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UBC Theses and Dissertations

Unsteady aerodynamics of stationary elliptic cylinders in subcritical flow Wiland, Erling 1968

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UNSTEADY AERODYNAMICS OF STATIONARY ELLIPTIC CYLINDERS IN SUBCRITICAL(FLOW by E. WILAND B.Sc. (Hons.), The University of Strathclyde, 1965 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 requi red standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1968 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t ish Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood.that copying or publication of this thesis for financial gain shall not be allowed without my written permission. E. Wiland Department of Mechanical Engineering The University of Br i t ish Columbia Vancouver 8, B.C. i ABSTRACT The aerodynamics of a set of two-dimensional e l l i p t i c cyl inders with eccent r i c i t y of 0.8 and 0.6 is studied experimentally during the organised wake condit ion. The dynamic ca l ib ra t ion of the transducer used for measurement of f luctuat ing pressures is described in d e t a i l . The data on Strouhal number, unsteady pressures and wake geometry are presented as a function of angle of attack during s t a t i c condition of the models. The e f fect of Reynolds number on the f luctuat ing pressure is also examined. The results indicate dependence of the unsteady forces on Reynolds number at zero angle of attack. Basing the Strouhal number on projected width appears to reduce i t s dependence on the angle of attack of the models. The existence of a large phase angle between the f luc tuat ing pressures is of i n te res t . The wake geometry study indicates a gradual reduction in the rat io o f : the la te ra l to the longitudinal spacing with increase in angle of attack. n TABLE OF CONTENTS Section Page 1 Introduction . . . . . . . . . . •••• 1 2 Purpose and Scope of the Investigation . . . . 5 3 Models and Supporting S y s t e m . . . . . . . . . . . . . . 6 4 Instrumentation and Cal ibrat ion 12 5 Test Procedures . . . . . . 32 6 Test Results and Discussion 38 7 Concluding Remarks . 62 Bibliography . . . . . . . . . - . . . . . 65 Appendix I .• 68 Appendix II 70 i i i LIST OF TABLES * Table Page 1 Model data . . . . . . . . . . 6 2 Effect of phase shift on fluctuating l i f t coefficient... 54 3 Spacing of vortices in-fully developed wake, Nr = 70,000 58 / i v LIST OF FIGURES Figure Page, 1 Constructional deta i ls of models - . . 8 2 Numbering of pressure taps 9 3 Wind tunnel out l ine . . . . 10 4 Wind tunnel test section with model 11 5 Schematic of Barocel pressure transducer • 13 6 Block diagram of pressure generator and set-up fo r comparison of dynamical response . . . . . . • 15 7 Response of ca l ib ra t ion system to d i f ferent input wave forms • • .' • • 18 8 Pressure attenuation as a function of tube length and frequency (tube diameter = 0.066 i n . ) . . . . . - . . 19 9 Cal ibrat ion plots for Barocel pressure t r a n s d u c e r . . . . . . . . . 20 10 Block diagram of the ca l ib ra t ion apparatus . . . . . . 22 11 Cal ibrat ion apparatus ... . . 23 12 Cal ibrat ion plots for Barocel pressure transducer with damping bott le ..• 24 13 Comparison between ca l ib ra t ion curves fo r Barocel pressure transducer with and without damping bot t le . . 25 14 Geometry of the disc probe ..- 27 15 Typical f luc tuat ing pressure s ignals . . . 28 16 R-C damping c i r c u i t . . . . . . . . . . . . . . 31 17 Block diagram of the f luc tuat ing pressure measuring set-up ... • 33 V Figure Page 18 Block diagram of the phase measuring system 34 19 Instrumentation set-up during a t yp ica l test run 35 20 Traversing gear with probe during wake measurements 37 21 Var iat ion of Strouhal frequency with Reynolds number- 39 22 Var iat ion of Strouhal number with angle of attack 40 23 Variat ion of f luc tuat ing pressure coef f i c ien t with Reynolds number, e = 0 . 6 . . • 42 24 Variat ion of f luc tuat ing pressure coe f f i c ien t with Reynolds number, e = 0.8 . . . . . . . . . . . . . . . . 43 25 Dis t r ibut ion of mean and f luctuat ing pressure c o e f f i c i e n t s , a = 0 45 26 Dis t r ibut ion of mean and f luc tuat ing pressure c o e f f i c i e n t s , a = 30° . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 27 Dis t r ibut ion of mean and f luc tuat ing pressure c o e f f i c i e n t s , a = 60° •.. 47 28 Dis t r ibut ion of mean and f luctuat ing pressure c o e f f i c i e n t s , a = 90° . . . . . . 48 29 Dis t r ibut ion of mean and f luc tuat ing pressure coef f ic ients on the surface of a c i r c u l a r cy l inder (e = 0) •.. . . . 49 30 Var iat ion of l i f t and maximum f luc tuat ing pressure coef f ic ients with angle of a t t a c k . . . . . . . . . . . . . . . . . 51 31 Amplitude modulation of the pressure s ignals on the surface of the model, e = 0.6 . . . . • • • • 52 32 Amplitude modulation of the pressure signals on the surface of the model, e = 0 . 8 - . . . . . . . . . . . . . 53 33 Phase s h i f t between pressure signals on the surface of the model, e = 0 . 6 , a = .0,-90° • • •• -.- 55 vi Figure Page 34 Phase s h i f t between pressure s ignals on the surface of the model, e = 0 . 8 , a = 0 , 90° and Nr = 67,000. . . . . 56 35 Representative results of wake measurements . . . . . . . 59 (a) Typical wake traverse (b) Typical amplitude decay in wake 36 Var iat ion of drag and moment coef f i c ients with angle of a t t a c k . . . . . . . . . . 61 ACKNOWLEDGEMENT The author wishes to express his appreciation for the helpful supervision and advice by Dr. V . J . Modi. Thanks are also due to the Department of Mechanical Engineering for use of t h e i r f a c i l i t i e s and to technicians of the department for t h e i r valuable assistance. F inancial support was received from the National Research Council of Canada, Grant A-2181. LIST OF SYMBOLS Drag c o e f f i c i e n t , P r e s % u r e d r a 3 1/2PV* 2 ac L i f t c o e f f i c i e n t , — L l f t l/2pV? 2 ac Fluctuating l i f t coe f f i c ien t (average amplitude) Moment c o e f f i c i e n t , — o m e n t 1/2PV? 4 a 2 c P-P Pressure c o e f f i c i e n t , — V2pV? P' Fluctuating pressure c o e f f i c i e n t , — 5 -1/2PV! Longitudinal spacing between vort ices V 2a Reynolds numbers Strouhal number, y— or — Percentage circumference, measured clockwise from tap 0 Free stream veloc i ty Lateral spacing between vort ices Semi major axis of model Semi mi nor axis of model Length of model Eccentri ci t y , Frequency of vortex formation, Strouhal frequency Projected height of model, 2 / a s in a + b cos a Distance downstream from centre of model Mean pressure Fluctuating pressure (average amplitude) about mean Free stream s t a t i c pressure Transverse distance from centre of model Angle of attack A i r density A i r kinematic v i scos i ty 1. INTRODUCTION It i s well known that under certain conditions e l a s t i c a l l y mounted b l u f f bodies, when exposed to a f l u i d stream, may exhib i t se l f - induced o s c i l l a t i o n s . The v ibrat ion of smoke s tacks , transmission l i n e s , periscopes, a i r c r a f t wings, br idges, launch veh ic les , e t c . , has been of in terest to engineers. The