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An investigation of the effects of periodic wake disturbances on flat-plate boundary layers Yip, Ronald S. K. 1985

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AN INVESTIGATION OF THE EFFECTS OF PERIODIC WAKE DISTURBANCES ON F L A T - P L A T E BOUNDARY LAYERS by RONALD S. K. YIP B A . S c . , University of British Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA January 1985 • RONALD S. K. YIP, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date ^ /ff^T HI ^  Abstract Flat plate turbulent boundary layers disturbed by periodic moving wakes have been observed in an experimental rig mounted in a low speed wind tunnel. The wakes are produced periodically by cylinders traversing in front of the leading edge of a flat plate on which the boundary layers are measured. This is to simulate the unsteady flow pattern generated by upstream blades on the downstream blade boundary layer in an axial flow turbomachine. Both the time-averaged and ensemble-averaged data are taken - from the free stream and boundary layer at different flow conditions. Free stream steady and unsteady wakes are compared and found to be similar to each other. The wake disturbance in the free stream is a function of time and distance from the cylinder. The periodic disturbance in the inner half of the boundary layer lags behind that in the free stream. This phase lag is due to the lower convection velocity near the solid surface. Similar to a steady wake, the velocity defect of an unsteady wake is higher in boundary layer than in free stream. This results in the maximum velocity defect amplitude in the inner half of the boundary layer. Phase lag and amplitude ratio profiles of the boundary layers are plotted and found to be similar to data obtained from axial flow turbomachines. Phase-averaged velocity and turbulence intensity profiles at different phase angles between two successive wakes are shown in a series of transparencies. ii Tab le of Con ten t s A b s t r a c t ii L is t of F igures v A c k n o w l e d g e m e n t s v i i i Nomenc la tu re ix 1. INTRODUCTION 1 1.1 Introductory Remarks 1 1.2 Scope of the Present Investigation 2 2. BACKGROUND MATERIAL AND REVIEW OF PREVIOUS WORK 5 2.1 Background Material on Flat Plate Boundary Layers 5 2.2 Previous Work on Boundary Layers in an Unsteady Free-Stream 9 2.2.1 S tud ies of F lu id F l o w in Exper imenta l A x i a l Tu rbomach ines 10 2.2.2 F l o w S tud ies U s i n g S i m p l i f i e d M o d e l s 12 2.2.3 Numer i ca l P r o g r a m s fo r Ca l cu l a t i ng Uns teady F l o w s 14 3. EXPERIMENTAL APPARATUS AND INSTRUMENTATION 15 3.1 Experimental Apparatus ; 15 3.1.1 W i n d Tunnel 15 3.1.2 The Rota t ing C y l i n d e r C a s c a d e and The Flat P la te 16 3.1.3 T rave rse M e c h a n i s m 17 3.2 Instrumentation 17 3.2.1 Boundary Layer M e a s u r e m e n t - Hot W i r e A n e m o m e t r y 17 3.2.1.1 Hot W i r e Ca l i b ra t i on 19 3.2.1.2 C o r r e c t i o n fo r W a l l P r o x i m i t y E f fec t 19 3.2.1.3 De te rm ina t i on of the Ve r t i ca l D is tance of Hot W i r e f r o m the Flat P la te 20 3.2.2 Phase Re fe rence - Hal l E f f e c t S w i t c h .20 3.3 Signal Conditioning and Data Acquisition 21 iii 3.4 An sEnsemble-Averaging Technique 22 3.5 Data Acquisition Program 23 4. EXPERIMENTAL RESULTS AND DISCUSSION 25 4.1 Time-Averaged Measurements of Steady and Unsteady Boundary Layers 25 4.1.1 Exper imen ta l De ta i l s 25 4.1.2 Expe r imen ta l Resu l t s 26 4.2 Ensemble-Averaged Measurements of Disturbed Free Stream and Boundary Layer .28 4.2.1 Expe r imen ta l De ta i l s .28 4.2.2 Expe r imen ta l Resu l t s 29 4.2.2.1 U n s t e a d y W a k e M e a s u r e m e n t s and C o m p a r i s o n w i t h S t e a d y W a k e M e a s u r e m e n t s 29 4.2.2.2 U n s t e a d y Boundary Layer M e a s u r e m e n t s 30 4.3 Discussion 32 5. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 35 5.1 Conclusions 35 5.2 Suggestions for Further Work 36 F I G U R E S 38 R E F E R E N C E S 70 A P P E N D I C E S 73 iv L is t o f F igures 1.1 T w o C o m m o n C o n s t r u c t i o n s o f A x i a l F l o w Turb ines 39 1.2 Layou t o f a C o n v e n t i o n a l L o w S p e e d Cascade Tunnel .40 1.3 S c h e m a t i c D iag ram o f P e r i o d i c D is tu rbance on Turb ine S ta to r B l a d e s 4 0 2.1 S c h e m a t i c D iag ram o f Flat P la te Boundary Layer D e v e l o p m e n t 41 2.2 T y p i c a l D i s t r i bu t i ons in Laminar and Turbulent Boundary Laye rs on a Flat P la te 41 2.3 Turbulent Boundary L a y e r V e l o c i t y P r o f i l e s in S e m i - L o g F o r m 42 2.4 O s c i l l o g r a m s of Bounda ry Layer V e l o c i t y R e c o r d s and Instantaneous V e l o c i t y P r o f i l e s on a C o m p r e s s o r S ta to r B l a d e , x / C = 0 . 5 43 2.5 O s c i l l o g r a m s of Bounda ry Layer V e l o c i t y R e c o r d s and Instantaneous V e l o c i t y P r o f i l e s on a C o m p r e s s o r S ta to r B l a d e , x / C = 0 . 7 .44 2.6 P r e s s u r e L o s s C o e f f i c i e n t s " A c r o s s a Turb ine Ro to r B lade and the S a m e B lade in a L inear C a s c a d e 45 2.7 A m p l i t u d e and Phase Lag P r o f i l e s f o r the Boundary Layers of a Turb ine Ro to r B lade S u c t i o n - S u r f a c e 46 2.8 S c h e m a t i c D iag ram of Exper imen ta l S e t - U p by P h e i l , Herbst and S c h r o d e r 47 2.9 M e a s u r e d M e a n V e l o c i t i e s o f W a k e and Boundary Layer Interact ing F l o w C o m p a r e d to Und i s tu rbed Flat P la te F l o w 48 2.10 D e c o m p o s i t i o n of Boundary L a y e r - W a k e In teract ion for A n a l y z i n g S e l f - S i m i l a r i t y of W a k e F l o w .49 2.11 C o m p a r i s o n o f M e a s u r e d M e a n V e l o c i t i e s w i th Resu l t s of F i n i t e - D i f f e r e n c e S o l u t i o n 49 3.1 S c h e m a t i c D iag ram o f the Exper imen ta l S e t - U p U s e d in this S tudy ...50 v 3.2 S c h e m a t i c D iag ram of the Uns teady F l o w P r o d u c e d by W a k e s 50 3.3 T y p i c a l Cons tan t Tempera tu re Hot W i re A n e m o m e t e r Br idge 51 3.4 Ca l i b ra t i on Curve of the Cons tan t Tempera tu re Hot W i re A n e m o m e t e r 51 3.5 S t i l l A i r W a l l C o r r e c t i o n fo r Hot W i r e A n e m o m e t e r 52 3.6 M e t h o d of De tec t i ng the Con tac t o f the Hot W i re Probe and the A l u m i n u m Flat P la te 52 3.7 A n I l lus t ra t ion of the E n s e m b l e - A v e r a g e d Procedure ; 53 3.8 Data A c q u i s i t i o n P rog ram F l o w Chart 54 4.1 N o n d i m e n s i o n a l T i m e - A v e r a g e d V e l o c i t y P r o f i l e s 58 4.2 D i m e n s i o n a l Turbu lence Intensi ty P r o f i l e s o f Und is tu rbed Boundary Layer 59 4.3 N o n d i m e n s i o n a l Turbu lence Intensi ty P r o f i l e s of Und is tu rbed Boundary Layer 59 4.4 Boundary L a y e r D e v e l o p m e n t 60 4.5 C lauser P l o t s of T i m e - A v e r a g e d V e l o c i t y P r o f i l e s of D i s tu rbed Boundary L a y e r s 61 4.6 E n s e m b l e - A v e r a g e d T i m e - V e l o c i t y Reco rd of a T rave rs i ng Wake 62 4.7 N o n d i m e n s i o n a l W a k e V e l o c i t y P r o f i l e s 63 4.8 N o n d i m e n s i o n a l W a k e Turbu lence In tens i ty P r o f i l e s 64 4.9 E n s e m b l e - A v e r a g e d T i m e - V e l o c i t y R e c o r d s 65 4.10 E n s e m b l e - A v e r a g e d Turbu lence I n t e n s i t y - T i m e R e c o r d s 66 4.11 N o n d i m e n s i o n a l P h a s e - A v e r a g e d V e l o c i t y P r o f i l e s at D i f fe ren t Phase A n g l e s : 67 4.12 N o n d i m e n s i o n a l P h a s e - A v e r a g e d Turbu lence Intensi ty P r o f i l e s at D i f fe ren t Phase A n g l e s 68 4.13 M a x i m u m A m p l i t u d e Rat io P r o f i l e s 69 vi 4.14 Phase Lag P r o f i l e s 69 C.I T y p i c a l C lause r P lo t 79 L IST OF P L A T E S P L A T E 1 The L o w S p e e d W i n d Tunnel 55 P L A T E 2 The C y l i n d e r C a s c a d e 55 P L A T E 3 The Dr i v ing m o t o r and S p e e d R e d u c t i o n Gears 56 P L A T E 4 Bear ing Suppo r t s and T rave rse M e c h a n i s m 56 P L A T E 5 Flat P la te and Suppor t 57 P L A T E 6 Hot W i r e P robe for Bounda ry Layer Measu remen t 57 v i i A c k n o w l e d g e m e n t s The author w i s h e s to exp ress his s i nce re grat i tude to P r o f e s s o r R.L. Evans fo r his inva luab le adv i ce and gu idance throughout al l phases o f the i n v e s t i g a t i o n . Thanks are due to P r o f e s s o r I.S. Gar t sho re fo r his he lp fu l d i s c u s s i o n and r e f e r e n c e s . Thanks are a l s o due to the techn ica l s ta f f o f the m e c h a n i c a l eng ineer ing depar tment fo r their a s s i s t a n c e in the c o n s t r u c t i o n o f the exper imen ta l appara tus . F inanc ia l suppor t f o r th is research by the Natural S c i e n c e s and Eng ineer ing Research C o u n c i l o f Canada is g ra te fu l l y a c k n o w l e d g e d . v i i i Nomenc la tu re C cons tan t in the. law of the w a l l (= 5 to 5.6) f l o w c o e f f i c i e n t (= U ^ / U g ) d cy l i nde r d iamete r E c hot w i re a n e m o m e t e r output v o l t a g e EQ hot w i re a n e m o m e t e r output vo l t age at ze ro v e l o c i t y K v o n Karmen cons tan t = 0.41 R e d R e y n o l d s number based on c y l i n d e r d iamete r R e v R e y n o l d s number b a s e d on d o w n s t r e a m d i s tance f r o m the A plate lead ing edge Rw hot w i re r es i s tance t t ime *1/2 h a l f - w i d t h of an uns teady w a k e * u t ime averaged turbu lence in tens i t y . * u e n s e m b l e ave raged tu rbu lence in tens i t y U ins tan taneous v e l o c i t y in x d i r ec t i on 0 t i m e averaged v e l o c i t y in x d i r ec t i on U e n s e m b l e ave raged v e l o c i t y in x d i rec t i on U M A X - m a x i m u m v e l o c i t y in a t i m e ave raged s teady w a k e U M I N m i n i m u m v e l o c i t y in a t ime ave raged s teady w a k e U M A X m a x i m u m v e l o c i t y in an e n s e m b l e - a v e r a g e d uns teady w a k e U M I N m i n i m u m v e l o c i t y in an e n s e m b l e ave raged uns teady w a k e °cyl' uB cy l i nde r tangent ia l v e l o c i t y °- t ime ave raged v e l o c i t y at i n f i n i t y in x d i r ec t i on u turbulent v e l o c i t y in x d i r ec t i on ix u T shear ing v e l o c i t y (= ( r w / p ) / ) A u e n s e m b l e averaged max imum w a k e amp l i tude V i ns tan taneous v e l o c i t y in y d i r ec t i on v turbulent v e l o c i t y in y d i rec t i on w turbulent v e l o c i t y in z d i rec t ion x d o w n s t r e a m d is tance f r o m the p la te lead ing edge x Q d o w n s t r e a m d i s tance f r o m a cy l i nde r in f ront of the p late x Q / d d i s tance f r o m cy l i nde r to c y l i n d e r d iamete r rat io y ver t i ca l d i s tance V-\/2 h a l f - w i d t h of a s teady wake y/~B n o n d i m e n s i o n a l i z e d boundary layer ver t i ca l d i s tance f r o m p la te v k i nemat i c v i s c o s i t y p f l u id dens i t y r t I w a l l shear s t ress w 7T parameter in C o l e s l aw of the w a k e 6 boundary layer t h i ckness at 0.90^, * 6 d i s p l a c e m e n t t h i ckness 7j t ime averaged boundary layer t h i c k n e s s 8 m o m e n t u m th i ckness co d i s tu rbance f r equency (Hz) x Chapter 1 INTRODUCTION 1.1 INTRODUCTORY REMARKS A x i a l f l o w t u r b o m a