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Autorotation of thin plates Andersen, Fabian 1970

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AUTOROTATION OF THIN PLATES  *>y  FABIAN ANDERSEN B.Sc,  University of Alberta, I965  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc.  i n the Department of Mechanical Engineering  We accept t h i s t h e s i s as conforming  to the  required standard  THE UNIVERSITY OF BRITISH COLUMBIA March, 1970  In  presenting  this  thesis  an a d v a n c e d d e g r e e the L i b r a r y I  further  for  agree  scholarly  by h i s of  shall  at  the U n i v e r s i t y  make i t  tha  written  thesis  freely  permission  for  It  gain  permission.  Q^MOACL  British for  for extensive  Department  Date  of  by  Columbia  /97Q  shall  the  requirements  Columbia, reference  copying of  I agree and this  that  not  copying  or  for  that  study. thesis  t h e Head o f my D e p a r t m e n t  is understood  financial  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  fulfilment of  available  p u r p o s e s may be g r a n t e d  representatives.  this  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t  my  li ABSTRACT  A v i s u a l i z a t i o n technique i s applied to the unsteady separated flow about an autorotatlng f l a t p l a t e .  An unsteady p o t e n t i a l model i s attempted  to predict the pressure loading on the plate as a function of angle of attack. To v i s u a l i z e the flow, a s t r e a k l i n e i s marked by low density a i r created by the wake o f a heated wire probe.  An off-center parabolic  Schlieren system detects the density gradient.  mirror  Due to the highly unsteady  nature ( i n t h i s project, the p l a t e rotates a t almost 1,000 r.p.m. f o r a 10-foot per second freestream  v e l o c i t y ) of the flow, high-speed 35mm. s i n g l e  lens r e f l e x shots or l6mm. movie films recorded the image. Timing marks on the movie f i l m provided  information  of the angular  speed of the plate during acceleration to the autorotation speed and a t autorotation. The two-dimensional unsteady attached  flow model attempts to duplicate  the e f f e c t s of separation by superimposing v o r t i c e s i n the wake, as shown i n the flow v i s u a l i z a t i o n , and by eliminating terms representing the freestream v e l o c i t y i n the range o f 60 to 150 degrees angle of attack.  iii TABLE OF CONTENTS Page No. ABSTRACT  i i  LIST OF FIGURES  iv  ACKNOWLEDGEMENTS  v  SYMBOLS  vi  ,1.  INTRODUCTION  1  II.  EXPERIMENTAL APPARATUS AND PROCEDURES  4  III. IV.  V. VI.  2.1  General Outline  4  2.2  Wind Tunnel and F l a t Plate  4  2.3  Schlieren System and Heated Wire Probe  4  2.4  High-Speed Photography  8  2.5  Procedure  9  EXPERIMENTAL RESULTS  15  THEORETICAL MODEL  31  4.1  General Outline  31  4.2  Complex P o t e n t i a l f o r a Rotating Translating F l a t Plate  32  4.3  Vortex Superposition  35  4.4  Model Pressure C o e f f i c i e n t  37  4.5  Vortex Strength and P o s i t i o n Assumptions  42  MODEL RESULTS  47  SUMMARY OF RESULTS  59  REFERENCES  61  APPENDIX  PROBE WAKE BUOYANCY  62  iv FIGURES Figure No.  Page No.  1  Plate Arrangement and Symbols  3  2  Wind Tunnel Working-Section Schematic  6  3  Wind Tunnel Working-Section Arrangement  7  4  Schlieren - Wind Tunnel Working-Section Schematic  1  2  5  Schlieren - Wind Tunnel Working-Section Arrangement  1  3  6  'HYCAM* High-Speed  7  Relative Zero Angle of Attack  8 - 1 1  1 2 - 1 5  1 6 - 18  Camera  14 1  5  3 5 m m . S.L.R. Pictures of One Cycle of Flow F i e l d U = 1 1 . 3 ft./sec.  18 - 2 1  l 6 m m . Movie Film Frames at Approximately 1 5 Degree Angle of Attack Intervals f o r One Cycle U = 1 1 . 3 ft./sec.  2 2 - 2 5  Non-Dimensionalized Angular Velocity-Time Curves f o r Acceleration Period f o r Three Freestream V e l o c i t i e s  2 6 - 28  1 9  T i p Velocity-Freestream Velocity Curves  2  20  Tip-Freestream V e l o c i t y Compared to Cheng's Results  21 22  Gonformal Transformation from Unit C i r c l e to E l l i p s e Plane  t  t+At  Cylinder at Origin at Time  24  Elimination Factor & C i r c u l a t i o n vs Angle of Attack and Vortex P o s i t i o n Pressure C o e f f i c i e n t - Plate Position f o r Model and Cheng's Results  32  33  34  3  23  31  0  Vortex Conformal Transformation from Unit C i r c l e to E l l i p s e Plane  25 - 30  3  9  and  4  6  5 46 5 0 - 5 5  Drag C o e f f i c i e n t - Angle of Attack f o r Model and Cheng*s Results  56  L i f t C o e f f i c i e n t - Angle of Attack f o r Model and Cheng's Results  5  7  Torque C o e f f i c i e n t - Angle of Attack f o r Model and Cheng's Results  5  8  V ACKNOWLEDGEMENTS The author wishes to express gratitude to Professor G. V .  Parkinson  f o r h i s assistance and optimism, p a r t i c u l a r l y during the t h e o r e t i c a l side of this project.  The suggestions of Professor Z. Rotem concerning the Schlieren  apparatus and photography saved many confused  hours.  Thanks are due Mr. P. Hurren and Mr. J . Hoar, Chief Technicians, t h e i r s t a f f and my fellow graduate students f o r helping to provide the hardware and f o r the a i d i n getting through the maze. Special thanks to my wife, Cherie, f o r typing and f o r playing the r o l e of a student's wife so w e l l .  SYMBOLS  E l l i p s e major semi-axis E l l i p s e minor semi-axis Probe diameter Analytic  function  Chord of t h e o r e t i c a l model  k units  Chord of experimental plate  1  5/16"  Pressure c o e f f i c i e n t Drag c o e f f i c i e n t L i f t coefficient Torque c o e f f i c i e n t Complex p o t e n t i a l Complex v e l o c i t y D i r e c t i o n normal to surface Direction tangent to surface Normal v e l o c i t y Tip velocity Freestream v e l o c i t y Pressure Vortex p o s i t i o n i n f plane Point i n  J" plane  Point i n z Reynolds  plane  no.  Grashof no. Time Physical  plane  Physical plane f i x e d with respect to r o t a t i n g t r a n s l a t i n g cylinder  vii  a  Angle of attack R e l a t i v e zero angle of attack  e *.  