v ibrat ion of a structure is undesirable fo r many reasons, one of which is the danger of i t s col lapse due to fat igue of the mater ia l . In general , the nature of the forc ing funct ion , wake geometry, and Strouhal number form three important parameters in an aeroe last ic i n s t a b i l i t y study. The determination of the corresponding information fo r a set of stat ionary e l l i p t i c cyl inders 4 5 in the Reynolds number range of 3 x 10 - 10 forms the subject of th is presentation. In general , the e l a s t i c a l l y supported b l u f f bodies exh ib i t two d i s t i n c t forms of aerodynamically induced v ibrat ion transverse to the flow d i r e c t i o n . The f i r s t type, commonly known as vortex resonance,' refers to the condition where the pe r iod i c i t y of the organized vortex shedding coincides with the natural frequency of the system. The second form of o s c i l l a t i o n arises due to negative slope in the t rans-verse, force versus angle of attack diagram. This causes the body to be unstable under transverse disturbances resu l t ing in o s c i l l a t i o n that grows in amplitude unt i l the energy extracted from the f l u i d stream balances that d iss ipated through various forms of damping. Strouhal'' was the f i r s t to correlate the per iodic vortex shedding with the diameter of a c i r c u l a r cy l inder and f l u i d ve loc i ty . 2 This was followed by the c l a s s i c a l study of wake geometry and s t a b i l i t y 2 by Von Karman . Ever s ince , academic and p rac t i ca l interest in the vortex shedding phenomenon has resulted in many theoret ica l and experimental inves t igat ions , e . g . , by Roshko, Kovasnay, T r i t t o n , 3 Bi rkhof f , Humphreys, Schaeffer , E s k i n a z i , and others. Marris has presented an excel lent review of th is l i t e r a t u r e . More recent ly , Grove 4 5 et al as wel l as Bishop and Hassan measured f luctuat ing forces on a stat ionary c i r c u l a r cy l inder over a range of Reynolds number. The corrresponding preliminary results for square and rectangular cyl inders were presented by Modi and Heine**. In contrast to t h i s , the unsteady pressure var iat ions and wake structure associated with the o s c i l l a t i n g two dimensional b l u f f cyl inders have hardly been invest igated. The study of Bishop and Hassan^ showed that the vortex frequency, over,a range of c i r c u l a r o cy l inder frequencies, i s cont ro l led ; while Ferguson and Parkinson measured f luc tuat ing pressure and wake pattern related to the cy l inder executing vortex induced o s c i l l a t i o n s . A study of the related but somewhat modified problem involv ing ca lcu lat ion of the boundary layer and i t s separation over an i n f i n i t e , g yawed cy l inder was carr ied out by Chiu who also measured i t s Strouhal number using a flume. It was suggested that the vortex shedding of a yawed cy l inder depends only on the cross component of the loca l ve loc i t y . Among the numerous papers wr i t ten on the aspects of f l u i d mechanics and dynamics of b l u f f bodies, only a few are concerned with the actual measurements of unsteady pressures. McGregor"^ and Gerrard^ measured f luctuat ing forces on a c i r c u l a r cy l inder using a condenser 3 microphone system b u i l t into the model. The pressure d i s t r ibu t ion was 12 obtained by turning the cy l inder . Pendergast obtained spanwise pressure corre lat ion r e s u l t s , fo r a stat ionary cy l inder , using a 13 modified form of McGregor's apparatus. Keefe carr ied out f l u c t u a -t ing force measurements with the help of a carefu l l y designed s t r a i n -14 gauge transducer which also acted as a test model. Molyneux has also described a low frequency strain-gauge type transducer mounted inside the model to measure pressures on o s c i l l a t i n g wings. He ine^ and Ferguson^ designed pressure transducers, using a p i e z o l e c t r i c crysta l and l i gh t sens i t i ve resistances respect ive ly , which were located external ly and connected to the pressure taps on the model by a series of polyethylene tubings. The devices exhibi ted several undesirable features. The former has a marginal s e n s i t i v i t y while the l a t t e r , though s e n s i t i v e , was substant ia l l y affected by the ambient temperature and humidity. The avai lable information concerning b l u f f body interact ion with the separated flow of stable vortex street type is not l imi ted to the cyl inders of c i r c u l a r c ross -sect ion . Investigations with square, rectangular, t r iangular and hexagonal cy l inders , s t ructura l H and angle sect ions , as well as several i r regu la r geometries are reported. But i t must be emphasized that the bulk of the l i t e r a t u r e is indeed devoted to the c i r c u l a r geometry. This point i s well emphasized by the fact that the previous work on e l l i p t i c a l cyl inders seems to be l imi ted 15 to rather preliminary unsteady pressure measurements by Heine and the Strouhal number study by Schramm^. 4 The aeroelast ic i n s t a b i l i t y of b l u f f bodies has been under invest igat ion in th is department since 1958. The review of the progress 18 19 made has been reported in two survey papers ' . The invest igat ion described here forms the part of th is continuing programme and intends to study, experimental ly , the ef fect of eccent r i c i t y of the c y l i n d r i c a l b l u f f body on the fundamental parameters l i s t e d before. 5 2. PURPOSE AND SCOPE OF THE INVESTIGATION As pointed out before, considerable information about the vortex excited motion of a c i r c u l a r cy l inder i s reported in l i t e r a t u r e . The long range aim of th is project i s to obtain corresponding i n f o r -mation for two dimensional cy l inders , of intermediate c ross -sect ions , obtained by systematical ly varying the eccent r i c i t y from zero ( c i r c u l a r cyl inder) to i n f i n i t y ( f l a t p la te ) . In general the accuracy of the measured data depends on the capabi l i t y of the instrumentation and the accuracy of i t s c a l i b r a t i o n . This being the case, the project studies in deta i l the dynamic c a l i b r a -t ion of a transducer used to measure the magnitude, phase r e l a t i o n , and frequency of the acoustic level pressure var ia t ions . The main aim of the project i s to study the aerodynamics of two dimensional, e l l i p t i c c y l i n d r i c a l models, of e c c e n t r i c i t i e s 0.6 and 0 . 8 , under the condition conducive to vortex exci ted o s c i l l a t i o n s . The thesis presents experimental results on: ( i ) the var iat ion of Strouhal number with Reynolds number; ( i i ) the mean and f luc tuat ing s t a t i c pressure d i s t r i b u t i o n ; ( i i i ) wake geometry as a function of angle of attack during s t a t i c condition of the model. 4 5 In most cases the Reynolds number range is confined to 3 x 10 - 10 . Since the influence of the wind tunnel walls on the measured parameters is not wel l es tab l i shed , the results presented are uncorrected for that e f fect (Appendix I ) . 