c h i n e s , both turb ines and c o m p r e s s o r s , are c o m p o s e d of a l ternate r o w s o f s ta tor and rotor b l ades in the axial d i r e c t i o n . T w o c o m m o n ax ia l f l o w turbine cons t r uc t i ons are as s h o w n in F ig .1 .1 . In tu rb ines , the s ta tor b l a d e s , or n o z z l e s , inc rease the f l u i d tangent ia l v e l o c i t y b e f o r e enter ing the f o l l o w i n g ro to r b lade r o w . The ro to r b l ades then conver t the f lu id ' s s tagna t ion en tha lpy to mechan ica l ene rgy through the ro ta t i on of the shaf t . A x i a l f l o w c o m p r e s s o r s are c o n s t r u c t e d in a s i m i l a r manner except the b lades are o r i en ted in such a w a y t o conve r t shaf t energy to f l u id en tha lpy . The b lade e f f i c i e n c y is usua l l y ob ta ined e x p e r i m e n t a l l y under s teady c o n d i t i o n s in a l inear c a s c a d e tunne l . The b lades are a r ranged in a c a s c a d e near the out let o f the tunnel as in Fig.1.2. A p i t o t - s t a t i c p robe in f ront o f the cascade measures the un i f o rm f l o w v e l o c i t y in the tunnel ups t r eam of the c a s c a d e . Ano the r p i t o t - s t a t i c p robe is t r a v e r s e d d o w n s t r e a m o f the cascade to measure the w a k e p r o f i l e s o f one or more b l a d e s . The average va lues of the to ta l p ressure l o s s and the f l u id out let angle at each inc idence angle w i t h i n the w o r k i n g range can be ca l cu la ted f r o m the measurements . (S tuar t 1952) In a real t u rbomach ines , the f l o w c o n d i t i o n is much mo re c o m p l e x than in a c a s c a d e tunne l . S i n c e 3 - d i m e n s i o n a l and uns teady e f f e c t s are not a c c o u n t e d fo r in cascade t es t i ng , e m p i r i c a l co r rec t i on f a c t o r s must be app l i ed to c a s c a d e resu l ts be fo re they are used for d e s i g n p u r p o s e s . 1 2 In order to i m p r o v e the accu racy of t u rbomach ine p e r f o r m a n c e p r e d i c t i o n , a bet ter unders tand ing of the e f f e c t s is requ i red . One o f the ma in d i f f e r e n c e s be tween a real t u rbomach ine and a c a s c a d e tunnel is the uns teady f l o w pat tern . Th i s uns tead iness is due to the imp ingemen t of the w a k e be ing shed f r o m the u p s t e a m b l a d e s onto the d o w n s t r e a m b lade s u r f a c e s . Because the ups t ream b lades p a s s in f ront of each d o w n s t r e a m b lade at a cons tan t f r e q u e n c y , the boundary layer on that b lade is p e r i o d i c a l l y d i s tu rbed as i l l us t ra ted in F ig.1.3. In the l i terature, the e f f e c t s of uns teady f ree s t ream f l o w s on boundary layer d e v e l o p m e n t has been a popu lar research t op i c . L i t t le wo rk has been repor ted on the e f f ec t o f c o n v e c t e d p e r i o d i c w a k e d i s tu rbances on boundary layer d e v e l o p m e n t , h o w e v e r . The o b j e c t i v e of th is research is to o b s e r v e e x p e r i m e n t a l l y the pe r i od i c w a k e - t u r b u l e n t boundary layer in te rac t ion in order to better unders tand the b a s i c f l o w m e c h a n i s m . A s i m p l i f i e d ' exper imen ta l m o d e l is d e s i g n e d and c o n s t r u c t e d to i so la te the e f f e c t s of the uns tead iness on the boundary layer f r o m other f l o w c o m p l i c a t i o n s . The resu l t s can h o p e f u l l y make an in i t ia l con t r i bu t i on t o w a r d s the u l t ima te goa l o f p r o v i d i n g a mo re accura te p rocedure fo r b lade e f f i c i e n c y e v a l u a t i o n . 1.2 SCOPE OF THE PRESENT INVESTIGATION In th is t h e s i s , the e f f e c t s o f pe r i od i c w a k e d is tu rbance on turbulent boundary layer d e v e l o p m e n t is s t u d i e d . The turbulent boundary layer is d e v e l o p e d by t r i pp ing the boundary layer near the lead ing edge of a zero pressure grad ient hor i zon ta l f lat p la te s u r f a c e . A hor i zon ta l f la t p late 3 is used to e l im ina te the e f f e c t o f p ressure grad ient . A pe r i od i c d i s tu rbance is genera ted by p a s s i n g hor izonta l c i rcu lar bars para l le l to the p la te w id th in f ront o f the lead ing edge at a cons tan t f r equency . M e a s u r e m e n t s o f the boundary layer are made near the no rma l opera t ing f l o w c o e f f i c i e n t , C ^ , w h i c h is app rox ima te l y equal to 0.5. Through the use o f th is s i m p l i f i e d exper imen ta l m o d e l , the pe r i od i c wake - tu rbu len t boundary layer in te rac t ion can be s tud ied w i thou t the added c o m p l i c a t i o n of 3 - d i m e n s i o n a l and p r e s s u r e - g r a d i e n t e f f e c t s . B a c k g r o u n d mate r ia l and a s u m m a r y of p rev i ous work are g i ven in Chapter 2. The f i rs t part inc ludes background mater ia l on boundary layer d e v e l o p m e n t and ca l cu l a t i on o f s t eady boundary layer p r o f i l e s on a zero p ressure gradient f lat p la te . The s e c o n d part r e v i e w s p rev ious wo rk in the re la ted f i e l d . Th is i nc ludes the c o m p a r i s o n s o f c a s c a d e tunnel m e a s u r e m e n t s w i t h the m e a s u r e m e n t s f r o m opera t ing ax ia l f l o w t u r b o m a c h i n e s , and boundary layer g rowth in pe r i od i c uns teady f ree s t r e a m s . A l s o , the e f f e c t o f the wake f r o m a s ta t i ona ry bar on f l a t - p l a t e boundary layer d e v e l o p m e n t is r e v i e w e d . C o m p u t a t i o n a l me thods fo r p red i c t i ng uns teady laminar and turbulent boundary layers are a l so b r i e f l y d i s c u s s e d . The exper imen ta l apparatus and ins t rumenta t ion are . d e s c r i b e d in Chapter 3. The mechan ica l equ ipment inc ludes a l o w s p e e d w i n d tunne l , a ro ta t ing cage fo r pe r i od i c d is tu rbance gene ra t i on , a f lat p late for genera t ing the boundary laye r , and a t raverse m e c h a n i s m fo r m o v i n g the hot w i r e p robe a c r o s s the boundary layer . The e lec t ron i c equ ipment inc ludes a hot w i r e a n e m o m e t e r fo r f l o w measu remen t , a Hal l E f f ec t s w i t c h f o r phase r e f e r e n c e , a s igna l cond i t i one r and the NEFF s y s t e m fo r data a c q u i s i t i o n . 4 In Chapter 4, the exper imen ta l c o n d i t i o n s and resu l ts are p r e s e n t e d . The resu l ts inc lude t ime ave raged v e l o c i t y measu remen ts of d i s tu rbed and und is tu rbed boundary l aye rs , and a l so e n s e m b l e - a v e r a g e d t i m e - v e l o c i t y m e a s u r e m e n t s o f the d is tu rbed f ree s t r e a m s and boundary l a y e r s . The e n s e m b l e - a v e r a g e d phase v e l o c i t y p r o f i l e s , the e n s e m b l e - a v e r a g e d phase turbu lence in tens i t y p r o f i l e s , the amp l i t ude ra t io and phase lag p r o f i l e s are ob ta ined f r o m the t i m e - v e l o c i t y m e a s u r e m e n t s . Chapter 5 s u m m a r i s e s the exper imen ta l resu l ts and s u g g e s t s fur ther wo rk requ i red fo r th is s tudy . Chapter 2 BACKGROUND MATERIAL AND REVIEW OF PREVIOUS WORK In order to unders tand and d i s c u s s the o b s e r v a t i o n on s teady and uns teady turbulent boundary l aye r s , s o m e backg round mater ia l and pub l i shed resu l t s are p resen ted in this chapter . 2.1 BACKGROUND MATERIAL ON FLAT PLATE BOUNDARY LAYERS In a f l o w f i e l d far a w a y f r o m any s o l i d bounda ry , the f l o w c o n d i t i o n can be m o d e l l e d b y us ing po ten t ia l f l o w theory in w h i c h v i s c o u s e f f e c t s are i ns ign i f i can t . H o w e v e r , in a thin layer , c l o s e to the s o l i d boundary , f lu id f r i c t i on s l o w s d o w n the f l o w . Th is th in layer in wh i ch the v e l o c i t y is u l t ima te l y r e d u c e d to zero at the w a l l is ca l l ed the boundary layer , ou ts ide of w h i c h the real f l u id behaves ve ry much l ike the ideal f l u i d . A t the ea r l y s tage of f lat p late boundary layer d e v e l o p m e n t , i.e. at l o w R e y n o l d s numbers ( R e x = U 0 0 x / v), there is no m ix i ng w i th in the boundary l aye r ; the f l u i d m o v e s a long para l le l s t r e a m l i n e s . Th is para l le l s t reaml ine bounda ry layer is ca l l ed a laminar boundary layer . A t high R e v , i.e. fur ther d o w n s t r e a m , the f l u id in the boundary layer have h igh ly i r regular m o t i o n s w h i c h cause m ix ing w i t h i n the boundary layer . A t modera te R e x , s p o t s o f f l u id w i th loca l i r regular m o t i o n s appear in the layer wh i ch is the t rans i t i on reg ion f r o m laminar to turbulent boundary layer . The bounda ry layer t h i ckness i nc reases sudden l y when the boundary layer changes f r o m laminar to turbulent . The p r o c e s s of boundary layer g rowth is as s h o w n in Fig.2.1. 5 6 There are many ana ly t i ca l and numer ica l s o l u t i o n s fo r s t eady laminar boundary layer d e v e l o p m e n t fo r d i f fe ren t p ressure g rad ien ts . The mater ia l to be d i s c u s s e d in th is s e c t i o n is on l y on f la t p la te boundary layers s i n c e it is the main c o n c e r n in th is t h e s i s . For R e v < 105, the s t e a d y laminar boundary layer can be ob ta ined A by s o l v i n g the B las i us equa t i on . f f ' ' + 2 f " ' = 0 2.1 where f ( T?) =f (y (— ) 1 / 2) w i t h boundary c o n d i t i o n s vx r?=0:f = 0,f'=0; and 77=°°:f' = 1 Th is equa t i on has been s o l v e d n u m e r i c a l l y , and the v e l o c i t y d i s t r i bu t ion o f the bounda ry layer is as p lo t t ed in Fig.2.2. For Re v>2x10 6, it is d i f f i cu l t to prevent t rans i t i on f r o m occur r ing even though the sur face o f the f lat p late is s m o o t h and the f l o w c o n d i t i o n in the main s t ream is laminar . Un l i ke in the laminar boundary laye r , the v e l o c i t y in turbulent layer is r a n d o m l y f l uc tua t ing . For a t w o - d i m e n s i o n a l l aye r , the ins tan taneous v e l o c i t y can be rep resen ted by equa t ion 2.2 as in Sch l i ch t i ng (1968). U j = U + u V j = V + v 2.2 W j = w whe re U j , Vj and Wj are the ins tan taneous v e l o c i t i e s in the x, y and z d i r e c t i o n s . U and V are mean v e l o c i t i e s and u, v and w are the 7 f l uc tua t ing c o m p o n e n t s in the x, y and z d i r e c t i o n s . Emp i r i ca l f o rmu la t i ons fo r the m e a n v e l o c i t y p r o f i l e s o f turbulent boundary laye rs on zero p ressure gradient su r f aces have been d e v e l o p e d f r o m exper imen ta l resu l t s . B y subs t i tu t ing equa t ions 2.2 into the N a v i e r - S t o k e s equa t ions and takng t ime ave rages an extra te rm k n o w n as R e y n o l d s s t r e s s is ob ta ined f r o m the equa t ions of m o t i o n . A s s u m i n g that near the w a l l , de r i va t i ves w i th respec t to y are much greater than those w i t h respec t to x, the shear s t r e s s te rm f r o m the equat ion of m o t i o n can be e x p r e s s e d a s ; On the R.H.S. o f equat ion 2.3, the f i rs t te rm is the R e y n o l d s s t r e s s and the s e c o n d te rm is the laminar shear s t r e s s . M o v i n g a w a y f r o m the wa l l w i th in the boundary layer , the turbulent R e y n o l d s s t r ess i nc reases but the laminar s t r e s s d e c r e a s e s . By a s s u m i n g the mean v e l o c i t y is a f unc t i on of r w / p , v and the w a l l d i s t a n c e , y , d i m e n s i o n a l a n a l y s i s g i v e s the " law of the w a l l " wh i ch is a un ive rsa l turbulent boundary layer p ro f i l e w i th in the cons tan t s t ress r e g i o n . 