Eliminator f a c t o r Stream function Velocity potential  f  Unit c i r c l e plane and a point i n that plane  7  The u n i t c i r c l e i n f  a  A point on 7  \  Constant = (a-b)/(a+b)  •w  Constant = (a+b)/2  r  C i r c u l a t i o n per u n i t span  p  Mass density  V  Viscosity  u  Angular v e l o c i t y Angular v e l o c i t y at autorotation  1. I.  INTRODUCTION  The type o f autorotation investigated i n t h i s project i s that which occurs when a f l a t plate, f r e e to r o t a t e about a spanwise mid-chord axis and i n the presence o f a uniform freestream perpendicular to the a x i s o f r o t a t i o n ( F i g , l ) , i s given an I n i t i a l angular v e l o c i t y above a minimum i n either direction.  The plate then undergoes an angular a c c e l e r a t i o n up to a  stable angular v e l o c i t y known as the a u t o r o t a t i o n a l speed. then the flow pattern i s repeated every h a l f r e v o l u t i o n .  At autorotation Circulation i s  produced, g i v i n g a net p o s i t i v e l i f t f o r every cycle or one-half r e v o l u t i o n . Recently, Cheng ( l ) measured the instantaneous pressure a t taps on the center l i n e chord o f the plate as a function of angle o f attack.  Using  a mutual inductance transducer, he also measured the angular v e l o c i t y and acceleration during the period o f a c c e l e r a t i o n up t o autorotation and a t a u t o r o t a t i o n a l speed.  Previously, Crabtree (2) and Neumark (3) reviewed and  explored the p o s s i b i l i t i e s of employing autorotating or powered (higher angular speeds) plates as l i f t producing devices i n themselves or as l i f t augmenting and c o n t r o l devices replacing the f l a p on a conventional a i r f o i l . In normal operation, the r o t a t i n g f l a p not only produces l i f t but augments the c i r c u l a t i o n about the a i r f o i l .  Rotated backwards, i t acts as a s p o i l e r .  When not i n use, the f l a p can be aligned with the flow to give a low drag, Crabtree and Neumark model the r o t a t i n g f l a p on a conventional a i r f o i l with a vortex acting upon a t h i n a i r f o i l i n p o t e n t i a l flow.  The  vortex assumption may be v a l i d f o r a powered r o t a t i n g f l a p , since a t higher angular v e l o c i t i e s the flow i s probably s i m i l a r to that produced by a r o t a t i n g c i r c u l a r c y l i n d e r , but the flow about an autorotating f l a p i s highly  unsteady, James and Stone (7) measured the forces and angular v e l o c i t i e s  2.  of autorotating plates of various aspect r a t i o s using a contact switch to record times a t each h a l f revolution and a three component balance.  They  also made an attempt to v i s u a l i z e the flow using fumes of titanium t e t r a c h l o r i d e with the plate a t autorotational speed, and recording the r e s u l t s by means of a camera and an e l e c t r o n i c f l a s h .  The v i s u a l i z a t i o n gave some  i n d i c a t i o n of the flow pattern close to the plate, but the fumes appear to d i f f u s e too quickly to give an i n d i c a t i o n of downstream development. S i m i l a r l y , Baird and Pick (8) measured the autorotational speed as a function of freestream v e l o c i t y and aspect  ratio.  This project includesi i)  Flow v i s u a l i z a t i o n during the a c c e l e r a t i o n period and a t autorotation.  The technique involved a Schlieren  system, a heated wire probe and high-speed photography. ii)  Angular velocity-time and t i p velocity-freestream v e l o c i t y curves from timing pulses on the movie f i l m .  iii)  An unsteady attached flow two-dimensional p o t e n t i a l model f o r t h i s unsteady separated  flow.  3.  FIG. 1  PLATE ARRANGEMENT AND SYMBOLS  4.  II.  EXPERIMENTAL APPARATUS AND PROCEDURES  2 . 1  GENERAL OUTLINE  The v i s u a l i z a t i o n technique, i n b r i e f , consisted of a semi-focusing Schlieren system to render v i s i b l e a hot s t r e a k l i n e behind a probe upstream of an autorotating f l a t p l a t e i n a wind tunnel.  Due to the highly unsteady  nature of the flow, high-speed photography was employed to record the r e s u l t s .  2.2  WIND TUNNEL AND  FLAT PLATE  An aluminum f l a t plate (Figs. 2 - 3 ) was mounted i n an open c i r c u i t wind tunnel (top speed approximately 2 0 feet per second) with a 6 " x l 2 " working s e c t i o n .  The f l a t p l a t e was 6 inches long to span the width of the  tunnel and 1 5 / l 6 " chord to give approximately the same blockage r a t i o (chord/height of working section) as Cheng ( l ) . The side walls of the tunnel were polished f l a t plate glass (to minimize o p t i c a l aberration) and diamond d r i l l e d to accept the bearings (Fig.  2 ) f o r the p l a t e . The magnitude of the uniform freestream was measured with a p i t o t -  s t a t i c tube which was c a l i b r a t e d i n a 3 6 " x 2 7 " wind tunnel instrumented  with  a Betz manometer accurate to . 0 2 millimeters of water. 2.3  SCHLIEREN SYSTEM AND  HEATED-WIRE PROBE  The Schlieren system employed was the one used by Claassen ( 4 ) . In b r i e f , i t i s a semi-focusing (implying a source s l i t ) off-center system (Figs. 4 - 5 ) employing 8 - i n c h diameter parabolic mirrors of f o c a l length 6 3 # 5 inches.  Due to the short exposure times required, i t was decided to  take black and white pictures instead of colour to take advantage of the higher f i l m speeds a v a i l a b l e .  Thus, a h o r i z o n t a l k n i f e edge was substituted  f o r the colour f i l t e r used by Claassen.  A heated , 0 1 0 - i n c h Nichrome-V c i r c u l a r cross-section wire (Figs. 2 - 3 ) approximately one plate chord i n length and one and a h a l f plate chords upstream and on the same h o r i z o n t a l plane as the r o t a t i o n a l axis illuminated a streakline.  The length of wire was l i m i t e d by o s c i l l a t i o n s and the one  chord length was s e t t l e d upon,  A smaller diameter ( . 0 0 2 " ) wire was t r i e d ,  but t h i s l i m i t e d the heat output and the q u a l i t y of the picture decreased. The center l i n e of the plate was aligned with the center l i n e of the Schlieren system by using a one m i l l i w a t t l a s e r as described by Glaassen  (4).  The e f f e c t of the buoyancy of the heated s t r e a k l i n e i s discussed i n the appendix. The supports f o r the probe were # 2 0 s o l i d core copper wire.  