Transverse curvature effects on turbulent boundary layers.

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1 University of Windsor Scholarship at UWindsor Electronic Theses and Dissertations Transverse curvature effects on turbulent boundary layers. M. C. Joseph University of Windsor Follow this and additional works at: Recommended Citation Joseph, M. C., "Transverse curvature effects on turbulent boundary layers." (1969). Electronic Theses and Dissertations This online database contains the full-text of PhD dissertations and Masters theses of University of Windsor students from 1954 forward. These documents are made available for personal study and research purposes only, in accordance with the Canadian Copyright Act and the Creative Commons license CC BY-NC-ND (Attribution, Non-Commercial, No Derivative Works). Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would require the permission of the copyright holder. Students may inquire about withdrawing their dissertation and/or thesis from this database. For additional inquiries, please contact the repository administrator via or by telephone at ext

2 TRANSVERSE CURVATURE EFFECTS ON TURBULENT BOUNDARY LAYERS A Thesis submitted to the Faculty of Graduate Studies in partial fulfilment of the requirements for the degree of Master of Applied Science in the Department of Mechanical Engineering University of Windsor by M.C.Joseph,B.E., Windsor, Ontario, April, 1969.

3 UMI Num ber: EC INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI UMI Microform EC52750 Copyright 2008 by ProQuest LLC. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 E. Eisenhower Parkway PO Box 1346 Ann Arbor, Ml

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5 i ACKNOWLEDGEMENTS The author wishes to express his sincere thanks to Prof. K.Sridhar and Prof. A.McCorquodale for patient guidance offered throughout this work, and to Prof. T.W. McDonald for helpful advices during this study. The author is grateful to Dr.A.K.Menon for his help in setting up computer programmes. Thanks to Messrs. 0.Brudy,P. Feimer D.K.Liebsch and G.Michalsczuk. The author is grateful to Dr.J. F.Leddy, President, University of Windsor and the National Research Council for the financial support given.

6 ABSTRACT il This p ap er p r e s e n ts th e r e s u l t s of a n e x p e rim e n ta l stu d y o f th e tr a n s v e r s e c u rv a tu re e f f e c t s on th e tu r b u le n t boundary la y e r s form ed on th e o u ts id e s u rfa c e o f c y l i n d e r s. Tubes of d i f f e r e n t d ia m e te rs were a lig n e d w ith t h e i r a x is p a r a l l e l to th e flow of a i r o r w a te r. V e lo c ity p r o f i l e s were m easured in th e boundary l a y e r a t d i f f e r e n t s t a t i o n s a lo n g th e a x is o f th e c y lin d e r s w ith p i t o t and p i t o t - s t a t i c tu b e s. The c y lin d e r s were t e s t e d in a wind tu n n e l a t txflo d i f f e r e n t flow sp e e d s, v iz i 26 f t. / s e c. and 57 f t. / s e c. A s im i l a r ex p erim en t was co n d u cted in a w a te r tu n n e l w ith flow speeds o f 2 and 2.7 f t. / s e c. The c y lin d e r d ia m e te rs v a r ie d from 1.25 i n. to 0.25 i n. and th e R eynolds numbers based on c y lin d e r d ia m e te rs were in th e ran g e o f 3560 to I t was found t h a t in th e shape o f th e v e l o c i t y p ro f i l e s no s i g n i f i c a n t d if f e r e n c e s e x is te d betw een th e boundary la y e r s t h a t form ed in w a te r and th o se t h a t form ed in a i r. A com parison o f th e d isp la c e m e n t th ic k n e s s e s and th e momentum th ic k n e s s e s of th e boundary la y e r s i n w ate r and a i r a ls o le d to th e same c o n c lu s io n s. I t was found t h a t th e v e l o c i t y p r o f i l e s of th e t u r b u le n t boundary la y e r s were a f f e c te d by tr a n s v e r s e c u rv a t u r e. A g e n e r a l power law o f th e form u/u = ( y / 6 ) 1 /n

7 131 where U = f r e e stream v e l o c i t y u = v e l o c i t y a t any p o in t in th e boundary la y e r y = d is ta n c e p e rp e n d ic u la r to th e flow m easured from th e s u rfa c e o f th e c y lin d e r 6 = th ic k n e s s of th e boundary la y e r was found to be v a lid f o r th e tu r b u le n t boundary l a y e r v e l o c i t y p r o f i l e. The exponent 1 /n i n th e above r e l a t i o n d e c re a se d w ith in c r e a s in g tr a n s v e r s e c u r v a tu r e. From th e d i f f e r e n t c y lin d e r s t e s t e d th e fo llo w in g r e l a t i o n betw een th e r a d iu s o f th e c y lin d e r and n was o b ta in e d! ( n - 7 ) r 1,2 6 = where r i s th e c y lin d e r r a d iu s in in c h e s. The boundary l a y e r c h a r a c t e r i s t i c s were a ls o found to be a f f e c t e d by tr a n s v e r s e c u r v a tu r e. The boundary l a y e r th ic k n e s s e s and d isp la c e m e n t th ic k n e s s e s d e c re a se d w ith in c r e a s in g c u r v a tu r e. Momentum th ic k n e s s e s and s h e a r s t r e s s c o e f f i c i e n t s in c r e a s e d w ith c u r v a tu re. T his stu d y a ls o p o in te d o u t t h a t a s in g l e in n e r law v e l o c i t y p r o f i l e i s n o t v a l i d f o r a l l c u r v a tu r e s. I t in d ic a te d t h a t each c u rv a tu re has i t s own in n e r law f o r th e boundary l a y e r.

8 CONTENTS List of illustrations Nomenclature page vi vi i 1 Introducti on 1 2 Survey of literature Experimental work Theoretical work Remarks 7 3 Experimental study Water tunnel set up Auxilliary equipment Wind tunnel set up Experimental procedure Precautions taken 15 4 Results and discussion Velocity profiles Displacement thickness Momentum thickness Skin fri ction Accuracy of the results 25 5 Conclusi ons 28 Table.1. Boundary layer thickness 29 Table.2. Values of n 30 Table.3. Displacement thickness 31 Table.4. Momentum thickness 32 Appendi x.1. 52

9 V page Appendix Appendix References 63 Vita auctoris 66

10 v1 LIST OF ILLUSTRATIONS Fig. page 1 Water tunnel set up 33 2 Water tunnel test section 34 3 Wind tunnel test section 35 4 Defining diagram 36 5 Velocity profile, 1.25in. di a. cylinder 37 6 Velocity profile, 0.75in. dia. cyl i nder 37 7 Velocity profile, 0.50in. dia. cyl i nde r 38 8 Velocity profile, 1.0 in. di a. cy1i n de r 39 9 Velocity profile, 0.50in. dia. cyli nde r Velocity profile, 0.25in. dia. cyl inder Curvature effect on boundary layer thickness Nondimensional velocity profiles of 0.5 in. dia. cylinder Log log plot of velocity profiles of 0.5 in. dia. cylinder Nondimensional velocity profiles n for different cylinders Relation of n to cylinder radius n vs. axial distance Curvature effect on displacement thickness Curvature effect on momentum thickness Local friction coefficient 51

