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The measurement of vortex induced effects in flow past stationary and oscillating circular and D-section… Feng, C.C. 1968

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THE MEASUREMENT OF VORTEX INDUCED EFFECTS IN FLOW PAST STATIONARY AND OSCILLATING CIRCULAR AND D-SECTION CYLINDERS by C. C. Feng B. Sc., National Taiwan University, 1963 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of M.A. Sc. in the Department of Mechanical Engineering We accept this thesis as conforming to the required s.tandard_ THE UNIVERSITY OF BRITISH COLUMBIA October 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n D e p a r t m e n t o f Mechanical Engineering The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e Janaury 31, 1969 i ABSTRACT Experiments were performed i n a wind tunnel on 3-inch diameter circular and D-section cylinders, A detailed investigation of the vortex shedding frequency, displacement amplitude, and the phase angle between the fluctuating pressure and the displacement signals of both circular and D-section cylinders was made i n the "capture" region. These pheno-mena were investigated under various damping levels using magnetic dampers. Fluctuating surface pressures on a circular cylinder were measured along one half of the circumference at 11 sections selected along the span. The resulting sectional fluctuating l i f t coefficients as well as the total l i f t coefficients were obtained by integration for several wind speeds for both stationary and oscillating cylinders. Interesting to note are the vortex line inclination angles obtained from fluctuating surface pressure correlation. Using linearized hot wire anemometers, spanwise wake velocity correlation functions were measured and correlation lengths computed. i i TABLE OF CONTENTS Section Page 1. INTRODUCTION 1 2. INSTRUMENTATION 3 2.1 Wind Tunnel 3 2.2 Models 3 2.3 Model Mounting System.. k 2.h Wake Traversing Gear.... 5 2.5 Displacement Transducer 6 2.6 Magnetic Damping...... 6 2.7 Pressure Transducer 7 2.8 Correlator....*.;..... * 8 • 2.9 Hot Wire Anemometers and Linearizers 10 2.10 Band Pass F i l t e r s . 11 2.11 Other Electronic Instruments 12 3. EXPERIMENTAL PROCEDURES 13 3.1 Frequency, Amplitude, and Phase Measurements 13 3.1.1 Frequency Measurements 13 3.1.2 Phase Measurements. 13 3.1.3 Amplitude Measurements 1^  3.2 Spanvlse Fluctuating Surface Pressure Measurements 15 3.3 Spanwise Wake Velocity Correlation Measurements 15 i i i Section Page 3.h Measurements of Non-Aero dynamic Viscous Damping l6 k. EXPERIMENTAL RESULTS... 18 k.l Frequency, Amplitude and Phase Measurements for a Circular Cylinder 18 k.2 Frequency, Amplitude and Phase Measurement for a D-section Cylinder 19 ^•3 Fluctuating Pressures on the Surface of a Circular Cylinder 19 k.h Spanvise Correlations for Circular and D-section Cylinders Using Hot Wire Anemometers 21 5- DISCUSSION OF RESULTS 23 5.1 Frequency, Amplitude and Phase Measurements 23 5.2 Fluctuating Pressures on the Surface of a Circular Cylinder 25 5.3 Spanvise Correlations for Circular and D-section Cylinders Using Hot Wire Anemometers ...28 6. ' SUMMARY OF RESULTS .. .. 30 BIBLIOGRAPHY 32 APPENDICES .;. 3 ^ A. Tunnel Corrections to Wind Speed 3^+ B. Correlator Phase Measurement 35 i v LIST OF FIGURES Figure Page 1. Wind t u n n e l o u t l i n e 38 2 . Wind t u n n e l t e s t s e c t i o n w i t h model (downstream d i r e c t i o n ) 39 3. Models kO k. Spanwise pressure tap p o s i t i o n s f o r the c i r c u l a r c y l i n d e r k l 5 • Arrangement of model mounting system k2 6. T r a v e r s i n g gear 4 3 7. Bl o c k diagram of the c a l i b r a t i o n apparatus. kk 8. C a l i b r a t i o n curves f o r B a r o c e l pressure transducer 4 5 9. C a l i b r a t i o n curve f o r 55A01 DISA Anemometer without a l i n e a r i z e r k-6 1 0 . C a l i b r a t i o n curve f o r 5 5 A 0 1 DISA Anemometer u s i n g l i n e a r i z e r h-T 11. Instruments and wind t u n n e l t e s t s e c t i o n 48 1 2 . Phase angle c a l i b r a t i o n set-up ..^9 13. B l o c k diagram of the f l u c t u a t i n g pressure measuring set-up 50 Ik. Coordinate axes f o r wake probe p o s i t i o n s 51 15. Block diagram of the spanwise c o r r e l a t i o n f u n c t i o n measuring set-up 52 16. Phase s h i f t and displacement phenomena f o r the c i r c u l a r c y l i n d e r a t va r i o u s damping l e v e l s 53 V F i g u r e Page 17(a) S t a b i l i t y diagram f o r c i r c u l a r c y l i n d e r . . 5^ 17(b) S t a b i l i t y diagram f o r D-section c y l i n d e r 5k 18. O s c i l l a t i o n phenomena f o r c i r c u l a r c y l i n d e r , I , = 0 ma 55 d 19- O s c i l l a t i o n phenomena f o r c i r c u l a r c y l i n d e r , T = 100 ma 56 d 20. O s c i l l a t i o n phenomena f o r c i r c u l a r c y l i n d e r , T = 160 ma •. 57 d 21. O s c i l l a t i o n phenomena f o r c i r c u l a r c y l i n d e r , I , = 250 ma 58 d 22. O s c i l l a t i o n phenomena f o r c i r c u l a r c y l i n d e r , I . = 3^0 ma 59 d 23. O s c i l l o s c o p e t r a c e s of f l u c t u a t i n g surface pressure and displacement s i g n a l s d u r i n g the abrupt changes 60 2k. Phase s h i f t and displacement phenomena f o r the D-section c y l i n d e r a t v a r i o u s damping l e v e l s 6 l 25. O s c i l l a t i o n phenomena f o r D-section c y l i n d e r , I, = 0 ma 62 d 26. O s c i l l a t i o n phenomena f o r D-section c y l i n d e r , r = 80 ma 63 d 27. O s c i l l a t i o n phenomena f o r D-section c y l i n d e r , I„ = ll<-5 ma 6)+ d 28. O s c i l l a t i o n phenomena f o r D-section c y l i n d e r , I = 222 ma 65 d 29. O s c i l l a t i o n phenomena f o r D-section c y l i n d e r , I = k60 ma 66 d 30. Beat phenomena f o r the o s c i l l a t i n g c i r c u l a r c y l i n d e r . . . . .67 v i F i g u r e Page 31. Cp d i s t r i b u t i o n on the surface of a stationary-c i r c u l a r c y l i n d e r , V = 11.8 fps ..... 68 32. Cp' d i s t r i b u t i o n on-the surface of a s t a t i o n a r y c i r c u l a r cylinder,,' -V =.:13..2 fps ' .. 69 33' Cp' d i s t r i b u t i o n on "the surface of an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 11.3' fps 70 3^ -. Cp' d i s t r i b u t i o n on the surface of an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 17-6 f p s . . . 71 35 • Cp7. d i s t r i b u t i o n on'- the surface of an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 13«6 fps.. 72 36. C_g d i s t r i b u t i o n f o r the s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 11.8 f p s . . . . . . 73 37. C£ d i s t r i b u t i o n for' the s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 13-2. fps'. Ik 38. C^ d i s t r i b u t i o n f o r the o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 11.3 fps 75 39. Cg d i s t r i b u t i o n f o r the o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 17-5 f p s 76 kO. C£ d i s t r i b u t i o n f o r the o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 13-9 f p s • •• 77 kl. Spanwise phase s h i f t o f f l u c t u a t i n g s e c t i o n a l l i f t on a s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 11.8 fps 78 k2. Spanwise phase s h i f t of f l u c t u a t i n g s e c t i o n a l l i f t on a s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 13.2 fps 79 k3. Spanwise phase s h i f t of f l u c t u a t i n g s e c t i o n a l l i f t on an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 11.3 f P s 80 kk. Spanwise phase s h i f t of f l u c t u a t i n g s e c t i o n a l l i f t on an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 17-5 fps 81 V l l F i g u r e Page k-5 . Span-wise phase s h i f t of f l u c t u a t i n g s e c t i o n a l . l i f t on an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 13*9 fps . . . . .82 h6. E f f e c t of end clearances on spanvise phase s h i f t of f l u c t u a t i n g s e c t i o n a l l i f t . o n a s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 13.9 f p s . . . . . . 83 hf. E f f e c t of f l o o r and c e i l i n g s l o t s on spanvise phase s h i f t of f l u c t u a t i n g s e c t i o n a l l i f t on a s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 11.8 fps 8h 48. Two-point f l u c t u a t i n g .-wake v e l o c i t y c o r r e l a t i o n s f o r an o s c i l l a t i n g ' c i r c u l a r c y l i n d e r ' 85 h9. Two-point fluctuating'.wake ; v e l o c i t y c o r r e l a t i o n s f o r a s t a t i o n a r y and an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 13.9 f p s . . . . . . . . . . 86 50. Two-point f l u c t u a t i n g ,vake..velocity c o r r e l a t i o n s , f o r a s t a t i o n a r y andian. o s c i l l a t i n g D-section c y l i n d e r , V = lk.1 fps......':.. ....... 87 V l l l • LIST OF SYMBOLS o 4.- n 4. 4. ' - n . ^ x • 4. rms of s e c t i o n a l l i f t S e c t i o n a l . f l u c t u a t i n g - l x f t c o e f f i c i e n t , -f P V h m , a , '• •< • . , " rms of t o t a l l i f t T o t a l f l u c t u a t i n g l i f t c o e f f i c i e n t , • .v •§- P v hi • - ' •: •• p , t . rms F l u c t u a t i n g pressure c o e f f i c i e n t , — — Mean peak value; - ^/^S7' (' ) Spanwise se p a r a t i o n of the two probes Insi d e diameter'of' the Polyethylene t u b i n g Dimensionless spanwise. s e p a r a t i o n of the'two probes, — L a t e r a l aerodynamic f o r c e on the c y l i n d e r . C y l i n d e r o s c i l l a t i n g frequency Vortex shedding frequency N a t u r a l frequency of an e l a s t i c system • L a t e r a l dimension'of the c y l i n d e r s e c t i o n Damping current Length o f t h e • c y l i n d e r ; l e n g t h of the- t u b i n g Mass of o s c i l l a t i n g system; inverse of the slope of the l i n e i n F i g u re 9 D „ n X e s s . s s ^ 2m The Nth increment where the c o r r e l a t i o n f u n c t i o n reaches a l o c a l l y a l g e b r a i c maximum va l u e , implying the. two s i g n a l s being i n phase w i t h delay time of N Time delay i n c o r r e l a t i o n function. Time i n seconds Cylinder Reynolds number, — v Hot wire probe c o l d 'resistance Probe operating resistance C o e f f i c i e n t of viscous damping f h v Strouhal number, v V Dimensionless v i n d v e l o c i t y , CD h n Wind v e l o c i t y Bridge d.c. voltage . , . Bridge voltage at zero v i n d speed Stream wise ;';eoordiriate • Transverse coordinate or displacement Spanwise coordinate Dimensionless transverse displacement, ^ Dimensionless transverse, amplitude, — . .'