6 3. MODELS AND SUPPORTING SYSTEM Two e l l i p t i c c y l i n d r i c a l models, 27 inches long, were designed to span the wind-tunnel cross -sect ion thus approximating the two-dimensional flow condi t ion. The constructional deta i ls are shown in Figure 1 and the physical parameters are l i s t e d below: Model e a/b Materi al Wei ght, Lb. Number of Bulkheads Skin Thi ckness, In. . 1 2 0. 6 0.8 2.5/2 2.5/1.5 t Aluminum Plexig lass 4.20 1.45 -7 7 0.02 0.02 Table 1. Model data The models were so constructed that they can be mounted on the wind tunnel balance or brackets attached to the tunnel , or may be supported by the ex is t ing a i r bearing system for measurements under v ibrat ing condit ion. At the central bulkhead of each model there are 32 pressure taps (d = 0.025 i n . ) equally spaced around the circumfer-ence (Figure 2 ) . Two taps were provided in the spanwise d i r e c t i o n , at a distance of 4.5 i n . and 9 i n . from the mid -sect ion , in the plane of the minor ax is . The pressure taps are connected to p l a s t i c tubes of inside diameter 0.066 i n . which are brought out from one end of the cy l inder (Figure 1) . In case of the plexig lass model i t was thought advisable, due to the thin s k i n , to ascertain the r i g i d i t y of the 7 panel. A simple test of def lect ion and natural frequency showed the panels to be of adequate s t i f fness with a maximum def lect ion of less than 0.0004 i n . and natural frequency of 180 cps during operating condit ions. The models were tested in a low speed, low turbulence, return type wind tunnel where the a i r speed can be varied from 4 - 150 ft/sec with a turbulence level less than 0.1%. The pressure d i f f e r e n t i a l across the contraction section of 7:1 rat io can be measured on a Betz micromanometer with an accuracy of 0.2 mm of water. The test sect ion ve loc i ty is ca l ibrated against the above pressure d i f f e r e n t i a l . The rectangular c ross - sec t ion , 36 i n . x 27 i n . , is provided with 45° corner f i l l e t s which vary from 6 i n . x 6 i n . to 4.75 i n . x 4.75 i n . to compensate part ly fo r the boundary layer growth. The spat ia l var iat ion of mean ve loc i ty in the test section is less than 0.25%. The tunnel i s powered by a 15 horsepower d i rect current motor dr iv ing a commercial axif low fan with a Ward-Leonard system of speed cont ro l . An outl ine of the tunnel i s shown in Figure 3. Figure 4 shows a model mounted in the wind tunnel during t e s t . gure 1 . Constructional deta i l s of models Figure 2. Numbering of pressure taps i—Turning vanes 5350 Figure 3. Wind tunnel outline 12 4. INSTRUMENTATION AND CALIBRATION 4.1 Pressure Transducer Recently a new pressure transducer, ca l led Barocel Modular Pressure Transducing System, has appeared on the market. An extensive experimentation with the unit showed i t to be quite su i table for the intended measurements. Consequently, two pressure transducers together with the necessary power and signal conditioner units were acquired. Developed by Datametrics Inc. of Waltham, Massachusetts, the Barocel i s a high p r e c i s i o n , stable capacit ive voltage d i v ide r , the var iable element of which i s a thin prestressed s ta in less steel diaphragm (Figure 5) . Posit ioned between f ixed capacitor p l a t e s , the diaphragm deflects proportional to the magnitude of the applied pressure. An a .c . c a r r i e r voltage at 10 Kc. is applied to the stat ionary capacitor p lates . The diaphragm attains a voltage level determined by i t s r e l a -t ive posi t ion between the f ixed capacitor p lates . With the Barocel appropriately arranged in a bridge c i r c u i t , the output voltage is determined by the rat io of capacitance of the diaphragm to each of the stat ionary electrodes. The c a r r i e r voltage is thereby amplitude modulated in accordance with the input pressure. The units have 8 ranges. 0-10 mm of mercury on the least s e n s i t i v e , and 0-0.001 mm of mercury on the most sens i t i ve range. Further data given by the manufacturer are: Output: 0-5 volts d.c f u l l scale L inear i t y : _ 0.1% in ranges used Accuracy of + S t a t i c Ca l ib rat ion : _ 0.5% of f u l l scale 13 Figure 5. Schematic, of Barocel pressure transducer 14 Transient response: Typ ica l ly less than 2 mil l iseconds to step input for l ine pressure of 750 mm of mercury S t a b i l i t y : (a) - 0.1% for t 15° ambient pressure change (b) t 0.01% for t 10 volts l ine voltage change For (a) and (b) constant, t 0.01% day to day The Barocel i s accurately ca l ibrated for s t a t i c pressures. For the f luctuat ing pressure s ignals transmitted through r e l a t i v e l y long tubes considerable attenuation i s to be expected. The attenuation of the signal w i l l depend on such variables as tube length, tube diameter, number and s i ze of constr ict ions (pressure taps ) , frequency and shape of the pressure s ignal and, in add i t ion , the basic frequency response of the Barocel i t s e l f . The inconsistencies in the r e s u l t s , obtained with the c a l i b r a -t ion methods previously developed in the department, were at t r ibuted to a resonance condition with in the system and/or er ror in the theo-r e t i c a l predict ion of pressure. It was therefore real i zed that a system had to be developed where no resonance condition would ex is t between the transducer with tube and pressure tap on one side and the volume where the ca l ib ra t ion s ignal was generated on the other s ide . This condition was obtained with s u f f i c i e n t accuracy by using as a pressure source a 5 gallon drum with a thin latex diaphragm on one end. The pressure signal was generated by a baf f le plate actuating the diaphragm. The baf f le plate was driven by a v ibrat ion generator which in turn received i t s impulse from a function generator as shown in Figure 6. The natural frequency of the drum with diaphragm, baf f le 15 Function generator i Amplifier Barocel wtMumnnnunnihiiinnnnininninni Diaphragm Baffle Sound level meter yf-n. 5 gallon drum 'h))l))))))))))))))))))})i))))hiiiiiiiiiiiiiiii7TTTm, Vibrator Barocel o o Signal conditioner Power supply i A / W W V J W A A A A Oscilloscope Figure 6. Block diagram of pressure generator and set-up for comparison of dynamical response 16 and v ibrat ion generator was found to be 60 cps. However, th is does not l i m i t the use of the apparatus to lower frequencies provided the natural frequency of the transducer used to measure the source pressure is s u f f i c i e n t l y high. To measure the source pressure a sound level meter could be used but these meters have a decibel s c a l e , and an accuracy of - 1 db is equivalent to about 25% var iat ion in the absolute pressure. A l so , most sound level meters have a poor,frequency response below 20 cps. This led to an invest igat ion of the open port frequency response of the Barocels with a view to use one of them for measuring the c a l i -bration pressure at, the source. The natural frequency of the Barocel diaphragm i t s e l f i s given by the manufacturer to be 2500 cps or higher* A lso , the t ransient ,response is spec i f ied to be less than 2 ms. The Helmholtz resonator frequency of the cavity and the connection on one side of the diaphragm was calculated to be 290 cps. The experimental