11 NOMENCLATURE v i i a C, k ^, k 2 Radius of cylinder Constants L Total length of cylinder r Radius measured from the axis of the cylinder u U x Velocity in the axial direction Free stream velocity Coordinate in the axial direction measured from the leading edge of the cylinder y Coordinate in the radial direction measured from the surface of cylinder, y = r-a v Velocity in y-di recti on -v Kinematic viscosity f Mass density of fluid ' f Shear stress in the boundary layer T a Shear stress at the surface of the cylinder b Rea Rex Boundary layer thickness Ua/y- Ux/-f u* (fc*/f)1/2 u+ u/u^ y+ yu r h c cf k Local friction coefficient Average friction coefficient Karman constant 1/n Index of the power law

12 1. INTRODUCTION Most flo w s t h a t o ccu r in p r a c tic e a re tu r b u le n t and th e t u r b u le n t boundary la y e r i s re s p o n s ib le f o r th e la r g e d ra g f o r c e s e x p e rie n c e d by moving o b je c ts. The a p p a re n t s t r e s s e s c r e a te d by tu rb u le n c e in f l u i d flo w c o n s t i t u t e m ost of th e r e s i s ta n c e en c o u n tere d by flow in a p ip e. The t u r b u le n t boundary la y e r has been a to p ic of g r e a t i n t e r e s t s in c e 192^ when B urgers (1) and Z ignen (2) p u b lish e d t h e i r works on boundary la y e r t r a n s i t i o n. The f l a t p l a t e boundary la y e r has been s tu d ie d q u ite e x te n s iv e ly and th e boundary la y e r s on many o th e r k in d s of s u rf a c e s a re b ein g s tu d ie d. T ra n sv e rse c u rv a tu re h ere d e n o te s th e c u rv a tu re in a p lan e norm al t o th e flo w d i r e c t i o n. Fundam ental i n f o r m a tio n ab o u t th e tr a n s v e r s e c u rv a tu re e f f e c t i s needed in th e c a l c u l a t i o n of f r i c t i o n on v a rio u s moving o b je c ts such as s h ip s and su b m arin es. In t h i s i n v e s t i g a t i o n th e tu r b u l e n t boundary la y e r s on c i r c u l a r c y lin d e r s of v a rio u s d i a m e ters were s tu d ie d e x p e rim e n ta lly.

13 2. SURVEY OF LITERATURE The s tu d ie s co n d u cted by e a r l i e r i n v e s t i g a t o r s a re c l a s s i f i e d a c c o rd in g to th e n a tu re of t h e i r w orks. Most of th e e a r l i e r works co u ld be a rra n g e d in two s e c tio n s * e x p e r i m en ta l s tu d ie s and t h e o r e t i c a l s t u d i e s. I n v e s tig a tio n s t h a t in c lu d e b o th t h e o r e t i c a l and e x p e rim e n ta l s tu d ie s a r e d i s cu ssed i n b o th s e c t io n s. 2.1, E x p e rim e n ta l work The f i r s t work on tr a n s v e r s e c u rv a tu re e f f e c t was p u b lish e d a s e a r l y a s 1924 by Kempf ( 3 ). He made an e x p e r i m e n ta l s tu d y of th e e f f e c t of tr a n s v e r s e c u rv a tu re u s in g c y lin d e r s of d i f f e r e n t le n g th s. He d e term in e d a v erag e s k in f r i c t i o n c o e f f i c i e n t s a t d i f f e r e n t flo w s p e e d s. K arhan (4) co n d u cted ex p erim e n ts w ith pontoons of d i f f e r e n t ro u g h n esses i n By com bining h is r e s u l t s w ith th e e a r l i e r r e s u l t s o b ta in e d by Kempf ( 3 ). he d e riv e d a c u rv a tu re c o r r e c t i o n f o r th e f r i c t i o n c o e f f i c i e n t. Hughes (5) s tu d ie d th e e f f e c t on a one in c h d i a m e te r c y lin d e r and p r e d ic te d an in c re a s e in r e s i s ta n c e w ith an in c re a s e i n th e le n g th r a d iu s r a t i o. H is ex p erim en ts were a f f e c te d by p re s s u re g r a d ie n ts of unknown m a g n itu d e. Richmond s ( 6 ) ex p e rim e n ts were perform ed on a one 2

14 in c h d ia m e te r c y lin d e r and a w ire of 0. 02^ in c h d ia m e te r. They were m ounted u n d e r te n s io n in a wind tu n n e l. The v e l o c i t y p r o f i l e s were m easured u s in g p i t o t s t a t i c tu b e s and h o t w ire an en o m eters, a t v a rio u s a x i a l d i s t a n c e s. The s k in f r i c t i o n v a lu e s were c a lc u la te d and an in c r e a s e over f l a t p la te s k in f r i c t i o n was p r e d ic te d. The in n e r law s f o r th e boundary l a y e r were d e riv e d and p l o t t e d a s u+ v s. y * ( l + y /2 a ). The c a l c u l a t i o n of s k in f r i c t i o n was v a lid o n ly when th e la m in a r s u b la y e r th ic k n e s s i s n e g lig ib le in co m parison to th e r a d iu s of th e c y l i n d e r. So c o n s id e ra b le e r r o r was in tro d u c e d in th e f r i c t i o n v a lu e s and th e in n e r law v e l o c i t y p r o f i l e s of th e w ir e. The v e l o c i t y m easurem ents in th e boundary l a y e r on th e w ire was c a r r i e d o u t u s in g a h o t w ire w hich was supposed to be c a l i b r a t e d f o r mean flo w. The v e l o c i t y p r o f i l e o b ta in e d was n o t found to obey a power law. Yu (7) u sed a two in c h d ia m e te r c y lin d e r f o r h is s tu d y. ^ 5, The r a d iu s R eynolds numbers were 15,0 0 0, 3 0,0 0 0, and He u sed a P re s to n tu b e f o r s k in f r i c t i o n m easu rem ents w ith th e c a l i b r a t i o n g iv e n by Landweber and S ia o (2 2 ). U sing th e r e s u l t s o b ta in e d th e in n e r law s and th e o u te r laws of th e boundary l a y e r were p r e d ic te d. 0 4 y+ < 30 u+ = y+ 30 ^ y+ 300 In n e r law s, (eq n. 10, r e f. 7.) y+ O u ter law s, (eq n. 12, r e f. 7.) Y asuhara (8) s tu d ie d th e e f f e c t of c u r v a tu re on