2 . 3 2 . 4 v where U T = ( T W / P ) 1/2 (shear ing v e l o c i t y ) In th is cons tan t s t ress reg ion right next to the w a l l , there is a s m a l l 8 reg ion k n o w n as the v i s c o u s sub laye r . Exper imen ta l o b s e r v a t i o n s h o w e d that ; U yu — = — L 2.5 U _ v Further out f r o m the v i s c o u s sub laye r but s t i l l ins ide th is cons tan t s t r e s s reg ion is the l oga r i t hm ic r e g i o n , the v e l o c i t y p ro f i l e can be f o rmu la ted by u s i n g , — = ~ l n ( 2L)+C 2.6 U r K v where K=0.41 (Von Ka rman ' s cons tan t ) C = 5.0 to 5.6 Further away in the boundary layer f r o m the loga r i t hm ic r e g i o n , C o l e s has p r o p o s e d a un i ve rsa l " law of the w a k e " to d e s c r i b e the outer layer ' s mean v e l o c i t y p r o f i l e . In th is r e g i o n , the boundary layer is a s s u m e d to behave l ike a wake or f ree shear layer . The c o m p l e t e mean p ro f i l e of a fu l l y d e v e l o p e d turbulent boundary layer can then be d e s c r i b e d by us ing the C o l e s p r o f i l e . ~ - { l n ( ^ ) + f ( W ( f ) ) + C 2.7 U T K v K 0 where w ( ^ ) = 2 s i n 2 ( ^ f ) 6 iro The mean v e l o c i t y p r o f i l e s of the three reg ions in a turbulent boundary layer are p lo t ted in s e m i - l o g f o r m in Fig.2.3. 9 For a f la t p la te turbulent boundary layer w i th Re around 10 s , the mean v e l o c i t y p r o f i l e can a l so be app rox ima ted by the P o w e r L a w f o r m . CP w i t h n = 1/7, w h i c h w a s ob ta i ned e m p i r i c a l l y by N iku radse (Sch l i ch t ing 1979), Fig.2.2 The a b o v e represen ta t ion o f the boundary layer v e l o c i t y p r o f i l e s is on l y fo r s t e a d y f rees t ream f l o w s . The e f f ec t of p e r i o d i c w a k e s on turbulent boundary layers w i l l be d i s c u s s e d in a later s e c t i o n and the t ime ave raged p r o f i l e s of the d i s tu rbed boundary layer w i l l be c o m p a r e d w i th the a b o v e s t e a d y v e l o c i t y p r o f i l e s to o b s e r v e the e f f e c t o f the d i s tu rbances on the mean boundary , layer d e v e l o p m e n t . 2.2 PREVIOUS WORK ON BOUNDARY LAYERS IN AN UNSTEADY FREE-STREAM There has been t remendous in terest in t ry ing to unders tand the behav io r o f b lade boundary layers in the h igh ly uns teady and turbulent f l o w pat tern in ax ia l f l o w t u r b o m a c h i n e s . A r e v i e w of the l i terature has found no numer i ca l or ana ly t i ca l p r e d i c t i o n me thods fo r uns teady w a k e d is tu rbed turbulent boundary l a y e r s . H o w e v e r , va r i ous authors have pub l i shed papers on re la ted t o p i c s . T h e s e papers are c l a s s i f i e d into three ma in g roups w h i c h i nc lude ; s t ud i es o f f l u id f l o w ins ide exper imen ta l t u r b o m a c h i n e s , s tud ies of f l u id f l o w in s i m p l i f i e d expe r imen ta l m o d e l s , and s tud ies us ing numer ica l c o m p u t a t i o n a l me thods fo r p red ic t i ng the 10 boundary layers in uns teady f r e e s t r e a m f l o w s . i 2.2.1 S T U D I E S OF FLUID F L O W IN E X P E R I M E N T A L A X I A L T U R B O M A C H I N E S Evans (1974) measu red turbu lence and uns tead iness of the f l o w d o w n s t r e a m of the m o v i n g b lades in a s ing le s tage 22 b lade exper imen ta l c o m p r e s s o r . B y us ing an e n s e m b l e averag ing techn ique , he w a s ab le to separa te the random turbu lence f r o m the pe r i od i c u n s t e a d i n e s s . In another paper , Evans (1977) used a c o m p r e s s o r wh i ch has a 24 b lade ro tor f o l l o w e d by a 15 b lade s ta tor to s tudy the e f f e c t s of the ro tor b lade w a k e s on the s ta tor b lade boundary layer d e v e l o p m e n t . The resu l ts we re c o m p a r e d to the resu l t s of s im i l a r b l ades in a c a s c a d e tunne l . The boundary layer integral * pa rame te rs , such as 8 , 8, and C f w e r e higher in the c o m p r e s s o r m e a s u r e m e n t s than in the c a s c a d e m e a s u r e m e n t s . The higher m o m e n t u m l o s s of the c o m p r e s s o r b lade in an opera t ing mach ine is thought to be caused by the w a k e d is tu rbance f r o m the ro tor b lades u p s t r e a m . By us ing the e n s e m b l e averag ing techn ique , ins tan taneous v e l o c i t y p r o f i l e s at the m a x i m u m ( 0 ° ) , and the m i n i m u m ( 1 8 0 ° ) f r ees t r eam v e l o c i t i e s w e r e ob ta ined by read ing the v e l o c i t i e s o f the requ i red ang le f r o m a s e r i e s of e n s e m b l e - a v e r a g e d v e l o c i t y r e c o r d s wh i ch we re taken at d i f f e ren t ve r t i ca l d i s t a n c e s f r o m the p l a t e , as s h o w n in Fig.2.4. In the l ower part o f the boundary l aye r , the f l uc tua t ion we re 180° out of phase w i t h the f ree s t r e a m . Th is p h a s e - s h i f t p h e n o m e n o n d i sappea red further d o w n s t r e a m w h e n the boundary layer w a s f u l l y turbulent , as can be 11 seen in Fig.2.5. H o d s o n (1982) made boundary layer measu remen ts on a ro tor b lade in a l o w s p e e d , s i ng le s t age , ax ia l f l o w turb ine. The resu l ts we re c o m p a r e d to t hose of a cascade w i th the s a m e b lades in a s teady f l o w . F i g . 2.6 s h o w s that the ro tor re la t i ve l oss c o e f f i c i e n t in the turbine is h igher than the m a s s - w e i g h t e d p ro f i l e l o s s c o e f f i c i e n t ob ta ined f r o m the c a s c a d e tunnel tes t . The integral pa ramete rs were found to be higher than the c a s c a d e resu l t s . In the boundary layer m e a s u r e m e n t s , he s h o w e d that the w a k e induced f l uc tua t ion has a m a x i m u m ampl i tude w i th in the boundary layer on the suc t i on s ide o f the ro tor b lade . F i g . 2.7 s h o w s the ampl i tude rat io ( = ^ 0 / 0 ^ ) against d i s tance f r o m p late s u r f a c e , y . Th is amp l i tude a l so i nc reases in the d o w n s t r e a m d i r ec t i on . The ampl i tude of the w a k e f luc tua t ion approaches zero near the su r face because of the damp ing of the su r f ace shear s t r e s s . S i m i l a r to the 180° out of phase o b s e r v e d by E v a n s (1977), he a l so o b s e r v e d a phase lag inc rease be tween the w a k e d is tu rbance and the f r ees t r eam towa rds the b lade sur face and f r o m the lead ing edge to the t ra i l ing edge , Fig.2.7. H o w e v e r , unl ike E v a n s ' resu l t , th is phase lag did not d i sappear further d o w n s t r e a m . A l t h o u g h the nega t i ve e f f e c t s of the uns teady d i s tu rbances on the a e r o d y n a m i c e f f i c i e n c y o f b lades in t u rbomach ines are o b v i o u s , the bas i c m e c h a n i s m of the w a k e - b o u n d a r y layer in te rac t ion is s t i l l not w e l l unde rs tood . Researche rs have t r ied to use s i m p l i f i e d m o d e l s to gain bet ter unders tand ing of th is bas i c m e c h a n i s m . 12 2.2.2 F L O W S T U D I E S U S I N G S IMPL IF IED M O D E L S M a n y expe r imen ts have been p e r f o r m e d to s tudy the e f f e c t s of d i f fe ren t uns teady f r e e s t r e a m s on f lat p la te turbulent boundary layer d e v e l o p m e n t . K a r l s s o n (1958) i nves t i ga ted the e f f e c t s of a f ree s t ream o s c i l l a t i o n w h i c h is a f u n c t i o n of t i m e o n l y on a f la t p la te boundary layer . The resu l t s s h o w that the n o n - l i n e a r in te rac t ion be tween the osc i l l a t i ng amp l i t ude and the boundary layer is s m a l l . The boundary layer of an ins tan taneous f ree s t ream v e l o c i t y in an o s c i l l a t i n g c y c l e does not dev ia te ve ry much f r o m that of a s t e a d y f ree s t ream of equal v e l o c i t y . Pa te l (1975) used an o s c i l l a t i n g w i n d tunne l n o z z l e ex tens ion to create c o n v e c t e d f r e e s t r e a m o s c i l l a t i o n s of the f o l l o w i n g f o r m , U(x,t)= U ( 1 + N s in w ( t - x / Q ) ) . 2.9 where U is the f l o w v e l o c i t y as a f unc t i on o f d o w n s t r e a m d is tance x and t. U ! is f r ees t ream o s c i l l a t i o n a m p l i t u d e . U Q is mean v e l o c i t y . N ( = U ! A J 0 ) is the rat io o f the w a v e amp l i t ude to mean f r ees t r eam v e l o c i t y . u> is the radian f r equency and Q is the t rave l l i ng w a v e c o n v e c t i o n v e l o c i t y . Flat p la te turbulent bounda ry l aye rs we re m e a s u r e d . A m p l i t u d e rat io and phase lag p r o f i l e s s i m i l a r to t hose of H o d s o n (1982) we re o b t a i n e d . It w a s o b s e r v e d that the m a x i m u m amp l i tude ra t io inc reases w i t h oo but d o e s not depend on U , . On the con t ra r y , the m a x i m u m phase lag d o e s not depend on the amp l i tude rat io on l y w h e n it is l ess than 0.5%. The e f f e c t s of co 13 and d o w n s t r e a m d is tance on m a x i m u m amp l i t ude rat io and phase lag angle can be c o l l a p s e d by us ing the reduced f r equency paramete r , c o x / U 0 . P f e i l , Herbst and S c h r o d e r (1982) d e v e l o p e d an ana ly t i ca l m o d e l to desc r i be the t rans i t i on p r o c e s s f r o m a laminar to a turbulent boundary layer under p e r i o d i c w a k e induced f l uc tua t i ons . Expe r imen ts we re car r ied out to c o n f i r m the m o d e l . In the e x p e r i m e n t , the ro tor b lades w e r e s i m p l i f i e d to bars o f a ro ta t ing cage and the s ta tor b lade to a f la t p la te as in F ig 2.8. The t rans i t i on m e c h a n i s m of a zero p ressu re gradient and a f a v o r a b l e p ressure gradient boundary layer w e r e s tud ied and f o u n d to agree w i th the m o d e l . M a r u m o , Susuk i and S a t o (1978) as w e l l as T s i o l a k i s , Krause and M i i l l e r (1983) s tud ied the e f f e c t s of the w a k e f r o m a s ta t i ona ry bar on a fu l l y d e v e l o p e d turbulent f la t p la te boundary layer . The bar w a s p l aced at d i f f e ren t ve r t i ca l d i s t a n c e s f r o m the p la te . A hot w i re a n e m o m e t e r p robe w a s t rave rsed through the boundary laye rs at d i f fe ren t d o w n s t r e a m p o s i t i o n s in both e x p e r i m e n t s . M a r u m o et al (1978) t o o k m e a s u r e m e n t s at x / d = 4 . 6 to 105 and T s i o l a k i s et al (1983) t ook m e a s u r e m e n t s at x / d = 2 0 to 86 . The resu l t s f r o m both repor ts s h o w that the nearwa l l reg ion of the d i s tu rbed boundary layer r e c o v e r s much fas te r than the outer r e g i o n . Fig.2.9, w h i c h is taken f r o m T s i o l a k i s (1983), s h o w s the d i f f e rence in p r o f i l e s of the und is tu rbed layer and the d is tu rbed layer w i th the s ta t i ona ry bar p l aced at t w o d i f fe ren t d i s t a n c e s f r o m the p la te . The resu l t s s h o w e d that the s e l f - s i m i l a r i t y o f a d i s to r t ed boundary layer can be m o d e l l e d by a s s u m i n g the boundary layer f l o w to be 14 c rea ted by a v e l o c i t y de fec t £,0, F ig.2.10. The f l o w of the w a k e of the c y l i n d e r , A 2U, can be s u p e r i m p o s e d to the f i rs t a p p r o x i m a t i o n to y i e l d the resul tant v e l o c i t y de fec t , AU. Th is mode l r e a s o n a b l y app rox ima ted the s e l f - s i m i l a r upper part o f the inner laye r , but it b roke d o w n in the nearwa l l reg ion because of the n o n - l i n e a r i t y and large v e l o c i t y de fec t c l o s e to the w a l l . 