•-'.;.-'•. • n ,'. Spanwise dimensionless distance from the mid-span, ^ Dimensionless damping c o e f f i c i e n t , 2 m co n Tap p o s i t i o n angle with respect to the up-stream d i r e c t i o n The angle between the vortex l i n e and the model z - axis Phase angle by which cy l i n d e r l i f t leads displacements; phase s h i f t angle with respect to No. 9 tap signals Normalized c o r r e l a t i o n function, defined i n Appendix B. C i r c u l a r frequency of transverse o s c i l l a t i o n Natural c i r c u l a r frequency,2 u f Cross-correlation function, defined i n Appendix B Auto-correlation function, defined i n Appendix B Correlation length, defined i n text x i ACKNOWLELXJEMEKT The author wishes to express his sincere appreciation for the guidance and encouragement given by Dr. G. V. Parkinson during his supervision of this investigation. Sincere appreciation i s also expressed to Mr. J. E. Slater, a fellow graduate student, for his helpful advice in many areas. Thanks are also due to the Department of Mechanical Engineering for use of the f a c i l i t i e s and to technicians of the Department for their valu-able advice and assistance. Financial support was received from the National Research Council of Canada, Grant A586. I . IMTEODUCTIOW I t i s w e l l known t h a t b l u f f c y l i n d e r s , when e l a s t i c a l l y mounted, exhib i t v a r i o u s forms of o s c i l l a t i o n . One important form i s the v o r t e x - e x c i t e d o s c i l l a t i o n . This o s c i l l a t i o n occurs when the v o r t e x shedding frequency ap proaches a n a t u r a l frequency of the e l a s t i c system. The p e r i o d i c character i s t i c of the f l o w f i e l d causes a f l u c t u a t i n g pressure d i s t r i b u t i o n on the c y l i n d e r surface and the r e s u l t i n g p e r i o d i c f o r c e s e x c i t e c y l i n d e r o s c i l -l a t i o n s over a d i s c r e t e range of wind speeds. T y p i c a l l y , a graph of c y l i n -der amplitude versus wind speed has a form not u n l i k e t h a t of a f o r c e d v i -b r a t i o n w i t h damping, w h i l e a graph of vortex frequency versus wind speed p o r t r a y s a 'capture' or ' l o c k i n g - i n ' phenomenon. V o r t e x - e x c i t e d o s c i l l a -t i o n s are of considerable engineering s i g n i f i c a n c e since they can a f f e c t systems such as t r a n s m i s s i o n l i n e s , smoke s t a c k s , submarine p e r i s c o p e s , and launch v e h i c l e s . Numerous i n v e s t i g a t i o n s of t h i s and other a s s o c i a t e d phenomena have been r e p o r t e d . Most of the i n v e s t i g a t i o n s r e l a t e t o the s t a t i o n a r y c i r c u -l a r c y l i n d e r , such as Keefe's^"^ d i r e c t measurement of f l u c t u a t i n g f o r c e s (2) (3.4) and McGregor's^ ' and G e r r a r d ' s w ' i n v e s t i g a t i o n s on f l u c t u a t i n g surface pressures and the s t r u c t u r e of the wake. For o s c i l l a t i n g c y l i n d e r s , very few d i r e c t measurements of c y l i n d e r surface l o a d i n g and c o r r e l a t i o n of wake (5) v e l o c i t i e s have been r e p o r t e d . Bishop and Hassan de s c r i b e d the measure-ment of f l u c t u a t i n g l i f t and drag on a mechanically o s c i l l a t i n g c i r c u l a r f c\ c y l i n d e r i n a water channel. Heine presented measurements of f l u c t u a -t i n g surface pressure on a c i r c u l a r c y l i n d e r i n free o s c i l l a t i o n i n a wind (7) tunnel. Den Hartog\ described the e f f e c t of the cyl i n d e r ' s o s c i l l a t i o n s on i t s vortex wake. Recently.Ferguson investigated wake and surface e f f e c t s on a c i r c u l a r c y l i n d e r i n vortex-excited o s c i l l a t i o n s and Koop-( 9 10) mamr ' reported some r e s u l t s on the vortex wakes of both mechanically and wind-excited v i b r a t i n g cylinders.. However, there i s s t i l l a la c k of information on some of the charac-t e r i s t i c features of vortex-excited o s c i l l a t i o n s i n the capture region, as observed on c i r c u l a r and D-section c y l i n d e r s , including d e t a i l e d measure-ments of frequencies, displacement amplitudes, phase s h i f t of the e x c i t i n g force with respect to the displacement, and spanwise c o r r e l a t i o n of wake v e l o c i t i e s and f l u c t u a t i n g surface pressures on o s c i l l a t i n g c y l i n d e r s . These are the t o p i c s investigated by the author as part of a contin-uing programi.-: i n t h i s laboratory to study the a e r o e l a s t i c i n s t a b i l i t y of b l u f f bodies. 3 I I . INSTRUMENTATION 2 . 1 Wind Tunnel The wind tunnel used i s a low speed, low turbulence, return type. The a i r speed can be varied through the range 4 ft/sec to 1 5 0 ft/sec with a turbulence l e v e l less than 0 . 1 $ . The pressure d i f f e r e n t i a l across the contraction section r of 7 : 1 r a t i o can be measured on a Betz micromanometer which gives a reading to 0 . 0 2 millimeter of water. The test section v e l o c i t y i s calibrated against the above pressure d i f f e r e n t i a l . The rectangular cross-section, 3 6 i n . x 2 7 i n . , i s provided with 4 5 ° corner f i l l e t s which vary from 6 i n . x 6 i n . to 4 . 7 5 i n . x 4 . 7 5 i n . to compensate for the boundary layer growth. The s p a t i a l v a r i a t i o n of mean v e l o c i t y i n the t e s t section i s less than 0 . 2 5 $ . The tunnel i s powered by a 15 horsepower direct current motor driving a commercial axiflow fan with a Ward-Leonard system of speed control. Figure 1 shows the outline of the tunnel and Figure 2 shows a model mounted i n the wind tunnel during t e s t . 2 . 2 Models Three models were used, two 3-inch diameter c i r c u l a r cylinders and one 3-inch D-section cylinder (Figure 3 ) « They are a l l 27 inches long. Polyethylene tubing of 0 . 0 6 6 inch inside diameter and 0 . 0 9 5 inch outside diameter was used • to convey the fluct u a t i n g pressure from the surface taps. 0 . 0 2 2 inch w a l l thickness aluminum tube and clear p l a s t i c provided the body of the models. Pressure tap holes i n the model surface were 0 . 0 2 5 inch i n diameter. For measurements of frequency, amplitude, and phase, the c i r c u l a r cylinder (8) and D-section cylinder both designed and used by Ferguson were used. For spanwise f l u c t u a t i n g surface pressure measurements, another 3-inch c i r c u l a r c y l i n d e r was made. P l a s t i c end f i t t i n g s which allowed the model t o he r o t a t e d about i t s own a x i s , yet remain attached t o the a i r b e a r i n g shaft b r a c k e t s , were secured by an epoxy adhesive t o the aluminum tube. Along the c y l i n d e r a x i s IT pressure taps-were e q u a l l y spaced h a l f a c y l i n d e r diameter apart w i t h the n i n t h tap l o c a t e d at midspan. The d i s t r i b u t i o n of taps and t h e i r angular d e f i n i t i o n are shown i n Figure h. Due t o the advantages of the symmetry of the s e c t i o n and the r o t a t a b i l -i t y of the model on the mounting system, the number of pressure taps was kept t o a minimum. A p l a s t i c block,'which was r a d i u s e d t o f i t the model i n s i d e s u r f a c e , was d r i l l e d t o e f f e c t 90 ° bends i n the pressure t u b i n g . An epoxy adhesive served t o bond the p l a s t i c b l o c k t o the polyethylene t u b i n g and t o the aluminum tube w i t h the b l o c k holes a l i g n e d w i t h the 0 . 0 2 5 - i n c h taps i n the aluminum s k i n . 2.3 Model Mounting. System The a i r b e a r i n g system -designed by S m i t h ^ V was used f o r the experiment. The models were c o n s t r a i n e d t o only the l a t e r a l plunging degree of freedom w i t h a minimum of damping from the mounting system. Adjustments were provided t o ensure the p a r a l l e l i s m of the two sets of bearings and the p e r p e n d i c u l a r i t y to the t u n n e l f l o o r of the bearing plane. S l o t s i n the top and bottom panels of the t e s t s e c t i o n allowed the model t o be attached t o the a i r bearing s h a f t s . To provide the e l a s t i c system f o r the models f o u r h e l i c a l t e n s i o n springs (8) designed by Ferguson were used. They were attached t o the shaft brackets 5 and t o the a i r h e a r i n g frame. A streamlined aluminum bar was used t o determine the damping due t o the spr i n g - b e a r i n g system. A i r supply f o r the bearings was produced by an Ingersoil-Rand 2-stage-com-p r e s s o r , model 1 1 3A x 7 x 8 VHB-2, v i a a 2 5 0 cubic f o o t storage tank. A i r pressure of 6 0 pounds per square i n c h was used f o r the a i r bearing system from the main supply a t a maximum of 1 1 8 pounds per square i n c h . A diagrammatic arrangement of the model, b e a r i n g s , s h a f t s and springs i s shown i n Fi g u r e 5• 2.h Wake Tr a v e r s i n g Gear To enable two hot wire probes t o be p o s i t i o n e d w i t h c o n t r o l o f movement i n ( 8 ) a l a t e r a l , v e r t i c a l , and l o n g i t u d i n a l sense, the e x i s t i n g t r a v e r s i n g gear was m o d i f i e d so t h a t a sequence o f d e s i r e d spanwise separations of the two