Holes  were d r i l l e d i n the copper wire with a # 8 0 twist d r i l l and the Nichrome wire was soldered i n place. O r i g i n a l l y , the power supply f o r the Schlieren tungsten l i g h t source was l l O v 6 0 - c y c l e .  Since pictures were being taken at a top rate of 2 , 0 0 0  per second, the power supply caused overexposed frames a t 1 2 0 cycles per second.  Consequently, the power supply was a l t e r e d to a v a r i a b l e amplitude  6 0 - c y c l e source feeding into a f u l l wave r e c t i f i e r with 9 , 0 0 0 microfarads across the l i n e .  6.  pitot-tube\  brass cone bearing silicon seated  plate  B E A R I N G  see brg. d e t a i l plan  x-section  D E T A I L solenoid ~7  power supply ^  4  0  start-up assembly  •§•" p o l i s h e d plate glass  probe  side view  front x-sectlon NOTEi  NOT TO SCALE  60"  MAT'Li  P L A T E  FIG. 2  •s  . 0 5 0 " aluminum  D E T A I L  WIND TUNNEL WORKING-SECTION SCHEMATIC  .010" diameter Nichrome V three wire probe plate .010" diameter Nichrome V single wire probe p i t o t - s t a t i c tube s t a r t i n g hook •§•*' plate glass and r e f l e c t i o n  WIND TUNNEL WORKING—SECTION ARRANGEMENT  8.  2.4  HIGH-SPEED PHOTOGRAPHY  Two cameras were used to record the Schlieren image. To record the entire image, a 35^> Pentax Spotmatic S.L.R. camera body was used with a three wire probe ( F i g , 3)»  The f i l m was TRI-X taken a t  l/l,000 seconds a t a rated 400 ASA. Pictures were taken at random with the plate autorotating at a freestream  v e l o c i t y of 11.3 f e e t per second.  To obtain a continuous record of the plate a c c e l e r a t i n g from the i n i t i a l angular v e l o c i t y up to autorotational speed and also a t autorotational speed, a 1 6 m m , r o t a t i n g prism high-speed camera body was used (brand name HYGAM) with a s i n g l e wire probe.  When the camera i s started, the motor  accelerates the f i l m from the supply r e e l (100-foot capacity) to a take-up r e e l up to a predetermined speed ( c o n t r o l l e d by the camera power supply). The r o t a t i n g prism moves the image a t f i l m speed during the exposure time (l/2.5 x l/frames per second = exposure time). The f i l m used was Eastman 4-X (ASA 400) type 7224 double perforation negative pan. To give the plate a c o n t r o l l a b l e i n i t i a l v e l o c i t y and synchronize t h i s with camera start-up, a solenoid, which was triggered by a microswitch i n the HYCAM, released a weight attached autorotation.  to a hook ( F i g s . 2-3) to i n i t i a t e  The HYGAM's microswitch closed a f t e r a set number of feet of  f i l m had l e f t the supply r e e l .  9. 2 . 5  i)  PROCEDURE The procedure f o r taking 35mm.. random ' s t i l l ' shots was as  follows.  a)  A three wire probe was placed one and a h a l f chord lengths  upstream of the plate axis with the middle wire on the h o r i zontal plane of the axis of r o t a t i o n .  The p i t o t - s t a t i c  tube  was lowered into place and the wind tunnel speed was set at 1 1 . 3 feet per second.  A reference l i n e was stretched p a r a l l e l to the  wind d i r e c t i o n and through the axis of r o t a t i o n to give a f r e e stream reference i n the picture. source were turned out.  Lights other than the Schlieren  The probe power supply was then turned  on u n t i l the wire reached a very dim red i n colour (experience indicated that a bright red probe d i d not l a s t very long).  b)  The 3 5 ' Pentax Spotmatic camera body was set such that m m  the image cast by the objective lens was i n focus.  The  h o r i z o n t a l k n i f e edge was r a i s e d or lowered to give a l i g h t gray background  with good d e t a i l on the s t r e a k - l i n e about the  stationary plate.  The proper exposure, as indicated on the  Pentax averaged-through-the-lens l i g h t meter, was obtained by adjusting the l i g h t source power supply. c)  The tunnel speed was then given a f i n a l check and the  p i t o t - s t a t i c tube r a i s e d .  Autorotation was i n i t i a t e d by  r e l e a s i n g the s t a r t i n g weight.  d)  Pictures were then taken at random with the p l a t e at  autorotation speed and the three wire probe i n place.  10. ii)  The procedure f o r taking high-speed movie films o f the plate  accelerating from r e s t was as follows.  a)  A s i n g l e wire probe was placed one and a h a l f chord lengths  upstream of the plate axis with the wire on the same horizontal plane as the r o t a t i o n a l a x i s .  The s t a r t i n g hook was then  attached to the plate and a predetermined weight (one which would just i n i t i a t e autorotation a t that tunnel speed) added. The p i t o t - s t a t i c tube was lowered and the tunnel speed set at either 1 1 . 3 ? 1 5 • 3 or 1 9 . 2 feet per second.  A reference l i n e  was stretched p a r a l l e l to the freestream d i r e c t i o n and through the axis of r o t a t i o n .  Lights other than the Schlieren source  were turned o f f . The probe power supply was then turned on and adjusted to give the probe a dim red colour.  b)  The 1 6 m m . 'HYCAM' camera body, with f i l m threaded from  supply to take-up r e e l , was set such that the image cast by the objective lens was i n focus on the f i l m i n the gate.  The camera  cross h a i r was l i n e d up with the reference l i n e and the l i n e removed.  The A.C. solenoid from the start-up mechanism was then  connected to the camera microswitch through a 6V A.C. power supply.  To set the exposure, the l i g h t source power supply was  adjusted by experience, since the 'HYCAM' exposure meter could not p h y s i c a l l y be used. For acceleration shots, the camera gear box was set i n low and the camera A.C. power supply set to take 3 0 0 p.p.s. (see Ref. 5 ) « The camera timing l i g h t was set to 1 , 0 0 0 pulses per second.  11. c)  The camera was  then started by turning on the power supply.  When a predetermined ( s e l e c t i o n on the camera) length of f i l m (approximately 20 f e e t ) had l e f t the supply r e e l , the solenoid closed, t r i g g e r i n g the plate to accelerate to autorotational speed.  