value obtained by actuating one side of the Barocel with a horn dr iver was found to be around 210 cps. Both these values are of the order of magnitude suggested by the t ransient response given above. It was estab l i shed , through preliminary experiments j that the required frequency range would be from 5 to 35 cps. Considering the lower value of 210 cps for the natural frequency, and a dr iv ing frequency of 40 cps, one would get an output within - 4% of the input i f a l i near system with damping c o e f f i c i e n t somewhere between zero and c r i t i c a l 1s assumed. This j u s t i f i e s the use of the s t a t i c a l l y ca l ibrated Barocel for measuring the pressure Inside the drum. Figure 6 i l l u s t r a t e s the 17 set-up of the pressure generator during the i n i t i a l evaluation of the Barocel response. It was found that for s inusoidal inputs the signals from the Barocel and the function generator could be superimposed on the osc i l loscope . Even in the extreme cases of square and t r iangular wave inputs the response of the ca l ib ra t ion system was quite good as indicated in Figure 7. The results of the i n i t i a l ca l ib ra t ion with i varying tube length and frequency are shown in Figure 8. The approxi -mate exponential decay of output amplitude with tube length i s as 20 expected . The appearance of "bumps" at shorter tube lengths, most i pronounced at 15 cps, cannot be at t r ibuted to a resonance condition between the drum and the Barocels. The same phenomena w i l l occur under actual measuring condition in the wind tunnel , and i t may be considered as a p e c u l i a r i t y of th is combination of tube, pressure tap and t rans -ducer. 1 Since noticeable attenuation occurred due to the presence of a const r ic t ion in the tube a l l ca l ibrat ions were performed with a pressure tap in the c i r c u i t . The ca l ib ra t ion plots for a Barocel when connected to a pressure tap through a tube of given length and diameter are shown in Figure 9. This also demonstrates the l i n e a r i t y of the system. For work in the wind tunnel i t was found impossible to use atmospheric pressure as reference. This is because the difference between the s t a t i c pressure at the tap and the atmospheric pressure was found to be so large as to throw the Barocel off scale at sens i t ive se t t ings . Moreover, surges in the base pressure affected the pressure f i e l d around the model and gave r ise to the same e f f e c t . Hence, the W i l l i • • j i ••••111 (b) Figure 7. Response of ca l ib ra t ion system to d i f ferent input wave forms (1 cps, 0.0016 p s i ) . (a) Tr iangular wave input (b) Square wave input 19 1.2 Figure 8. Pressure attenuation as a function of tube length and.frequency (tube diameter = 0,066 1n.) 0 500 1000 1500 2000 Input, mV Ffgsarae -SL- Eallfiforatiioni plots for Barocel pressuire transducer 21 f i n a l ca l ib ra t ion set-up incorporated a damping volume between the pressure ports of the Barocel , thus using the s t a t i c pressure of the tap in question as reference. A block diagram and a photograph of the ca l ib ra t ion set-up are shown in Figure 10 and 11 respect ive ly . The e f fec t of amplitude and frequency on output, with the damping bott le in the c i r c u i t , i s shown in Figure 12,. For convenience the ca l ib ra t ion curves in Figures 9 and 12 are plotted in Figure 13 as a rat io of out-put to input. It can be noticed that above 15 cps there is no attenua-t ion due to the damping b o t t l e . The s p e c i f i c tube length and diameter were chosen fo r p rac t i ca l reasons. Sinusoidal s ignals were used throughout the c a l i b r a t i o n . 4.2 Manometer An i n c l i n e d Lambrecht manometer with ethyl alcohol was used to measure s t a t i c pressure on the surface of the model. The manometer can be read with an accuracy of l/10th of a mi l l imeter . It was found necessary to reduce the f luctuat ions of the l i q u i d column caused by the pressure surges previously mentioned. The constr ict ions in the tubing formed by four hypodermic needles (#19) in series gave adequate damping. 4.3 Wake Probe The wake survey was carr ied out using a disc probe constructed 16 21 by Ferguson and described in deta i l by Bryer et al . As pointed out by these inves t igators , the probe i s r e l a t i v e l y insens i t i ve to 22 Function generator Amplifier Barocel Vibration generator Polyethylene tube 1=5', dj = 0-066' R.M.S. Voltmeter Oscilloscope Figure.10. Block diagram of the ca l ib ra t ion apparatus 23 Figure 11. Calibration apparatus 1000 > E 3 o. O 500 500 1000 1500 2000 f igure 12. Input, mV Cal ibrat ion plots for Barocel pressure transducer with damping bott le ro Figure 13. Comparison between ca l ib ra t ion curves for Barocel pressure transducer with and without damping bott le 26 pitch (- 4°) and yaw (- 20°). The main dimensions of the probe are given in Figure 14. 4.4 Band Pass F i l t e r There are several sources of disturbances, e .g . fan dr i ve , .o ther laboratory equipment, surges in the tunnel , e t c . , which superpose undesirable pressure var iat ions on that created by the shedding vor t ices . In general , the intens i ty of the 'noise ' may be considered constant. On the other hand the pressure f luctuat ions due to shedding vort ices depend on locat ion of the tap and att i tude of the model. This being the case there were s i tuat ions where the noise had a tendency to overshadow the vortex-generated pressure var ia t ions . It was, there-fo re , necessary to introduce a band pass f i l t e r in the pressure measuring system to el iminate the undesirable noise. The t yp ica l pressure traces of f i l t e r e d and unf i l te red s ignals are compared in Figure 15. At a = 0 , where the signals are weak in re lat ion to no ise , the necessity of using a f i l t e r i s obvious. At a = 90°, where the s ignals are more powerful, the noise level becomes r e l a t i v e l y i n s i g -n i f i c a n t and the f i l t e r is no longer e s s e n t i a l . It was also observed that the pressure signal from the separated flow region was s l i g h t l y more<irregular than that from the laminar f i e l d . This may be due to higher turbulence level in the wake. During measurements the f i l t e r was ca l ibrated for every change af fect ing the vortex shedding frequency. Operating the f i l t e r at mid-band frequency and with the high and low cut -o f f sett ings separated by a factor of 1.5 gave an attenuation between 0.8 and 0 . 9 . This was 27 Figure 14. Geometry of .the d i s c probe e = 0.6 e = 0.8 (b) Figure 15. Typical f luc tuat ing pressure traces (not to the same scale) (a) a = 0 , (b) a = 90° Upper trace represents f i l t e r e d s ignal 29 found by feeding a s inusoidal s ignal from the function generator to the rms voltmeter and measuring the difference in output with and without the f i l t e r . 