probes c o u l d be made. Two probe mounting brackets were c a r r i e d by f o l l o w e r nuts on the v e r t i c a l \ i n c h - 2 0 NC end screws which were s e p a r a t e l y enclosed i n t h e i r r e s p e c t i v e guide tubes .each soldered s o l i d l y i n p a r a l l e l w i t h the model a x i s on the l a t e r a l l e a d screw f o l l o w e r p i e c e . The l a t e r a l l e a d screw w i t h 5/8 i n c h 1 0 acme double thread spanned the t e s t s e c t i o n . The e n t i r e assembly was mounted on a h o r i z o n t a l r i g i d frame which w i t h i t s grooved wheels c o u l d be p o s i t i o n e d l o n g i t u d i n a l l y along the r a i l s on the e x t e r i o r of the t u n n e l side panels. Hand wheels and f l e x i b l e s h a f t s enabled the v e r t i c a l l e a d screws to r o t a t e . Each probe was e l e c t r i c a l l y i n s u l a t e d from the e n t i r e t r a v e r s i n g gear assembly so t h a t separate probe c i r c u i t s were maintained. The transverse d e v i a t i o n i n p o s i t i o n i n g the probe was c a l i b r a t e d t o be 6 0.177 inch or about 5'9$ of cylinder diameter. Since the probes were always positioned at the flat portion of the probe signal amplitude vs. transverse lateral distance (section 3»3)> this deviation gave negligible probe signal error. Figure 6 shows the modified"traversing gear. 2.5 Displacement Transducer A signal corresponding to model amplitude was obtained from an air core transformer designed by S m i t h - T h e coaxial cylindrical construction a l -lowed the air bearing shaft to be inserted between the primary and secondary windings, thus varying the magnetic coupling. A 10 kc frequency signal of k rms volts supplied by a Hewlett-Packard 200 CD oscillator was modulated by the shaft oscillations and this signal was in turn rectified to give the re-sulting displacement signal which was displayed either on a storage oscillo-scope, or fed into a Honeywell Visicorder to record time-amplitude traces, or fed into the correlator to give the phase value between fluctuating surface pressure and cylinder displacement. The displacement transducer was mounted on the top channel of the air bearing mounting system as shown in Figure 5. For calibration of the transducer a wooden scale was mounted close to the shaft under the floor of the tunnel, and a calibration was performed dur-ing each series of tests. 2.6 Magnetic Damping In addition to the inherent damping of the springs and air bearing sys-tem, magnetic damping was produced by means of electromagnetic eddy-current dampers designed by Smitl/ 1 1^. The air bearing shafts passed through the mag-7 netic field created by the damper and eddy currents induced in the shafts dis-sipated energy from the oscillating system. An appreciable amount of undesirable residual magnetism built up on the damper was removed periodically by switching the damper coils over to a var-iable a.c. source. The a.c. voltage is raised to give a greater magnetic field than that produced by the d.c. source, this effectively erasing the re-sidual magnetism. Positions of the dampers are shown in Figure 5-2.7 Pressure Transducer A Barocel Modular Pressure Transducing System developed by Datametrics Inc. of Waltham, Massachusetts was used for measuring surface fluctuating pressures. The Barocel is a high precision, stable capacltive voltage divid-er, the variable element of which is a thin prestressed stainless steel dia-phragm. Positioned between fixed capacitor plates, the diaphragm deflects proportionally to the magnitude of the applied pressure. An a.c. carrier volt-age at 10 kc is applied to. the stationary capacitor plates. The diaphragm attains a voltage level determined by its relative position between the fixed capacitor plates. With the Barocel appropriately arranged in a bridge circuit, the output voltage is determined by the ratio of capacitance of the diaphragm to each of the stationary electrodes. The carrier voltage is thereby amplitude modulated in accordance with the input pressure. Since the frequency range required for this investigation also f e l l in the (12) range between 5 and 35 cps used by Wilandv , the basic calibration system he 8 devised was used. No resonance c o n d i t i o n e x i s t e d between the transducer w i t h tube and pressure tap on one side and the volume where the c a l i b r a t i o n s i g n a l was generated on the other s i d e . The b l o c k diagram of the c a l i b r a t i o n appara-tu s i s shown i n F i g u r e 7, and the c a l i b r a t i o n curve together w i t h Wiland's i s shown i n Figure 8. The s m a l l d i f f e r e n c e i s due t o s l i g h t l y larger tube i n s i d e diameter i n the present c a l i b r a t i o n . 2.8 C o r r e l a t o r A c o r r e l a t i o n f u n c t i o n computer, Model 1 0 0 , produced by P r i n c e t o n A p p l i e d Research C o r p o r a t i o n , was a c q u i r e d during the i n v e s t i g a t i o n and was used t o measure c o r r e l a t i o n f u n c t i o n s and phase angles ( s e c t i o n s 3.2 and 3'3)« The s i g n a l c o r r e l a t o r i s designed-to compute the c r o s s - c o r r e l a t i o n f u n c t i o n , R p ( r ) , of two e l e c t r i c a l s i g n a l s d e f i n e d by l , d T R J T ) = l l m - f f , Ct) f 0 ( t - T ) dt 1,2V ' T-+ oo T J o r  1 2> ' and the a u t o - c o r r e l a t i o n f u n c t i o n , R.. - (T), of two i d e n t i c a l s i g n a l s d e f i n e d by. 1,1 T R 1 M = m 3 J I 1 W \ f l ( t ) f n ( t " 0 d t 1.1 T-*00 T J N 1 1 U t i l i z i n g both analog and d i g i t a l techniques, these instruments operate as h y b r i d computers t o solve e i t h e r of the two i n t e g r a l s f o r one hundred incremen-t a l l y i n c r e a s i n g values of the time delay, A T • The "n"th p o i n t on the c o r -r e l a t i o n f u n c t i o n computed i s c l o s e l y represented by. t - t ' -t R i p C * ) = b \ e R C M * ' ) E p C * ' - t w o i t X,d RC J-oo 1 d 9 where "RC" i s the time constant of the averaging c i r c u i t , "nAx" defin e s the time coordinate of the computed p o i n t , and t ' represents the past h i s t o r y of computation, or by d e f i n i t i o n , t ' > t . Computation a t each p o i n t i n v o l v e s three b a s i c operations: sampling the input wave form and de l a y i n g the samples, m u l t i p l y i n g the delayed samples by e i t h e r the o r i g i n a l input wave form ( a u t o - c o r r e l a t i o n ) o r by a second wave form ( c r o s s - c o r r e l a t i o n ) , and averaging the lagged products i n an RC i n t e g r a t o r . The r a t e of time s h i f t i n g determines the incremental value of time delay, A T , and thus se t s the time base against which each po i n t of the c o r r e l a t i o n func-t i o n i s computed. M u l t i p l i c a t i o n - o f the two input s i g n a l s i s performed auto-m a t i c a l l y a t each p o i n t . I n t e r n a l RC networks perform the i n t e g r a t i o n w i t h a time constant of kO seconds i n t h i s c o r r e l a t o r . F i v e times t h i s constant, or 200 seconds, i s r e q u i r e d f o r the f u n c t i o n t o grow t o w i t h i n one percent of i t s f i n a l v a l u e . Because a l l the mathematical operations i n v o l v e d i n e v a l u a t i n g each p o i n t are performed simultaneously i n r e a l time, the l e n g t h of time r e -q u i r e d t o compute the complete f u n c t i o n depends only on t h i s averaging time constant. As i t i s computed, the c o r r e l a t i o n f u n c t i o n i s s t o r e d i n the 100 channel analog memory. V e r s a t i l e readout c i r c u i t r y a l l o w s the f u n c t i o n t o be d i s p l a y e d on an o s c i l l o s c o p e as i t i s being computed, and t o be n o n - d e s t r u c t i v e l y read out e i t h e r a f t e r or during computation. As w i t h any computer, the computed f u n c t i o n i s o n l y a good approximation of an i d e a l r e s u l t . Conformity of the computed and the i d e a l f u n c t i o n i s ex-1 0 c e l l e n t f o r Model 1 0 0 . I t s d i f f e r e n c e t y p i c a l l y does not exceed one percent at any p o i n t . According t o standard p r a c t i c e , the normalized c r o s s - c o r r e l a t i o n f u n c t i o n , ot^r^i) , i s d e f i n e d and computed as f o l l o w s : R 1 2 ( T ) a l 2 ( T ) = • R ^ O ) R 2 ( 0 ) To measure the phase angle "between two p e r i o d i c s i g n a l s , the f o l l o w i n g formula was developed i n Appendix A: <t> = 3.6 N T f v 2.9 Hot Wire Anemometers and L j n e a r i z e r s Two hot-wires were used. Each was made of plat i n u m - p l a t e d tungsten and was of 0 . 0 0 5 mm. diameter and approximately 1 . 