15 la m in a r and t u r b u le n t boundary la y e r s t h a t form ed on a 20 mm. k d ia m e te r c y l i n d e r. The stu d y of th e t u r b u le n t boundary l a y e r in d ic a te d th e g e n e r a l tre n d of th e boundary la y e r c h a r a c t e r i s t i c s a lth o u g h th e r e s u l t s co u ld n o t be u sed f o r q u a n t i t a t i v e p r e d i c t io n. B onsignore ( 9 ), from h is s tu d ie s of a 1.25 in c h d ia m e te r c y lin d e r, p r e d ic te d a change in th e shape of th e v e l o c i t y p r o f i l e from t h a t of th e f l a t p l a t e. He in d ic a te d a v a r i a t i o n of th e in d ex of th e power law w ith c u r v a tu r e. S ingh (10) in h is s tu d y of w a ll j e t s u sed c y lin d e r s of 10, 3.5. and 1 in c h d ia m e te rs. U sing a P re s to n tu b e th e f r i c t i o n c o e f f i c i e n t s were d e term in e d and th e in n e r law s were c a lc u la te d and p l o t t e d. S h i r t l i f f e (11) m easured th e tu r b u le n t boundary l a y e r v e l o c i t y p r o f i l e s on 1/ ^, 3/ 8, 1 /2 and 1 in c h d ia m e te r c y lin d e r s of two f e e t le n g th, m ounted in a wind tu n n e l. L o cal f r i c t i o n c o e f f i c i e n t s were p r e d ic te d assum ing th e r e s u l t s of Sparrow (2 0 ). V enkatasubram anian (12) found from h is e x p e rim e n ta l s tu d y t h a t th e u n iv e r s a l v e l o c i t y p r o f i l e i s n o t a c c e p ta b le f o r th e c u rv a tu re c a s e, 2,2. T h e o r e tic a l work Landweber (13) assumed a o n e -se v e n th power law v e l o c i t y d i s t r i b u t i o n and th e B la s iu s e m p iric a l s h e a r s t r e s s fo rm u la f o r th e c a l c u l a t i o n of c u rv a tu re e f f e c t. The v a r i a t i o n of power law w ith c u rv a tu re was n o t known a t t h a t tim e

16 5 and no e f f o r t f o r such a c o r r e c t i o n was made. The momentum th ic k n e s s f o r th e boundary l a y e r on th e c y lin d e r was d e f in e d and a d i f f e r e n t i a t i o n of th e momentum th ic k n e s s w ith th e assum ed power law le d to th e c o e f f i c i e n t of f r i c t i o n. The f l a t p la te s k in f r i c t i o n was a l s o c a lc u la te d s i m i l a r l y and th e r a t i o s of s k in f r i c t i o n s were o b ta in e d. In h is s tu d y, E c k e rt (1^) made th e same assu m p tio n s a s Landweber (13)«His work was an e x te n s io n of th e work of Landweber i n t o th e c o m p re ssib le flo w r e g io n. Boundary l a y e r th ic k n e s s, d isp la c e m e n t and momentum th ic k n e s s e s, and s k in f r i c t i o n c o e f f i c i e n t s were c a lc u la te d and p l o t t e d f o r d i f f e r e n t c u r v a tu re s and d i f f e r e n t Mach num bers. K arhan ( 1 5 ) s e le c te d a g e n e r a liz e d power law of th e form u/u = ( y / i ) 1^ 11 f o r th e v e l o c i t y p r o f i l e and c a lc u l a t e d s k in f r i c t i o n in th e same way a s Landweber (13) and E c k e rt ( 1 4 ), He a l s o o v erlo o k ed th e change in th e power law due to c u r v a tu r e. However, th e assu m p tio n s of Landweber, E c k e rt and K arhan c o u ld be j u s t i f i e d t o some e x te n t s in c e t h e i r s tu d ie s were o r ie n te d tow ard th e c a l c u l a t i o n of d rag f o r c e s on s h ip s where th e c u rv a tu re i s to o sm a ll to cause any a p p re c ia b le change in th e power law. T e lf e r (16) d e riv e d th e fo rm u la f o r a v erag e s k in f r i c t i o n = [ ^ " ] f c H V 3 By t h i s fo rm u la cf d e c re a s e s w ith le n g th a t f ix e d r a d iu s and in c r e a s e s w ith c u rv a tu re f o r f ix e d le n g th.

17 G in e v s k ii and S o lo d k in (17) assumed a s h e a r s t r e s s d i s t r i b u t i o n of th e form 6 7 " - r f o r th e e n t i r e boundary l a y e r - Af c "/ dy ^ o r th e la m in a r s u b la y e r I «lu and r t = - i f " kv* 4 y du dy f o r th e tu r b u le n t co re I t was f u r t h e r assum ed t h a t th e la m in a r s u b la y e r ex ten d ed up to y+ = V e lo c ity p r o f i l e s were e s ta b li s h e d f o r convex and concave s u r f a c e s. S asjim a, T a k a ji and Tanaka (18) s tu d ie d th e c u rv a tu r e e f f e c t w ith th e same assu m p tio n s a s G in e v s k ii and S o lo d k in (17)» e x c e p t t h a t th e y assum ed th e s h e a r s t r e s s to be c o n s ta n t f o r any r a d iu s w ith in th e la m in a r s u b la y e r. The la m in a r s u b la y e r was assum ed to e x te n d o n ly up to y+ = The e r r o r in tro d u c e d by th e assu m p tio n of a l i n e a r r e l a t i o n betw een u and y was lim ite d by assum ing a s m a lle r th ic k n e s s? b u t a t th e same tim e more e r r o r s were in tro d u c e d by t h i s, s in c e th e tu r b u le n t co re was assum ed to ex ten d t o y+ = V e lo c ity p r o f i l e s and f r i c t i o n c o e f f i c i e n t s were c a lc u la te d and com pared w ith f l a t p la te v a lu e s. R eid and W ilson (19) a l s o u sed th e K arm an -P ran d tl m ixing le n g th th e o ry w ith 0.4 a s th e von Karman c o n s ta n t k. They assum ed th e d i s t r i b u t i o n of s h e a r s t r e s s» r =» r. and p r e d ic te d v e l o c i t y p r o f i l e s f o r th e la m in a r s u b la y e r and th e tu r b u le n t c o r e. The r e s u l t s in c lu d e d f r i c t i o n

18 7 c o e f f i c i e n t s f o r smooth and rough c y l i n d e r s. Sparrow, E c k e rt and Minkowycz (20) p r e d ic te d th e f r i c t i o n c o e f f i c i e n t s by assum ing a d i s t r i b u t i o n f o r th e eddy d i f f u s i v i t y. They assum ed d i f f e r e n t d i s t r i b u t i o n s f o r re g io n s n e a r th e w a ll and away from th e w a ll. They a l s o assum ed th e r e l a t i o n The in n e r law was c a l c u l a t e d. Rao (21) assum ed y t - CL = /*- f o r th e Lam inar s u b la y e r and th e v e l o c i t y p r o f i l e f o r t h i s re g io n was o b ta in e d a s u/ ut = ^ lo g ( r / a ) From t h i s f in d in g he i n f e r r e d t h a t th e v e l o c i t y p r o f i l e f o r th e tu r b u le n t co re was«5.7 5 The la m in a r s u b la y e r was supposed t o e x ten d up to y+ = 5 V enkatasubram anian (12) (se e S e c tio n 2.1 ) in h is t h e o r e t i c a l a n a ly s is of tr a n s v e r s e c u rv a tu re assum ed th e u n iv e r s a l v e l o c i t y p r o f i l e, and p r e d ic te d s k in f r i c t i o n. The f r i c t i o n c o e f f i c i e n t was shown a s a f u n c tio n of Rea and Rex Remarks Landweber (1 2 ), E c k e rt (1*0, and Karhan (15)» in t h e i r works assum ed t h a t th e power law was u n a f fe c te d by c u r v a tu r e. T h is a ssu m p tio n was n o t su p p o rte d by any p re v io u s

19 work, and I f th e power law i s a f f e c t e d by c u r v a tu r e, th e c a l c u l a t i o n s of s k in f r i c t i o n made by th e s e i n v e s t i g a t o r s xiould n o t be e x a c t. The s tu d y of B onsignore (9) in 1966 in d ic a te d such an e f f e c t a lth o u g h h is work d id n o t e x p la in 8 c o m p le te ly th e c u r v a tu re e f f e c t s on th e power law. The p r e s e n t i n v e s t i g a t i o n was u n d e rta k e n to d eterm in e th e e x a c t e f f e c t s of d i f f e r e n t c u r v a tu re s on th e power law.

