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High frequency pressure fluctuations on a two dimensional square cylinder in smooth and turbulent flows Vasanji, Zainulabdeen 1978

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HIGH FREQUENCY PRESSURE FLUCTUATIONS ON A TWO DIMENSIONAL SQUARE CYLINDER IN SMOOTH AND TURBULENT FLOWS by ZAINULABDEEN VASANJI B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department o f Mechanical Engineering) We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA January 19 78 (c) Zainulabdeen V a s a n j i , 1978 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Zainulabdeen V a s a n j i Department o f Mechanical E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia 20 75 Wesbrook Pl a c e Vancouver, B.C., Canada V6T 1W5 Date: AjxA^t . 1978 . i i ABSTRACT There i s a need t o develop s t a t i s t i c a l d e s i g n procedures f o r g l a s s and c l a d d i n g panels i n b u i l d i n g s exposed t o s t r o n g winds. Account must be taken o f the h i g h frequency, h i g h amplitude p r e s s u r e f l u c t u a t i o n s observed o c c a s s i o n a l l y on f u l l s c a l e b u i l d i n g s and b u i l d i n g models i n si m u l a t e d atmospheric flows, and the non-Gaussian p r o b a b i l i t y d i s t r i b u t i o n o f these p r e s s u r e amplitudes. Most p r o b a b i l i t y d i s t r i b u t i o n s o f the p r e s s u r e f l u c t u a -t i o n s can be modelled over the low-pressure range where the danger of damage i s the h i g h e s t , by a decaying e x p o n e n t i a l curve o f the form F(x) = 1 - e " a x b Using t h i s type o f approximation, which f i n a l l y r e s t s on the det e r m i n a t i o n of the two e m p i r i c a l s t a t i s t i c a l parameters "a" and "b", plus the u s u a l l y measured mean and root-mean square pressure c o e f f i c i e n t s , the numerical r i s k of occurrence o f any s p e c i f i e d low pressure can be p r e d i c t e d f o r a p a r t i c u l a r time i n t e r v a l over which the h i g h winds p e r s i s t . The simple a n a l y s i s developed here a l s o r e q u i r e s as an i n p u t a non-dimensional S t r o u h a l number which r e l a t e s the l e n g t h s c a l e (panel s i z e ) and v e l o c i t y s c a l e ( r e f e r e n c e v e l o c i t y ) to the h i g h e s t frequency of p r e ssure f l u c t u a t i o n which i s l i k e l y t o a f f e c t the panel i n Question. Wind t u n n e l measurements have been made o f the p r e s s u r e f l u c t u a t i o n s on a s i m p l i f i e d b u i l d i n g model, a square p r i s m a t i c c y l i n d e r , i n smooth and t u r b u l e n t uniform flows. S p e c t r a and p r o b a b i l i t y d i s t r i b u t i o n s have been o b t a i n e d f o r v a r i o u s l o c a -t i o n s a t v a r i o u s angles o f the i n c i d e n t wind, where p o s s i b l e g l a s s and c l a d d i n g f a i l u r e c o u l d o c c u r . C a r e f u l c a l i b r a t i o n o f the p r e s s u r e measuring system ensured t h a t a broad frequency range of pre s s u r e f l u c t u a t i o n s was sensed. The measured p r o b a b i l i t y d i s t r i b u t i o n s o f the pre s s u r e f l u c t u a t i o n s have been f i t t e d by a Gaussian model f o r l o c a t i o n s where the mean p r e s s u r e c o e f f i c i e n t s are p o s i t i v e - g o i n g , and by a curve o f the e x p o n e n t i a l type over o t h e r l o c a t i o n s where the mean-pressure c o e f f i c i e n t s are d e f i n i t e l y n e g a t i v e . The two s t a t i s t i c a l parameters r e q u i r e d t o f i t the d i s t r i b u t i o n s i n the l a t t e r case do not c o r r e l a t e w e l l with the measured mean pre s s u r e c o e f f i c i e n t s . Once e m p i r i c a l v a l u e s o f the two parameters are obt a i n e d however, from experience or t e s t , d e sign of b u i l d i n g elements t o i n c l u d e the extreme val u e s o f nega-t i v e (suction) p r e s s u r e s can be r e a d i l y accomplished. Examples are g i v e n t o demonstrate t h i s procedure. TABLE OF CONTENTS i v Page ABSTRACT i ACKNOWLEDGEMENTS X LIST OF TABLES , v LIST OF FIGURES v: NOMENCLATURE .. X] CHAPTER 1. INTRODUCTION 1 1.1. Background 1.2. O b j e c t i v e s DESCRIPTION OF THE APPARATUS AND EXPERIMENTS .. 