2 mm. lo n g . Two DISA constant temperature anemometers s u p p l i e d the b a s i c c i r c u i t s f o r the transducers. The p r i n c i p l e of measurement i s based on the convective heat l o s s i n an e l e c t r i -c a l l y heated wire ( o r f i l m ) by the f l o w of f l u i d surrounding the w i r e . Fun-damentally, what i s measured i s the amount of power r e q u i r e d t o keep the tem-perature constant. The r e l a t i o n between fl o w v e l o c i t y V and anemometer out-put voltage E can be represented by 2 n ,,, E = A + BV (1) where A , B , and n are constants whose values depend on the probe con-nected t o the anemometer. The output voltage of a constant temperature anemometer i s thus a non-11 l i n e a r f u n c t i o n of the f l o w v e l o c i t y V under measurement. This n o n l i n e a r i t y i s u n d e s i r a b l e i n c o r r e l a t i o n f u n c t i o n measurement.' Therefore two l i n e a r i z e r s , DISA Type 55D10, were a c q u i r e d and used t o e l i m i n a t e d i s t o r t i o n . The l i n e a r -i z e r i s an e l e c t r o n i c analog computer whose b a s i c t r a n s f e r f u n c t i o n at constant s e t t i n g s of the o p e r a t i n g c o n t r o l s can be w r i t t e n as: ,2 2 .m E . = K (E. - E. ) (2) out i n mo ^ ' where i s the output voltage at zero f l o w v e l o c i t y and i s a con-s t a n t , as i s K . P u t t i n g the anemometer output 1 voltage E as being equal t o the l i n e a r i -z e r input voltage E. , we have., on s u b s t i t u t i n g ( l ) and (2), i n 2 'm . E . = K (A ,* BV11'- E. ) 2 1 Thus, f o r E. = A and m = — the l i n e a r i z e r output voltage w i l l be ' mo n p r o p o r t i o n a l t o the v e l o c i t y V • Figure 9 shows the c a l i b r a t i o n curve w i t h -out u s i n g a l i n e a r i z e r and" F i g u r e 10,:'.shows c a l i b r a t i o n curves u s i n g l i n e a r i -z e r s . The d i f f e r e n t slopes" Of the two curves i n Figure 10 a r i s e from d i f f e r e n t s e t t i n g s of g a i n adjustment. Both the anemometers and the l i n e a r i z e r s show n e g l i g i b l e phase s h i f t f o r f requencies below 300 kc. 2.10 Band Pass F i l t e r s To e l i m i n a t e the random wake turbulence and permit c o n c e n t r a t i o n on the d i s c r e t e v o r t e x shedding phenomena as f a r as p o s s i b l e , a band pass f i l t e r , Krohn-Hite, Model 330B , i n the pressure measuring system was introduced. During measurements the f i l t e r was c a l i b r a t e d f o r every change a f f e c -t i n g the v o r t e x shedding frequency.- The f i l t e r was operated at mid-band frequency and the a t t e n u a t i o n f a c t o r was formed, by feeding a s i n u s o i d a l s i g n a l from a f u n c t i o n , generator t o the o s c i l l o s c o p e and measuring the d i f -ference i n output w i t h and without,the f i l t e r . 2 . 1 1 Other E l e c t r o n i c Instruments F o l l o w i n g i s a l i s t of other e l e c t r o n i c apparatus used i n the e x p e r i -mental work: Voltmeters: Hewlett Packard, HP^3^00A tr u e rms v o l t m e t e r , and HP-412 vacuum tube voltmeter. F u n c t i o n Generators: Hewlett Packard, low frequency f u n c t i o n generator, Model 212A, and Heathkit audio generator, Model IG - 7 2 . V i b r a t i o n Generator: Goodmans, Type V 4 7 . O s c i l l o s c o p e s : T e k t r o n i x , Type 5 6 4 , dual t r a c e storage o s c i l l o s c o p e . Chart Recorder: Honeywell, 906 C v i s i c o r d e r Low frequency a m p l i f i e r w i t h power s u p p l i e s : B u i l t i n the Department ( 1 2 ) RC Damping C i r c u i t : B u i l t i n the Departments \ V a r i a b l e Transformer: Ohmite, Cat. No. VT8-F. F i l t e r e d D.C Power Supply: E l e c t r o , Model D - 6 1 2 T Most e l e c t r o n i c instruments together w i t h a side view of the t u n n e l t e s t s e c t i o n are shown i n F i g u r e 1 1 . 13 I I I . EXPERIMENTAL PROCEDURES 3.1 Frequency, Amplitude and Phase Measurements For each of the f o l l o w i n g experiments, the non-aerodynamic damping l e v e l was set f o r the o s c i l l a t i n g system and the wind speed increased i n s m a l l s t e p s . At each wind speed the d e s i r e d measurement was made. With the c y l i n d e r s t i l l o s c i l l a t i n g , the wind speed was increased by the next step, and the process was repeated. I t was continued t o the highest wind speed and then repeated f o r decreasing wind speeds w i t h the wind speed decrement being set w h i l e the c y l i n d e r was s t i l l o s c i l l a t i n g . 3.1.1 Frequency Measurements Vortex shedding f l u c t u a t i n g pressure s i g n a l s or v e l o c i t y s i g n a l s or c y l -i n d e r displacement s i g n a l s were d i s p l a y e d on a storagescope. The v o r t e x shedding frequency was measured from the pressure s i g n a l from a surface tap at the t r a n s v e r s e diameter f o r the c i r c u l a r c y l i n d e r , and from the tap near-es t the edge f o r the D-section c y l i n d e r . The time base of the storagescope was c a l i b r a t e d a g a i n s t a Type l8k Time-mark generator ( T e k t r o n i x I n c . ) . The e r r o r i n v o l v e d was c a l c u l a t e d t o be l e s s than 2$. 3.1.2 phase Measurements The phase between two p e r i o d i c s i g n a l s was obtained e a r l y i n the program by fe e d i n g them t o the V i s i c o r d e r and measuring the average phase s h i f t over 15 c y c l e s , and l a t e r i n the program by feeding them t o the c o r r e l a t o r and c a l -c u l a t i n g the phase value ( s e c t i o n 2.8) which was averaged over 200 seconds or e q u i v a l e n t l y over 1800 c y c l e s . For spanwise phase measurements the e f f e c t of Ik any phase s h i f t i n the i n s t r u m e n t a t i o n was n u l l i f i e d by measuring a l l phase (8) s h i f t s from a permanent reference d i s c probe constructed by Ferguson and mounted behind one side of the model a t the midspan. For measurement of the phase between the surface f l u c t u a t i n g pressure s i g n a l from a tap at the mid-span and at 90° and the negative c y l i n d e r displacement, a c o r r e c t i v e phase value was i n t r o d u c e d i n the r e s u l t by measuring the phase between the model surface tap and the output of the f i l t e r . To achieve t h i s , the c a l i b r a t i o n apparatus r e f e r r e d t o i n s e c t i o n 2.7 was used by feeding a s i n u s o i d a l s i g n a l of the same v o r t e x shedding frequency i n t o the system and then t a k i n g the re a d i n g from e i t h e r a V i s i c o r d e r or a c o r r e l a t o r . Figure 12 shows the c a l i -b r a t i o n set-up. the schematic diagram of which i s the same as Figure 10 ex-cept a V i s i c o r d e r or the c o r r e l a t o r was used t o measure the phase angle be-tween the two output s i g n a l s . As can.be seen, the phase r e s u l t from the c o r r e l a t o r i s much more repre-s e n t a t i v e of the time average value than t h a t from the V i s i c o r d e r . When the capture r e g i o n was reached, a phase angle <t> between the pres-sure and the displacement s i g n a l s c o u l d be measured. This phase angle <t> was the angle by which the s u c t i o n at the tap mentioned above leads the d i s -placement i n the d i r e c t i o n of t h a t t a p . ( T h i s i s the same: as the phase angle by which the t r a n s v e r s e e x c i t i n g f o r c e leads the displacement.) 3.I.3 Displacement Amplitude Measurements The s ,signal was d i s p l a y e d on a storagescope and during each s e r i e s of t e s t s , a c a l i b r a t i o n was made. For the minimum damping l e v e l o n l y f o r each 1 5 c y l i n d e r , the measurement was made again over the complete wind speed range w i t h the c y l i n d e r s t a r t i n g from r e s t at each wind speed. For i n v e s t i g a t i n g the s t a b i l i t y of the o s c i l l a t i n g systems a t various damping l e v e l s , the non-aerodynamic damping was set at va r i o u s l e v e l s up t o the maximum a v a i l a b l e by the magnetic dampers, and the range of wind speeds over which each c y l i n d e r o s c i l l a t e d w i t h Y > .01 was determined, as was Y i n the range. max 3-2 Spanwise F l u c t u a t i n g Surface Pressure Measurements Figure 13 shows the schematic arrangement of the instrumentation f o r span-wise f l u c t u a t i n g surface pressure measurements. For each angular p o s i t i o n of the surface t a p s , measurements were taken s u c c e s s i v e l y . Then another tap an-g u l a r p o s i t i o n was assumed by r o t a t i n g the c y l i n d e r 15° and another s e r i e s of measurements along the span was taken. Due t o symmetry, o n l y h a l f of the an-g u l a r t ap p o s i t i o n s were assumed. For each reading the averaging time was 200 seconds or more. 3.3 Spanwise Wake V e l o c i t y C o r r e l a t i o n Measurements For each wind v e l o c i t y and at a f i x e d spanwise p o s i t i o n , t r a v e r s i n g the hot w i r e probe from the center across the wake and p l o t t i n g the rms value of the s i g n a l gave a peak a t the vor t e x c e n t e r l i n e . To reduce the reading d e v i a -t i o n due t o tra n s v e r s e movement ( s e c t i o n 2.k) of the probe at various spanwise p o s i t i o n s i t was d e s i r a b l e t o put the probe near a f l a t t e r peak. This p o s i t i o n was found f o r each wind speed by measuring the rms value of the f l u c t u a t i n g v e l o c i t y s i g n a l amplitude across the wake and at s e v e r a l downstream p o s i t i o n s 16 and p i c k i n g out the d e s i r a b l e one. The coordinate axes f o r wake probes p o s i -t i o n are shown i n Figure Ik. Figure 1 5 shows the schematic set-up of the instrumentation f o r spanwise wake v e l o c i t y c o r r e l a t i o n measurements. Since o n l y the fundamental vortex shedding frequency was of i n t e r e s t , two band pass f i l t e r s were introduced i n the c i r c u i t t o screen the u n d e s i r a b l e noise and thus give a c l e a r e r s i g n a l . T h i s , however, a l s o introduced phase s h i f t s due t o the f i l t e r s . Time delay i n the c o r r e l a t o r c o u l d have been used w i t h some time saving f o r each s e r i e s of t e s t s by measuring the phase of the system i n terms of -the amount of delay time t o achieve no phase s h i f t c o n d i t i o n when two probes are very c l o s e l y p l a c e d , and then t a k i n g the reading of the c o r r e l a t o r output corresponding t o the same amount of delay time during the measurement. However due to t e c h n i -c a l d i f f i c u l t y i n p l a c i n g two probes very c l o s e l y , the f o l l o w i n g a l t e r n a t i v e was used. The two f i l t e r s were ad j u s t e d f o r each v o r t e x shedding frequency so t h a t each gave the same phase s h i f t . Before every s e r i e s of measurements, hot w i r e r e s i s t a n c e was measured w i t h i t s operating r e s i s t a n c e s e t , the bridge c i r c u i t of the anemometer was balanced and the l i n e a r i z e r temperature compensation and zero adjustment made. For each probe p o s i t i o n two a u t o - c o r r e l a t i o n s and one q r o s s - c o r r e l a t i o n were measured. Since only the normalized c o r r e l a t i o n f u n c t i o n was of i n t e r e s t , s i g -n a l amplitude a t t e n u a t i o n s i n the two measuring systems were not balanced. 