20 3. EXPERIMENTAL STUDY E xp erim en ts were co n d u cted to d e te im in e th e boundary- l a y e r T e lo c ity p r o f i l e s in a w a te r tu n n e l a s w e ll a s in a wind tu n n e l. The w a te r tu n n e l was f a b r i c a t e d f o r th e purpose of t h i s s tu d y and th e wind tu n n e l was th e e x i s t i n g low speed wind tu n n e l of th e M echanical E n g in e e rin g L a b o ra to ry W ater tu n n e l s e t up The w a te r tu n n e l had two m ain s e c tio n s i an i n l e t s e c tio n and a t e s t s e c t io n ( F ig. 1 ). The i n l e t s e c tio n was made out of 3 /4 " th ic k plywood and was 8 f t. lo n g and 11" x 11" I n s i d e. The f o u r c o rn e rs of t h i s s e c tio n were made w a te r tig h t by a p p ly in g r e s o r c in o l g lu e. The c o n n e c tin g p ie c e shown i n F ig. 1, was 15 x 20" and was u sed to c o n n ec t th e i n l e t s e c t io n of th e tu n n e l to th e o u t l e t of a head ta n k. Two s c re e n s were f ix e d in s id e th e tu n n e l so a s t o make th e v e l o c i t y p r o f i l e more u n ifo rm. The s c re e n 1 c o n s is te d of a two in c h th ic k r e s i l a t e x s c re e n s u p p o rte d b o th s id e s by two m e t a l l i c s c re e n s. S creen 2 was a t h i n m e ta llic s c re e n. N ear th e downstream end of t h i s s e c tio n a flow s t r a i g h t e n e r was f ix e d a s shown in F ig. 1. The flow s t r a i g h t e n e r made o u t of s h e e t m e ta l had 1" x 1" sq u a re c e l l s, 2 in c h e s lo n g. T h is p ie c e se rv e d t o s to p any 9

21 10 se co n d ary flow s t a r t i n g in th e i n l e t s e c t io n. The t e s t s e c tio n (P ig. 1. ), made o u t of th ic k p l e x i g l a s s, was 6 f t, lo n g and 11" x 11" in s id e ( s t a t i c p re s s u re ta p s were lo c a te d e v e ry 6" on one s i d e ). There was a window a t th e u p stream end of t h i s s e c t io n. The window gave a c c e s s i n t o t h i s p a r t of th e tu n n e l. A w a te r tig h t s h u t t e r was p ro v id e d f o r th e window. The in s id e s u rfa c e of t h i s s h u t t e r was even w ith th e in s id e s u rfa c e of th e tu n n e l w a ll when th e window i s c lo s e d. A s l o t wide was c u t a t th e to p of th e w orking s e c t io n. The s l o t was f t. lo n g and was lo c a te d e x a c tly a t th e c e n te r a s shown in F ig. 2, The s l o t was p ro v id e d w ith a c o v e r made out of 6 in c h lo n g p l e x ig la s s p ie c e s. The lo w er s u rfa c e of th e c o v e r p ie c e s was f l u s h w ith th e in s id e tu n n e l s u r f a c e. An a d ju s ta b le g a te was f ix e d a t th e downstream end cf th e t e s t s e c t io n. T h is wooden g a te had two p a r t s, a f ix e d g a te and a m ovable g a t e. Each was made o u t of r e c ta n g u la r b a rs p la c e d h o r iz o n ta ll y on a fram e w ith gaps betw een them. I n th e f u l l y open p o s it i o n th e b a rs of th e f ix e d and m ovable g a te s c o in c id e d le a v in g a b o u t h a lf of th e a r e a open. When th e m ovable g a te was p u lle d upw ard, th e open a r e a s were g r a d u a lly c lo s e d by th e b a rs i n th e movable g a te, th e re b y a c h ie v in g a c o n tin u o u s v a r i a t i o n of th e a r e a open f o r th e flow. The w a te r f o r th e ex p erim en t was s u p p lie d from th e sump below th e f l o o r by a c e n t r i f u g a l pump w ith a n a p p r o x i m ate c a p a c ity of 3000 g.p.m. The pump was d r iv e n by an

22 11 in d u c tio n m otor of ^0 H.P. The w a te r from th e pump was c a r r i e d in a one f o o t d ia m e te r p ip e w ith an a d ju s ta b le v a lv e. A m ag n etic flo w m e te r re c o rd e d th e flo w r a te i n g.p.m. The p ip e d is c h a rg e d th e w a te r i n t o a head ta n k. The flo w d i a gram is shown in F ig. 1. The ta n k was V x 2 ' x 5 ' I t was made of wood. There was a 11" x 11" opening a t th e bottom w hich se rv e d a s th e o u t l e t, A rounded o u t l e t was p ro v id e d t o g iv e a smooth flo w i n t o th e tu n n e l. The p ip e w hich d isc h a rg e d w a te r i n t o th e ta n k ex te n d ed to th e bottom to red u ce a i r e n tra in m e n t Auxiliary equipment Two r i g i d m e t a l l i c s u p p o rts were e r e c te d a t th e downstre am end of th e t e s t s e c t io n and th e s e two s u p p o rts h e ld th e t e s t c y lin d e r f ir m ly w ith th e c y lin d e r a x is c o in c id in g w ith th e c e n te r l i n e of th e tu n n e l. Thin w ire s of 25 th o u, d ia m e te r were f ix e d to th e t e s t tu b e s a t th e le a d in g ed g e. The w ire s p a sse d th ro u g h th e tu n n e l w a ll and th e ends were c o n n ected to f o u r t u r n - b u c k le s w hich were f ix e d on one e n d. The tu rn b u c k le s p e r m itte d m inor a d ju stm e n ts of th e t e s t c y lin d e r a lig n m e n t. The t e s t c y lin d e r was a lig n e d a lo n g th e c e n te r of th e tu n n e l w ith r e fe r e n c e t o th e tu n n e l w a lls. The t r a v e r s i n g mechanism shown in F ig. 2. in c lu d e d a runway made of alum inium and a c a r r ia g e f o r th e p i t o t tube w hich c o u ld be moved a lo n g th e runw ay. The runway was f ix e d to th e to p of th e tu n n e l in such a way t h a t th e p i t o t tube