6 2.1. Wind Tunnel 2.2. The Model 2.3. Turbulence Generating G r i d 2.4. L o c a l i s e d Turbulence 2.5. Pressure Measuring System 2.6. Data A c q u i s i t i o n 2.7. S c a n i v a l v e Measurements 3. RESULTS AND DISCUSSION I 16 3.1. Mean P r e s s u r e s , Forces 3.2. F l u c t u a t i n g Pressure Measurements 3.3. Bandwidth A n a l y s i s V Page CHAPTER 4. RESULTS AND DISCUSSION I I : PROBABILITY CONCEPTS AND ANALYSIS . .. 2 6 4.1. A p p l i e d Pressures 4.2. Extreme Value S t a t i s t i c s As A p p l i e d to Pressure Loading ("Theory") 4.3. A p p l i c a t i o n s 4.3. (a) P o s i t i v e - G o i n g Cp R e s u l t s 3 rmean 4.3. (b) Negative Cp R e s u l t s mean 5. CONCLUSIONS 43 BIBLIOGRAPHY 47 APPENDIX I 49 v i LIST OF TABLES Page TABLE I P o s i t i o n s o f a f o r Pressure Tap #9. where P o s s i b l e C l a d d i n g Damage Could Occur 19 I I P o s i t i o n s o f a f o r Pres s u r e Tap #6 Where P o s s i b l e C l a d d i n g Damage Could Occur 19 I I I Comparison of C p m e a n Values Measured With Two D i f f e r e n t Systems 24 IV Comparison o f C p r m s Values Measured With Two D i f f e r e n t Systems 25 V (a) Comparison of the G r a p h i c a l Estimates of u N and <% wit h the Observed Values f o r Pressure Tap #9 34 V (b) Comparison o f the Observed C p m i n and the C a l c u l a t e d Expected Values 36 VI Summary of R e s u l t s f o r the E x p o n e n t i a l Curve F i t t i n g i n the Low-Pressure Range v i i LIST OF FIGURES Page 1. Flow around a b u i l d i n g 3 2. O u t l i n e of the U.B.C. Aerodynamics Wind Tunnel 51 3. Diagram showing the Model Dimensions, P o s i t i o n and Numbering of the P r e s s u r e Tappings 52 4. (a), D i s t r i b u t i o n o f U/U Across the Tunnel Downstream of the 9" Mesh G r i d 53 4. (b) D i s t r i b u t i o n o f u'/U Across the Tunnel Downstream o f the 9" Mesh G r i d 54 5. C a l i b r a t i o n of the Dynamic Head f o r the 9" Mesh G r i d 55 6. The V e l o c i t y P r o f i l e s U/U 0 10 and 16 Rod Diameters Downstream o f the Vortex Generating Rod 56 7. The R e l a t i v e Turbulence u 1 P r o f i l e s 10 and 16 Rod Diameters Downstream of the Vortex Generating Rod 57 7. (a) L o n g i t u d i n a l Turbulence S p e c t r a 10 and 16 Rod Diameters Downstream o f the Vortex Generating Rod 58 8. The High Frequency Pressure Measuring System 59 9. C a l i b r a t i o n Apparatus f o r the Pressure Measuring System 60 10. Frequency Response Curves f o r V a r i o u s Lengths o f P l a s t i c Tubing o f I.D. 0.045 i n . and O.D. 0.062 i n . 61 11. Frequency Response Curve f o r the Pressure Measuring System 62 12. S t a t i c C a l i b r a t i o n of the Pressure Measuring System 6 3 13. C a l i b r a t i o n Curve f o r the PDP 11/10 I/O 6 4 v i i i Page 14. Mean Pressure D i s t r i b u t i o n for Various Angles of Incident Smooth Flow 6 5 15. Mean Pressure D i s t r i b u t i o n for Various Angles of Incident Grid Turbulent Flow 66 16. Variation of the. Base Pressure C o e f f i c i e n t , Cpbf with Angle of Incidence, a, i n Smooth and Uniform Turbulent Flows 67 17. Variation of the Co e f f i c i e n t of Drag, Co, with Angle of Incidence i n Smooth and Grid Turbulent Flows 6 8 18. Comparison of the Corrected Mean Pressure D i s t r i b u t i o n with Parkinson's Wake Source Model Results at a = 0 i n Smooth Flow 69 19. Variation of C p m e a n with a For Pressure Tap #9 i n the Three Types of Flow 7 0 20. Variation of Cp m ean with a For Pressure Tap #6 i n the Three Types of Flow 71 21. Spectra Showing High Energy Levels at Higher Frequencies for Pressure Tap #9 7 2 22. Spectra Showing High Energy Levels at Higher Frequencies for Pressure Tap #6 7 3 23. Some Typical Pressure Spectra for Pressure Tap #9 "74 Variation of C p m i n with Frequency and Pressure Spectrum for: 24. Pressure Tap #9 at a = 0° i n Grid Trubulent Flow 7 5 25. Pressure Tap #9 at a = 160° i n Grid Turbulent Flow . 