2.2.3 N U M E R I C A L P R O G R A M S FOR C A L C U L A T I N G U N S T E A D Y F L O W S Evans (1974) d e v e l o p e d a numer ica l c o m p u t a t i o n p rog ram to ca lcu la te the d e v e l o p m e n t of turbulent boundary layers in d i f fe ren t f r e e s t r e a m turbulent i n tens i t i es . The p red ic ted resu l ts c l o s e l y agree w i t h exper imen ta l resu l ts of c o m p r e s s o r c a s c a d e boundary l aye rs . H o w e v e r , th is d o e s not g i ve adequate p red i c t i on on boundary layer under pe r i od i c w a k e d i s tu rbance . A compu te r p rog ram w a s wr i t ten by C e b e c i and Carr (1978) to ca lcu la te laminar and turbulent boundary laye rs fo r t w o d i m e n s i o n a l t i m e - d e p e n d e n t f ree s t ream f l o w s . Pa te l (1975) c l o s e l y p red i c ted the phase lag and ampl i tude ra t io p r o f i l e s of boundary l aye rs in quas i s teady and high f requency f ree s t r eam o s c i l l a t i o n s o f the f o r m o f equat ion 2.10. T s i o l a k i s , Krause and Mu ' l l e r (1983) s o l v e d the boundary layer equat ion w i th an imp l i c i t f i n i t e - d i f f e r e n c e m e t h o d d e v e l o p e d at the A e r o d y n a m i s c h e s Inst i tute of Techn i ca l U n i v e r s i t y , A a c h e n , to ob ta in mean v e l o c i t y p r o f i l e s of d i s to r t ed boundary l a y e r s , as s h o w n in F ig .2 .11. G o o d agreement be tween the c o m p u t e d m e a n v e l o c i t y p r o f i l e s and exper imen ta l data w a s o b t a i n e d . Chapter 3 EXPERIMENTAL APPARATUS AND INSTRUMENTATION 3.1 EXPERIMENTAL APPARATUS The s tudy of the e f f e c t s of p e r i o d i c d is tu rbance on b lade p e r f o r m a n c e in opera t ing t u rbomach ines is ve ry d i f f i cu l t because of the c o m p l i c a t e d f l o w pa t te rn . In order to i so la te th is par t icu lar f l o w c o n d i t i o n , a su i tab le exper imen ta l m o d e l must be s e l e c t e d . A m o n g the exper imen ta l s t ud ies on uns teady f l o w s , the exper imenta l s e t - u p used by P f e i l et al (1982), Fig.2.8, w a s c o n s i d e r e d to be the m o s t c o m p a t i b l e w i th the ava i l ab le w i n d tunnel in the labora to ry . H o w e v e r , in order to a v o i d the f l o w be ing d i s tu rbed t w i c e , the f lat p late is i ns ta l l ed ins ide the ro ta t ing c a s c a d e in th is expe r imen t , as s h o w n in F ig .3 .1 . W a k e s are genera ted p e r i o d i c a l l y and car r ied d o w n s t r e a m a long the p la te at the c o n v e c t i o n v e l o c i t y w h e n the c y l i n d e r s pass in f ront of the lead ing edge of the p la te at a cons tan t f r e q u e n c y , Fig.3.2. 3.1.1 WIND T U N N E L A l o w - s p e e d open w i n d tunne l , P la te 1, in the a e r o d y n a m i c s l abo ra to ry of the U n i v e r s i t y o f B r i t i sh C o l u m b i a is used to generate the air f l o w requi red in th is expe r imen t . The d i m e n s i o n s o f the tes t s e c t i o n is 45.7 c m x 45.7 c m and 190.5 c m l ong . T w o pa i rs of w i n d o w s are on the tunnel w a l l s of the test s e c t i o n . The pair o f w i n d o w s at the ups t ream is r e m o v e d fo r the i ns ta l l a t i on of the appara tus . The tunnel 's con t rac t i on ra t io be tween the intake and the 15 16 tes t s e c t i o n is high and the e m p t y tunnel turbulence in tens i t y is about 0.3%. The air f l o w v e l o c i t y ranges f r o m 0 m / s e c to 20 m / s e c and the mean c o n v e c t i o n v e l o c i t y p ro f i l e is s l i gh t l y higher near the c e i l i n g of the tunnel . H o w e v e r , the e f f e c t of th is s l ight v e l o c i t y va r i a t i on is a s s u m e d to be neg l i g ib le s i n c e the m e a s u r e m e n t s are made o n l y near the center of the c r o s s s e c t i o n . 3.1.2 THE R O T A T I N G C Y L I N D E R C A S C A D E A N D THE F L A T P L A T E The ro ta t ing cy l inder c a s c a d e is c o m p o s e d o f t w o c i rcu lar end p la tes o f 43.2 cm in d iamete r and a lum inum cy l i nd r i ca l rods o f 0.635 c m in d iamete r . The rods are ar ranged at a d i s tance of 28.4 c m apart f r o m each o ther , P la te 2. The c a s c a d e is d r i ven by a va r iab le s p e e d direct current m o t o r w i th a s p e e d reduc t ion sp rocke t and cha in d r i ve . The c a s c a d e ro ta t iona l s p e e d ranges f r o m 0 to 1000 r p m . The cascade is c o n n e c t e d to the d r i v ing s y s t e m at one s ide o f the tunne l , P la te 3. The other s i de of the c a s c a d e is s u p p o r t e d by three externa l bea r ings . The a l ignment of the c a s c a d e and the sp rocke t is done by ad jus t ing the d i s tance of the bear ings f r o m the center of r o t a t i o n , P la te 4. On the s a m e s ide as the externa l bea r i ngs , there is a p l a t f o r m fo r moun t ing the t rave rse m e c h a n i s m and and the f la t p la te ins ide the ro ta t ing c a s c a d e . The p la te u s e d is a luminum of 0.635 c m th ick and has a chord length o f 33 c m w i th a round lead ing edge . A t r ipp ing w i re is g lued on to the p la te near the l ead ing edge to ensure the boundary layer is turbulent . The plate is r e i n f o r c e d by a channe l under it to e l im ina te the v i b ra t i on induced b y the ro ta t i on o f the mo to r and the w i n d 17 l o a d . H o l e s are cut f r o m the f l anges of the channel to reduce its o b s t r u c t i o n in the air f l o w , P la te 5. 3.1.3 T R A V E R S E M E C H A N I S M A m e c h a n i s m is used to t raverse the hot w i re p robe fo r f lu id v e l o c i t y measu remen t . The ver t i ca l d i s p l a c e m e n t o f the probe is measu red by a M i t u t o y o 197-201 m i c r o m e t e r head wi th" a n o n - r o t a t i n g sp ind le . The m i c rome te r has a 2 inches t rave l capac i t y w i t h a s m a l l e s t d i v i s i o n o f 0.0002 inches on the t h i m b l e . A n e x t e n s i o n is connec ted to the end of the sp ind le to h o l d the p robe ins ide the tunne l . P recau t i on has been taken to p lace the p robe far a w a y f r o m both end p la tes s o that their e f fec t on the boundary layer can be n e g l e c t e d . The hor izon ta l m o v e m e n t is ob ta ined by s l i d i ng the m i c r o m e t e r a l ong a ho r i zon ta l rec tangular bar moun ted on the p l a t f o r m . A c lear ac r y l i c w i n d o w box is used to c o v e r the ent i re a s s e m b l y dur ing the exper imen t . The a c c e s s to the m i c r o m e t e r ins ide the box is by a s t e m w h i c h pro t rudes the top of the w i n d o w box f r o m the m i c r o m e t e r . 3.2 INSTRUMENTATION 3.2.1 B O U N D A R Y L A Y E R M E A S U R E M E N T - HOT WIRE A N E M O M E T R Y L o w f r e q u e n c y r e s p o n s e d e v i c e s , such as p i t o t - s t a t i c t u b e s , is not su i tab le fo r measur ing high f requency p e r i o d i c v e l o c i t y and are not 18 used in th is exper iment . H o w e v e r , hot w i re a n e m o m e t r y , w h i c h is a s tandard techn ique to measure uns teady f lu id f l o w , is used here because of its high f requency r e s p o n s e . A cons tan t tempera tu re a n e m o m e t e r is used in th is expe r imen t . The schema t i c c i rcu i t is as s h o w n in Fig.3.3. The vo l t age at C , E c , " i nc reases as the air s p e e d inc reases and dec reases as air speed d e c r e a s e s . The v e l o c i t y - v o l t a g e re la t ionsh ip is acco rd i ng to the f o l l o w i n g e x p r e s s i o n . E C J = A 2 + B U n 3.1 w i th A , B and n are cons tan t s to be ob ta ined f r o m the ca l i b ra t i on of the hot w i r e . The f r equency r e s p o n s e of a hot w i re s y s t e m can be inc reased by inc reas ing the overheat rat io wh ich is g i ven by the e x p r e s s i o n , { R w h e a t B d - R w a m b i e n t ) / R w a m b i e n t . H o w e v e r , if the s y s t e m is t oo s e n s i t i v e , i.e. the overheat ra t io is too h igh, it t ends to burn w i r e s . The hot w i re used in th is exper iment is a D i s a 55P15 boundary layer p robe w i th a 5 um d i amemte r and 1.25 m m long p la t inum p la ted tungsten w i r e , P la te 6. The br idge is D i s a 56C16 general pu rpose br idge. The f r equency response of the hot w i re br idge w i thou t the w i re is about 100 kHz wh ich is a s s u m e d to be high enough fo r the pe r i od i c uns teady f requency , wh i ch is in the order o f 200 Hz in this exper imen t . The spat ia l r e s o l u t i o n based on the length o f the w i re and f ree s t r eam v e l o c i t y g i v e s the c u t - o f f f r equency at about 8000 Hz . The c o n v e r s i o n f r o m v o l t a g e to 19 v e l o c i t y of equa t ion 3.1 is done in a data acqu i s i t i on p r o g r a m . 3.2.1.1 Hot W i r e Ca l i b ra t i on The hot w i re ca l i b ra t i on is done in the empty w i n d tunnel b e f o r e the rest o f the apparatus is i ns ta l l ed . The mean air f l o w v e l o c i t y is measu red by a p i t o t - s t a t i c tube. The cons tan t A in equa t ion 3.1 is the vo l t age output at zero v e l o c i t y . T w o se ts o f v e l o c i t i e s and v o l t a g e s f r o m the upper range and l o w e r range of v e l o c i t i e s are requ i red to de te rmine the cons tan t s B and n in the v o l t a g e - v e l o c i t y ca l i b ra t i on e x p r e s s i o n , equa t ion 3.1. The ca l i b ra t i on curve is c o m p a r e d w i t h the v e l o c i t y - v o l t a g e exper imen ta l data po in t s in Fig.3.4. The ambient temperature va r ia t ion in the labora to ry is a s s u m e d to be t oo s m a l l to have any e f f ec t on the hot w i re c a l i b r a t i o n . 3.2.1.2 C o r r e c t i o n fo r W a l l P r o x i m i t y E f f e c t B e c a u s e the heat t rans fe r rate o f the hot w i re i nc reases when it is c l o s e to a s o l i d boundary , the hot w i re s igna l must be c o r r e c t e d w h e n it is used in boundary layer measu remen t . The s t i l l - a i r c o r r e c t i o n techn ique used by H o d s o n (1982) is used here. The hot w i r e output v o l t a g e s in s t i l l air at va r ious d i s t a n c e s f r o m the s u r f a c e are m e a s u r e d . The ca l i b ra t i on curve is then m o d i f i e d to g ive the f o l l o w i n g e q u a t i o n ; E c ^ A ' - ( E 0 ' ( y ) - E 0 ' ( « « . ) ) = B U n 3 2 where E D ( y ) is the hot w i r e vo l t age output at the ve r t i ca l d i s t a n c e , y , f r o m the su r face in s t i l l a ir . S i n c e A = E 0 ( ° ° ) , equat ion 3.2 can be 20 s i m p l i f i e d to E c > - E 0 ' ( y ) = B U n 3.3 E Q ( y ) is measu red and a cub ic sp l i ne f i t t ed b e f o r e any boundary layer m e a s u r e m e n t s . The c o e f f i c i e n t s f r o m the cub ic sp l ine curve f i t t i ng of the data are s t o r e d in the data acqu i s i t i on p rog ram for co r rec t i ng the hot w i re s i g n a l s . The va r ia t ion o f the s t i l l air hot w i r e output , E Q , w i th the ve r t i ca l d i s t a n c e , y , f r o m the a luminum p la te su r face is s h o w n in Fig.3.5. The data s h o w that the s o l i d su r face a f f e c t s the hot w i re output o n l y when the w i r e is l ess than 3 m m f r o m the plate s u r f a c e . 