3.h Measurements of Mon-Aerodynamic Viscous Damping The non-aerodynamic damping was set at various l e v e l s up t o the maximum IT a v a i l a b l e by the magnetic dampers. These l e v e l s corresponded r e s p e c t i v e l y t o those f o r every s e r i e s of frequency, amplitude and phase measurements. For each damping l e v e l the model was p u l l e d t o one side and then r e l e a s e d . T his was done w i t h wind o f f . A V i s i c o r d e r was used t o rec o r d the amplitude decay. To determine the e f f e c t of the s t i l l - a i r aerodynamic damping of the model, a streamlined aluminum bar of the same weight as the model was used i n s t e a d and the same procedures were a p p l i e d . IV. . EXPERIMEEEAL RESULTS k.l Frequency, Amplitude and Phase Measurements f o r a C i r c u l a r C y l i n d e r A t o t a l of f i v e damping l e v e l s were.set f o r measuring frequency, amplitude and phase s h i f t . These f i v e ; l e v e l s correspond t o damping currents of 0, 100, l 6 0 , 250 and 3^0 ma. Figure l 6 shows a summary of the c h a r a c t e r i s t i c s of vor-t e x - e x c i t e d o s c i l l a t i o n phenomena. They a l l d i s p l a y the capture, over a d i s -c r e t e range of wind speed, of the vortex frequency by the c y l i n d e r frequency, which remains n e a r l y constant and clo s e t o the n a t u r a l frequency of the e l a s t i c system. The phase angle ; <t> increases w i t h t h e wind v e l o c i t y during capture. At a wind speed beyond.that corresponding t o the maximum displacement, the vor-t e x frequency r e v e r t s a b r u p t l y t o i t s value f o r the s t a t i o n a r y c y l i n d e r at the end of the capture range.. Figure 17(a) shows the e f f e c t of damping on v o r t e x - e x c i t e d o s c i l l a t i o n "" "" (Ik) phenomena i n the form of the s t a b i l i t y diagram introduced by Scruton . The f i g u r e shows the upper a^^&vef: -botpdaries Y = 0.01 p l o t t e d w i t h 2rfU and Y a g a i n s t 2TT . max n •''••"•„. •  . . Fi g u r e s 18, 19, 20, 21 and 22 show the d e t a i l s of the above phenomena at each damping l e v e l . , I n these f i g u r e s Y , f / f , f / f and <t> are p l o t t e d ' v n c n ag a i n s t U . A reference l i n e corresponding t o the known S t r o u h a l number f o r the s t a t i o n a r y c y l i n d e r i s i n c l u d e d on each f i g u r e . (S = O.198 f o r the c i r -k c u l a r c y l i n d e r i n the range of Reynolds, number under c o n s i d e r a t i o n , from 10 t o 5(io)\) 19 The behavior of f l u c t u a t i n g surface pressure and the c y l i n d e r o s c i l l a -t i o n during the abrupt decrease of Y from Y^^. i s i n t e r e s t i n g . F i g u r e s 2 3(a), (b) and ( c ) show s u c c e s s i v e l y the phenomena.. I t i s seen t h a t a 5 percent wind speed increase r e s u l t s i n 80 percent f l u c t u a t i n g surface pres-sure decrease and 60 percent displacement decrease. Figure 2 3(d), on the other hand, shows the phenomena when wind speed i s decreased u n t i l the abrupt increase of Y takes p l a c e . I t i s seen t h a t a 3-6 percent wind speed decrease b r i n g s 60 percent f l u c t u a t i n g surface pressure increase but o n l y 11 percent displacement i n c r e a s e . k.2 Frequency, Amplitude and Phase Measurement f o r a D-section C y l i n d e r S i m i l a r t o the c i r c u l a r c y l i n d e r measurements as above, a t o t a l of f i v e damping l e v e l s were set which correspond t o damping currents of 0, 80, 1^5, 222, and k6o ma. F i g u r e 2k shows a s i m i l a r summary of the c h a r a c t e r i s t i c s of v o r t e x - e x c i t e d o s c i l l a t i o n phenomena. Fig u r e 17(b) shows the phenomena i n terms of the s t a b i l i t y diagram. Comparing w i t h F i g u r e 17(a) i t i s seen t h a t D-section c y l i n d e r o s c i l l a t i o n s are much harder t o suppress by i n c r e a s e d damping than those of the c i r c u l a r c y l i n d e r . F i g u r e s 25, 26, 27, 28 and 29 show the d e t a i l s of the phenomena at v a r i o u s damping l e v e l s mentioned before. 4.3 F l u c t u a t i n g Pressures on the Surface of a C i r c u l a r C y l i n d e r For both a s t a t i o n a r y and a v o r t e x - e x c i t e d o s c i l l a t i n g c i r c u l a r c y l i n d e r , 20 the f l u c t u a t i n g surface pressures at each s e c t i o n experienced amplitude modu-l a t i o n which was i n phase around the c y l i n d e r . These f l u c t u a t i n g pressures at the fundamental frequency were approximately i n phase over one side of the c y l -i n d e r and l80 ° out ,of phase w i t h the opposite s i d e . For a s t a t i o n a r y c y l i n d e r , the f l u c t u a t i n g pressure s i g n a l experienced a random amplitude modulation. For an o s c i l l a t i n g c i r c u l a r c y l i n d e r , an a m p l i -tude modulation was experienced by the c y l i n d e r a t wind speeds i n i t i a t i n g c y l -i n d e r o s c i l l a t i o n . The modulation showed a beat phenomenon. I t s frequency was roughly the d i f f e r e n c e between the f l u c t u a t i n g pressure frequency and the c y l -i n d e r o s c i l l a t i o n frequency. Representative o s c i l l o s c o p e t r a c e s are shown i n F i g u r e 3 0 ' This amplitude modulation disappeared at higher wind speeds i n the 'capture 1 range. S i m i l a r phenomena were a l s o observed by Ferguson f o r a ( 1 5 ) v o r t e x - e x c i t e d c y l i n d e r and by Toebes f o r a mechanically v i b r a t i n g c y l i n -der. F i g u r e s 31 t o 35 i n c l u s i v e show the f l u c t u a t i n g pressure c o e f f i c i e n t d i s -t r i b u t i o n around one h a l f of the c y l i n d e r circumference at v a r i o u s successive s e c t i o n s along the c y l i n d e r . F l u c t u a t i n g pressure c o e f f i c i e n t C ' i s defined as the r a t i o of f l u c t u a t i n g pressure root mean square amplitude, P ' , t o the dynamic pressure, \ pv of the f r e e stream. The value of i s t h e r e f o r e a l s o of rms value i n a l l the subsequent f i g u r e s unless otherwise i n d i c a t e d . C ' d i s t r i b u t i o n corresponding t o the wind v e l o c i t y producing n e a r l y the maximum model displacement and that corresponding to the same wind v e l o c i t y but w i t h the model h e l d s t a t i o n a r y are shown r e s p e c t i v e l y i n Figures 35 and 32. 21 F i g u r e s 33 and 34 show the -C ' d i s t r i b u t i o n of an o s c i l l a t i n g c y l i n d e r cor-responding r e s p e c t i v e l y t o the. wind v e l o c i t y i n i t i a t i n g c y l i n d e r o s c i l l a t i o n and a wind v e l o c i t y somewhere a f t e r i the resonant wind v e l o c i t y . Another C ' d i s t r i b u t i o n f o r . a s t a t i o n a r y c y l i n d e r at V = 11.8 fps i s shown i n Figure 31. With the f l u c t u a t i n g pressures around each s e c t i o n known, the f l u c t u a t i n g s e c t i o n a l l i f t c o e f f i c i e n t and subsequently, the t o t a l l i f t c o e f f i c i e n t may be i n t e g r a t e d n u m e r i c a l l y . For each of the f i v e cases i n the previous paragraph, the l i f t c o e f f i c i e n t d i s t r i b u t i o n i s shown i n F i g u r e s 36, 37, 38, 39 and 40. With the l i n e of spanwise taps f i x e d at 9 = 90°, the center tap ( l l o . 9 tap) was taken as the reference and the phase s h i f t between the f l u c t u a t i n g pressure s i g n a l from the c e n t r a l tap and t h a t from one of the others was mea-sured f o r the f i v e cases mentioned i n the above. The r e s u l t s are shown i n F i g -ures 4 l , h2, U3, UU and 45. P o i n t s w i t h d i f f e r e n t symbols on F i g u r e s U l , 43 and UU were obtained on a d i f f e r e n t day t o show the r e p e a t a b i l i t y of the e x -periment . As a p r e l i m i n a r y approach t o understanding the causes of the spanwise phase s h i f t , the gap between the model end and e i t h e r the t u n n e l c e i l i n g or t u n n e l f l o o r was v a r i e d and the phase s h i f t measurements taken. The r e s u l t i s shown i n F i g u r e U6. The s i z e s of the s l o t s i n the t u n n e l c e i l i n g and f l o o r were a l s o a l t e r e d and the r e s u l t , i s shown i n Figure U7. U.U Spanwise C o r r e l a t i o n s f o r C i r c u l a r and D-section C y l i n d e r s Using Hot  Wire Anemometers For an o s c i l l a t i n g c i r c u l a r c y l i n d e r , three wind speeds were s e l e c t e d f o r 22 spanwise c o r r e l a t i o n func t ion measurements: the wind speed i n i t i a t i n g o s c i l -l a t i o n , a wind speed near the resonant wind speed, and a wind speed somewhat beyond the resonant wind speed. For the s ta t ionary c i r c u l a r c y l i n d e r , the wind speed producing near l y imximum model displacement was se l e c t ed . The r e s u l t s are shown i n F igures 48 and 49-For the D-section c y l i n d e r , the spanwise c o r r e l a t i o n func t ion was measured at on ly one wind speed, f o r both the s ta t ionary and o s c i l l a t i n g c y l i n d e r , which produced near l y maximum displacement f o r the o s c i l l a t i n g model. The r e su l t s '-are shown i n F igure 50. 