d)  The procedure f o r taking a f i l m when the plate was  at  autorotational speed was s i m i l a r except that the plate autorotating when the camera was gear box was  started.  was  Also, the camera  i n high and the power supply adjusted  (see Ref.  5)  to give 2,000 p.p.s.  Generally, better r e s u l t s were obtained at night.  i f the movie f i l m s were run  I t i s believed that t h i s was due to a decrease i n outside  disturbances  which vibrated the k n i f e edge to give varying exposures  throughout the f i l m . Films were analyzed f o r angle of attack-time r e l a t i o n s h i p s by mounting a fabricated p l e x i g l a s s gate on a L e i t z s l i d e projector with supply and take-up r e e l s from an e d i t o r . an orange f i l t e r was was  To prevent damage to the f i l m from heat,  placed between the l i g h t source and the f i l m .  Thus, i t  possible to see four or f i v e frames complete with timing marks at  time and i t was  convenient to handle the 100-foot lengths of f i l m .  and time were tediously measured with protractor and r u l e r on the image. A l l data was reduced on a d i g i t a l computer.  one  The angles projected  3 5 mm. S.L.R. or 'HYCAM' hi-speed movie camera  objective lens f 5 . 6 f.l.-210mm  wind tunnel t e s t section  8 " parabolic mirror f.l.-63.5 M  parabolic mirror f . 1 . - 6 3 . 5 "  source s l i t .006"x.5" condensing lens f 3 . 5 f.1.-5" 5 0 0 watt fan cooled tungsten lamp" p l a n  FIG. 4  v i e w  SCHLIEREN - WIND TUNNEL WORKING-SECTION SCHEMATIC  1. 2. 3. k, 5. 6. 7. 8.  FIG. 5  8" diameter parabolic mirror tungsten l i g h t source and cooling tube wind tunnel probe power supply tungsten source power supply plane mirror knife-edge and objective lens 'HYCAM' hi-speed camera body  SCHLIEREN-WIND TUNNEL WORKING-SECTION ARRANGEMENT  1.  8" diameter parabolic mirror  2.  plane mirror  3.  horizontal knife-edge  4.  objective lens  5.  'HYCAM* hi-speed camera body  6.  timing l i g h t pulse generator  7.  plate start-up solenoid power supply  8.  camera power supply  FIG. 6  f5.6 f.l.-210mm  'HYGAM' HIGH-SPEED CAMERA  15. III.  EXPERIMENTAL RESULTS The r e s u l t s obtained from the v i s u a l i z a t i o n technique were 3 5 ™ .  s t i l l photographs of the entire 8 - i n c h diameter Schlieren f i e l d with three streaklines illuminated and l 6 m m . high-speed movie close-up f i l m s of the plate accelerating and at autorotational speed with one s t r e a k l i n e illuminated. Velocity-time curves and t i p velocity-freestream v e l o c i t y curves were obtained from the movie films by measuring plate angles and time from l / l , 0 0 0  second  timing marks on the edge of the f i l m . The sequence (Figs. 8 - 1 1 ) of random 3 5 m m . shots at U = 1 1 . 3 feet per second show the flow f i e l d developing through one cycle, that i s from zero to 1 8 0 degrees angle of attack.  Flow i s  from the r i g h t p a r a l l e l to the reference l i n e and the plate i s r o t a t i n g a n t i clockwise, as depicted i n f i g u r e 7 . Separation i s delayed u n t i l the plate reaches a p o s i t i v e angle of attack close to the r e l a t i v e zero angle of attack ( 0( see f i g u r e 7 ) , due to the e f f e c t o  of the t i p v e l o c i t y . O( = 0  From the v e l o c i t y curves, as w i l l be discussed l a t e r ,  3 ^ degrees f o r U = 1 1 . 3 feet per second.  A separation bubble begins  to form at approximately 3 0 degrees angle of attack on the upstream t i p . It grows i n s i z e , appears to develop into a vortex at about 4 5 degrees, then s t a r t s to shed r a d i a l l y outwards as the plate passes 9 0 degrees. degrees, the t i p i s beginning to pass t h i s vortex.  By 1 2 0  The influence of t h i s  vortex can be seen by comparing the flow pattern with Cheng's measured pressure c o e f f i c i e n t s (Figs. 25-30), p a r t i c u l a r l y f o r angles of attack of 6 0 and 9 0 degrees.  I t i s seen that the largest suction peaks measured during  16. the complete cycle occur near the position of t h i s 'dominant' vortex. This vortex continues to influence the flow pattern near the plate during the next cycle as another vortex i s formed at t h i s t i p .  It moves  downstream and i s deflected downwards by the generated c i r c u l a t i o n .  The  d e f l e c t i o n of the wake can best be seen by noting that the position of the lower s t r e a k l i n e f o r a l l angles of attack i s lower than at the marking point. The wake i s deflected i n the same d i r e c t i o n as the plate r o t a t i o n i r r e s p e c t i v e of the angle of attack. Not so v i s i b l e i s a vortex forming at the other t i p and becoming v i s i b l e as the plate approaches 9 0 degrees.  I t appears to s t a r t shedding at  near 120 degrees as the t i p moves upstream.  An unstable shear layer t r a i l i n g  t h i s vortex forms at t h i s t i p u n t i l a p o s i t i v e angle of attack i s reached where the 'dominant* vortex forms.  Again, Cheng measured r e l a t i v e l y high  suctions (Figs. 2 5 - 3 0 ) near t h i s vortex. The plate appears bent i n the 3 5 " i m . pictures near the zero and  180  degree angles of attack, since the S.L.R. has a f o c a l plane shutter which allows a d i s c r e t e time difference as the exposure s l i t t r a v e l s from the l e f t of the image to the r i g h t . The movie f i l m s of the plate accelerating from r e s t are too lengthy to reproduce here, since the acceleration period i s i n the order of f i v e seconds and 3 0 0 p.p.s. were taken.  The i n i t i a l angular v e l o c i t y appears to  delay separation at small p o s i t i v e angles of attack, thus s e t t i n g up the autorotation flow pattern. As the plate accelerates, the angle of r e l a t i v e zero angle of attack increases u n t i l the autorotation speed i s reached. Enlargements of l 6 m m , frames at angles of attack closest to 15 i n t e r v a l s from zero to 180 degrees are shown i n figures 12-15  with the plate  at autorotational speed and the freestream at 1 1 . 3 feet per second. twelve frames of approximately s i x t y f o r one cycle are shown.  degree  Only  These pictures  17. show a l i m i t e d part of the Schlieren f i e l d to give a better indication of the flow development near the plate.  