3.2.1.3 De te rm ina t i on of the Ve r t i ca l D i s tance of Hot W i re f r o m the Flat P la te The d i s tance be tween the hot w i re and the f lat p late is measu red by the M i t u t o y o m i c r o m e t e r . The zero o f f s e t va lue is the read ing on the m i c r o m e t e r head when the hot w i re p rongs touch the p la te s u r f a c e . The con tac t o f the p rongs and the a luminum f lat p la te is de tec ted by an o h m m e t e r w h i c h is c o n n e c t e d be tween the p rongs and the s u r f a c e , as in Fig.3.6. The res i s tance d rops f r o m in f in i te to s o m e f in i te va lue w h e n the p rongs touch the p la te . The t w o p rongs are a l igned on the s a m e p lane by e y e . The zero p o s i t i o n read ing is sub t rac ted f r o m the m i c r o m e t e r read ing at each p o s i t i o n to ob ta in the true d i s tance f r o m the p la te . 3.2.2 P H A S E R E F E R E N C E - H A L L E F F E C T S W I T C H B e c a u s e of the con t i nuous ro ta t ion of the c a s c a d e , a re fe rence 21 point on the c a s c a d e is requ i red to ind ica te the s tar t ing point o f each revo lu t i on fo r data acqu i s i t i on pu rpose . A U G N - 3 0 1 9 7 Hal l E f f e c t D ig i ta l s w i t c h is used in th is a p p l i c a t i o n . Th is dev i ce is a 3 - p i n s ing le output in tegrated c i rcu i t . The f i rs t pin is the input p in to w h i c h 9 V D.C. is a p p l i e d . The s e c o n d p in is the g round . The th i rd p in is the output w h i c h is n o r m a l l y high but d rops to zero w h e n it s e n s e s a magne t i c f i e l d . The vo l t age drop is near ly i ns tan taneous so that the r e s p o n s e t ime is a s s u m e d to be z e r o . The s w i t c h is moun ted f a c i n g the s p r o c k e t on the c a s c a d e . A p i ece o f magnet is g lued on the s p r o c k e t . S i n c e the c a s c a d e and the sp rocke t are ro ta t ing at the s a m e angular s p e e d , the s w i t c h vo l t age d rops at the s a m e instant in each revo lu t i on when the magnet p a s s e s in f ront of the s w i t c h . The s w i t c h output vo l t age is f e d into the c o m p u t e r to be s a m p l e d by a data acqu i s i t i on p rog ram to s igna l the start o f a data a c q u i s i t i o n c y c l e . 3.3 SIGNAL CONDITIONING AND DATA ACQUISITION A high s p e e d data acqu i s i t i on s y s t e m , N E F F , is used to f i r s t d ig i t i ze the input s i g n a l s , and then send the d i g i t i zed s i gna l s to a PDP 11 compu te r fo r fur ther p r o c e s s i n g . The input s i g n a l s are f r o m the hot w i r e a n e m o m e t e r and the Hal l E f f e c t s w i t c h through t w o channe ls . The m a x i m u m s a m p l e s i ze is 4096 and the m a x i m u m s a m p l i n g f r equency is 20kHz us ing m u l t i - c h a n n e l s . B e c a u s e the N E F F ' s m a x i m u m range of input v o l t a g e is 1 V , a D i s a 55D25 s igna l cond i t i one r is used to subtract an o f f s e t v o l t a g e , 1.3 V , f r o m the hot w i re output s igna l and then l inear ly 22 a m p l i f y the d i f f e r e n c e . The m i n i m u m and the m a x i m u m hot w i r e output s igna ls are m o n i t o r e d by us ing a Tek t ron i x 434 s to rage o s c i l l o c o p e throughout the exper imen t to make sure the s igna l is w i th in the vo l tage range. 3.4 AN ENSEMBLE-AVERAGING TECHNIQUE The NEFF s y s t e m enab les one con t inuous reco rd of hot w i re output in a pe r iod of t ime to be read- by a data acqu i s i t i on p rogram at a t i m e . H o w e v e r , these r e c o r d s are the sum of the random turbu lence f l uc tua t ion and the pe r i od i c u n s t e a d i n e s s , the e f f ec t o f the pe r i od i c uns tead iness on the boundary layer cannot be seen c l e a r l y un less the r e c o r d s are e n s e m b l e - a v e r a g e d . The e n s e m b l e - a v e r a g i n g techn ique is used to supp ress the n o n - p e r i o d i c f l uc tua t ion in the data to ob ta in the ' c l e a n ' pe r i od i c uns tead iness in the boundary layer . The e n s e m b l e - a v e r a g i n g techn ique requ i res a number o f hot w i re reco rds wh i ch are taken at the same l oca t i on and s a m p l e d w i th in the s a m e per iod o f t i m e in each revo lu t i on as s h o w n in Fig.3.7.a. S i n c e the t i m e r - a v e r a g e o f the random c o m p o n e n t of a f luc tua t ing v e l o c i t y approachs z e r o , the random v e l o c i t y f l uc tua t ion can be e l im ina ted f r o m the pe r i od i c u n s t e a d i n e s s by ca l cu la t i ng the mean v e l o c i t y of a large number of r e c o r d s taken at the s a m e instant of each c y c l e . The e n s e m b l e - a v e r a g e d v e l o c i t y as a func t i on o f t ime is then g i ven b y ; N U ( t ) = ( Z U . ( t ) ) / N 3.4 i 23 The number of r e c o r d s , N , is c h o s e n s o that a further i nc rease in N w i l l not fur ther reduce the random f l uc tua t i on in the e n s e m b l e ave raged r e c o r d s . S i m i l a r to the e n s e m b l e a v e r a g e d v e l o c i t y , a r eco rd of r a n d o m turbu lence in tens i ty as a f unc t i on o f t i m e at the s a m e l o c a t i o n can a l s o be ob ta ined af ter the e n s e m b l e - a v e r a g e d v e l o c i t y has been o b t a i n e d . T u ( t ) = ( ( Z ( U i ( t ) - U ( t ) ) 2 ) / N ) ° - 5 / U ( t ) 3 . 5 i U ^ ( t ) - U ( t ) is the random v e l o c i t y f l uc tua t i on in the f l o w at t i m e , t and is des igna ted as u in Fig.3.7.b. B e c a u s e each e n s e m b l e - a v e r a g e d w a k e d is tu rbance is iden t i ca l and p e r i o d i c , each instant w i t h i n one c y c l e can be re fe r red to as a phase ang le w h i c h va r ies f r o m 0 ° to 3 6 0 ° . E n s e m b l e - a v e r a g e d v e l o c i t y p r o f i l e s and tu rbu lence in tens i ty p r o f i l e s at a par t icu lar phase angle can be o b t a i n e d b y s tack ing a number o f e n s e m b l e - a v e r a g e d r e c o r d s , wh i ch are ob ta i ned f r o m d i f fe ren t ve r t i ca l d i s t a n c e s f r o m a s u r f a c e , and " s l i c i n g " at the requ i red phase ang le . The a b o v e s igna l p r o c e s s i n g is car r ied out in a data acqu i s i t i on p rog ram as d e s c r i b e d in the f o l l o w i n g s e c t i o n . 3.5 DATA ACQUISITION PROGRAM The data acqu i s i t i on p rog ram is s h o w n by the f l o w chart s h o w n in Fig.3.8. A tes t subrout ine is i nc luded t o check if bo th channe ls are w o r k i n g p rope r l y by s a m p l i n g t w o k n o w n s igna l s through the t w o channe ls . The p rogram c o n v e r t s d i g i t i zed hot w i r e output back to v e l o c i t y 24 unit. In order to do the w a l l p r o x i m i t y c o r r e c t i o n , E Q ( y ) in equat ion 3.4 is ca l cu la ted f r o m the cub ic sp l i ne f i t t ed curve for w a l l p r o x i m i t y e f f ec t w h i c h has been d i s c u s s e d in s e c t i o n 3.2.1.b. Because of the huge amount of data requ i red , the t ime to ca r ry out the c o n v e r s i o n must be cut as short as p o s s i b l e dur ing the expe r imen t . Instead of repeat ing the above c o n v e r s i o n for each d ig i t i zed input po in t , the co r respond ing v e l o c i t y is ob ta ined f r o m a " l o o k - u p " tab le w h i c h c o n s i s t s of the p reca l cu la ted v e l o c i t y va lues for d ig i t i zed v o l t a g e b e t w e e n 0 and 1 V. Th is p rocedure can s p e e d up the c o n v e r s i o n by a f a c t o r o f 10. M o r e o v e r , in order to do the e n s e m b l e - a v e r a g i n g p rocedure as d i s c u s s e d in s e c t i o n 3.4, al l r eco rds have to start at the s a m e instant dur ing each r e v o l u t i o n . The r e c o r d s are s y n c h r o n i z e d by s c a n n i n g the Hal l E f f ec t s w i t c h output f o r the jump f r o m the l o w to high s ta te . Each hot w i re reco rd is added to the p r e v i o u s o n e , s o on l y the e n s e m b l e - a v e r a g e d v e l o c i t y co rd is left in the data array at the end . The e n s e m b l e - a v e r a g e d turbu lence in tens i t y reco rd is ob ta ined by repeat ing the data a c q u i s i t i o n c y c l e . But i ns tead of add ing to the p rev ious hot w i re r e c o r d , the e n s e m b l e - a v e r a g e d v e l o c i t y ob ta ined above is sub t rac ted f r o m the n e w hot w i re r eco rd be fo re add ing , as s h o w n in equat ion 3.5. The e n s e m b l e - a v e r a g e d r e c o r d s are then s t o r e d in f l o p p y d i s k e t t e s . Chapter 4 EXPERIMENTAL RESULTS AND DISCUSSION TIME-AVERAGED MEASUREMENTS OF STEADY AND UNSTEADY BOUNDARY LAYERS 4.1.1 E X P E R I M E N T A L D E T A I L S Dur ing the expe r imen t , f ree s t ream v e l o c i t i e s of 8.4 m / s and 11.7 m / s we re used to p roduce boundary laye rs on a f la t p la te . The und is tu rbed and d i s tu rbed boundary layer p r o f i l e s w e r e measured at 100 m m and 200 m m d o w n s t r e a m f r o m the lead ing edge of the p la te . The R e y n o l d s n u m b e r s , R e x , c o r r e s p o n d i n g to the above c o n d i t i o n s we re a p p r o x i m a t e l y at 0.5x10 5 , 0 .7x10 J , I .OxlO 5 and 1.4x10 5. The uns teady boundary layer w a s d is tu rbed by the c a s c a d e cy l i nde rs at a f r equency of 50 Hz (i.e. tangent ia l v e l o c i t y , U g , o f the cy l i nde r is at 14 m/s . ) . The f l o w c o e f f i c i e n t , C ^ ( = 0 o o / U g ) , w a s at 0.57 and 0.84 r e s p e c t i v e l y fo r the t w o cases of f ree s t ream v e l o c i t i e s . The hot w i re p robe w a s t r ave rsed ac ross the boundary layer to obta in the t i m e - a v e r a g e d v e l o c i t y and turbu lence in tens i ty p r o f i l e s . The t i m e - a v e r a g e d hot w i r e vo l t age output w a s ob ta ined by us ing a D i s a 56N22 M e a n Va lue Uni t w i th an integral t ime o f 10 s e c o n d s . The t i m e - a v e r a g e d vo l t age w a s then c o n v e r t e d to v e l o c i t y uni ts as d e s c r i b e d in s e c t i o n 3.2.1. The t i m e - a v e r a g e d turbu lence in tens i ty w a s ob ta ined by f eed ing hot w i re output vo l t age to a D i s a 55D35 R M S v o l t m e t e r . S i n c e the hot w i re a n e m o m e t e r w a s not l i nea r i zed , the vo l tme te r read ing f r o m the R M S meter w a s ca l ib ra ted acco rd ing 25 26 to append ix A to ob ta in the true tu rbu lence in tens i t y . • 4.1.2 E X P E R I M E N T A L R E S U L T S The n o n - d i m e n s i o n a l t i m e - a v e r a g e d v e l o c i t y p r o f i l e s of d i s tu rbed and und is tu rbed boundary layers are s h o w n in Figs.4.1.a and 4.1.b. W i thou t any d i s tu rbance , the boundary layer is not qui te turbulent even w i t h the t r ipp ing w i r e . Th is is because of the R e y n o l d s numbers f o r the short p la te . For the c a s e in wh ich the f ree s t ream is d i s tu rbed by the m o t i o n of the c a s c a d e c y l i n d e r s , the turbu lence l eve l is i n t e n s i f i e d . The d is tu rbed bounda ry laye rs are c l o s e r but not qui te equal to the fu l l y d e v e l o p e d turbulent boundary layer wh i ch can be rep resen ted by the 1 /7- th P o w e r L a w as d i s c u s s e d in s e c t i o n 2.1. Figs.4.2 s h o w s the turbu lence in tens i t y p r o f i l e s of the und is tu rbed boundary layer . In F i g . 