o ro 01 ro o c o r r e c t e d f o r 6% f i t ter band width uncor rec ted reg ion of uncertainty •7 o bo ro H O 9£L 11 rv ? o \ ov _1' S£ « 8 0 I 3 —» Q CO o JO =J- 80 CM O O o 6 70 o or JQ X» PL: 60 > CD _ J 50 0> KB . 3 CO CO Pre 40 XJ c 3 o CO 30 overall levels 420 theory experiment r e s o . 0 6 5 R r=-2 0.Ft/sec Uc=47.2 Ft/sec R o / R ^ 0 . 8 2 840 1260 Frequency, Hz 1680 2100 2520 Figure 40 Comparison of Theoretical and Experimental Far F ie ld Fan Noise Spectra (Negative C i rcu la t ion Sense) OJ 9 0 1 -J Q 0> > _ J l_ 3 to (/> Q> 8 0 70 6 0 X J c 3 • O cn o 501 c o E i_ x 401 ' T= -30. Ft /sec r = 0.078 Ft Uc= 49.1 F t / s e c Overall 420 Hz • -840 Hz 1260Hz 1680 Hz 2100 Hz 2520 Hz 1 0.4 0.5 0.6 07 Radius of Vortex Entry, R / R t 0.8 0.9 R t = 0 .792 Ft Figure 41 Var iat ion of SPL with Radial Pos i t ion of Vortex Entry (Theore t i ca l , Negative C i r cu la t i on Sense) (Compare with Figure 29 for Experimental Curves). CO 00 90 I 1 _ I I I I 0.4 0.3 0.6 0.7 0.8 0.9 Radius of Vortex Entry, R / R t Rt= 0.792 Ft Figure 4 2 Var iat ion of SPL with Radial Pos i t ion of Vortex Entry (Theore t i ca l , Pos i t i ve C i r cu la t i on Sense) (Compare with Figure 28for Experimental Curves). 8000 o> - 6 0 0 0 E r=-30. Ft/sec rc= 0.078 Ft U c s49.i Ft/sec Circumferential Angle, £ , Radians o Figure 43 Change in dL/dt Curves as Vortex is Moved Radia l ly Outwards <© 0 5 10 15 2 0 25 30 35 40 4 5 50 HJ cr c F t 4 / sec 2 Figure '4.4 Parametric Dependence of Vortex Interact ion Noise (Pos i t i ve C i r cu la t i on Sense, Overal l Level) 80 co O CM O O o 0 > or T J _1 0_ 00 70 60 o> > _J 4> w. ~ Z3 CO to 0) k. 0. "O c O CO m 50 40 30 _______ T varies ; r c, U c const. — —- - rc var ies ;T , U c const. _. U c varies; rc , T const. 1 -5 -10 -15 -20 -25 -30 -35 4 2 r Ucrc Ft/sec -40 -45 -50 Figure 45 Parametric Dependence of Vortex Interact ion Noise (Negative C i r cu la t i on Sense, Overal l Level) Sound Pressure L e v e l , S P L , d b Re o .ooo2^bars OI 4* (j\ O) ->l 09 o o o o o o Sound Pressure Leve l , S'PL,db Re o . 0 0 0 2 / i bars CO 0> w. 3 to CO <3> 50 C 3 co 40 0 r varies ;U c , r const-/ / / / rc varies; r,U cconst. U c varies,r c ,r const. 1 10 15 20 25 30 rLj.rc R%ec 2 35 4 0 45 50 Figure .48 Parametric Dependence of Vortex Interact ion Noise (Pos i t i ve C i r cu la t i on Sense, 1680 Hz) Sound Pressure Level ,SPL,db Re 0 0 0 0 2 /ibars o o O 0) O o 00 o 01 H» O I cm I ro o I O) cn 1 » o 1 cn I cn O \ X \ W V I I < a < a < a -1 cn K-J cT O O o o 3 3 cn C o o o 3 \ 25 20 ^ 15 _ l £ IO o c o 0 ) CO — IT .21 2 theoretical lift T = 2 0 . Ft%ec r = 0 .065 Ft U c s 25. Ft/sec R 0 /R=0.82 R/R f =0 .86 experimental-natural vortex pressure from Figure 30 C R/R t =0.94 Circumferential An gle, ^ , Radians w .0.02 .0.01 L o CO CL v. tO CO U o U-O.OI 5 to L-0.02 „ -a o Figure 50 Circumferential Var iat ion of Sectional L i f t and Blade Pressure fo r Blade Sections 

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