4.5 R.M.S. Voltmeter Due to considerable, seemingly random, amplitude modulations of the f luc tuat ing pressure i t was necessary to present the results as time-average values. Under certain conditions several minutes of averaging was required to get reproducible resu l t s . This led to the necessity of using a true rms voltmeter converting the f luc tuat ing pressure s ignal to an equivalent d . c . s i g n a l . An external r -c damping c i r c u i t was used to reduce the var iat ions caused by the amplitude modulations. The f i n a l steady d .c . level was then measured on the vafcuum tube voltmeter. Because of the extremely high input resistance of the l a t t e r no measurable voltage drop occurred over the damping res istors (Figure 16). 4.6 E lect ron ic Instruments Following is the l i s t of the e lec t ron ic apparatus used in the experimental work: F i l t e r s : Krohn-Hite, band pass var iable f i l t e r , models 330B & 330A. Voltmeters: Hewlett Packard, HP-3400A true rms voltmeter, and HP-412 vacuum tube voltmeter. Function Generator: Hewlett Packard, low frequency function generator, model 202A. Vibrat ion Generator: Oscil1oscope: Chart Recorder: Low frequency ampli -f i e r with power suppl ies : R-C damping c i r c u i t : 30 Goodmans, type V47. Tectronix , type 564, dual trace storage oscil1oscope. Honeywell, 906c v i s i co rder . 22 B u i l t in the department . B u i l t in the department (Figure 16). 31 H P - 3 4 0 0 A R.M.S. Voltmeter Output resistance 1 KO O 0 H P - 4 1 2 A V.T. Voltmeter Input resistanc 200 M O O O *-A/WW Figure 16, R-C damping c i r c u i t 32 5. TEST PROCEDURES 5.1 Fluctuating Pressure Measurements ; The diagrammatic lay-out of the apparatus used for f luc tuat ing pressure measurements is given in Figure 17. During measurements the r -c damping c i r c u i t was set to give minimum var iat ion of the voltmeter reading. The f luctuat ing pressure signal was also displayed on the osci l loscope to determine the maximum amplitude and Strouhal frequency. 5.2 Phase Measurements The phase between the f luc tuat ing pressures at d i f fe rent taps was obtained by feeding the s ignals to the V is icorder and measuring the average phase s h i f t over 10-cycles.. The e f f e c t of any phase s h i f t in the instrumentation was n u l l i f i e d by. measuring a l l phase s h i f t s from a permanent reference tap at 90° to the wind d i r e c t i o n . The data showed considerable scat ter which increased with the distance from the reference tap. Figure 18 shows, schematical ly , the arrangement of the instrumentation for phase measurements. The set-up during a t yp ica l test run is shown in Figure 19. 5.3 Mean S t a t i c Pressure D is t r ibut ion Mean pressure on the model was measured using a Lambrecht manometer. One leg of the manometer was "connected to a tota l head tube 1n the s e t t l i n g chamber and the other to the pressure tap on the model. 33 \ Power supply Damping bottle Voltmeter R -C damping R. M. S. circuit voltmeter Oscilloscope Figure 17. Block diagram of the f luc tuat ing pressure measuring set-up 34 Barocel Signal conditioner E7 Damping bottle Barocel Power supply Signal conditioner Filter Visicorder Filter Figure 18. Block diagram of the phase measuring system Figure 19. Instrumentation set-up during a t yp ica l test run w 36 5.4 Wake Measurements The determination of wake geometry was accomplished using the method described by He ine^ and Ferguson^. The transverse distance between the vortex centrel ines was measured by the wake probe connected to the pressure transducer in a manner s i m i l a r to that shown in Figure 17. A photograph of the wake probe and the pressure transducers connected to the damping bott les is given in Figure 20. Moving the probe across the wake and p lo t t ing the rms value of the s ignal gave two peaks at the vortex centre l ines . The distance-between these peaks represents the l a t e r a l spacing between the vortex core l ines at that s t a t i o n . The longitudinal spacing between the consecutive vort ices was obtained by using a pressure tap as reference and moving the probe downstream on the same side of the model unt i l the signals were 180° out of phase. From th is pos i t ion the probe was moved further downstream so that the the s ignals were in phase. The process was repeated un t i l l imi ted by the t ravel of the t ravers ing gear. Twice the distance between two successive measurements gave the desired longitudinal spacing between the vor t i ces . The instrument arrangement was s i m i l a r to the one used for phase measurements (Figure 18), except that the signals were d i s -played on the osc i l loscope . Figure 20. Traversing gear with probe during wake measurements 38 6. TEST RESULTS AND DISCUSSION 6.1 Strouhal Number The Strouhal frequency was measured in the Reynolds number 4 5 range of 2 x 10 - 10 for three d i f ferent angles of attack, and i t was observed to be l i n e a r with increasing wind speed as shown in Figure 21. The e f fec t of angle.of attack on the Strouhal number was also obtained in the same Reynolds number range. These variat ions based on the minor axis as well as the projected height are shown in Figure 22. The Strouhal number based on projected height (h) showed comparatively less dependence on the angle of attack (a) in the Reynolds number range invest igated. An observation concerning the Strouhal number var iat ion at low angles of attack i s pert inent here. For the th icker e l l i p s e a slow r ise in the projected height with a small drop in shedding frequency leads to an almost uniform Strouhal number in this range. On the other hand, for the thinner e l l i p s e the very sharp drop in shedding frequency o f f rsets the r e l a t i v e l y greater r ise in projected height thus leading to a decrease in Strouhal number. The s i m i l a r 17 tendency was also observed by Schramm who carr ied out Strouhal number measurements for the e l l i p s e s of four d i f ferent e c c e n t r i c i t i e s (e = 0.662, 0.866, 0.969, 0.998). In a l l cases except one the measure-ments were l imted to 0 - 45°, range due t o , as reported by him, "lack of stable vortex s t r e e t " . This is in contrast to the strong and wel l defined f luc tuat ing s ignals observed in the test results presented here. 40 0-22 0 30 60 90 o c ° Naurs l l . Variation.-©f Stf§uhaT number with -angle ©f.attaek 41 Schramm does not give any deta i ls of the wind tunnel test section geometry and the measuring equipment,used. A modulation of the vortex shedding frequency .was present at a l l times. It amounted to as much as - 5%, and may be re la ted , to some extent , to the amplitude modulation., The frequency:modulation was too small to af fect the f i l t e r c a l i b r a t i o n . 6.2 Fluctuating Pressure D is t r ibut ion For both e l l i p s e s , the e f fec t of varying the Reynolds number in the range 3 x 10 4 t o . 1 0 5 was investigated for a = 0 , 30°, 60°, 90°. The f luc tuat ing pressure measurements were taken at four taps , two in the laminar and other two in the separated flow regions. The results are p lotted in Figures 23, 24. These curves represent the percentage var iat ion about the mean of the pressure coef f i c ien t at each of the ports in question. No s i g n i f i c a n t Reynolds number e f fec t was noticed except for the abrupt increase in the f luctuat ing pressure coef f i c ien t at zero angle of attack and low wind speed. The cause of th is phenomenon is not quite c l e a r , but i t should be emphasized that i t i s not associated with any noticeable change in base pressure. Th is , together, with the Reynolds number at which the behaviour occurred, makes i t unl ike ly to be due to e i ther proximity to the c r i t i c a l Reynolds number or any other change in character of the separat ion . , However, i t should also be mentioned that f luc tuat ing signals in this region are very.weak and distorted by noise. 