23 12 la y In th e v e r t i c a l p lan e of th e c e n t e r l i n e of th e t e s t c y l i n d e r. The p i t o t tu b e p assed th ro u g h a c o v e r p ie c e f i t t e d in th e s l o t of th e tu n n e l w a ll. T urning of th e h a n d le, shown in F ig. 2., cau sed th e p i t o t tu b e to move up o r down, and th e p i t o t tu b e p o s it i o n c o u ld be re a d a c c u r a te ly to , The p i t o t tube p o s itio n s were re a d w ith re fe r e n c e to an i n i t i a l re a d in g ta k e n when th e p i t o t tu b e to uched th e c y lin d e r s u r f a c e. The s l o t and th e runway h e lp e d to f i x th e p i t o t tube a t any l o c a t i o n a lo n g th e a x is of th e t e s t tu b e, 3.3. Wind tu n n e l s e t up The wind tu n n e l u sed was th e c lo s e d c i r c u i t low speed wind tu n n e l of th e M echanical E n g in e e rin g L a b o ra to ry. The w orking s e c t io n was 30" x 30" and 25 f t. lo n g. The to p w a ll of th e wind tu n n e l was p r e a d ju s te d to g iv e z e ro p re s s u re g r a d i e n t. The tu n n e l was ru n by a 20 H.P. v a r ia b le speed in d u c tio n m o to r. Two r i g i d m e t a l l i c s u p p o rts, s im i l a r to th e s u p p o rts u sed in th e w a te r tu n n e l, were u sed to h o ld th e t e s t c y l i n d e r s. The s u p p o rts were one f o o t a p a r t and were f ix e d to th e tu n n e l w a ll a t th e dow nstream end of th e t e s t s e c tio n of th e tu n n e l. The ov erh an g in g p o r tio n s of th e t e s t c y lin d e r s were su p p o rte d by sm a ll w ire s of 25 th o u, d ia m e te r. The arran g em en t is shown in F ig. 3. The tu rn b u c k le s shown in th e f ig u r e were h e lp f u l in sm a ll a d ju stm e n ts of th e t e s t c y lin d e r w h ile a lig n in g th e s e tu b e s w ith th e c e n t e r l i n e of th e tu n n e l.

24 The t r a v e r s in g mechanism c o n s is te d of th e same ru n 13 way and c a r r ia g e u sed in th e w a te r tu n n e l. A v e r t i c a l stem was f ix e d on th e c a r r ia g e on w hich th e probe co u ld be su p p o rte d and th e t r a v e r s in g c o u ld be done. In t h i s s e t up a p i t o t - s t a t i c tu b e was h e ld h o r iz o n ta ll y, in th e p lan e of th e a x is of th e t e s t c y l i n d e r. The p i t o t - s t a t i c tu b e had an in s id e d ia m e te r of 20 th o u, f o r th e t o t a l p re s s u re ta p p in g. To a v o id any e f f e c t s of th e t r a v e r s in g mechanism on th e r e a d in g s, a p a r a l l e l b a r was f ix e d to th e p i t o t - s t a t i c tube and t h i s b a r was h e ld betw een th e c o l l a r s of th e t r a v e r s in g m echanism. By d o in g so, th e p i t o t - s t a t i c tube was p la c e d s e v e r a l in c h e s u p stream of th e stem of th e c a r r i a g e, a f l e x i b l e s h a f t was f ix e d to th e h andle of th e tr a v e r s in g m echanism, a s shown in F ig. 3*> and th e o th e r end of t h i s s h a f t p assed th ro u g h th e tu n n e l w a ll. From t h i s end t r a v e r s in g c o u ld be done, and th e e x a c t p o s it i o n of th e probe c o u ld be re a d th ro u g h th e tr a n s p a r e n t p l e x ig la s s windows of th e wind tu n n e l. The t o t a l and s t a t i c p re s s u re ta p s of th e probe were co n n ected to a Lam brecht m anom eter. The m anom eter d i r e c t l y in d ic a te d th e dynamic p r e s s u r e. The t r a v e r s i n g mechanism and th e t e s t c y lin d e r were c o n n ected w ith a f l a s h l i g h t c e l l and a m i l l i v o l t m e ter in s e r i e s. When th e probe was in c o n ta c t w ith th e s u rf a c e of th e c y lin d e r, th e v o l t m ete r r e g i s t e r e d an e l e c t r i c p o te n t i a l. This was used to ta k e th e i n i t i a l re a d in g when th e probe was to u c h in g th e c y lin d e r.

25 E x p e rim en tal P ro ced u re A. W ater tu n n e l ex p erim en t The c y lin d e r s t e s t e d in th e w a te r tu n n e l were 1,2 5 ", 3 /4 ", and 1/2" i n d ia m e te r. They were 6 f t, lo n g alum inium tu b e s. The o u ts id e s u rfa c e of th e tu b e s was p o lis h e d and a t th e le a d in g edge th e in s id e of th e tu b e was ta p e re d w hich gave a s h a rp le a d in g e d g e. The in s id e of th e tu b e was k e p t open f o r th e flo w and th e s h a rp le a d in g edge d iv id e d th e flo w w ith o u t s e p a r a tio n from th e o u ts id e s u rfa c e a t th e le a d in g ed g e. F ig. 4. s p e c i f i e s th e c o o rd in a te system u sed f o r t h i s e x p e rim e n ta l s tu d y. V e lo c ity p r o f i l e s were m easured a t d i f f e r e n t s t a t i o n s lo c a te d a lo n g th e a x i s. The s t a t i o n s were from x = 1 f t. t o x = 5 f t, a t 1 f t. i n t e r v a l s. A p i t o t tu b e of 50 th o u, in s id e d ia m e te r was u sed f o r m easu rin g th e t o t a l p r e s s u r e. The s t a t i c p r e s s u r e s in d ic a te d by a l l th e w a ll ta p s were found to a g re e w ith th e s t a t i c p re s s u re obta in e d from a p i t o t - s t a t i c tu b e, and so one of th e w a ll ta p s was used f o r m easu rin g th e s t a t i c p r e s s u r e. The p i t o t tube was b ro u g h t i n c o n ta c t w ith th e c y lin d e r f o r th e f i r s t r e a d in g. L a te r on re a d in g s were ta k e n a t 25 th o u, i n t e r v a l s t i l l 0.1 in c h o r d in a te and th e n a t 50 th o u, i n t e r v a l s. E xperim ents were co n d u cted a t two d i f f e r e n t flo w sp eed s of a p p ro x im a te ly 2.2 and 2.7 f t. / s e c. H igher v e lo c i t i e s c o u ld n o t be o b ta in e d s in c e th e h e ig h t of th e head ta n k was l i m i t e d. The w a te r l e v e l in th e ta n k was shown by a g la s s tu b e on th e ta n k w a ll. The ex p e rim e n t was