4 .3, the y d i s tance is n o n d i m e n s i o n a l i z e d by the t i m e - a v e r a g e d boundary layer t h i c k n e s s . The ex t raord inar i l y high turbulence i n tens i t y near the su r f ace requi res mo re ca re fu l c o n s i d e r a t i o n because the h ighest tu rbu lence in tens i ty is usua l l y about 0.4 w i th in a short d i s tance f r o m a t r ipp ing w i re and the h ighest read ing ob ta ined in th is exper iment is around 0.7. The d i m e n s i o n a l p lo t s h o w s that b e c a u s e o f the m ix i ng o f the h igh ly turbulent f l u id pa r t i c l es in the inner boundary layer and the l o w turbulent f lu id in the f ree s t r e a m , the tu rbu lence in tens i t y i nc reases in the outer layer and d e c r e a s e s in the inner layer as the f l u i d m o v e s d o w n s t r e a m . The n o n d i m e n s i o n a l p lo t s h o w s that a l though the high turbu lence in tens i t y f l u i d d i f f u s e s in to the f ree 27 s t r e a m , the rate is not as fas t as the g row th of the boundary layer t h i c k n e s s . C o n s e q u e n t l y , the turbu lence i n t e n s i t y - a t any po in t on a n o n d i m e n s i o n a l p ro f i l e is l o w e r than that of a c o r r e s p o n d i n g point on a p r o f i l e of l ower R e y n o l d s number . The d i s p l a c e m e n t and m o m e n t u m t h i c k n e s s , 5 and 8, are de te rm ined a c c o r d i n g to append ix B. Fig.4.4 s h o w s the inc rease of the d i s p l a c e m e n t and m o m e n t u m th i ckness of the d i s tu rbed and und is tu rbed boundary l aye rs . The resu l ts at 100 m m and 200 m m are j o i ned by stra ight l ines to s h o w the genera l t rend. The integral pa ramete rs o f the und is tu rbed case start at a s m a l l e r va lue when c o m p a r e d to the d is tu rbed boundary layer but they g r o w a lot f as te r . A higher m o m e n t u m t h i c k n e s s i nd i ca tes a higher m o m e n t u m l o s s in the boundary layer . A n e s t i m a t i o n o f su r face shear s t ress by us ing C lause r p lo t s has been a t t emp ted . Th is techn ique is d i s c u s s e d in append ix C . H o w e v e r , as s h o w n in Fig.4.5a and 4.5b, the l inear reg ion is not apparent in the d i s tu rbed p r o f i l e s because of the l o w R e y n o l d s numbers used in th is exper imen t . T h e r e f o r e , it is c o n c l u d e d that the boundary layer p r o f i l e s are not f u l l y turbulent and are d i f f e ren t f r o m "the L a w of the W a l l " . A s a resu l t , the e s t i m a t e d va lues of the su r face shear s t r e s s e s are not accura te and are not s h o w n . 28 ENSEMBLE-AVERAGED MEASUREMENTS OF DISTURBED FREE STREAM AND BOUNDARY LAYER 4.2.1 E X P E R I M E N T A L D E T A I L S The f lat p late w a s r e m o v e d f r o m the exper imen ta l r ig fo r tak ing the uns teady f ree s t r e a m m e a s u r e m e n t s . The c o n d i t i o n s we re as d i s c u s s e d in s e c t i o n 4.1.1. The x Q / d ra t ios (where x Q is the d o w n s t r e a m d is tance f r o m the bar and d is the cy l i nde r d iamete r ) w e r e 25 and 4 1 , c o r r e s p o n d i n g to the 2 p o s i t i o n s , 100mm and 2 0 0 m m d o w n s t r e a m o f the lead ing edge . The R e y n o l d s number based on the cy l i nde r d iamete r , R e d , w a s 3600 and 5000 r e s p e c t i v e l y . The v e l o c i t y s i gna l s w e r e e n s e m b l e - a v e r a g e d as the bars p a s s e d in f ront o f a s ta t i ona ry hot w i re p robe us ing the techn ique in s e c t i o n 3.4. The w a k e f r o m a s ta t i ona ry c y l i n d e r w a s a l so m e a s u r e d to c o m p a r e w i th the m o v i n g w a k e w h i c h w a s c rea ted by the ro ta t ing cy l i nde r c a s c a d e . The exper imen ta l c o n d i t i o n s fo r uns teady boundary layer m e a s u r e m e n t s were a l so as d e s c r i b e d in s e c t i o n 4.1.1. The hot w i re p robe w a s t rave rsed a c r o s s the boundary layers and the s i gna l s w e r e e n s e m b l e - a v e r a g e d to g ive the average v e l o c i t y and tu rbu lence in tens i t y at each instant in a c y c l e . 175 reco rds w e r e used for each e n s e m b l e - a v e r a g e d reco rd in both the f ree s t ream and boundary layer m e a s u r e m e n t s . The p h a s e - a v e r a g e d v e l o c i t y and turbu lence in tens i t y p r o f i l e s at the requ i red phase angle we re then ob ta ined as d i s c u s s e d in s e c t i o n 3.4 29 4.2.2 E X P E R I M E N T A L R E S U L T S 4.2.2.1 U n s t e a d y Wake M e a s u r e m e n t s and C o m p a r i s o n w i th S t e a d y W a k e M e a s u r e m e n t s Fig.4.6 s h o w s the e n s e m b l e - a v e r a g e d v e l o c i t y r eco rd fo r the m o v i n g w a k e s wh i ch w e r e caused by the c y l i n d e r s p a s s i n g in f ront of the s ta t ionary hot w i re p robe . The random turbu lence is s u p p r e s s e d and the pe r i od i c v e l o c i t y d e f e c t s c rea ted by the m o t i o n of the cy l i nde rs are c lea r l y s h o w n . S i n c e the c a s c a d e c y l i n d e r s t rave rse in f ront of the hot w i re p robe , each v e l o c i t y de fec t is a w a k e f r o m a pass ing cy l i nde r . S i m i l a r to s ta t i ona ry w a k e s , the m o v i n g w a k e s are s y m m e t r i c a l as w e l l . Fig.4.7 s h o w s the n o n d i m e n s i o n a l s t e a d y and t rave rs ing w a k e p r o f i l e s . The exper imenta l resu l ts are c o m p a r e d w i t h the theore t i ca l w a k e p ro f i l e as d e s c r i b e d by Reichardt (Sch l i ch t ing 1979). The v e l o c i t y ax is is n o n d i m e n s i o n a l i z e d as s h o w n on the graph. The w id th of the s teady w a k e s are n o n d i m e n s i o n a l i z e d by their h a l f - w i d t h s , y ^ , and t r , e t ravers ing w a k e s are by the h a l f - t i m e s , t 1 / 2 ' w m c n ' s t n e t ' m e f ° r t n e w a k e s to m o v e f r o m the m i n i m u m v e l o c i t y to half w a y b e t w e e n the m a x i m u m and m i n i m u m v e l o c i t y . S e l f - s i m i l a r i t y of the w a k e s is not e x p e c t e d because of the s m a l l x Q / d ra t io . H o w e v e r , the n o n d i m e n s i o n a l w a k e p r o f i l e s are p lo t ted to s h o w that the t ravers ing w a k e s do r esemb le the s t e a d y w a k e s . Th is s h o w s that the cy l i nde r ' s tangent ia l v e l o c i t y de te rm ines h o w fas t the w a k e s t raverse a c r o s s the hot w i r e probe but has ve ry l i t t le e f f e c t on the shape of the w a k e . Fig.4.8 s h o w s the turbu lence in tens i ty in both the n o n d i m e n s i o n a l t rave rs ing and s teady w a k e s . The turbu lence in tens i ty 30 d i s t r i bu t ions in the t ravers ing w a k e s are s i m i l a r ' H p those of s t e a d y w a k e s . S i m i l a r to the v e l o c i t y d i s t r i bu t ions of the t rave rs ing w a k e s , there is an unexpec ted inc rease of tu rbu lence be fo re each ma in w a k e . The turbu lence in tens i ty of the t rave rs ing w a k e s are about 25% higher than the s teady o n e s . 4.2.2.2 U n s t e a d y Boundary Layer M e a s u r e m e n t s Fig.4.9 s h o w s the e n s e m b l e - a v e r a g e d v e l o c i t y r eco rds at d i f fe ren t p o s i t i o n s a c r o s s a d is tu rbed boundary layer . The w a k e s are l ess s y m m e t r i c a l than the m o v i n g w a k e s in the f ree s t r e a m . The v e l o c i t y d e c r e a s e s soone r and more ' s l o w l y than it i n c r e a s e s . In the inner part o f the boundary layer , the w a k e s arr ive at the measu r i ng point later than those in the outer layer due to the l o w e r c o n v e c t i o n v e l o c i t y in the inner laye r . Th is de lay is quan t i f i ed in te rms of degrees o f phase lag re la t i ve to the f ree s t r e a m d i s tu rbance . A fu l l 360° is de f i ned as the pe r i od be tween t w o s u c c e s s i v e w a k e s , as s h o w n in Fig.4.9. Fig.4.10 s h o w s the e n s e m b l e - a v e r a g e d turbu lence i n tens i t y r e c o r d s at the s a m e p o s i t i o n s at w h i c h the v e l o c i t y reco rds w e r e o b t a i n e d . The turbu lence leve l be tween the wake d i s tu rbances is s im i l a r to that of s t eady turbulent boundary layers as s h o w n in Figs.4.2 and 4.3. When a w a k e p a s s e s the measur ing p r o b e , the tu rbu lence in tens i ty r i ses qu i ck l y . A s the p robe is m o v e d c l o s e r to the s u r f a c e , the rms turbu lence in tens i t y b e t w e e n w a k e s i n c r e a s e s , and f i n a l l y , the w a k e d is tu rbance d i sappea rs w i th in the o v e r a l l tu rbu lence in tens i t y . The phase lag o f the w a k e s in the inner layer can a l s o be seen f r o m these turbu lence in tens i t y r e c o r d s . 31 Figs.4.11 and 4.12 s h o w the p h a s e - a v e r a g e d v e l o c i t y and tu rbu lence p ro f i l e s at eve ry 60° phase ang le in a c y c l e f r o m 0 ° to 360° as des igna ted in Fig.4.9. The p r o f i l e s are ob ta ined b y read ing o f f v e l o c i t y va lues f r o m each reco rd at the requi red phase ang le . The p r o f i l e s are s h o w n in a se r i es of ove r l app ing t r anspa renc ies so that the change in p r o f i l e s at each 6 0 ° phase angle can be c lea r l y s e e n . A t 0 ° , the v e l o c i t y of the p ro f i l e is the l owes t b e c a u s e the c y l i n d e r has just t r ave rsed in f ront of the boundary layer . A t 6 0 ° , the w a k e is ou ts ide the boundary layer and the f ree s t ream v e l o c i t y . p ro f i le has i n c r e a s e d to i ts m a x i m u m va lue . H o w e v e r , the inner layer cannot r espond as fas t as the outer layer , and the inner layer v e l o c i t y d o e s not reach the m a x i m u m value unti l 1 2 0 ° . Th is is the c o n s e q u e n c e of the phase lag cha rac te r i s t i c s o f w a k e d is tu rbance in the inner layer . The phase lag p h e n o m e n o n has a l ready been o b s e r v e d f r o m Fig.4.9 and 4.10. For the s a m e r e a s o n , the outer layer v e l o c i t y s ta r ts to dec rease soone r than the inner layer at 180° as another w a k e s tar ts to m o v e through the boundary layer . Th is reduc t ion in the outer layer p ro f i l e con t i nues to 240° phase ang le , and then at 3 0 0 ° , the inner layer a l so s ta r ts to s h o w a reduced v e l o c i t y . A t 3 6 0 ° , the v e l o c i t y p ro f i l e is aga in fu l l y re tarded in the m idd le of the w a k e , and is e s s e n t i a l l y the s a m e as the 0 ° p r o f i l e . In Fig.4.12, the p h a s e - a v e r a g e d turbu lence in tens i t y p r o f i l e s are s h o w n at i nc remen ts of 6 0 ° phase ang les on a se r i es of t r anspa renc ies . A t 0 ° , the turbu lence leve l is high and e v e n l y d i s t r i bu ted s ince the w a k e has just m o v e d a c r o s s the boundary l aye r . A s the w a k e has m o v e d a w a y , the in tens i t y d e c