42 Figure 23, VariaticW of f l u c t u a t i n g pressure c o e f f i c i e n t with Reynolds number, e = 0 o 6 43 2 4 6 8 10x10 Nr Figure 24. Var iat ion of f luc tuat ing pressure coef f i c ient .with Reynolds number, e = 0.8 In general , the best s ignals were obtained at a wind speed around 28 ft/sec ( N r ^ 68,000). It was, therefore , decided to carry out f luc tuat ing pressure measurements at th is wind speedc The unsteady pressures on the surface of the two e l l i p t i c models were recorded for a = 0 , 30°, 60°, 90°, These results are plotted in Figures 25-28 together with the mean pressure d i s t r i b u t i o n . For comparison the corresponding data for a c i r c u l a r cy l inder are presented in Figure 29. Based on these results the fo l lowing remarks can be made: ( i ) There are two points where the f luc tuat ing pressure tends to vanish. They occupy posit ions which are approximately 180° from the stagnation points . One would expect this due to cancel lat ion of pressures which are 180° out of phase. As shown in the f igures th is e f fect i s less complete at the rear of the cy l inder , probably due to i r r e g u l a r i t i e s in the wake. ( i i ) The f luc tuat ing pressure increases as the mean pressure decreases and, in general , the var iat ions can be represented by curves fol lowing a s i m i l a r trend. ( i i i ) The mean pressure increases negatively with angle of attack. The same is true for the unsteady pressure* But, while the mean pressure coe f f i c ien t approximately doubles in the range a = 0 - 90° the corresponding increase in f luc tuat ing pressure coe f f i c ien t i s as high as 10 to 20 times. ( iv) As expected, at zero angle of attack, the f luc tuat ing pressure coef f i c ien t fo r the slender e l l i p s e is considerably less than that for the th icker e l l i p s e , but at 90° they are p r a c t i c a l l y equal . s 0 25 50 75 100 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 Tap number Figure 27. D is t r ibut ion of mean and f luctuat ing pressure c o e f f i c i e n t s , a = 60° 49 Figure 29. D is t r ibut ion of mean and f luctuat ing pressure coef f ic ients on the, surface of a c i r c u l a r cy l inder , (e = 0) 50 Var iat ion of the maximum f luc tuat ing pressure coef f i c ien t with angle of attack i s shown in Figure 30. To i l l u s t r a t e the amount of amplitude modulation present in the f luc tuat ing pressure s i g n a l s , the rat io between the maximum and the average amplitude has been plotted in Figures 31 and 32. The maximum amplitude used i s a representative value observed during a two minute per iod. The results showed considerable s c a t t e r , but the increase in maximum/average ra t io towards the rear "stagnation point" was quite d i s t i n c t . There was also a trend towards a reduction of the rat io i n . the laminar flow region. As stated by several i n v e s t i g a t o r s 1 0 , 1 1 » ^ , 1 the amplitude modulations were found to be in phase around the models. 6.3 Phase S h i f t Gerrard 1 1 was probably the f i r s t one to measure phase re lat ion between the f luc tuat ing pressure s ignals on a c i r c u l a r cy l inder . He used two b u i l t - i n pressure transducers which could be rotated i n d i v i d -ua l ly and concluded that the pressures were e s s e n t i a l l y in phase over one side of the model and 180°.,out of phasewith that on the other s ide . Any deviation from th is was at t r ibuted to the spanwise separation between the pressure taps. The present experiments confirmed the 180° phase difference between pressures on two sides of the model. But contrary to Gerrard's observation i t indicated substant ia l phase s h i f t between the signals from the neighbouring pressure taps. The phase measurement .was carr ied out on both e l l i p t i c models at a = 0 , 90°, and the results are plotted 51 Figure 30% 1 Va'fialion oT l i f t and rfiaxjRIUm- fiU§|U&tjfifj. pressure coefficients With angle of attack 52 25 S 50 75 100 ^ 2 x o o E 4 i | 'Pff / IV ' & • J v • - • r-w—w .,„ — W • w w a = 0° i > • • • 4 » • . • " • * 30 A • • • . / > « • • ) 9 a = 60° \ / \ / 1 • m • * K a — 90 " • • — • • • 4 » i • • • I 0 4 8 Figure 31 24 28 12 16 20 Tap position Amplitude modulation of the pressure s ignals on the surface of the model, e = 0.6 53 Figure 32. Amplitude modulation of the pressure s ignals on the surface of the model, e = 0.8 54 in Figures 33, 34. It is apparent that for the models at a = 0 the signals from upstream and downstream pressure taps are lagging and leading respective-ly with respect to the reference located at the 90° position (tap 16 or 3 2 ) . On the other hand at a = 90° a l l signals lag the reference (tap 8 or 2 4 ) . The phase difference between two pressure signals was found to be as large as 60°. 6.4 Fluctuating L i f t The fluctuating l i f t coefficients for both ell ipses at a.='0, 30°, 60°, and 90° were calculated from the pressure data given ear l ie r . The results are plotted in Figure 30. For either e l l ipse the maximum fluctuating l i f t coefficient ( 1 . 0 ) f i = 0 > 6 , ( 0 . 7 2 ) e 0.8) was found at 90° while the minimum value occurred at a = 0. The fluctuating l i f t coefficient for a circular cylinder as obtained by McGregor*0 was 0.6. It may be pointed out that the maximum and minimum values given above were obtained without taking phase shif t into account. With phase angle .between the pressure signals the corresponding values are modified as shown in Table 2. e a, Deg. V a phase % change in C-, 0.6 0 90 0.156 1.007 . 0.152 1.001 2.5 0.5 0.8 0 90 0.064 0.719; 0.060 0.710 6.0 1.0 Table 2. Effect of phase sh i f t on fluctuating l i f t coefficient 55 10 -8 o -5-10 a-20 -30 S 25/75 50 i I ^ - ^ - V q > a c 4 c • V k D i A"' J • i • L 1 • N r = 33,0C - A Nr = 67,0C • Nr = 100,00 • K) u 0 0 2/30 4/28 6/26 8/24 10/22 12/20 14/18 Tap position (a) 16 75 S 0/50 25 20 10 «T 0 O) a> -o s - i o o f - 2 0 -30 • c ^— ) 1 A i L r T ( ] n \ ^ — i - VCD a i 3 ^ • 4 i 1 k. A^V^ L • *—J 1 a • N r = 33,000 - A Nr - 67,000 • N r = 100,000 i i a J M -24 26/22 28/20 30/18 0/16 2/l4 4/12 6/10 Tap position (b) Figure 33. Phase s h i f t between pressure s ignals on the surface ef the model,. e = O.i, a = 0, I 0 § 8 56 75 S 0/50 25 40 30 * 20 10 0 - 10 - 2 0 1 1 1 0 voo \ • • Top surface • A Bottom surface > < i > 24 26/22 28/20 30/18 0/]6 2/14 4/12 6/10 Tap position (b) Figure 34. Phase shi f t between pressure signals on the surface of the model, e = 0.8, a = 0 , 90° and Nr = 67,000 8 57 For f luc tuat ing drag and moment coef f ic ients the influence of phase angle.may be greater , but no attempt has been made to evaluate these parameters since they are more sens i t i ve to errors in pressure and phase measurements, which occur near the stagnation points . 