26 co n d u cted o n ly when th e w a te r le v e l i n th e ta n k rem ained c o n s ta n t and th e m ag n etic flo w m e te r in d ic a te d c o n s ta n t 15 flow r a t e i n t o th e ta n k. In o th e r w ords, a c o n s ta n t flo w r a t e th ro u g h th e tu n n e l was m a in ta in e d d u rin g th e e x p e rim e n t. B. Wind tu n n e l e x p erim en t Three c y lin d e r s of 1", and d ia m e te rs were t e s t e d in th e wind tu n n e l. They were 10 f t. lo n g alum inium tu b e s. The tu b e s were p re p a re d i n th e same way a s th e tu b e s u sed i n th e w a te r tu n n e l and were w ith th e same type of le a d in g e d g e. The v e l o c i t y p r o f i l e s were m easured w ith a p i t o t - s t a t i c tube a t s t a t i o n s lo c a te d a t a x i a l d is ta n c e s of 1 to 8 f t. The s t a t i o n s were a t 1 f t. i n t e r v a l s a s i n th e p re v io u s c a s e. R eadings were ta k e n a t 15 th o u, i n t e r v a l s t i l l 60 th o u., 25 th o u, i n t e r v a l s t i l l 0.1 in c h and 50 th o u, i n t e r v a l s f o r th e r e s t of th e boundary l a y e r th ic k n e s s. The flo w sp eed s u sed f o r th e ex p erim en t were a p p ro x i m a te ly 26 f t. / s e c, and 57 f t. / s e c. S in ce the maximum speed allo w ed f o r th e d r iv in g m otor of th e tu n n e l was lim ite d to r.p.m., h ig h e r v e l o c i t i e s c o u ld n o t be a c h ie v e d f o r th e e x p e rim e n t. The te m p e ra tu re rem ained more o r l e s s cons t a n t a t 75 F w h ile th e e x p erim e n ts were c a r r ie d o u t. 3.5* P re c a u tio n s ta k e n B efore th e ex p erim en ts were c o n d u cted, th e w a te r tu n n e l was ru n w ith o u t t e s t c y lin d e r s and the v e l o c i t y p r o f i l e was m easured from to p to b o tto m. The p r o f i l e was found to

27 16 be sym m etrical ab o u t the c e n t e r l i n e and th e th ic k n e s s o f th e boundary la y e r s on th e w a lls was a b o u t 1 3/^"* A h o r iz o n ta l tr a v e r s i n g from w a ll to w a ll was n o t p o s s ib le and th e p r o f i l e was n o t te s te d in t h i s d i r e c t i o n. I t was found by th e v e r t i c a l tr a v e r s i n g t h a t a p a r t from th e boundary la y e r s on th e w a lls, th e r e s t o f th e a r e a had an a p p ro x im a te ly uniform v e l o c i t y. The w a te r tu n n e l was a ls o t e s te d f o r any m ajor e d d ie s. A dye was in je c te d from th e s id e a t th e u p stream end and th e flow p a tte r n was o b se rv e d. The t e s t confirm ed t h a t th e r e were no e d d ie s in th e flo w. Symmetry o f flow around th e c y lin d e r was checked by m easuring th e v e l o c i t y p r o f i l e a lo n g th re e d i f f e r e n t r a d i a l p a th s. T his was done in th e wind tu n n e l. No such check co u ld be made in th e w a te r tu n n e l ex p e rim en t. A m easurem ent o f th e wind tu n n e l tu rb u le n c e was conducted a t th e two speeds a t w hich th e ex p erim en ts were c a r r ie d o u t. The p e rc e n ta g e o f tu rb u le n c e was c a lc u la te d f o r each speed from h o t w ire anemometer re a d in g s and th ey w ere : f o r 27 f t. / s e c. 0, 62^ f o r 58 f t. / s e c. To av o id th e e r r o r in y - o rd in a te due to th e c le a ra n c e betw een th e d i f f e r e n t screw th re a d s in th e tr a v e r s in g m echanism, th e h an d le was r o ta te d o n ly in one d i r e c t i o n w h ile t r a v e r s i n g.

28 4. RESULTS AND DISCUSSION 4.1. V e lo c ity p r o f i l e s A. V e lo c ity p r o f i l e s f o r f l a t p la te The f l a t p la te v e l o c i t y p r o f i l e s were n o t s tu d ie d i n t h i s i n v e s t i g a t i o n s in c e th e l / 7t h power law i s g e n e r a lly a c c e p te d f o r th e range of Reynolds numbers u s e d. So f o r th e f l a t p la te th e l / 7t h power law was assum ed. 3. V e lo c ity p r o f i l e s f o r th e c y lin d e r s The v e l o c i t y p r o f i l e s o b ta in e d f o r d i f f e r e n t c y lin d e r s a t d i f f e r e n t s t a t i o n s a re shown in F ig u re s 5 t o 10. F o r th e v e l o c i t y p r o f i l e s th e boundary l a y e r t h i c k n ess was f ix e d a s th e y - o rd in a te when u/u = 0.99«This l i m i t was ta k e n from an a v erag e cu rv e computed from th e observ ed d a t a. The boundary l a y e r th ic k n e s s e s a r e e n te r e d in Table 1. When th e boundary l a y e r th ic k n e s s e s were com pared w ith th o se of th e f l a t p la te d e riv e d from l / 7t h power law, i t was found t h a t th e tr a n s v e r s e c u rv a tu re cau sed a r e d u c tio n in boundary la y e r th ic k n e s s. T his e f f e c t i s shown in F ig. 11. The f l a t p la te boundary l a y e r th ic k n e s s was th e maximum. The boundary l a y e r th ic k n e s s w ent on d e c re a s in g w ith c u rv a tu re and d ia m e te r c y lin d e r had th e l e a s t th ic k n e s s. The v e l o c i t y p r o f i l e s o b ta in e d f o r each c y lin d e r 17

29 18 were found to b e a r s i m i l a r i t y. That i s, th e n o n d im en sio n al v e l o c i t y - p r o f i l e s rem ained unchanged f o r each c y lin d e r i r r e s p e c t i v e o f th e a x i a l d is t a n c e. T h is is shown in P ig, 12 and F ig. 13* In m ost c a se s th e s i m i l a r i t y s ta r te d from an a x i a l d is ta n c e o f 4 f t, from th e le a d in g edge, even though in a few c a se s s i m i l a r i t y was o b ta in e d even a t 3 I t. d i s t a n c e. 1/n A ll v e lo c ity f r o f i l e s obeyed th e r e l a t i o n u/u = (y /$ ). For each t e s t c y lin d e r a v a lu e o f n was found to rem ain c o n s ta n t a s shown in F ig. 13, i r r e s p e c t i v e o f flow v e l o c i t y o r a x i a l d is ta n c e. At t h i s p o in t i t i s i n t e r e s t i n g to n o te th e paper (2*0 by N ik u rad se, He r e la te d n to R eynolds num ber. The Reynolds number e f f e c t on n, which he p re d ic te d was v e ry sm all and th e v a r i a t i o n o f flow speed ach iev ed in th e ex p erim en ts was n o t s u f f i c i e n t enough to in d ic a te any such e f f e c t. In F ig. 1*+ th e n o n d im en sio n al v e l o c i t y p r o f i l e s a re shown f o r d i f f e r e n t c y lin d e r s. The f l a t p la te v e lo c ity p r o f i l e a c c o rd in g to th e 1 /7 th power law is a ls o shown. The v e lo c ity p r o f i l e s o f th e c y lin d e r s co u ld be seen to have l a r g e r v e l o c i t y g r a d ie n ts a t sm all y - o r d in a te s and s m a lle r g r a d ie n ts a t la r g e y - o r d in a te s compared to th e f l a t p la te p r o f i l e. Tt was observed t h a t in th e re g io n o f s i m i l a r i t y th e v alu e o f n depended v e ry much on th e r a d iu s o f th e tube (See F ig. 1 5 ). n in c re a s e d w ith c u r v a tu r e. The e x p erim e n ts conducted b o th in th e w a te r tu n n e l and th e wind tu n n e l su p p o rted t h i s e f f e c t and a ls o th e v a lu e s o f n o b ta in e d in b o th c a se s