a y s s l o w l y . In 32 b e t w e e n w a k e s , at 180° and 2 4 0 ° , the p h a s e - a v e r a g e d turbu lence in tens i t y p r o f i l e s are s i m i l a r to t hose of the und is tu rbed boundary l aye rs as s h o w n in Fig.4.2 and 4.3. The tu rbu lence in tens i ty i nc reases again as the next w a k e is approach ing at 3 0 0 ° . A t 3 6 0 ° , the in tens i t y p ro f i l e is aga in f u l l y i nc reased to the p r o f i l e at 0 ° . Fig.4.13 is p lo t ted to s h o w h o w the m a x i m u m v e l o c i t y d e f e c t s of w a k e s change w i t h ve r t i ca l d i s tances f r o m the p la te . The de fec t amp l i tude is the d i f f e r e n c e be tween the e n s e m b l e - a v e r a g e d m a x i m u m and m i n i m u m v e l o c i t y in a w a k e . The resul t s h o w s , as e x p e c t e d , that the amp l i t ude ra t ios are c l o s e to cons tan t in the f ree s t ream and zero at the p late s u r f a c e . There is a m a x i m u m amp l i tude ra t io in the l ower half o f the boundary layer . B e c a u s e of the d e c a y of the w a k e s , the amp l i tude ra t ios dec rease w i t h x Q / d . The phase lag p h e n o m e n o n wh i ch has been o b s e r v e d p r e v i o u s l y in Figs.4.9 and 4.10 is p lo t ted in Fig.4.14 to s h o w h o w the phase lag changes w i t h d i s t a n c e s f r o m the p la te . The phase lag m e a s u r e d is re la t i ve to the reco rd near the f ree s t r e a m . The phase lag s ta r ts to occur at about half w a y d o w n the boundary layer and i nc reases t o w a r d the s u r f a c e . The phase lag p r o f i l e s a l s o appear to i nc rease w i t h R e Y , i.e. the phase lag is higher as the boundary layer p r o f i l e is measu red far ther a w a y f r o m the lead ing edge . 4.3 DISCUSSION E v e n in th is s i m p l i f i e d m o d e l , the f l o w pat tern is c o m p l i c a t e d due to the p e r i o d i c d i s tu rbance o f the w a k e s genera ted by the m o v i n g ba rs . The 33 f ree s t ream measu remen ts s h o w that the m o v i n g w a k e is s im i l a r to the s teady w a k e . The f ree s t ream v e l o c i t y at any po in t i s , t he re fo re , a fun t ion of t i m e , t, beacuse o f the pe r i od i c nature of the w a k e and the d i s tance f r o m the cy l i nde r , x because o f the decay o f the max imum w a k e amp l i t ude w i t h d i s tance f r o m the bar. C o n s i d e r the o b s e r v a t i o n of a w a k e genera ted by a par t icu lar bar s tar ts at t i m e , t = 0 , when the bar is i m m e d i a t e l y a b o v e the p la te . The w a k e is ca r r ied d o w n s t r e a m by the ma in f l o w . H o w e v e r , as the w a k e enters the boundary layer , the v e l o c i t y s l o w s d o w n o n ' the p la te su r face and the w a k e is d i s to r t ed by the v i s c o u s shear s t r e s s . A short t ime later , A t , the bar has m o v e d a s m a l l d i s t ance far ther a w a y f r o m the plate and the w a k e at th is po in t is car r ied d o w n s t r e a m at a higher v e l o c i t y s i nce the f l u id pa r t i c les are far ther a w a y f r o m the su r f ace . A s a resu l t , the w a k e d is tu rbance in the inner layer lags beh ind the outer layer d i s tu rbance , as s h o w n in Fig.4.9 and the phase lag re la t i onsh ip w i th d i s tance f r o m the plate is as s h o w n in F ig.4.13. Th is phase lag behav io r has a l so been o b s e r v e d by H o d s o n (1982), Fig.2.7. E v e n at l o w R e y n o l d s numbers , the p resence of the pe r iod i c w a k e s make the t i m e - a v e r a g e d boundary layer p r o f i l e s c l o s e r to a fu l l y d e v e l o p e d turbulent boundary layer because of the pe r i od i c inc rease of the tu rbu lence in tens i t y . The integral boundary layer parameters agree w i t h the data ob ta ined by Evans (1974) and H o d s o n (1982) s h o w i n g the m a x i m u m l o s s in the w a k e d is tu rbed boundary layer to be higher than in the und is tu rbed c a s e . S i n c e the w a k e s are v e l o c i t y d e f e c t s , their pe r i od i c ex i s t ence in the boundary layer d e c r e a s e s the t i m e - a v e r a g e d v e l o c i t y and th i ckens the boundary laye r , e s p e c i a l l y at the ear l y part o f the boundary layer whe re the w a k e has high v e l o c i t y de fec t . 34 A t cons tan t f ree s t r e a m v e l o c i t y and cy l i nde r d i ame te r , the w a k e amp l i tude ra t io is a f u n c t i o n o f x Q / d o n l y . The amp l i t ude ra t io in the f ree s t ream at each x Q is cons tan t wh i le on the s u r f a c e , the amp l i tude ra t io is zero because the v e l o c i t y is zero due to the n o - s l i p c o n d i t i o n . W i th i n the boundary l aye r , the in terac t ion b e t w e e n a w a k e f r o m a s ta t i ona ry bar and the bounda ry layer can be d e s c r i b e d by the theore t i ca l m o d e l as s h o w n in Fig.2.10. A s measured by T s i o l a k i s et al (1983), when the cen te r l i ne o f the s t e a d y w a k e is in the boundary l aye r , the m a x i m u m v e l o c i t y de fec t is larger than the f ree s t ream v e l o c i t y de fec t as can be o b s e r v e d in Fig.2.9. A s a resu l t , there is a m a x i m u m w a k e amp l i tude s o m e w h e r e w i th in the boundary layer . The m a x i m u m amp l i t ude has been found to be w i th in the inner hal f o f the boundary layer f r o m the exper imen ta l data repor ted here . H o d s o n (1982) has o b s e r v e d a s im i l a r uns teady w a k e and boundary layer in terac t ion on a turb ine ro tor b lade s u r f a c e , Fig.2.7. M o r e o v e r , the ove ra l l amp l i tude rat io p r o f i l e d e c r e a s e s w i t h d i s tance f r o m the c y l i n d e r because o f the decay o f the w a k e . Further, because each p h a s e - a v e r a g e d boundary layer p r o f i l e is a p ro f i l e at a par t icu lar phase in the d i s t u rbance , one w o u l d expec t the p r o f i l e to look s i m i l a r to Fig.2.9 w h i c h s h o w s the e x i s t e n c e o f the w a k e in the boundary layer . H o w e v e r , due to the phase lag in the inner l aye r , the k ind o f p r o f i l e ob ta ined in the s teady expe r imen ts of T s i o l a k i s et al d o e s not ex i s t . Chapter 5 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 5.1 CONCLUSIONS S o m e ins ight in to the foundamen ta l behav io r o f a boundary layer in a f ree s t r eam w h i c h is p e r i o d i c a l l y d is tu rbed by t rave rs ing w a k e s have been ob ta i ned . The uns tead iness causes the t i m e - a v e r a g e d boundary layer p ro f i l e s to be c l o s e r to fu l l y turbulent even at l o w R e y n o l d s numbers . The p e r i o d i c e x i s t e n c e o f the w a k e d e f e c t s in the f lat p late boundary layer c a u s e s the d ramat i c inc rease of d i sp lacemen t and m o m e n t u m t h i c k n e s s e s , e s p e c i a l l y in the ear l y part o f the boundary layer . The d i f f e rence is l e s s far ther d o w n s t r e a m because o f the decay of the w a k e . The m o v i n g w a k e s c rea ted by the ro ta t ing cy l i nde r c a s c a d e have been o b s e r v e d and f o u n d to be c l o s e l y s i m i l a r to the s teady w a k e s . The d is tu rbance at any po in t in the f ree s t r eam is a func t i on of t ime a c c o r d i n g to the shape of the w a k e s and d i s tance f r o m the cy l i nde r wh i ch is sou rce o f the w a k e . The w a k e ' s e f f e c t s on the boundary layer d e v e l o p m e n t have been s h o w n in a s e r i e s of p h a s e - a v e r a g e d v e l o c i t y and turbu lence in tens i t y p ro f i l e s at eve ry 6 0 ° phase angle b e t w e e n t w o s u c c e s s i v e w a k e s . The shape o f these p r o f i l e s are de te rm ined by the re la t i ons o f the phase lag and m a x i m u m w a k e ampl i tude ra t io w i t h the d i s tance f r o m the su r face wh i ch have a l s o been o b s e r v e d in th is s tudy . The phase lag and m a x i m u m w a k e amp l i tude rat io p r o f i l e s are s i m i l a r to those ob ta ined in opera t ing t u rbomach ines by p rev ious r esea rche rs . The phase lag is b e l i e v e d to be due to the change o f c o n v e c t i o n v e l o c i t y of the f lu id in the d i s tance perpend icu la r to the p la te . The high amp l i tude 35 36 rat io w h i c h appears in the inner half o f the boundary layer is thought to be the s a m e m e c h a n i s m as the in terac t ion be tween a boundary layer and the w a k e of a s ta t ionary c i rcu lar cy l i nde r wh i ch has been s tud ied by T s i o l a k i s et al (1983). In c o n c l u s i o n , the p h a s e - a v e r a g e d v e l o c i t y and turbu lence in tens i ty p ro f i l e s in a boundary layer w h i c h is p e r i o d i c a l l y d is tu rbed by m o v i n g w a k e s have been o b s e r v e d . They are the resul t o f the c o m b i n a t i o n of the m o v i n g w a k e - t u r b u l e n t boundary layer in te rac t ion and the phase lag c h a r a c t e r i s t i c s . The f l o w behav iour of the p e r i o d i c a l l y d is tu rbed f la t p late boundary layer has a l so been found to be s i m i l a r to that measu red by Evans (1974) and H o d s o n (1982) in opera t ing t u rbomach ines . H o w e v e r , th is exper iment has not c o n s i d e r e d the e f f e c t s o f p ressure gradient and the angle o f at tack f l uc tua t ion in each w a k e on the d o w n s t r e a m b lade boundary layer t h i c k n e s s . Th is e f f e c t is a l so impor tant in co r rec t i ng the c a s c a d e tes t resu l ts of tu rbomach ine b l a d e s . 5.2 SUGGESTIONS FOR FURTHER WORK Further wo rk in th is f i e l d of s tudy shou ld inc lude the e f f e c t s of p ressure gradient and the f l uc tua t ion o f angle o f at tack at the d o w n s t r e a m su r face . S i n c e the f l o w in tu rbomach ines ,is h igh ly turbulent , the boundary layer on b lades shou ld be near ly turbulent p r o f i l e s . The use o f longer p la te or l owe r k i nema t i c v i s c o s i t y f l u i ds are n e c e s s a r y to create fu l l y turbulent boundary l a y e r s . No ca l cu l a t i on p rocedure has been found to desc r i be the behav iour of a boundary layer in a f ree s t ream w h i c h is d is tu rbed by pe r i od i c 37 \ . t r ave rs ing w a k e s . A n a l y t i c a l or numer ica l p rocedu res must be d e v e l o p e d to c o m p a r e w i t h exper imenta l da ta , such as t hose ob ta ined here and in the fu ture. 38 a) DISK b) DRUM Fig.1.1 TWO COMMON CONSTRUCTIONS OF AXIAL FLOW TURBINES VO 40 Fig.1.2 LAYOUT OF A CONVENTIONAL LOW SPEED CASCADE TUNNEL (DIXON 1978) Fig. 1.3 SCHEMATIC DIAGRAM OF PERIODIC DISTURBANCE ON TURBINE STATOR BLADES 41 FIG.2.1 SCHEMATIC DIAGRAM OF FLAT PLATE BOUNDARY LAYER DEVELOPMENT (MASSEY 1968) FIG.2.2 TYPICAL DISTRIBUTIONS IN LAMINAR AND TURBULENT BOUNDARY LAYERS ON A FLAT PLATE 42 FIG.2.3 TURBULENT BOUNDARY LAYER VELOCITY PROFILES IN SEMI-LOG FORM (EVANS 1973) FIG.2.A OSCILLOGRAMS OF BOUNDARY LAYER VELOCITY RECORDS AND INSTANTANEOUS VELOCITY PROFILES ON A COMPRESSOR STATOR BLADE, x/C=0.5 (EVANS 1977) y/4 =-oi6 U (ft/sec) FIG.2.5 OSCILLOGRAMS OF BOUNDARY LAYER VELOCITY RECORDS AND INSTANTANEOUS VELOCITY PROFILES ON A COMPRESSOR STATOR BLADE, x/C=0.7 (EVANS 1977) FIG.2.6 PRESSURE LOSS COEFFICIENTS ACROSS A TURBINE ROTOR BLADE AND THE SAME BLADE IN A LINEAR CASCADE (HODSON 1983) FIG. 2.7 AMPLITUDE AND PHASE PROFILES FOR THE BOUNDARY LAYERS OF A TURBINE ROTOR BLADE SUCTION-SURFACE (HODSON 1983) WIND TUNNEL TEST SECTION CYLINDER CASCADE PLATE FIG.2.8 SCHEMATIC DIAGRAM OF EXPERIMENTAL SET UP USED BY PFEIL, HERBST AND SCHRODER (1982) 48 FIG.2.9 MEASURED MEAN VELOCITIES OF WAKE AND BOUNDARY LAYER INTERACTING FLOW ( ) COMPARED TO UNDISTURBED, FLAT PLATE FLOW ( ) (TSIOLAKIS, KRAUSE AND MULLER 1983) 49 FIG.2.11 COMPARISON OF MEASURED MEAN VELOCITIES WITH RESULTS OF FINITE-DIFFERENCE SOLUTION (TSIOLAKIS, KRAUSE AND MULLER 1983) 50 WIND T U N N E L T E S T S E C T I O N R O T A T I N G C Y L I N D E R C A S C A D E FLOW D I R E C T I O N F I G . 