Separation at the upstream t i p begins  to occur at an angle of approximately 28 degrees.  A separation bubble forms  and the 'dominant' vortex i s formed and shed, as described before.  Here the  vortex at the other t i p becomes v i s i b l e at about 7 5 degrees and appears to shed with the development of a shear layer feeding into i t . Velocity-time (Figs. 1 6 - 1 8 ) and tip-freestream v e l o c i t y (Figs. 1 9 - 2 0 ) curves were obtained from measurements of angles and times on both the acceleration and at-speed movie f i l m s .  It i s seen that even though the  difference between f i n a l t i p speed and the i n i t i a l t i p v e l o c i t y i s greatest for the highest freestream v e l o c i t y , the time taken to reach autorotational speed i s lowest f o r the highest freestream v e l o c i t y .  That i s , f o r the  range investigated, the higher the freestream v e l o c i t y the lower (considering minimum start-up v e l o c i t y runs) the acceleration period. . This i s i n agreement with Cheng's ( l ) r e s u l t s . Within the error involved i n measuring angles and times from the f i l m s , the angular v e l o c i t y was constant during a cycle at autorotation. The t i p v e l o c i t y varies l i n e a r l y with the freestream v e l o c i t y and the equation of the curve i s  V a. 814 U -2.S>4r  This r e s u l t i s compared to Cheng's i n f i g u r e 20.  The l i n e a r portion of  Cheng's curve f o r the case using end plates i s given by  Cheng's model had a larger thickness r a t i o ( 4 . 7 % ) and a smaller aspect r a t i o (3.0) than the plate i n t h i s project (3-8$ and 4 . 5 ) .  In t h i s project, the  minimum i n i t i a l v e l o c i t y required to s t a r t autorotation would appear to increase s l i g h t l y with freestream v e l o c i t y .  Cheng's r e s u l t s showed the  opposite trend and t h i s v e l o c i t y i s most probably a function of the type of bearings used.  PIG. 8  35mm. RANDOM PICTURES U = 11.3 f t . / s e c .  FIG. 9  35mm. RANDOM PICTURES U = 11.3 ft./sec.  FIG 1 0  3 5 m m RANDOM PICTURES U = 1 1 . 3 It./sec  FIG. 1 1  3 5 m m . RANDOM PICTURES U = 1 1 . 3 ft./sec.  FIG. 12  16mm.  MOVIE FILM FRAMES  U - 11.3 ft./sec.  FIG. 1 3  1 6 m m . MOVIE FIIM FRAMES U - 1 1 . 3 ft./sec.  FIG. 1 6  NON-DIMENSIONALIZED ANGULAR VELOCITY-TIME CURVES FOR ACCELERATION PERIOD U = 1 1 . 3 ft./sec.  to ON  1.  2.  3.  4.  5.  6.  7.  8.  9.  10.  ro FIG. 18  NON-DIMENSIONALIZED ANGULAR VELOCITY-TIME CURVES FOR ACCELERATION PERIOD U = 1 9 . 2 ft./sec.  v  29.  FIG. 19  TIP VELOGITY-FREESTREAM VELOCITY CURVES AT AUTOROTATION AND INITIAL VELOCITY  30.  U [  FIG. 2 0  ^  /  s  e  c  ]  TIP-FREESTREAM VELOCITY AT AUTOROTATION COMPARED TO CHENG'S RESULTS  31. IV.  THEORETICAL MODEL  4.1  GENERAL OUTLINE  The main purpose f o r t h i s two-dimensional  i r r o t a t i o n a l incompressible  flow model i s to predict as accurately as possible the pressure loadings on the p l a t e ;  that i s , to make the model agree as c l o s e l y as possible with  Cheng's measured instantaneous pressure c o e f f i c i e n t s (uncorrected f o r tunnel wall e f f e c t ) . As a s t a r t i n g point, a technique f o r obtaining the f i e l d complex p o t e n t i a l from a boundary condition on a r o t a t i n g t r a n s l a t i n g cylinder (a s p e c i f i c example i s an e l l i p s e , or i n one l i m i t ,  a f l a t plate) i s given  by Milne-Thomson (6). Thus, we are able to obtain the complex p o t e n t i a l f o r an attached flow model of a r o t a t i n g t r a n s l a t i n g f l a t plate.  This gives reasonable  r e s u l t s f o r regions of the a c t u a l flow which are attached, but of course when the flow separates (points of separation f i x e d at the sharp edges) the model prediction i s unreasonable. To improve t h i s , three stationary (with respect to the plate) v o r t i c e s are superimposed on the attached flow model i n the wake region. To better simulate a wake region f o r the purpose of c a l c u l a t i n g pressure c o e f f i c i e n t s on the plate i t s e l f , those terms which represent the e f f e c t of the freestream are gradually eliminated on the separated s i d e above the r e l a t i v e zero angle of attack (0<o) and below 180 degrees angle of attack. Thus, an unsteady modified attached flow (therefore predicting i n f i n i t e pressure c o e f f i c i e n t s a t the sharp edges) model i s constructed f o r an unsteady separated flow s i t u a t i o n .  32. 4.2  COMPLEX POTENTIAL FOR A ROTATING TRANSLATING FLAT PLATE  As stated before, the method f o r obtaining the complex p o t e n t i a l f o r any r o t a t i n g t r a n s l a t i n g c y c l i n d e r i s given by Milne-Thomson (6). B r i e f l y , one set of co-ordinate axes (Z') i s f i x e d with respect to a t r a n s l a t i n g r o t a t i n g c y l i n d e r which i s moving with respect to a f i x e d set of co-ordinate axes ( Z ) . The s i t u a t i o n i s considered at an instant when the two sets of axes coincide and the observer i s stationary with respect to the Z plane.  Referring to figure 2 1 , the c y l i n d e r i s at the o r i g i n r o t a t i n g  with angular v e l o c i t y CO, t r a n s l a t i n g with v e l o c i t y U  at an angle of  attack Of . A boundary r e l a t i o n i s obtained from the normal v e l o c i t y at the surface of the c y l i n d e r  since the normal v e l o c i t y of the f l u i d at the boundary must match the normal v e l o c i t y of the boundary. Integrating the expression f o r V n ,  The function of the normal co-ordinate and time i s a constant since the r e l a t i o n i s v a l i d only on the boundary and i s considered at an instant of time when the c y l i n d e r axis i s coincident with the f i x e d frame of reference. Therefore, disregarding a constant, the boundary r e l a t i o n i s  2/> = Uc' z - uc**z -icozi icl  on X  Define  crzc * L  ar= /ar l  i  33. Since 3(<r) i s an a n a l y t i c function, i t can be expanded i n a Laurent series. i.e.  BC<r)*B (<r)*B (<r) t  where  B,C&)  contains a l l negative powers of  Therefore, B,(&)  powers.  