23 V. DISCUSSION OF RESULTS 5.1 Frequency, Amplitude and Phase Measurements As shown i n ' F i g u r e s 16 and 2k, i r r e s p e c t i v e of the damping l e v e l s f o r the o s c i l l a t i n g c i r c u l a r c y l i n d e r , the v a r i a t i o n of the phase angle, <t> w i t h U i s n e a r l y the same, i n c r e a s i n g g r a d u a l l y from around 0° and jumping t o around 100° a f t e r Y i s reached. This a l s o a p p l i e s t o the D-section c y l i n d e r , the max ' v a r i a t i o n of 0 being from around 15°to 30°without any jump except f o r the '• lowest damping l e v e l f o r which there does e x i s t an abrupt increase of <t> near Y max The 'capture' or ' l o c k i n g - i n ' r e g i o n , over which the vortex shedding f r e -quency remains the same as the c y l i n d e r o s c i l l a t i o n frequency, and Y both become sm a l l e r as the damping l e v e l i s increased f o r both c y l i n d e r s . However, f o r the c i r c u l a r c y l i n d e r , both the center of the capture r e g i o n and the l o c a -t i o n of Y occur at lower U values as the damping current i s increased; max on the other hand, f o r the D-section c y l i n d e r , they-both occur at higher U values as the damping is- stepped up. . For the c i r c u l a r c y l i n d e r , the Y vs. U curves seem t o share the same r i s i n g s ide and then s e p a r a t e l y and s u c c e s s i v e l y take the r e s p e c t i v e Y msix values and t u r n t o the descending side w i t h approximately the same slopes. For the D-section c y l i n d e r , on the other hand, Y v s . U curves seem t o share the same descending s i d e . For the c i r c u l a r c y l i n d e r amplitude v a r i a t i o n w i t h U at low damping 2k l e v e l s ( F i g u r e s 18 and 19) a clockwise o s c i l l a t i o n h y s t e r e s i s loop r e s u l t s ; t h a t i s , w i t h the c y l i n d e r s t i l l o s c i l l a t i n g , as the wind speed i s increased beyond t h a t f o r Y , Y drops suddenly t o the value t h a t was reached from max r e s t a t that wind speed. I f the wind speed i s then decreased, the ' r e s t ' v a l -ues of Y are obtained as i n d i c a t e d by the arrows i n the f i g u r e s . The phase angle v a r i a t i o n at these low damping l e v e l s a l s o d i s p l a y s s i m i l a r h y s t e r e s i s l o o p , but counter-clockwise. There i s no h y s t e r e s i s loop f o r e i t h e r <t> or Y data f o r the higher damping l e v e l s ( F i g u r e s 20, 21 and 22). For the D-section c y l i n d e r on the other hand, the clockwise o s c i l l a t i o n h y s t e r e s i s loop e x i s t s a t a l l the f i v e damping l e v e l s . Except f o r the lowest damping l e v e l , <t> was not measured w i t h decreasing wind speed. The r e l a t i v e p o s i t i o n of the capture range i s worthy of comparison. For the c i r c u l a r c y l i n d e r capture f i r s t occurs as f f o r the s t a t i o n a r y c y l i n d e r c l o s e l y approaches f i r r e s p e c t i v e of the damping l e v e l s , and Y occurs c max i n the middle of the capture range, For the D-section, however, capture occurs when f i s only from J&fo at the lowest damping l e v e l t o 91$ at the highest damping l e v e l of f , and Y occurs at the end of the range. c max Although both models have the same mass, w i t h the maximum magnetic damping (3^0 ma.) the c i r c u l a r c y l i n d e r gives a maximum dimensionless amplitude of only 0.087; while the D-section c y l i n d e r , w i t h k60 ma. damping c u r r e n t , gives maxi-mum dimensionless amplitude of 0.311. I t i s seen t h a t the D-section c y l i n d e r o s c i l l a t i o n i s much harder t o suppress, r e i n f o r c i n g the co n c l u s i o n by P a r k i n -son^"*"^ t h a t the f i x e d f l o w separation l i n e s at the edges of the f l a t face of 25 the D-section c y l i n d e r make grea t e r e f f e c t i v e s t r e n g t h of the wake v o r t i c e s . T h i s i s f u r t h e r strengthened by the r e s u l t s i n S e c t i o n 4 .4 . 5.2 F l u c t u a t i n g Pressures on.the Surface of a C i r c u l a r C y l i n d e r In g e n e r a l , C d i s t r i b u t i o n s are f a i r l y symmetrical w i t h respect t o the JO center s e c t i o n . The unsymmetrical shape of Figure 37 i s not ex p l a i n e d . Spanwise, C ' d i s t r i b u t i o n s are f a i r l y d i s p e r s e d , and they present the same g e n e r a l d i s t r i b u t i o n p a t t e r n and range of v a r i a t i o n . However, at la r g e o s c i l l a t i o n amplitude ( F i g u r e 35) the maximum C ' jumps t o more than three times the value f o r very low amplitude or f o r the s t a t i o n a r y c y l i n d e r case, and the range of spreading along the span i s a l s o g r e a t l y a m p l i f i e d . The amplitude modulation of the surface pressures f o r both a s t a t i o n a r y and an o s c i l l a t i n g c y l i n d e r and the beat phenomena have been observed i n an (8) e a r l i e r ' i n v e s t i g a t i o n by Ferguson . I n h i s experiment on the wind-induced (9) v i b r a t i o n s of c i r c u l a r c y l i n d e r , Koopmamr a l s o n o t i c e d short b u r s t s of p e r i -o dic motion of the c y l i n d e r i n the plane normal t o the d i r e c t i o n of the wind at a wind speed i n i t i a t i n g the o s c i l l a t i o n . The beat phenomena can be regarded as a repeated e f f o r t of f t o cat c h up w i t h f ^ . When at a greater wind speed i t does succeed i n the e f f o r t , , t h e beat phenomena disappear. The governing mechanism probably i n v o l v e s the alignment of v o r t e x - l i n e s or the spanwise cor-r e l a t i o n s i n c e , as noted b e f o r e , a t la r g e c i r c u l a r c y l i n d e r o s c i l l a t i o n a mpli-tude, the vortex l i n e s are a l i g n e d p a r a l l e l t o the c y l i n d e r i n s t e a d of i n c l i n e d w i t h i t as when the c y l i n d e r i s s t a t i o n a r y or o s c i l l a t i n g at very low amplitude ( F i g u r e s 4 l , 43, 44 and 45). • 26 From the phase angle d i s t r i b u t i o n along the c i r c u l a r c y l i n d e r span, the i n c l i n a t i o n w i t h respect t o the c y l i n d e r a x i s of the f i r s t v ortex l i n e l e a v i n g the model may be c a l c u l a t e d since i n t h i s case a time h i s t o r y and s p a t i a l r e -p r e s e n t a t i o n are s i m i l a r . As shown i n Fig u r e s 4 l , 42 , k-3, 44 and 45 , the av-erage i n c l i n a t i o n f o r the s t a t i o n a r y c y l i n d e r and the o s c i l l a t i n g c y l i n d e r w i t h v e r y low displacement amplitude i s from 7° t o 9° • This i s compared w i t h ap-p r o x i m a t e l y 17° and 25° r e s p e c t i v e l y f o r the f o r c e d and wind-induced v i b r a t i o n s of c i r c u l a r c y l i n d e r s at low Reynolds numbers i n the i n v e s t i g a t i o n s by Koop-mann^' 1 0^. On the other hand, G e r r a r d ^ ^ report e d the presence of vortex l i n e s almost s t r a i g h t and p a r a l l e l t o the s t a t i o n a r y c y l i n d e r measured three 4 diameters down stream of the c y l i n d e r a x i s at Re = 2 x 10 but showed that the v o r t e x l i n e appear t o t i l t backwards and forwards between the l i m i t s of + 15° .• At Re = 85 he r e p o r t e d the i n c l i n a t i o n t o be 14° measured at 17.2 d i -ameters down stream of the c y l i n d e r a x i s . However, the i n c l i n a t i o n of the f i r s t v o r t e x l i n e l e a v i n g the model, as represented by the r e s u l t of the pres-ent i n v e s t i g a t i o n , i s expected t o be c o n s i d e r a b l y l e s s than 14° and + 15° and t o be nearer t o the present r e s u l t . As shown i n Koopmann's^'1'^ photograph us-i n g smoke v i s u a l i z a t i o n technique, a t Re = 200 : the i n c l i n a t i o n of the vortex l i n e nearest t o the c y l i n d e r i s approximately 17-5° w h i l e about IT diameters down stream of the c y l i n d e r a x i s the i n c l i n a t i o n increases to 65 0 . At l a r g e o s c i l l a t i n g amplitude, however, the average i n c l i n a t i o n i s near-( 3 ) l y 0° as shown i n Figure 45 . S i m i l a r r e s u l t s were reporte d by Gerrard and ( 9 ) Koopmann . I t i s i n t e r e s t i n g t o note t h a t large o s c i l l a t i o n caused the 27 alignment of v o r t e x l i n e s p a r a l l e l t o the c y l i n d e r a x i s and thus enhanced two-d i m e n s i o n a l i t y of the e a r l y wake f l o w . This i s a l s o t r u e f o r f o r c e d v i b r a t i o n (15) (10) as r e p o r t e d by Toebes and Koopmann In the above d i s c u s s i o n the p e c u l i a r end p o r t i o n s of the vortex l i n e s were not considered. Considering t h i s p a r t i c u l a r model, i t i s not s u r p r i s i n g t h a t low aspect r a t i o e f f e c t s were present at the ends. As shown i n Fi g u r e s hi, k2, k-3 and kk the data p o i n t s near both ends i m p l i e d curved vortex l i n e s . Figure U6 shows the e f f e c t of the gaps between model ends and t u n n e l c e i l i n g and f l o o r on the phase angle change.