30 19 showed v e ry good a g re em e n t. D iam eter in in c h e s 1. 25* * * 0.25 n (* T e sts were co n d u cted i n w a te r tu n n e l) From th e r e l a t i o n u/u = (y /j ) * /n th e v a lu e of n was c a lc u la te d f o r each observ ed d a ta and th e a v erag e n was found f o r each s t a t i o n. These v a lu e s a re shown in T able 2. The a v e rag e of th e s e v a lu e s of n was c a lc u la te d f o r s t a t i o n s lo c a te d in th e re g io n of s i m i l a r i t y f o r each c y lin d e r and th e new v a lu e s a re shown i n th e ta b le a b o v e. The above ta b le in d ic a te d th e e x is te n c e of a r e l a t i o n betw een n and r a d iu s a. S in ce the e x p erim e n ts f a i l e d to in d ic a te any e f f e c t of flo w speed on th e v a lu e of n, in no way co u ld th e r a d iu s of th e c y lin d e r be e x p re sse d non- d im e n s io n a lly. So a r e l a t i o n betw een n and r a d iu s was obta in e d k eep in g a i n d im e n sio n a l fo rm. A p r e lim in a r y in v e s t i g a t i o n le d to th e r e la tio n * (n -7 ) ak l = k 2 The r e l a t i o n had th e se obvious advan tag es* ( 1 ) a» eo n» 7 / F l a t p la te and l / 7 t h power la w / (2) a * O n eo /The o b je c t v a n ish e s and th e boundary l a y e r a l s o v a n is h e s / U sing th e e x p e rim e n ta l v a lu e s of n and a th e b e s t v a lu e s of kj_ and k2 were d ete rm in e d s im u lta n e o u s ly by u s in g th e p ro gram of Appendix 1. in an I.B.M co m puter. The f i n a l

31 2 0 r e l a t i o n was* ( n - 7 ) a 1 *26 = 0.62 a in in c h e s In F ig. l 6. th e f i n a l r e l a t i o n of n and ra d iu s a shows very- good ag reem ent w ith th e e x p e rim e n ta l p o in ts. A few v a lu e s of n a v a ila b le from o th e r e x p e rim e n ta l i n v e s t i g a t o r s lik e Yu (7) and Richmond ( 6 ) a l s o in d ic a te d good ag re em e n t. The v e l o c i t y p r o f i l e s m easured by o th e r i n v e s t i g a t o r s co u ld n o t be com pared due t o n o n a v a i l a b i l i t y of d a ta. The v e l o c i t y p r o f i l e s o b ta in e d a t s t a t i o n s i n th e f i r s t ^ f t. of a x i a l d is ta n c e showed a v a r i a t i o n of n w ith x - d is t a n c e. I t was observ ed t h a t n g r a d u a lly d e c re a se d w ith a x i a l d is t a n c e. But th e r e s u l t s were n o t c o n v in c in g enough to make p o s it i v e c o n c lu s io n s. However th e o b ta in e d v a lu e s of n a re shown in F ig. 1?. ^.2. D isp lacem en t th ic k n e s s The d e f i n i t i o n of th e d isp la c e m e n t th ic k n e s s u sed f o r th e f l a t p l a t e was n o t a p p lic a b le h ere s in c e a s th e r a d iu s ch an g es, a v a r i a t i o n of c ir c u m f e r e n tia l a r e a i s ta k in g p la c e. So th e new d e f i n i t i o n used f o r th e d isp la c e m e n t th ic k n e s s i s j = J (1 - u /u ) (1 + y /a ) dy (see A ppendix 2.) T able 3«shows th e d isp la c e m e n t th ic k n e s s e s o b ta in e d f o r each c y lin d e r a t d i f f e r e n t s t a t i o n s. The d isp la c e m e n t th ic k n e s s e s a re found to d e c re a se w ith c u rv a tu re a s se en in F ig. 18. T his d e c re a se was cau sed by th e c u rv a tu re e f f e c t on th e v e l o c i t y p r o f i l e. A lth o u g h th e e f f e c t of th e c u rv a tu re

32 term y /a in th e d e f i n i t i o n was to in c re a s e th e v a lu e, th e n e t r e s u l t was a d e c r e a s e, , K ome ntum t h i ckne s s J u s t as in s e c tio n 4.2. th e new d e f i n i t i o n u sed f o r th e momentum th ic k n e s s i s i = X ( u / u ) ( l - u/u) (1 + y /a ) dy Table 4, shows th e momentum th ic k n e s s e s c a lc u la te d by th e above d e f i n i t i o n. In momentum th ic k n e s s th e e f f e c t of th e c u rv a tu re term y /a was q u ite s i g n i f i c a n t. The c u rv a tu r e e f f e c t on th e v e l o c i t y p r o f i l e cau sed th e momentum th ic k n e s s to d e c re a se from th e v alu e of f l a t p l a t e. But t h i s d e c re a se i s g r a d u a lly overcome by th e e f f e c t of th e c u rv a tu re term i n th e d e f i n i t i o n. F ig. 19. shows th e t o t a l c u rv a tu re e f f e c t on th e momentum th ic k n e s s s k in F r i c t i o n M easurem ents of l o c a l s k in f r i c t i o n c o e f f i c i e n t s were conducted by Yu (7) and S ingh (1 0 ), u s in g P r e s t o n 's method of u s in g a c a l i b r a t e d s ta g n a tio n tu b e. The c a l i b r a tio n used by Yu was o b ta in e d from Landweber and S ia o (22) w hich was based on th e lo g a rith m ic law of th e f l a t p la te boundary l a y e r. The assu m p tio n t h a t th e lo g a rith m ic law of th e f l a t p la te was v a lid f o r th e boundary la y e r w ith tr a n s v e r s e c u rv a tu re was q u e s tio n a b le. The l o c a l s k in f r i c t i o n c o e f f i c i e n t s were p r e d ic te d and as e x p ec ted th e in n e r law s c a lc u la te d u s in g th e s e f r i c t i o n v a lu e s tu rn e d

33 22 ou t to be d i f f e r e n t from th e lo g a rith m ic law. However th e e r r o r s were n o t c o n s id e ra b le s in c e th e c u r v a tu re s in v o lv e d were sm all* From t h i s i t was p la in t h a t f o r h ig h e r c u rv a tu r e s th e e r r o r s co u ld be c o n s id e r a b le. M oreover th e e r r o r s cau sed co u ld be d i f f e r e n t in m agnitude f o r d i f f e r e n t c y l i n d e r s. So P re s to n tu b e w ith th e a v a ila b le c a l i b r a t i o n s c o u ld n o t be u sed in t h i s s tu d y. Many p re v io u s i n v e s t i g a t o r s based t h e i r c a l c u l a t i o n s of s k in f r i c t i o n on some form of in n e r law. But a g a in, com paring th e in n e r law s o b ta in e d by d i f f e r e n t p eople by d i f f e r e n t m eans, one m ight i n f e r t h a t a s in g le g e n e ra l in n e r law was n o t a p p lic a b le f o r a l l c u r v a tu r e s. At l e a s t one co u ld d e f i n i t e l y q u e s tio n th e id e a of a s in g le in n e r law v e lo c ity p r o f i l e. In t h i s l i g h t th e c a l c u l a t i o n of f r i c t i o n based on a v e lo c ity p r o f i l e was a ls o q u e s tio n a b le. The o n ly way open was to c a l c u l a t e f r i c t i o n u s in g momentum th ic k n e s s. The av erag e f r i c t i o n c o e f f i c i e n t co u ld be o b ta in e d from th e momentum th ic k n e s s d eterm in e d from th e e x p e rim e n ta l v e l o c i t y p r o f i l e. But th e lo c a l f r i c t i o n c o e f f i c i e n t s found by th e slo p e of th e momentum th ic k n e s s a t each p o in t a re n o t a c c e p ta b le s in c e th e m easurem ent of a v ery sm a ll slo p e can n o t be v ery a c c u r a te. T h is d i f f i c u l t y was overcome by u s in g th e power law o b ta in e d by ex p erim en t w ith th e fo llo w in g a n a l y t i c a l a p p ro ach. u/u = ( y /6 ) 1 /n 1 u T = 2