2  CT  and  B (cr) 2  contains a l l other  i s regular outside }f ,  By applying Cauchy's i n t e g r a l formula and noting that there are no s i n g u l a r i t i e s i n the flow f i e l d outside  X,  FC?)*B,cr) where J i s a point outside 2f. Now, f o r a r o t a t i n g t r a n s l a t i n g e l l i p s e , as i n f i g u r e 19, from relations  1  and 3  .*. B,C&)- ~0(6CQS<X + La sincn)J&. - ceo  (a -b )^x 2  x  FCS) --UCbcos* tcasinc*)'/? - Leo C<*-&)Mf  x  For a f l a t  plate  "W= 1  and  ~X = 1  or  b = 0  and  a = 2.  34.  z=w($+A/£) where;  W=(a+b)/  2  A = ( " )/(a+b) a  Flat  FIG. 21  b  plate: a=2 & b =0 / . *W=1 & A =1  G0NF0RMAL TRANSFORMATION FROM UNIT CIRCLE TO ELLIPSE PLANE  4.3  VORTEX SUPERPOSITION  By r e f e r r i n g to figure 22 and keeping the Milne-Thomson c i r c l e theorem (6) i n mind, the complex potential f o r three fixed vortices i n the presence of a unit c i r c l e i s  Dropping  constants  Now  t h i s represents, f o r example i n the case of P, , a clockwise  vortex at S,e  an anti-clockwise vortex at •jj-e  at the o r i g i n , a l l of strength H .  •, and a clockwise vortex  Therefore, f o r a zero net c i r c u l a t i o n  an anti-clockwise c i r c u l a t i o n of strength P, i s added at the o r i g i n .  The  f i n a l complex p o t e n t i a l then f o r three v o r t i c e s i n the presence of a unit c i r c l e with zero net c i r c u l a t i o n i s  +15[A. Cf - S e *) -JU LS  4  *  A  it- ^  a  ,  C? - -fee  J  By conformal transformation to the e l l i p s e plane ( F i g . 22), e l l i p s e becomes a streamline i n the presence of external v o r t i c e s . the boundary r e l a t i o n , as derived i n section 4.2,  the Since  was obtained from an  integration involving the known normal v e l o c i t y on a r o t a t i n g t r a n s l a t i n g e l l i p s e , the v o r t i c e s may  be superimposed, since t h e i r contribution to the  normal v e l o c i t y at the boundary i s zero.  Z = W  w h e r e :  X = W ( S + ^ / S ) c o s 5 1  1  1  1  " W = ( a + b ) /  A = ( a - b ) / ( a + b )  Y ^ W t S p ^ S i l s i n * ,  FIG.  22  2  VORTEX CONFORMAL TRANSFORMATION FROM UNIT CIRCLE TO ELLIPSE PLANE  37. 4.4  MODEL PRESSURE COEFFICIENT  The unsteady B e r n o u l l i equation f o r a t r a n s l a t i n g body at the o r i g i n is  As stated before, f o r c a l c u l a t i n g the pressure c o e f f i c i e n t on the plate and to better simulate a separated region (that i s r = 1 a n d o i  (p£l&cf),  those terms representing the e f f e c t of the freestream are gradually eliminated on the separated side above JO degrees and below 1 8 0 degrees angle of attack. Those terms affected are underlined by a dashed l i n e i n the following work and f o r the elimination f a c t o r €  as a function of angle of attack, r e f e r to  f i g u r e 24. The lower l i m i t of 3 0 degrees i s the zero r e l a t i v e angle of attack f o r a t i p v e l o c i t y to freestream v e l o c i t y of 0 . 5 .  This approximation i s  shown i n f i g u r e 20. At 1 8 0 degrees angle of attack, the pressure d i s t r i b u t i o n as predicted by the model must be the same as at zero degrees angle of attack. Freestream  e f f e c t s s t a r t to become important again on the separated side  below 6 0 degrees and above 1 5 0 degrees angle of attack and  6 = 1 at 1 8 0  degrees to make the pressure loading equivalent to that at zero degrees angle of attack. Thus, between the above l i m i t s the only e f f e c t of the freestream on the separated side i s to c o n t r o l the magnitude of the generated  vorticity.  38. a)  complex v e l o c i t y magnitude lW(Z){  Now  = a Cbcasoi + La tincO^  + i§?  __£> x  Separating r e a l and imaginary parts and introducing £ gives  ifzTE^  2irr  39.  +  { H J ] Cos 2qS +. J  m  {Hi]  2<f>)  where  ft. fir? = 6. (r] | _ 5  s.'-*.s_  fzj = JU s,-^s,  40.  b)  d  ^/dt  By definition (refer to figure Zj)  ' Jt^al  A  J  t  To obtain the velocity potential att-*4t"and at point P, consider the rotating translating ellipse at the origin, as shown in figure 2 3 , The point P moves to point P' with respect to the ellipse in time At.  ~*t+o L "  2Tt  J  a x d y A t r  aexar  a^^t  arjdt  Af  zc* 2*  ar;at  dfi^c  y  5r|at ar\3r  now  ct)At]  - coy - §jUca>tfi  At =. - cox - 6. (JsiVio<  now  «*3>  =  - £  ^bs)no(cos+ a  ozso(sin  42. 4.5  VORTEX STRENGTH AND POSITION ASSUMPTIONS  Three v o r t i c e s are superimposed i n the wake region to represent the e f f e c t of separation. Since the model v o r t i c e s are fixed with respect to the plate and have only one d i r e c t i o n of r o t a t i o n , the strengths as a function of angle of attack must be zero and have a zero slope a t 1 8 0 degrees and zero degrees angle of attack to give a continuous  cycle.  The strength as a function of angle of attack must be continuous. The  'dominant' vortex (#l) begins to form as the upstream t i p passes  through the r e l a t i v e zero angle of attack ( 3 0 degrees f o r a t i p v e l o c i t y to freestream v e l o c i t y r a t i o of 0 . 5 ) . The flow separates at t h i s point and then reattaches to form a separation bubble.  As the angle of attack  increases, the bubble becomes a vortex, and a t approximately 9 0 degrees the f u l l y developed vortex begins to shed.  In Cheng's pressure measurements,  the maximum suction f o r the 'dominant' vortex occurred at tap # 1 2 (x = I . 4 3 6 ) at an angle of attack of 7 5 degrees.  Assumptions made f o r the 'dominant'  vortex are that the x co-ordinate equals I . 4 3 6 and that the strength i s zero up to the r e l a t i v e zero angle of attack and then negative (clockwise p o s i t i v e ) . Since the vortex i s fixed, the magnitude of P, must increase up to 9 0 degrees to simulate growth of the vortex, and then decrease as the vortex i s shed. The f i n a l form of Pt t/s<X i s given i n f i g u r e 24 and was determined  by t r i a l  and error computer runs, In an attempt to approximate Cheng's measured pressure loadings. V i s u a l information on the 'small' vortex ( # 3 ) formed on the other t i p Is f a i r l y l i m i t e d , which suggests that the strength i s less than the 'dominant* vortex.  This i s supported by Cheng's measurements, i n that the  maximum suction peak at the 'dominant' vortex i s approximately 2 . 