- I t i s seen t h a t the b l o c k i n g of end gap causes the phase angle of the corresponding end p o r t i o n of the vortex l i n e t o delay. I n Figure kj i t seems t h a t a moderate change i n s l o t s i z e does not show a d i s t i n c t i v e i n f l u e n c e on the phase angle. In g e n e r a l , however, presence of t u n n e l - s l o t s , s i z e of end gaps, and the t u n n e l w a l l boundary l a y e r retarded f l o w p l a y r o l e s i n shaping the end p o r t i o n s of the vortex l i n e s f o r the o s c i l -l a t i n g c y l i n d e r i n t h i s experiment. At each s e c t i o n of the o s c i l l a t i n g c i r c u l a r c y l i n d e r , a phase s h i f t be-tween the s i g n a l s from the neighboring pressure taps on the same side e x i s t e d . The maximum phase s h i f t w i t h respect t o the s i g n a l s from the tap at 90° at each s e c t i o n i s l a r g e r at the s e c t i o n s near the ends w i t h the average being approx-i m a t e l y 4-5° . The data i s not i n c l u d e d . S i m i l a r phase s h i f t s were a l s o r e -(12) p o r t e d by Wiland on h i s e l l i p t i c c y l i n d e r s . As a r e s u l t of measuring surface f l u c t u a t i n g pressure along and around the c i r c u l a r c y l i n d e r , C , , = 0.1+13 and 0.kk5 at V = 11.9 fps and 13.2 fps L(mpv) 28 4 4 r e s p e c t i v e l y (Re = 1.8 x 10 and 2 x 10 r e s p e c t i v e l y ) f o r the s t a t i o n a r y c y l i n d e r . Without c o n s i d e r i n g spanwise e f f e c t Ferguson found C , N = . L(mpv) 4 (2) 0.42 at Re = 1.5 x 10 w h i l e McGregor v ' found C , N = 0.58 at Re = L( mpv) 4 5 x 10 . I t i s apparent from the present i n v e s t i g a t i o n t h a t the spanwise e f f e c t s g i v e , a lower value of o v e r a l l C because of the drop i n s e c t i o n a l C near L £ both ends as shown i n F i g u r e s 36 t o 40. At l a r g e o s c i l l a t i n g amplitude, C , . = 1 . 9 1 at V = 13'9 f p s . Com-L(, mpv; p a r i s o n w i t h the s t a t i o n a r y value of C , v = 0.44-5 at V = 13-2 fps i n d i -L( mpv) cates a h i g h degree of spanwise c o r r e l a t i o n , as f u r t h e r v e r i f i e d by the subse-quent c o r r e l a t i o n measurements. 5-3 Spanwise C o r r e l a t i o n s f o r C i r c u l a r and D-section C y l i n d e r s Using Hot  Wire Anemometers Due t o the l i m i t e d aspect r a t i o , end gaps, presence of t u n n e l s l o t s and the boundary l a y e r r e t a r d e d f l o w , the c o r r e l a t i o n curves obtained do not ap-proach zero as the s e p a r a t i o n between the two hot w i r e s increases t o the max-imum a v a i l a b l e . Since the c o r r e l a t i o n f u n c t i o n must go t o zero at l a r g e separ-a t i o n s due t o the random nature of the two s i g n a l s , the c o r r e l a t i o n curves have been extended u n t i l they meet the h o r i z o n t a l a x i s . They were extended by ne-g l e c t i n g the data p o i n t s near the model ends since those data p o i n t s were under probable e f f e c t s of end clearance and w a l l boundary l a y e r . The c o r r e l a t i o n l e n g t h i s d e f i n e d as the e quivalent l e n g t h over which v e l -o c i t y f l u c t u a t i o n s i n the wake may be d escribed as p e r f e c t l y c o r r e l a t e d . The 29 c o r r e l a t i o n length i s then obtained by integ r a t i n g the area under the c o r r e l -a t i o n curve. I t i s seen that for both c i r c u l a r and D-section c y l i n d e r s , the corre-l a t i o n lengths for the o s c i l l a t i n g models at resonant wind speeds are much higher than those at other wind speeds or those for stationary models. For stationary models, A = 5.85h for the D-section cylinder while for the c i r c u l a r c y l i n d e r A = 4.56h only. For the c i r c u l a r c y l i n d e r the value i s comparable i n magnitude to those obtained by other i n v e s t i g a t o r s . Measure-ments of two-point c o r r e l a t i o n by P r e n d e r g a s t a n d e l B a r o u d i ^ ^ resulted (19") i n a A of about 3.5h and those by Vickery , 5.6h. Interesting are the high values of A for both c y l i n d e r s o s c i l l a t i n g at t h e i r resonant wind speeds. For. the c i r c u l a r c y l i n d e r t h i s r e s u l t i s i n close agreement with the r e s u l t from the f l u c t u a t i n g l i f t c o e f f i c i e n t c a l c u l a t i o n s discussed previously. 30 V I . SUMMARY OF. RESULTS Based on the experimental r e s u l t s the f o l l o w i n g may he concluded: (1) There are l a r g e r o s c i l l a t i n g amplitudes and much l a r g e r wake v e l o c i t y c o r r e l a t i o n lengths f o r a D-section c y l i n d e r o s c i l l a t i n g a t near maximum am-p l i t u d e than f o r a c i r c u l a r c y l i n d e r . (2) For "both c i r c u l a r and D-section o s c i l l a t i n g c y l i n d e r s , the v a r i a t i o n of the phase angle <t> w i t h U i s n e a r l y the same i r r e s p e c t i v e of the magnetic damping l e v e l s . (3 ) I r r e s p e c t i v e o f the damping level s . , f o r the c i r c u l a r c y l i n d e r "capture" f i r s t occurs as f f o r the s t a t i o n a r y c y l i n d e r c l o s e l y approaches f ^ , and Y occurs i n the middle of the capture range. For the D-section c y l i n d e r , max however, "capture" occurs when f i s only from 78$ t o 91$ of f and Y v c max occurs a t the end of the range. (4) The amount of the gap between the model end and the t u n n e l c e i l i n g or f l o o r d e f i n i t e l y a f f e c t s the r e l a t i v e p o s i t i o n s of the end p o r t i o n s of the vort e x l i n e s . (5) At each s e c t i o n of the o s c i l l a t i n g c i r c u l a r c y l i n d e r , a phase s h i f t be-tween the s i g n a l s from the neighboring pressure taps on the same side e x i s t s . (6) Wake v e l o c i t y c o r r e l a t i o n l e n g t h i s higher f o r a D-section c y l i n d e r than f o r a c i r c u l a r c y l i n d e r . A l s o , f o r a given c y l i n d e r , the c o r r e l a t i o n l e n g t h i s higher f o r l a r g e amplitude o s c i l l a t i o n s than f o r low amplitude o s c i l l a t i o n s or when the c y l i n d e r i s s t a t i o n a r y . (7) In general the vortex wake i s h i g h l y three-dimensional both f o r the s t a -31 t i o n a r y and the o s c i l l a t i n g c i r c u l a r c y l i n d e r . I f the e f f e c t s due t o boundary l a y e r flow a t t u n n e l w a l l s and the s l o t s i n the t u n n e l c e i l i n g and f l o o r are n e g l e c t e d , the vortex l i n e s are n e a r l y s t r a i g h t but i n c l i n e d a t about 7° t o 9° w i t h respect t o the c y l i n d e r a x i s f o r both the s t a t i o n a r y c i r c u l a r c y l i n d e r and the o s c i l l a t i n g c i r c u l a r c y l i n d e r w i t h low amplitude. For the o s c i l l a t i n g c i r c u l a r c y l i n d e r w i t h n e a r l y the maximum o s c i l l a t i n g amplitude, the vortex l i n e s form i n n e a r l y s t r a i g h t l i n e s p a r a l l e l t o the c y l i n d e r . 32 1. Keefe, R. T. 2. McGregor, D. M. 3. G e r r a r d , J . H. 4. G e r r a r d , J . H. 5. Bishop, R. E. Hassan, A. T. 6. Heine, W. 7« Den Hartog, J . P. 8. Ferguson, K. 9- Koopmahn, G. H. 10. Koopmann, G. H. • BIBLIOGRAPHY "An I n v e s t i g a t i o n of the F l u c t u a t i n g Forces A c t i n g on a S t a t i o n a r y C i r c u l a r C y l i n d e r i n a Subsonic Stream and of the A s s o c i a t e d Sound F i e l d " , U.T.I.A. Report 76, September 1961. "An Experimental I n v e s t i g a t i o n of the O s c i l l a t i n g Pressures on a C i r c u l a r C y l i n d e r i n a F l u i d Stream"., I n s t i t u t e of Aerophysics, U n i v e r s i t y of Toronto, U.T.I.A. T e c h n i c a l Note No. 14, June 1957-"The Three-dimensional St r u c t u r e of the Wake of a C i r c u l a r C y l i n d e r " , J . F l u i d Mech., V o l . 25, 1966, pp. 143-164. "An Experimental I n v e s t i g a t i o n of the O s c i l l a t i n g L i f t and Drag of a C i r c u l a r C y l i n d e r Shedding Turbulent V o r t i c e s " , J.F.M., V o l . 11, 196l , pp. 244-256. "The L i f t and Drag Forces on an O s c i l l a t i n g C y l i n d e r " , Proceedings of the Royal S o c i e t y of  London, S e r i e s A, V o l . 277, 1964, pp. 51-75-"On the Experimental I n v e s t i g a t i o n of Vortex E x c i t e d Pressure F l u c t u a t i o n s " , M.A. Sc. Th e s i s , U n i v e r s i t y of B r i t i s h Columbia, 1964. "Recent T e c h n i c a l M a n i f e s t a t i o n s of Von Kantian's Vortex Wake", Proceedings of the N a t i o n a l Academy  of Sciences of USA, V o l . 40, No. 3, 1954. "The Measurement of Wake and Surface E f f e c t s i n the S u b c r i t i c a l Flow Past a C i r c u l a r C y l i n d e r at Rest and i n Vo r t e x - E x c i t e d O s c i l l a t i o n " , M.A. ;Sc. The s i s , U n i v e r s i t y of B r i t i s h Columbia, September 1965. "On the Wind-Induced V i b r a t i o n s of C i r c u l a r C y l i n d e r s " , M.A. Sc. Thesis, C a t h o l i c U n i v e r s i t y , March 1967. "The Vortex Wakes of V i b r a t i n g C y l i n d e r s at Low Reynolds Numbers", J. F l u i d Mech., V o l . 28, 1967, pp. 501-512. 33 Smith, J.D. "An Experimental Study of the Aeroelastic I n s t a b i l i t y of Rectangular Cylinders", M.A. Sc. Thesis, University of B r i t i s h Columbia, August 1962. Wiland, E. "Unsteady Aerodynamics of Stationary E l l i p t i c Cylinders i n S u b c r i t i c a l Flow", M.A. Sc. Thesis, University of B r i t i s h Columbia, A p r i l 1968. Cheng, S. "An Experimental Investigation of the Autorotation of a F l a t Plate", M.A. Sc. Thesis, University of  B r i t i s h Columbia, 1966. Scruton, C. "On the Wind-Excited O s c i l l a t i o n s of Stacks, Towers and Masts", Proc. Int. Conf. on Wind E f f e c t s on Buildings and Structures, N. P. L., London, 1965. Toebes, G. H. " F l u i d e l a s t i c Features of Flow Around Cylinders", Proc. Int. Res. Sem. Wind E f f e c t s on Buildings and Structures, vol.2, Ottawa, September, 1967. Parkinson, G.V. Ferguson, N. Feng, C.C. "Mechanisms of Vortex-Excited O s c i l l a t i o n of B l u f f Cylinders", Proc. Symp. Wind E f f e c t s on Buildings and Structures, vol.2, Loughborough, April,1968. Prendergast, V. "Measurements of Two-Point Correlations of the Surface Pressure on a C i r c u l a r Cylinder", U.T.I. Tech. Note 23, July 1958. e l Baroudi, M.Y. "Measurement of Two-Point Correlations of Velocity Near a C i r c u l a r Cylinder Shedding a Karman Vortex Street", U.T.I.A. Tech. Note 31, January 1960. Vickery, B.J. Pankhurst, R.C. Holder, D.W. "Fluctuating L i f t and Drag on a Long Cylinder of Square Cross-Section i n a Smooth and i n a Turbulent Stream", NPL Aero Rep. No. 1146, A p r i l 1965. "Wind Tunnel Technique", Pitman, 1948. 