34 E q u a tio n 1 was n o n d im e n sio n aliz ed u s in g e q u a tio n 2 to g e t e q u a tio n s 3 and from s e c tio n 4-3 u /u» = c 3 _ J c - 2n /(l+ n ) y2n / ( 1+n) ^Vj2/( l + n ) ^ 6 = J. Q x / v ) ( 1-u/U ) (1 + y /a ) dy 5 t a _ Cy _ d <f? 6 f l? * dx com paring 5 and 6 ^ = ^ U2 -i(u /u ) (1-u/U ) ( 1+y/a) dy 7 The i n t e g r a t i o n in e q u a tio n 7 was perform ed u s in g e q u a tio n 1 (see A ppendix 3.)» The r e s u l t i n g e q u a tio n i s a s i = f u 2 ( n. n 6 ) &6 8 ( (n+2 )(n + 1) (S n + l)(n + i) a ) dx The e x p re s s io n s f o r a in e q u a tio n s 8 and 4 were eq u ated and th e r e s u l t i n g e x p re s s io n s in te g r a te d w ith r e s p e c t to x to g e t (A ppendix 3,)i J tm ( n + 3 ) - o -2 n /(n + 3 ) ^ - 2 / n + 3 ) n n(n + 1 )/(n + 3 ) d(1+2(2tst) a> ~ C - X X[u 4 2)(n+$j? u s in g e q u a tio n s 4 and 6 Cj, _ 2c 2n/^1+n^ u 2/ d + n ) ^_v_j2/(l+n) 1Q E q u a tio n s 9 and 10 were used to g e th e r w ith e x p e rim e n ta l

35 24- r e s u l t s to c a l c u l a t e th e lo c a l f r i c t i o n c o e f f i c i e n t s. The f a c t o r s i n e q u a tio n s 9 and 10 were d eterm in e d to a good l e v e l of a c c u ra c y by e x p e rim e n t. So by t h i s m ethod th e m easurem ents of th e s lo p e s of momentum th ic k n e s s w hich a re sx ib ject to la r g e e r r o r s were a v o id e d. The r e s u l t s o b ta in e d a r e shown in F ig, 20, I t co u ld be seen t h a t th e l o c a l f r i c t i o n c o e f f i c i e n t s of d i f f e r e n t c y lin d e r s a rra n g e d th em selv es i n s e p a ra te p a t t e r n s. T h is f u r t h e r s tre n g th e n e d th e id e a of d i f f e r e n t in n e r law s f o r d i f f e r e n t c u r v a tu r e s. F or th e f l a t p l a t e, th e l / 7 t h power law a g re e s v e ry w e ll w ith th e u n iv e r s a l v e l o c i t y p r o f i l e. u /u * = 5.75 lo g y in th e r e g io n 30 4 y+ < 300. In th e p lace of 1 /7 th power law o th e r power law s w ith s u c c e s s iv e ly d im in ish in g in d ex 1 /n w ith c u rv a tu re were o b ta in e d in S e c tio n T h is f a c t a l s o p o in te d ou t t h a t th e id e a of a u n iq u e in n e r law f o r a l l c u r v a tu re s was n o t a c c e p ta b le.

36 25 ^ 5* A ccuracy o f th e r e s u l t s A com plete e s tim a tio n o f th e a c c u ra c y o f a l l r e s u l t s i s n o t a tte m p te d in t h i s s e c tio n. The e r r o r s from d i f f e r e n t so u rc e s a re s ta te d and th e ap p ro x im ate m agnitude o f each is in d ic a te d. A c c u r a c y o f m a n o m e t e r r e a d i n g s....+$% A l i g n m e n t o f t e s t c y l i n d e r in wind t u n n e l inch in w a te r t u n n e l inch T em perature f lu c t u a t i o n s + 3 F C o rresp o n d in g f lu c t u a t i o n s in a b s o lu te v i s c o s i t y ( fo r a i r... ( f o r w a te r., # F lu c tu a tio n s in p re s s u re f o r a i r % f o r w a t e r......v e ry sm all T o leran ce o f c y l i n d e r s inch c a l i b r a t i o n e r r o r s,,,...± 2% A rough e s tim a tio n o f th e e r r o r s in some o f th e d e riv e d r e s u l t s co u ld be made a s shown: e r r o r in & y/6 = (u/u)n It I = H J + nr I A lthough th e d i a l in th e tr a v e r s i n g mechanism 5s marked up to O.OOl", due to to le r a n c e s on d i f f e r e n t p a r ts o f th e mechanism th e same a c c u ra c y co u ld n o t be cla im e d.

37 :100 in ch a c c u ra c y co u ld be s a f e l y assumed and c o n s id e rin g th e boundary l a y e r th ic k n e s s to be ro u g h ly one in c h d y /y i s e s tim a te d to be The maximum v alu e of n co u ld be ta k e n a s 16. u 2 = kh 2 d u /u = d h /h f o r a maximum manometer re a d in g of 100 mm. and an u n c e r t a i n t y of 0.5 mm. d u /u = 1/2 d h /h =.0025 = x.0025 =.05 = 5 % In t h i s a n a ly s is n i s assumed to be a c c u r a te. T his co u ld be j u s t i f i e d because m ost of th e e r r o r s in th e v alu e of n a re random e r r o r s and a mean of a sample of 20, w i l l have a v e ry sm all e r r o r. The on ly e r r o r in n w hich i s n o t rand cm in n a tu re i s due to th e e r r o r in boundary la y e r th ic k n e s s i t s e l f. T h is e r r o r i s c o n s ta n t f o r a l l re a d in g s w hich belong t o one v e l o c i t y p r o f i l e. So f o r a c y lin d e r w ith 5 s t a t i o n s in th e s i m i l a r i t y re g io n and v e l o c i t y p ro f i l e s a v a ila b le f o r 2 flo w sp ee d s, t h i s e r r o r a l s o co u ld be t r e a t e d as a random e r r o r w ith a sam ple s iz e of 10. However a rough e s tim a te of th e a c c u ra c y of n w i l l be i n d ic a te d by th e r o o t mean sq u are v alu e of th e d e v ia tio n s in n from th e av erag e v a lu e. F o r 0.5 in c h d ia m e te r c y lin d e r* th e av erag e v alu e of n = 10.9 Mean of th e sq u are of d e v ia tio n s from t h i s (see T able 2) = ( f o r wind tu n n e l)

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