5 times that under the 'small' vortex.  The maximum suction f o r the 'small' vortex  43. occurs at tap #2  (x = - I . 4 3 6 ) and, therefore, the x co-ordinate of the  'small' vortex i s f i x e d at t h i s value.  The pressure d i s t r i b u t i o n , as  predicted by the attached flow model, i s degraded by a c i r c u l a t i o n generated in the 'small* vortex below the r e l a t i v e zero angle of attack.  Therefore,  the strength of the 'small' vortex ( F i g . 24) i s zero up to 3 0 degrees and returns to zero again at 1 8 0 degrees to comply with the r e s t r i c t i o n of continuity, as mentioned e a r l i e r . Another vortex (#2)  i s superimposed to represent the e f f e c t of the  downstream wake; i . e . p r i n c i p a l l y the 'dominant' vortex which formed oneh a l f c y c l e before.  Thus, the maximum magnitude of the 'old' vortex must be  assumed to be l e s s than or equal to the maximum magnitude of the 'dominant' vortex.  This 'old* vortex appears to move (to an observer on the p l a t e )  from a p o s i t i o n r e l a t i v e l y close to the t i p forming the 'small' vortex to a p o s i t i o n somewhat more downstream and near 9 0 degrees r e l a t i v e to the plate, while the angle of attack progresses from zero to 9 0 degrees.  It  then moves downstream and i t s influence to the flow near the plate drops. I t was found that the pressure d i s t r i b u t i o n on the vortex side of the p l a t e was just as dependent on the slope of the P**0( curves as the magnitude.  Thus, i f the slope of one of the c i r c u l a t i o n per u n i t span  curves i s high over a range of angles of attack compared to other angles, a sharp change i n the predicted pressure d i s t r i b u t i o n on the vortex side can be expected over that range. The f i n a l form of the c i r c u l a t i o n per u n i t span as a function of angle of attack and the y co-ordinate of each vortex ( F i g . 24) was determined  then  by t r i a l and error computer runs, attempting to duplicate Cheng's  pressure c o e f f i c i e n t s .  The assumed functions arei  60_c<'</50  6= 0  o±o(±30 1 7 = 0 X= (.4-34  X = o.  X = -/.436  _ /.o  FIG. 2 3  CYLINDER AT ORIGIN AT TIME t AND t * A t  FIG. 24  ELIMINATION FACTOR & CIRCULATION VS ANGLE OF ATTACK AND VORTEX POSITION  47. V.  MODEL RESULTS  The model w i l l be compared to Cheng's ( l ) pressure loadings and the t o r q u e - l i f t - d r a g curves derived from these measurements.  Cheng obtained  these l a t t e r curves by assuming that the pressure measured at a tap acted over a f i n i t e area around that tap.  Thus, the l i f t , drag and torque a c t i n g  on the plate could be found as a function of angle of attack.  The model  was treated i n the same manner; that i s , the pressure prediction of the model at Cheng's tap positions was used over a f i n i t e area to determine the integrated curves shown i n figures 3 1 - 3 3 • Pressure c o e f f i c i e n t s versus position on the plate determined  from  the model and compared to Cheng's measurements are shown i n figures 2 5 - 3 0 f o r 3 0 degree i n t e r v a l s i n angle of attack from zero to 1 8 0 degrees. Since the model i s attached flow, i n f i n i t e pressure c o e f f i c i e n t s are given at the edges.  At zero degrees, the model predicts a symmetric pressure  loading about the axis of r o t a t i o n .  The magnitude of the pressure and  general trend i s represented quite well, but the smaller v a r i a t i o n s are not given.  Above zero degrees, the symmetry of the model prediction disappears.  In f i g u r e 2 6 (0(=  JO degrees), the loading trend and magnitude i s predicted  quite well except as the position approaches zero. pressure i s given f o r experimentally determined  Here, a p o s i t i v e  suction on the vortex side  and a suction prediction f o r p o s i t i v e pressure on the attached side.  As the  angle of attack increases further, the trend of the model prediction i s accurate but the magnitude of the pressures i s not represented, as can be seen i n f i g u r e 2 7 (P( - 6 0 degrees).  At 6 0 degrees, a flow d e t a i l which i s  not given by the model i s the maximum i n the suction pressure near the 'dominant' vortex.  The  term_j?_C 1  which represents the time rate of  change of the 'dominant' vortex strength, does give t h i s but the e f f e c t i s  48. completely masked by the term  jWCZ)f  approaching i n f i n i t y at the edge.  At 9 0 and 120 degree angles o f attack, the model again gives the trend of the pressure c o e f f i c i e n t but not the magnitude.  As the angle increases  further, the magnitude p r e d i c t i o n again becomes more accurate, as can be seen i n f i g u r e JO. At 180 degrees, of course, the pressure loading i s equal to that a t zero degrees. In f i g u r e 3 1 »  the drag c o e f f i c i e n t curve f o r the model i s compared  to Cheng's experimental curve.  In both cases, the maximum drag occurs near  90 degrees, but r e f l e c t i n g the model's f a i l u r e to reach the pressure magnitudes a t high angles of incidence the magnitude i s not predicted. This i s true also of the l i f t c o e f f i c i e n t ( F i g . 3 2 ) . Here, however, the maximum l i f t i s predicted a t zero degrees, whereas i n f a c t i t occurs near 4 5 degrees. The torque c o e f f i c i e n t ( F i g . 3 3 ) i s quite well represented except f o r a spike near 3 0 degrees.  This occurs since, as mentioned e a r l i e r , the model  gives the wrong sign f o r the pressure c o e f f i c i e n t a t one end of the plate on both sides. negative.  This produces a p o s i t i v e torque contribution rather than  The torque a t zero degrees i s zero, since the model pressure  d i s t r i b u t i o n i s symmetric about the a x i s of r o t a t i o n a t 0( = 0. I t must be emphasized that t h i s s o l u t i o n i s a compromise based on assumptions of vortex p o s i t i o n and strength and the elimination f a c t o r £  .  