6.5 Spanwise Effects. The v a l i d i t y of assuming two dimensional flow in the present 1 o case may be questionable . Measurements at two spanwise taps (4.5 i n . ' and 9 i n . from center s e c t i o n , taps 33 and 34) indicated substant ial phase d i f ference. Occasionally the phase s h i f t between the signals was observed to remain steady for a short per iod , but essent ia l l y i t varied randomly. Th is , in par t , may be at t r ibuted to end condit ions. It was not p rac t i ca l with avai lable instrumentation to measure, q u a n t i t a t i v e l y , the time dependent phase s h i f t . However, the presence of spanwise phase difference may not necessari ly influence the f l u c t u a -t ing l i f t too much. The calculat ions showed a maximum drop of 10% in f luc tuat ing l i f t for a,phase s h i f t up to 50° at a = Oj or up to 30°.at a = 90°. This holds for both e l l i p s e s . Measurements showed the amplitude modulation to be essent ia l l y i n . phase. 6.6 Wake Geometry Typical average amplitude s ignals across the wake are given in Figure 35a. It was found that a f ter a downstream distance of nearly three major axes the spacings between the vortices remained approximately constant. The wake signals at a = 0 became too weak 58 to be recorded at 20-30 i n . downstream. On the other hand more power-f u l s ignals at higher angles of attack could be observed down to 45 i n . which was the l i m i t for the measuring r i g . For a = 30° and 60° the wake was s l i g h t l y unsymmetrical, the vortex from the rearmost separa-t ion point being the stronger. Typical decay of pressure amplitude with downstream distance is given in Figure 35b. Average values for the spacing between vort ices are given in Table 3. a e = 0.6 e = 0.8 e = 0* L, In. W, In. W/L L, In. W, In. W/L W/L 0 16.00 5.0 0.31 10.00 4.3 0.42 0.32 30 19.25 5.2 0.27 16.50 5.1 0.31 60 21.75 5.7 0.26 20.75 5.7 0.28 90 22.75 5.8 0.26 22.25 5.9 0.26 *Ref erence 16 Table 3. Spacing of vort ices in f u l l y developed wake, Nr = 70,000 ; No quant i tat ive pressure measurements were made in the wake as the probe was.not ca l ib ra ted , but the maximum pressure s ignals immediately behind the model were estimated to be around 5 times higher than that on the model surface. Within the Range of the . 4 5. Reynolds number invest igated (3 x 10 < N < 10 ) , no s i g n i f i c a n t change in wake geometry occurred. 60 6.7 S t a t i c L i f t , Drag and Moment The stat ionary values of l i f t , drag and moment were calculated from the s t a t i c pressure results given e a r l i e r (Figures 25-28) . The var iat ion of s t a t i c l i f t coe f f i c ien t with angle of attack is given in Figure 30. The drag and moment .coef f ic ients are plotted in Figure 36. The measured results are uncorrected for wal l ef fects and therefore are somewhat higher than the values corresponding to the unconfined stream. Approximate corrections to the drag coef f i c ients may be obtained 23 from the expressions given by Whitbread . It amounts to 12% for both e l l i p s e s at a = 90°. At a = 0 i t i s 10% for e = 0.6 and 8% for e = 0 . 8 . As suggested by Whitbread the same corrections may be applied to the l i f t and moment coef f i c ients (Appendix I ) . 61 Figure 36. Variat ion of drag and moment coef f i c ients with angle of attack 7. CONCLUDING REMARKS Based on the experimental results the fo l lowing general remarks can be made concerning unsteady aerodynamics of the e l l i p t cyl inders tested: ( i ) The Strouhal frequency, which shows s l i g h t modulation, ! increases l i n e a r l y with wind speed. The var iat ion of ' Strouhal number with angle of attack is considerably less when based on projected height. ( i i ) The f luc tuat ing pressure coef f ic ients tend to increase with angle o f .a t tack . In the range a = 0 - 90°, C-, increased from 0.1 to 0.8 for e = 0.6 and from 0 . 0 4 m a X to 0.8 for e = 0 . 8 . Marked dependency on the Reynolds number appears to be l im i ted to zero angle of attack condit ion. The pressure signals are always amplitude modulated and the extent of modulation, : as expressed by is of the order 2 - 4. p' ( i i i ) Pressure signals on the surface of the models have consider-able phase d i f ferences . At times th is can be as high as 60°. ( iv) S imi la r to the f luc tuat ing pressure c o e f f i c i e n t s , the l i f t coef f i c ients are affected by the att i tude of the models. The maximum l i f t coef f ic ients (based on the major axis) were found to be.1,0 and 0.7 for e = 0.6 and 0.8 respect ively . The phase s h i f t has only small e f fect on these values. (v) The quant i tat ive observations indicate the unsteady flow condition to be f a r from two dimensional. Based on the observation in ( iv) the spanwise var iat ions in phase between the pressure s ignals is not l i k e l y to af fect the f luctuat ing l i f t s u b s t a n t i a l l y . (v i ) The rat io of the transverse to the longitudinal spacing decreases with increasing angle of attack. This was observed to be true fo r both e l l i p s e s but the reduction was more pronounced for the thinner e l l i p s e . ( v i i ) For a = 0 the separation points for the boundary layer as indicated by the s t a t i c pressure d i s t r ibu t ion occurs at approximately the same angular posi t ion ( « 75° - 80°) fo r both e l l i p s e s . * This appears reasonable compared to the experimentally measured value of 82° for a c i r c u l a r cy l inder . It may be pointed out that the var iat ion of separation points with e c c e n t r i c i t y , and angle of at tack, does not seem to be reported in l i t e r a t u r e . * Appendix II A few suggestions concerning the future studies may be appropriate here: ( i ) For better appreciation o f : t h e t r a n s i t i o n a l ef fects from c i r c u l a r cy l inder to f l a t p l a t e , further invest igat ions of cyl inders with d i f fe rent eccent r i c i t y should be under^ taken. ( i i ) The present technique for phase measurements involves considerable e f fo r ts to get meaningful resu l t s . Some d i rect method of obtaining the time average of the phase s h i f t i s cer ta in ly des i rable . ( i i i ) The Reynolds number ef fects at zero angle of attack should be studied in d e t a i l . ( iv ) A study of the three-dimensional nature of the flow should be of i n te res t . A ref ined phase measuring device might prove useful in such a study. (y) The experimental measurements of separation points on cyl inders of d i f fe rent eccent r i c i t y and i t s corre lat ion with theory should prove to be a valuable study. (vi) Of course, the study of aerodynamics and dynamics of the models during s e l f - e x c i t e d motion would be the l o g i c a l extension of th is work. 65 BIBLIOGRAPHY 1. St rouhal , V , , "fiber eine Besondere Art der Tonerregung," Wied.  Ann. Physik u. Chem,, Neue Folge, Vo l . .V, 1878, pp. 216-2FT 2. Karman, Th . , Von, "F luss igke i t u. Luftwiderstand," Phys. Z . , Vol . 13, 1911, p.49. 3. Marr is , A.W., "A Review on Vortex S t reets , Per iodic Wakes, and Induced Vibrat ion Phenomena;" J: Basic Engng., V o l . , 8 6 , 1964, •pp. 185-196. 4. Grove; A . S . , Sha i r , F .H . , Petersen, .E.E-. and Acr ivos , A . , "An Experimental Investigation of : the Steady Separated Flow Past a C i rcu la r Cy l inder , " J . F lu id Mech., Vol . 