34 APPENDIX A TUNNEL CORRECTIONS TO WIND SPEED Wind speeds were c o r r e c t e d according t o Reference 20. In the absence of b e t t e r data, c o r r e c t i o n s t o wind speed f o r the o s c i l l a t i n g c y l i n d e r were the same f o r the s t a t i o n a r y c y l i n d e r . S o l i d Blockage: V = V uncorr. 1 + CX, (|) where C = 0.822 f o r a c l o s e d t u n n e l X = 1.0 (model shape f a c t o r ) h = model widt h H = t u n n e l w i d t h Wake Blockage: V = V uncorr. i + 0.25 ( 5) c d where C = Measured drag c o e f f i c i e n t (assumed 1.25) d Therefore 3 3 V = V j 1 + 0.82 ( ^ ) + 0.25 (1.25) uncorr. L 3° 3° = 1.032 V uncorr. 35 APPENDIX B CORRELATOR PHASE MEASUREMENT The s i g n a l s from e i t h e r the B a r o c e l pressure transducer or the hot wire anemometer during the experiments have a strong fundamental frequency but are random i n amplitude. For a n a l y t i c a l purposes, the mean amplitude w i l l replace the random one. The c r o s s - c o r r e l a t i o n f u n c t i o n , R (T) , of two v a r i a b l e s i s de f i n e d by where the time delay, T , i s independent of t . I f f^(t) and fg(t) are i d e n t i c a l s i g n a l s , we o b t a i n the a u t o - c o r r e l a -t i o n f u n c t i o n of one v a r i a b l e , **-]_-]_(T) T R-MC-O = ^ £ [' f n ( t ) f . ( t - T ) at ( 2 ) 11 T-*°° T J 0 1 1 S i n c e , f o r the purpose of t h i s i n v e s t i g a t i o n , o n ly the fundamental s i g n a l frequency i s of i n t e r e s t and since the mean amplitudes replace the random ones, we assume ( 1 ) A n s i n ait (3 ) f 2 = A 2 S ± n ^ + * " < x i N A T ) where OD = fundamental angular frequency = mean s i g n a l amplitudes * = phase angle between the two s i g n a l s N = the Nth increment where the.top p o i n t of:the cosine f u n c t i o n i s reached • A T = I5o T OSN AT = the amount of phase angle delayed by the c o r r e l a t o r . As a r e s u l t of the approximated p e r i o d i c i t y , ( l ) and (2) become . T l / 2 1 -T-i./ x/2 T V 2 R U ( T ) = ±- J fx(-t) f 1 ( t - T ) d t (6) where - i s the p e r i o d of the f u n c t i o n i n seconds,.and 2TT T = — 1 ^ Define the normalized c r o s s - c o r r e l a t i o n f u n c t i o n as R1 2 ( T ) a i 2 ( r ) = V ^ i o ) R 2 (o) With the help of (3) and (4), then T l / T l / A 2 1 r '2 2 1 r 12 ? 2 i R i i ( 0 ) = T" J f i ( t ) d t = T~ J A i s i n d t = T 1 - T 1 ; 1 " T l / "72 i/2 S i m i l a r l y , A 2 = T and 37 i rV R 1 2 ( T ) = — J A I A2 S I N °^ S I N ( A I F C + * - CJCMAT) dt A l A2 = — — — cos(<t> - aiNAx) Q ; 1 2 ( T ) = co's(0 - O IN A T ) (7) Without d e l a y i n g time, T , a l 2 ( 0 ) = cos 0 , showing the c o r r e l a t i o n i s the great e s t i f 0 = 0 . On the other hand, by u s i n g the de l a y i n g mechanism i n the c o r r e l a t o r the Nth increment, where the top p o i n t of the cosine f u n c t i o n i s reached, may he counted. The phase angle between the two s i g n a l s i s , according t o (7) 0 = OJNAT T But since CD = 2TT f v and A T = z~z^r , 0 = 3-6 N T f v Figure 1. Wind tunnel outline Figure 2. Wind t u n n e l t e s t s e c t i o n w i t h model (downstream d i r e c t i o n ) C O Figure 3> Models Tap No. z ( l n c h ) z/h 1 -12.0 -4 .0 2 -10.5 -3-5 3 -9-0 -3-0 4 -7-5 -2.5 5 -6.0 -2.0 6 -4.5 -1.5 7 -3-0 -1.0 8 -1-5 -0.5 9 0.0 0.0 10 +1.5 +0.5 11 +3.0 +1.0 12 +1.5 13 +6.0 +2.0 i 4 +7-5 +2.5 15 +9.0 +3-0 16 +10.5 +3-5 17 +12.0 +4.0 Figure 4. Spanwise pressure tap p o s i t i o n s f o r the c i r c u l a r c y l i n d e r k2 Figure 5» Arrangement of model mounting system hh Barocel Vibration generator Polyethylene tube Jt=5', di = 0.066' R.M.S. Voltmeter Oscilloscope Figure 7. Block diagram of the calibration apparatus Figure 8. C a l i b r a t i o n curves f o r B a r o c e l pressure transducer 4=-o Flow velocity, m/sec Figure 9 . C a l i b r a t i o n curve f o r 55AQLDISA Anemometers without a l i n e a r i z e r -p-12 10 co o 8 co O) 03 •<— o > 6 o TJ CD O) m 4 — © — Upper —-A---- Lowei probe, R at 77 °F I ' probe, R at 80°F I 3.56 ohm, R0."6.4C 153 ohm, R0I6.33 1 ohm i ohm > St* Eo I 5.4 volts m I 2.03 0 10 20 3 0 4 0 51 D Wind speed, fps Figure 1 0 . C a l i b r a t i o n curves f o r 5 5 A 0 1 DISA Anemometers u s i n g l i n e a r i z e r s 4=-Figure 11. Instruments and wind t u n n e l t e s t s e c t i o n CO 50 Cylinder top view RMS voltmeta l - C damping circuit Oscilloscope Figure 13. Block diagram of the f l u c t u a t i n g pressure measuring set-up Figure Ik. Coordinate axes f o r wake probe p o s i t i o n s Model Hot wire probes Linearizer DISA anemometer Krohn-Hite filter DISA anemometer Linearizer Krohn-Hite filter F i g u r e 15. Block diagram of the spanwise c o r r e l a t i o n f u n c t i o n measuring set-up F i g u r e 16. Phase s h i f t and displacement phenomena f o r the c i r c u l a r c y l i n d e at v a r i o u s damping l e v e l s 8 27TU "1 u - * 0 -i o u Ymax \ % \ \ \ \ \ \ • \ \ \ \ o '£ \ \ * 'r^  .8 rmax .2 Figure 17(a). S t a b i l i t y diagram f o r the c i r c u l a r cylinder 8 27TU u — D - o U Ymax 27rfi/n 10 8 rmax Figure 17(b). S t a b i l i t y diagram f o r D-section c y l i n d e r Figure 19• O s c i l l a t i o n phenomena f o r c i r c u l a r c y l i n d e r , I, = 100 ma F i g u r e 21. O s c i l l a t i o n phenomena f o r c i r c u l a r c y l i n d e r w i t h a damping current of 250 ma. Fig. 22 Oscillation Phenomena for Circular Cylinder, 1^  = 3^ 0 ma. 6o f .11 firwmm L i j m • • • • • • • 1 (a) V = 14.4 fps (b) During the change E i i i i i i i i i • i (c) V - 15-2 f p s (&) T r a n s i t i o n from V = 13-9 i\P6 to V = 13.4 fps Top trace, pressure s i g n a l from No. 9 tap at 9 Bottom t r a c e , aisplacement Time base, 2 sec/div. 90° Figure 23. Oscilloscope traces of f l u c t u a t i n g surface pressure and aisplacement signals auring the abrupt changes. Figure 2k. Phase s h i f t and displacement phenomena f o r the D-section c y l i n d e r a t Various damping l e v e l s F i g . 25 O s c i l l a t i o n Phenomena for D-section Cylinder, I, = 0 ma Figure 26. O s c i l l a t i o n phenomena f o r D-section c y l i n d e r , I = 80 ma Figure 2J. O s c i l l a t i o n phenomena f o r D-section c y l i n d e r , I = 1^ 5 ma 65 F i g u r e 2 8. O s c i l l a t i o n phenomena f o r D-section c y l i n d e r , I = 2 2 2 ma 66 i l l . nun JftdL i IliilHIil I n I I P H V = 1 1 . 3 fps Top t r a c e , surface pressure at 0 = 9 0 ° f = 9 . 0 4 cps Bottom t r a c e , displacement c f = 8 . 7 3 cps Time base, 1 s e c / d i v . Figure 3 0 . Beat phenomena Legend Tap No. • 1 to 3 cr 5 a. 7 X 9 to 11 £> 13 Q 15 O 17 o 0 D CD 4-X I • o X CD ft * . # 9 ^ V J-*l cr to to D o g 60" Re : 18100 ir-30 120 150 180 Figure 31. c' d i s t r i b u t i o n on the surface of a s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 1 1 . 8 fps Legend Tap No. o p D cr a, x t» P Q cr o 1 2 3 5 7 9 11 13 15 16 17 cr o 4 -_CE_ a T o o. a x> P cr Q b p ca i cr t i cr x D § P 30 O, 60 Re : 19600 er 90 120" Figure 3 2 . C ' d i s t r i b u t i o n on the surface of a s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 13«2 fps Legend Tap No. 0«r • 1 p 2 t) 3 d 5 a 7 X 9 D 11 P 13 CX 15 cr 16 o 17 ft 9 ft X tl 4 -X I f cr .P a o 3 0 •8-b x I P ar ti a - X . B Q O I a Re I 17000 6 0 0° 9 0 120 P-t» o • Figure 33. C ' d i s t r i b u t i o n on the surface of an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 11.3 fps 0.3 Legend Tap No. 0.2 o l p 2 tl 3 cr 5 a 7 X 9 o 11 c 13 Q 15 cr 16 o 17 + * 9 3-*1 Figure 3^. .C ' d i s t r i b u t i o n on the surface of an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 17.6 fps Figure 35. c ' d i s t r i b u t i o n on the surface of an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 13-6 fps 78 79 Figure k2. Spanwise phase s h i f t of f l u c t u a t i n g s e c t i o n a l l i f t on a s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 13.2 fps 80 Figure 43. Spanwise phase s h i f t o f f l u c t u a t i n g s e c t i o n a l l i f t on an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V =; 11.3 i p s 8 1 Figure kk. Spanwise phase s h i f t of f l u c t u a t i n g s e c t i o n a l l i f t on an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 17•5 fps 82 Figure k-5. Spanwise phase s h i f t of f l u c t u a t i n g s e c t i o n a l l i f t on an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V = 1 3 • 9 f p s Figure k6. E f f e c t of end clearances on spanwise phase s h i f t of f l u c t u a t i n g s e c t i o n a l l i f t on a s t a t i o n a r y c i r c u l a r c y l i n d e r , V = 13•9 fps 84 * • I 17900 < B O — A — o A 6 -- 6 A -> B—U O A O O • i r o Same s l o t s sizes as "before • S l o t s lengthened and widened A S l o t s lengthened and widened Traversing gear moved downstream o—o -A & • D -40 - 2 0 2 0 4 0 6 0 8 0 100 Figure kf. E f f e c t of f l o o r and c e i l i n g s l o t s on spanwise phase shift of f l u c t u a t i n g s e c t i o n a l l i f t on a stationary c i r c u l a r c y l i n d e r , V = 11.8 fps V : 11.3 fps, Y = 0.025, Re : 16900 V : 17.0 fps. Y = 0.270, Re Z 26200 i Z 5.6 I k 1 1 I 1 X:4.22h * — , ( :: • r D 1 ^^^^ u -— A ' o " ' > 1 D diameters 8 Figure 48. Two-point fluctuating wake velocity correlations for an oscillating circular cylinder Figure 49. Two-point f l u c t u a t i n g wake v e l o c i t y c o r r e l a t i o n s f o r a s t a t i o n a r y and an o s c i l l a t i n g c i r c u l a r c y l i n d e r , V=13-9fps - e — Oscillating, — Stationary Y : 0.58, Re 121200 D diameters Figure 50. Two-point f l u c t u a t i n g wake v e l o c i t y c o r r e l a t i o n s f o r a s t a t i o n a r y and an o s c i l l a t i n g D-section c y l i n d e r , V = lk.1 fps oo -3 

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