3 . 1 S C H E M A T I C D I A G R A M O F T H E E X P E R I M E N T A L S E T - U P U S E D I N T H I S S T U D Y C Y L I N D E R ! V F I G . 3 . 2 S C H E M A T I C D I A G R A M O F T H E U N S T E A D Y FLOW P R O D U C E D B Y WAKES 51 SERVO AMPLIFIER , HOT WIRE RESISTANCE FIG. 3 . 3 TYPICAL CONSTANT TEMPERATURE HOT WIRE ANEMOMETER BRIDGE 2.2 2-o 1.8 o < > 1.6-1.4-1.2 LEGEND x EXPERIMENTAL DATA — EQUATION 3.1 . 1 1 1 1 i 0 5 10 15 20 25 VELOCITY, U (m/s) FIG. 3 . 4 CALIBRATION CURVE OF THE CONSTANT TEMPERATURE HOT WIRE ANEMOMETER 52 1.30 1.25 1.20-1.15 1 i 1 I 1 l 0 5 10 15 20 25 30 DISTANCE FROM PLATE, y (mm) FIG. 3 . 5 STILL AIR WALL CORRECTION FOR HOT WIRE ANEMOMETER PRONGS .OHMMETER 0 ALUMINUM PLATE, //*///////////////// //> 5T7T////// /////// FIG. 3 . 6 METHOD OF DETECTING THE CONTACT OF THE HOT WIRE PROBE AND THE ALUMINUM FLAT PLATE 53 u x(t) u N(t) TIME, t TIME, t a) TYPICAL VELOCITY RECORDS AT ONE LOCATION STARTING AT THE SAME INSTANT OF TIME A INSTANTANEOUS VELOCITY, U AT TIME, t AVERAGE VELOCITY, U OF N RECORDS AT TIME, t j U^(t),INSTANTANEOUS VELOCITY U(t),ENSEMBLE-AVERAGED VELOCITY U.MEAN VELOCITY t l TIME, t b) TYPICAL ENSEMBLE-AVERAGED VELOCITY RECORD FIG. 3 . 7 AN ILLUSTRATION OF THE ENSEMBLE AVERAGING PROCEDURE 54 L ^ YES W SAMPLE TWO KNOWN SIGNALS INPUT WALL PROXIMITY CORRECTION DATA AND COEFFICIENTS FY7 INPUT CAGE SPEED, SAMPLE SIZE FOR CALCULATING SAMPLING FREQUENCY SO THAT THE PROGRAM WILL SAMPLE AT LEAST 1 . 5 REVOLUTIONS I PREPARE NEFF FOR DATA REQUISITION INPUT DISTANCE FROM SURFACE STORE IN FILE r | STOP \ PREPARE A CONVERSION TABLE WHICH CONSISTS OF THE CORRESPONDING VELOCITY FOR EACH DISCRETE STEP BETWEEN 0 TO 1 V INPUT VOLTAGE SAMPLE HOT WIRE AND HALL EFFECT SWITCH OUTPUTS FOR ENSEMBLE AVERAGED VELOCITY RECORDS /s STORE RECORD (ON FLOPPY IDISKETTE YES SAMPLE HOT WIRE AND HALL EFFECT SWITCH FOR ENSEMBLE AVERAGED TURBULENCE INTENSITY RECORDS X STOP FIG. 3 . 8 DATA ACQUISITION PROGRAM FLOW CHART PLATE 1 THE LOW SPEED WIND TUNNEL PLATE 2 THE CYLINDER CASCADE PLATE 3 THE DRIVING MOTOR AND SPEED REDUCTION GEARS PLATE 4 BEARINGS AND THE TRAVERSE MECHANISM PLATE 6 THE HOT WIRE PROBE FOR BOUNDARY LAYER MEASUREMENT 1.5 0.5 0 * 0 LEGEND 1/7-th POWER LAW PROFILE ._ UNDISTURBED ~ DISTURBED b) Re =1.4x10" x 0.2 0.4 FIG. 4.1a&b NONDIMENSIONAL TIME AVERAGED VELOCITY PROFILES, C^- 0.84, UJ= SoHz 1 . 2 59 30 25->* 20-w H 15-o OS fn W 10-U < H W M P 5-DISTANCE FROM LEADING EDGE X= 100 mm 200 mm 0.2 0.3 0.4 0.5 0.6 TURBULENCE INTENSITY, T 0.7 0.8 FIG. 4.2 DIMENSIONAL TURBULENCE INTENSITY PROFILES OF UNDISTURBED BOUNDARY LAYER, U = 11.75 m/s 2.5 2-1.5-0.5 V V V LEGEND Re - 0.7x10" x c = 1.4x10" i i i i i i - i -0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 TURBULENCE INTENSITY, T FIG. 4.3 NONDIMENSIONAL TURBULENCE INTENSITY PROFILES OF UNDISTURBED BOUNDARY LAYER 60 3-r 2.5 1.5-80 -1— 100 T T LEGEND A x UNDISTURBED DISTURBED — I — 180 T 220 120 HO 160  200 DISTANCE FROM LEADING EDGE, X (mm) FIG. 4.4 BOUNDARY LAYER DEVELOPMENT, C^ = 0.84>u;r5oHz 61 100 i i i i i 11 1000 10000 100000 YU / V CO 0 . 0 0 0 .01 0 .02 0 . 0 3 0 .04 0 .05 0 .06 T I M E , t (s) F I G . 4 .6 ENSEMBLE-AVERAGED T I M E - V E L O C I T Y RECORD OF A TRAVERSING WAKE AT Um= 11.75 m / s , U = 14 ro/s AND X = 26 cm °° B o <79 65 6.5H -J 1 1 1 I 1 1 1 1 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 TIME, t (sec.) FIG. 4.9 ENSEMBLE-AVERAGED TIME-VELOCITY RECORDS AT U«,= 11.75 m/s, <n= 50 Hz AND X= 200 mm ' 66 0.200-0.175 A ii 0.150 0.025-0.000 i i i A Al 1.65 ONE PERIOD (360 )• 3.18 4.45 9.53 H 12.07 17.15 22.23 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 TIME, t (SEC.) FIG. 4.10 ENSEMBLE-AVERAGED TURBULENCE INTENSITY-TIME RECORDS AT U = 11.75 m/s,co= 50 Hz AND X= 200 mm m -20 0 20 40 60 80 100 120 140 DEGREE FIG, 4.14 PHASE LAG PROFILES, C = 0.84 70 71 1. B inde r , A . , Fo r s te r , W. , K ruse , H. and R o g g e , H. (1984) A n exper imen ta l i n ves t i ga t i on into the e f f e c t s of w a k e s on the uns teady turbine ro tor f l o w . A . S . M . E . paper 8 4 - G T - 1 7 8 2. Ca r r , L.W. (1981) A r e v i e w o f uns teady turbulent b o u n d a r y - l a y e r expe r imen t s . N . A . S . A . T M 81297 3. C e b e c i , T. and Car r , L.W. (1978) A compu te r p rog ram for ca l cu la t i ng laminar and turbulent boundary laye rs fo r t w o - d i m e n s i o n a l t i m e - d e p e n d e n t f l o w s . N . A . S . A . T M 78470 4. D i x o n , S.L . (1978) T h e r m o d y n a m i c s and f l u id m e c h a n i c s of t u rbomach ine ry . 3rd ed . , P e r g a m o n P r e s s 5. E v a n s , R.L. (1973) Turbulent boundary layers on a x i a l - f l o w c o m p r e s s o r b l a d e s . Ph.D. d i s s e r t a t i o n , U n i v e r s i t y of C a m b r i d g e . 6. E v a n s , R.L. and H o r l o c k , J . H . (1974) C a l c u l a t i o n o f the d e v e l o p m e n t o f turbulent boundary layers w i th a turbulent f r e e - s t r e a m . A . S . M . E . paper 7 4 - F E - 2 4 7. E v a n s , R.L. (1977) B o u n d a r y - l a y e r d e v e l o p m e n t oh an a x i a l - f l o w c o m p r e s s o r s ta tor b lade A . S . M . E . paper 7 7 - G T - 1 1 8. H o d s o n , H.P. (1983) U n s t e a d y boundary layers on a x i a l - f l o w turbine ro tor b l a d e s . Ph.D. d i s s e r t a t i o n , U n i v e r s i t y of C a m b r i d g e . 9. K a r l s s o n , S.K.F . (1958) A n uns teady turbulent boundary layer . 72 J . F . M . , V o l . 5, pp .622 -636 10. L a n g s t o n , L .S. and B o y l e , M.T. (1982) A new s u r f a c e - s t r e a m l i n e f l o w - v i s u a l i z a t i o n techn ique . J . F . M . , V o l . 125, pp .53 -57 11. M a s s e y , B .S. (1968) M e c h a n i c s of f l u i d . 2nd ed. , Van Nos t rand Re inho ld c o . 12. M a r u m o , E., S u z u k i , K. and S a t o , T. (1977) A turbulent boundary layer d is tu rbed by a cy l i nde r . J . F . M . , V o l . 87 , part 1, pp .121-141 13. P a t e l , M.H. (1977) On turbulent boundary laye rs in o s c i l l a t o r y f l o w . P r o c . R o y . S o c . (London) A 3 5 3 pp .121 -144 14. P f e i l , H., Herbs t , R. and S c h r o d e r , T. (1982) Inves t iga t ion of the laminar - tu rbu len t t rans i t i on o f boundary l aye rs d is tu rbed by w a k e s . A . S . M . E . paper 8 2 - G T - 1 2 4 15. S c h l i c h t i n g , H. (1979) Boundary layer t heo ry . 7th ed. , M c G r a w - H i l l 16. S c h l i c h t i n g , H, and D a s , A . (1967) On the in f luence of tu rbu lence leve l on the a e r o d y n a m i c l o s s e s of axial t u r b o m a c h i n e s . in , F l o w research on b lad ing ed . , L .S. Dzung , E l se rv i e r pub l i sh ing c o . 17. T s i o l a k i s , E.P. K rause , E. and M u l l e r , U.R. (1983) Turbulent boundary l a y e r - w a k e i n te rac t i on . 4th S y m . on turbulent shear f l o w , S e p t . 1 2 - 1 4 , 1983, Kar l s ruhe , F.R. G e r m a n y . A P P E N D I C E S 73 74 A P P E N D I X A T U R B U L E N C E INTENSITY F R O M U N L I N E A R I Z E D C O N S T A N T  T E M P E R A T U R E A N E M O M E T E R (CTA) O U T P U T The c o n n e c t i o n b e t w e e n the i ns tan taneous v e l o c i t y va r i a t i on , u, and the A C c o m p o n e n t of the C T A output is u = — , — A.1 dE c/dU and thus the r o o t - m e a n - s q u a r e re la t i onsh ip i s , i / e 2 / u 2 = — 1 / 0 A.2 v u dE /dU E„ is the C T A mean vo l t age and U is the mean f l o w v e l o c i t y . dE / d U i s , t he re fo re , the s l ope of the ca l i b ra t i on cu rve , E c = f ( U ) , w h i c h is a c c o r d i n g to the ca l i b ra t i on e x p r e s s i o n ; E c 2 = A 2 + BU n A.3 F r o m A . 3 , the de r i va t i ve of E w i th respec t to U is (dE c/dU)= (nBU n 1 ) / 2 E C A.4 A s a resu l t . /u 2= ( ( 2 E c ) / ( n B U n 1 ) ) / i 2 A.5 75 The turbu lence in tens i ty can > be ob ta ined f r o m the un l inear ized hot w i re ouput a c c o r d i n g to the f o l l o w i n g equa t i on . ( i/u 2/U)= ( . ( 2 E c ) / ( n B U n ) ) / i 2 76 A P P E N D I X B C A L C U L A T I O N S OF I N T E G R A L • B O U N D A R Y L A Y E R  P A R A M E T E R S : D I S P L A C E M E N T T H I C K N E S S A N D M O M E N T U M  T H I C K N E S S * The d i s p l a c e m e n t t h i c k n e s s , o , is the d i s tance w h i c h the sur face w o u l d have to be d i s p l a c e d ou twards to reduce the to ta l f l o w of a f r i c t i o n l e s s f lu id by the same amount reduced by a real boundary layer . S i m i l a r l y , the m o m e n t u m t h i c k n e s s , 8, is the t h i c kness through w h i c h the to ta l reduc t ion o f f lu id m o m e n t u m under f r i c t i o n l e s s c o n d i t i o n s is equal to that o f a real boundary layer . They are ca l cu la ted f r o m the boundary layer data a c c o r d i n g to the f o l l o w i n g equa t i ons , 5*= ( 1 - ( U / U m ) ) dy B.1 and , 0 = So ( U / U m ) ( l - ( U / U m ) ) dy B.2 U and U m are mean v e l o c i t i e s in the boundary layer and f ree s t r e a m . The in tegra t ions of the exper imen ta l data are ca r r ied out by a U B C c o m p u t e r l ib rary in tegra t ion rou t ine , Q I N T 4 P , wh i ch is f o r in tegra t ion of unequa l ly s p a c e d data po in t s us ing the quadrature m e t h o d . 77 A P P E N D I X C E S T I M A T I O N OF B O U N D A R Y S U R F A C E S H E A R S T R E S S  C O E F F I C I E N T S B Y C L A U S E R P L O T S Th i s techn ique is to es t ima te the su r face shear s t r e s s c o e f f i c i e n t , C f , f r o m the shape of a v e l o c i t y p r o f i l e . H o w e v e r , the shape of the v e l o c i t y p ro f i l e has to agree w i t h the shape of the "Law of the W a l l " p r o f i l e , s e c t i o n 2.1, in order to use th is techn ique . o i l n ( Z ! ! t ) + c u K v T w h e r e u T = ( r w / p ) 1 / 2 C.2 and C f = ( ^ / ( ^ p U V ) ) C.3 f r o m C.3 , rw 1 " 7 - I U » 2 C f C.4 subs t i t u te C.4 into C.2 and then into C .1 , U U w ( C f / 2 ) 1 / 2 K f r o m C .5 . l n ( ( y U o e ( C f / 2 ) l / 2 ) / v ) + C C.5 j U ( ^ , 1 / 2 l l n ( Z ^ + ( ^ , 1 / 2 ( C 4 I l n ( ^ ) 1 / 2 ) U „ 2 K v 2 K 2 C.6 w h i c h can be s i m p l i f i e d to 78 U = m(ln yu, CC )+b C.7 where m= ( C f / 2 ) b= ( C f / 2 ) 1 / 2 ( C + l n ( C f / 2 ) l / 2 / K ) By subs t i tu t ing d i f fe ren t va lues into equa t ion C.7 , the l inear re la t i onsh ip be tween U/U m and lnCyU^,/v) can be p lo t ted as s h o w n in F ig .C .1 . A v e l o c i t y p ro f i l e wh ich agrees w i t h the "Law o f the W a l l " shou ld have a l inear reg ion wh ich is para l le l to the s t ra ight l ines w h e n it is p lo t ted a c c o r d i n g to equat ion C.7. A s a resu l t , the su r face shear s t r e s s c o e f f i c i e n t s can be e s t i m a t e d . 

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