19, part 1, 1964, pp. 60-80. 5. Bishop, R.E.D. and Hassan, A . Y . , "The L i f t and Drag Forces on a C i rcu la r Cylinder 1n a Flowing F l u i d , " Proc. Roy. Soc . , Series A, Vol . 277, 1964, pp. 32-50. 6. Modi, V . J . and Heine, W., "On the Pressure Fluctuations and Wake Geometry Associated with Several B lu f f Bodies," Proc. 15th  Japan Nat. Cong. Appl. Mech., Japan Soc. Mech. Engrs. 1965, pp. 7-18. 7. Bishop, R.E.D. and Hassan, A . Y . , "The L i f t and Drag Forces on a C i rcu la r Cyl inder O s c i l l a t i n g 1n a Flowing F l u i d , " Proc. Roy. S o c , Series A, Vol . 277, 1964, pp, 51-75. 8. Ferguson, N. and Parkinson, G.V.» "Surface and Wake Flow Phenomena of the Vortex-Excited O s c i l l a t i o n of a C i rcu la r Cylinder,'" ASME Vibration Conference, Paper 6 7 - V 1 b r . - 3 l , 1967. 9. Ch1uB W.S., ."The Boundary Layer Formation and Vortex Shedding on Yawed'Cylinders," Washington State Un ivers i ty . College of  Engng,. B u l l e t i n 299, 1966, 10. McGregor^ D.M.( "An Experimental Investigation of the O s c i l l a t i n g Pressures on a . C i r c u l a r Cyl inder 1n a F lu id Stream," University  of Toronto, Inst i tute of Aerophyslcs, Tech. Note 14, 1957. 11. Gerrard, J .H . . , "An Experimental Investigation of . the O s c i l l a t i n g Pressures on a C i rcu la r Cyl inder Shedding Turbulent Vor t i ces , " Journal of F lu id Mechanics, Vo l . 11; 1961, pp. 244-256. 12. Prendergast, V . , "Measurement of two-Point Correlations of the Surface Pressure on a C i rcu la r Cy l inder , " University of Toronto, Ins t i tu te of Aerophysics, Tech. Note 23, 1958. 66 13. Keefe, R.T., "An Investigation of the Fluctuating Forces Acting on a Stationary C i rcu la r Cylinder in a Subsonic Stream, and of the Associated Sound F i e l d , " University of Toronto, Inst i tute  of Aerophysics, Report 76, 19FT 14. Molineux, W.G., "Measurement of the Aerodynamic Forces on O s c i l l a t i n g A i r f o i l s , " AGARD Report. 35, 1956. 15. Heine, W., "On the Experimental Investigation of Vortex Excited Pressure F luctuat ions , " Universi ty of B r i t i s h Columbia, M.A.Sc. Thesis , 1964. 16. Ferguson, N . , "The Measurement of Wake and Surface Effects in the S u b c r i t i c a l Flow Past a C i rcu la r Cylinder at Rest and in Vortex Excited •Oscil lations, '" University of B r i t i s h Columbia, M.A.Sc. Thesis , 1965. 17. Schramm, W.;, Wirbelfrequenzmessungen an umstromten Baute i len , " I f L - M i t t . , Vol . 5 , 1966, pp. 308-318. 18. Parkinson, G.V., "Aeroelast ic Galloping in.One Degree of Freedom," Proc. F i r s t Int . Conf. on Wind Effects on Bldgs. and  S t r u c t s . , NPL. t London, Vo l . I I , 1965, pp. 581-609. 19. Parkinson, G.V. and Modi, V . J . , "Recent Research on Wind Effects on B l u f f Two-Dimensional Bodies," Int. Research Seminar: Wind  Effects on Bldgs. and S t r u c t s . , NRC, Ottawa, 1967. 20. Morse, P . M . , "Vibration and Sound," McGraw-Hi l l , New York, 1948, pp. 233-265. 21. Bryer, D.W., Walshe, D.E. and Garner, H.C., "Pressure Probes Selected for Three-Dimensional.Flow Measurement," Aeronautical  Research Counci l , R. and M. No. 3037, 1958. 22. Cheng, S . , "An Experimental Investigation of the Autorotation of a F lat P l a t e , " University of B r i t i s h Columbia, M.A.Sc. Thesis , 1966. 23. Whitbread, R .E . , "Model Simulation of Wind Effects on Structures ," Proc. F i r s t Int. Conf. on Wind Effects on Bldgs. and S t r u c t s . , NPl , London, Vol . 2 , 1965, pP7 581-610. 24. Pankhurst, R.C. and Holder,' D.W;, "Wind-Tunnel Technique," Pitman & Sons L t d . , London, 1952, chapter 8. 25. Maske l l , E .C . , '!A Theory of the Blockage Effects on B lu f f Bodies and S t a l l e d Wings in a Closed Wind Tunnel," RAE_ Report No. Aero 2685, Nov. 1963. 67 26. S c h l i c h t i n q , H. , "Boundary - Layer Theory," McGraw-Hi l l , New York, 1968, pp. 21 , 202-206, 475. 27. Meksyn, D., "New Methods in Laminar Boundary - Layer Theory," Pergamon Press, London, 1961, chapter 11. 28. Gor t le r , H. , "A New Series for the Calculat ion of Steady Laminar Boundary Layer Flows," Journal of Mathematics and Mechanics, Vol . 6 , No. 1, 1957, pp. 1-66. APPENDIX I REMARKS CONCERNING WIND TUNNEL WALL CORRECTIONS • There is considerable information avai lable for predict ion of equivalent free f l i g h t results from wind tunnel measurements as far as steady flow is concerned.. Unfortunately, the same cannot be sa id for unsteady flow around a body. 24 As indicated by Pankhurst and Holder the interference from wind, tunnel walls during steady flow conditions may be subdivided i n t o : ( i ) So l id blockage ( i i ) Wake blockage ( i i i ) L i f t e f fect ( iv) Interference due to s t a t i c pressure gradient (v) Wall boundary-layer interference on model spanning a closed tunnel For an e l l i p s e at a = 0° and 90° ( i i i ) would not e x i s t . Neglecting ( iv) and (v) gives a correction to wind speed as fo l lows: V corr . + n c h d w where K = constant x = shape factor n = wake blockage factor W 4 . = tunnel width 69 This expression gives a correction s i m i l a r to the one obtained using 23 Whitbread's equation, which is a pa r t i cu la r form of a more general 25 expression given by Maskell . For f luc tuat ing pressures apparently no corrections are a v a i l -ab le , probably because the s i tua t ion is considerably more complex. In addit ion to blockage inf luencing both f luc tuat ing pressure and vortex frequency, i t is l i k e l y that the boundaries w i l l impose a ve loc i ty f i e l d due to the wake, vor t i ces . Moreover, the periodic, nature.of the flow'may lead to an expression for wall correction which i s also p e r i o d i c a l . APPENDIX II LOCATION OF SEPARATION POINTS ON ELLIPTIC CYLINDERS The avai lable information on the laminar boundary layer separation on b l u f f bodies is somewhat scarce and unre l iab le . The bulk of the information is obtained a n a l y t i c a l l y using boundary layer separation c r i t e r i a with external flow d i s -t r i b u t i o n assumed p o t e n t i a l . This information is summarised in the fo l lowing chart . The posit ions of minimum pressure as given by the present set of experiments are also included. X Laminar separation as calculated by Schlichting and 111 r i c h 2 6 A Laminar separation as measured by.Schubauer and calculated by.Meksyn2' • Laminar separation as calculated by Polhausen26 B Laminar separation as calculated by Gortler' 2i ® , Minimum pressure as measured by Flachsbart O Minimum pressure as given by present measurements 1 2 4 6 8 Dependence of points of separation and minimum pressure on bluffness, a = 0; x = distance along circumference, V = semi circumference. Ah! v o i l a quatre-vingts volumes de recuei ls d'une academie des sciences! s ' e c r i a Mart in , II se peut q u ' i l y a i t l a du bon. - - II y en a u r a i t , d i t Pococurante, s i un seul des auteurs de ces fatras avait invente seulement T a r t de f a i r e des epingles; mais i l n'y a , dans tous ces l i v r e s , que de vains systemes, et pas une seule chose u t i l e . VOLTAIRE 

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