Thus, although considerable computing time was spent searching f o r the best assumptions to bring the model into the best possible agreement with the a c t u a l flow s i t u a t i o n , a better set of assumptions may e x i s t . w i l l probably i l l u s t r a t e the compromise involved quite well. attempts a t assumptions,  the strength /jj was equal to  An example In early  up to 9 0 degrees  angle of attack, and then decreased sharply but continuously to 120 degrees. The computer output was being examined at JO degree i n t e r v a l s from zero to 180 degrees.  With t h i s assumption,  the suction magnitudes were reached a t  49. 6 0 and 9 0 degrees and were good a t 1 2 0 degrees.  However, when computer  output f o r the pressure c o e f f i c i e n t was produced a t 1 0 degree i n t e r v a l s , the prediction  a t 1 0 0 and 1 1 0 degrees showed very high p o s i t i v e  rather than suction.  pressures  This was due to the r e l a t i v e l y high value of  a  ^  these angles. To improve upon the model, the f i r s t step might be to l e t the vortex p o s i t i o n be a function of time, as shown by the flow v i s u a l i z a t i o n .  Also,  the time dependent vortex c i r c u l a t i o n might be solved f o r a n a l y t i c a l l y or by i t e r a t i o n to force separation a t the edges.  FIG. 2 5  PRESSURE COEFFICIENT - PLATE POSITION ANGLE OF ATTACK = 0 DEGREES  FIG. 2 6  PRESSURE COEFFICIENT - PLATE POSITION ANGLE OF ATTACK = 3 0 DEGREES  FIG. 2 7  PRESSURE COEFFICIENT - PLATE POSITION ANGLE OF ATTACK = 6 0 DEGREES  FIG. 2 8  PRESSURE COEFFICIENT - PLATE POSITION ANGLE OF ATTACK = 90 DEGREES  FIG. 2 9  PRESSURE COEFFICIENT - PLATE POSITION ANGLE OF ATTACK = 1 2 0 DEGREES  FIG. 3 0  PRESSURE COEFFICIENT - PLATE POSITION ANGLE OF ATTACK = 1 5 0 DEGREES  FIG. 31  DRAG COEFFICIENT - ANGLE OF ATTACK FOR MODEL AND CHENG'S RESULTS  Ox ON  FIG. 32  LIFT COEFFICIENT - ANGLE OF ATTACK FOR MODEL AND CHENG'S RESULTS  oi  FIG. 33  TORQUE COEFFICIENT - ANGLE OF ATTACK FOR MODEL AND CHENG'S RESULTS  59. SUMMARY OF RESULTS  Results of t h i s project are summarized as follows. 1.  The high suction peak near the upstream t i p on the separated side of the p l a t e i s due to the formation of a large vortex at t h i s t i p every h a l f - r e v o l u t i o n ,  2.  Flow separation a t the upstream t i p a t a u t o r o t a t i o n a l speed i s delayed up to a p o s i t i v e angle of attack which i s l e s s than but near the r e l a t i v e zero angle of attack,  3.  A vortex, which appears to be smaller than the vortex formed at the upstream t i p , forms at the downstream t i p .  4.  The vortex shed from the upstream t i p i s the dominant feature of the downstream wake.  5.  The wake undergoes angular d e f l e c t i o n i n the same d i r e c t i o n as plate r o t a t i o n f o r a l l angles of attack.  6.  For the freestream range investigated, the t i p v e l o c i t y i s l i n e a r l y dependent upon the freestream v e l o c i t y .  7.  The angular v e l o c i t y during autorotation i s constant within experimental error.  8.  The higher the freestream v e l o c i t y , the lower the a c c e l e r a t i o n period to autorotation from start-up,  9.  For the given assumptions of vortex position, vortex strength and the elimination f a c t o r € , t h i s model predicts the sign and trend of surface pressure f o r a c y c l e with reasonable accuracy.  The magnitude of these curves, p a r t i c u l a r l y a t high  angles of incidence, i s not accurate, 10.  The trend of\the model drag c o e f f i c i e n t versus angle of attack curve i s accurate. the c y c l e . .  The magnitude i s not accurate f o r most of  6o. 11,  The model l i f t c o e f f i c i e n t versus angle of attack i s not veryaccurate i n trend or magnitude.  Maximum l i f t  zero degrees and i t a c t u a l l y occurs at k5 12,  i s predicted at  degrees.  The model torque c o e f f i c i e n t versus angle of attack i s accurate i n trend and magnitude, except f o r a spike at 3 degrees.  0  61. REFERENCES  1.  Cheng, S.  2.  Crabtree,  3 .  Neumark, 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, November, 1966, L.F.  "The Rotating Flap as a H i g h - L i f t Device", Aeronautical Research Council Technical Report, Current Paper No. 4 8 0 , i 9 6 0 . "Rotating A i r f o i l s and Flaps", Journal of the Royal Aeronautical Society, January, 1 9 6 3 ,  pp. 4 7 - 6 1 .  4.  Claassen, L.  "Combined Free and Forced Convection from Horizontal Plates", M.A.Sc. Thesis, University of B r i t i s h Columbia, A p r i l , 1968.  5.  HYCAM Manual  Model K30S1, Red Lake Labratories, Inc., 2971 Corvin Drive, Santa Clara, C a l i f o r n i a , 95051.  6.  Milne-Thomson, L.M.  "Theoretical Hydrodynamics", MacMillan, 1962, pp. 243-253, and pp. 154.  7.  James, D.B. Stone, J.W.  "The C h a r a c t e r i s t i c s of Thin Wings Autorotating About a Spanwise Axis", Department o f Aeronautical Engineering, University of B r i s t o l , Undergraduate Report No. 6 3 , June, I 9 6 I .  8.  Baird, H. Pick, R.  "Autorotation of F l a t Plates", Senior Year Project Report, Department of Mechanical Engineering, U n i v e r s i t y of B r i t i s h Columbia, A p r i l , 1964.  9.  S c h l i c h t i n g , H.  "Boundary Layer Theory", McGraw-Hill, New York, Fourth E d i t i o n , pp. 2 9 4 - 3 0 2 .  Marks, L.S. Baumeister, T.  "Standard Handbook f o r Mechanical Engineers", McGraw-Hill, New York, Seventh E d i t i o n , pp. 15-103,  10.  London,  62. APPENDIX Since the a i r i n the wake of the heated probe i s at a higher temperature than the ambient conditions, the e f f e c t of buoyancy must be considered. It can be neglected ( 9 ) , i f  Considering the l i m i t i n g case, then  Solving, using the lowest freestream v e l o c i t y ( 1 1 . 3 feet per second)  = 4.9 Oo'*)  3 /ff*  s/u  s  This corresponds to a much higher temperature than the maximum safe working temperature f o r Nichrome-V of 1 , 1 0 0 degrees Centigrade, according to reference 1 0 . It i s i n t e r e s t i n g to note that the shear layers developed at the probe (Figs. 8 - 1 1 ) are not unsteady, even though the Reynolds number f o r the probe based on bulk conditions i s greater than f o r t y . The power input to the probe was constant within the measurement error at 7 . 7 5 watts ( 2 . 5 v o l t s r.m.s., 6 0 cycle) f o r the range of v e l o c i t i e s ( 1 0 - 2 0 f.p.s.) i n the experiment.  1  

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