76 26. Pressure Tap #9 at a = 290° i n Grid Turbulent Flow 7 7 27. Pressure Tap #6 at a = 130° i n Grid Turbulent Flow 7 8 28. Pressure Tap #6 at a = 70° i n Rod Turbulence 79 i x Page Some Examples o f Gaussian D i s t r i b u t i o n s o f the Pressure F l u c t u t a t i o n Amplitudes: 29. Pressure Tap #9 a t a = 290° i n Smooth Flow ... 80 30. Pre s s u r e Tap #9 a t a = 0° i n G r i d T u r b u l e n t Flow 1 81 31. Pressure Tap #9 a t a - 290° i n G r i d T u r b u l e n t Flow 82 32. Pressure Tap #9 a t a = 15° i n Rod Turbulence .. 83 33. Pressure Tap #9 a t a = 290° i n Rod Turbulence . 84 34. P r e s s u r e Tap #6 a t a = 0 i n Rod Turbulence ... 85 35. Pressure Tap #6 at a = 70° i n Rod Turbulence . . 8 6 36. Pressure Tap #6 a t a = 0° i n G r i d T u r b u l e n t Flow w i t h 3 1-Second Samples o f 2000 Readings 87 Some Examples o f Non-Weibull P r o b a b i l i t y D i s t r i b u t i o n s of the Pressure F l u c t u a t i o n Amplitudes: 37. Pressure Tap #9 at a = 90° i n the Three Types o f Flows 88 38. Pressure Tap #9 a t a = 130° i n the Three Types o f Flows 89 39. Pressure Tap #6 a t a - 90° i n the Three Types of Flows 9 0 40. Pressure Tap #6 at a = 130° i n the Three Types o f Flows 9 1 41. Pressure Tap #3 at a = 90° i n the Three Types of Flows 9 2 42. Pressure Tap #3 a t a = 130° i n the Three Types o f Flows 9 3 X Page Some Examples of the Non-Log-Normal P r o b a b i l i t y D i s t r i b u t i o n s of the Pressure F l u c t u a t i o n Amplitudes: 43. Pressure Tap #9 at a = 0 ° i n Smooth Flow 9 4 44. Pressure Tap #9 at a = 90° i n Smooth Flow .... 9 5 45. Pressure Tap #9 at a = 130° i n Smooth Flow ... 9f> 46. Pressure Tap #6 at a = 90° i n Smooth Flow .... 9 7 47. Pressure Tap #6 at a = 90° i n G r i d Turbulent Flow 9 8 48. Pressure Tap #6 at a = 160° i n G r i d Turbulent Flow 9 9 49. Pressure Tap #9 at a = 180° i n Rod Turbulence . 1 0 0 50. Pressure Tap #9 at a = 90° i n Rod Turbulence .. 1 0 1 Some Examples of the Exponential Curve-Fits i n the Low-Pressure Range f o r : 51. Pressure Tap #9 at a = 90° and a = 130° i n Smooth Flow . .. 1 0 2 52. Pressure Tap #9 at a = 90°, a = 110°, a = 130° and a = 160° i n G r i d Turbulent Flow 10 3 53. Pressure Tap #9 at a = 90°, a = 130° and a = 180° i n Rod Turbulence 104 54. Pressure Tap #6 at a = 90°, a = 130° and a = 180° i n Smooth Flow 105 55. Pressure Tap #6 at a = 90° and a = 130° i n G r i d Turbulent Flow 106 56. Pressure Tap #6 at a = 90° and a = 130° i n Rod Turbulence 107 57. A P l o t of " a u vs. ( C p m a v " Cp) .. 108 x i APPENDIX P a g e Appendix 1: The E v a l u a t i o n o f the Extreme Value Type I Model Parameters f o r a Parent Normal P r o b a b i l i t y D i s t r i b u t i o n 49 x i i NOMENCLATURE Cp P o P P U Cp. mean Cp cp rms min max cp a IT q Re d v St f G(+) G(-) Pressure c o e f f i c i e n t s ( p - p 0 ) / J 5 P U 2 S t a t i c P r e s s u r e P r e s s u r e on the model s u r f a c e Density o f a i r Freestream v e l o c i t y Mean p r e s s u r e c o e f f i c i e n t Root-mean square p r e s s u r e c o e f f i c i e n t Minimum p r e s s u r e c o e f f i c i e n t Maximum p r e s s u r e c o e f f i c i e n t Angle o f i n c i d e n c e o f the oncoming wind w.r.t. the model face A Average wind v e l o c i t y T o t a l RMS. v e l o c i t y f l u c t u a t i o n o f the l o n g i t u d i n a l v e l o c i t y component Dynamic head = JspU2 Reynolds Number = (Ud/v) C h a r a c t e r i s t i c model dimension Kinematic v i s c o s i t y S t r o u h a l number = fd/U Frequency o f the p r e s s u r e f l u c t u a t i o n - L a r g e s t p o s i t i v e Gust f a c t o r = (Cp -Cp )/Cp 3 c max rmean rrms Lowest n e g a t i v e Gust f a c t o r = (Cp . -Cp )/Cp min rmean rms - Drag c o e f f i c i e n t C o r r e c t e d drag c o e f f i c i e n t x i i i c p t mm |Cp . | -1 'min 1 ^ C^min^ bw Cp rms •Cp. mean L f (L) F(L) n • n ( L ) Base pr e s s u r e c o e f f i c i e n t . C o r r e c t e d base p r e s s u r e c o e f f i c i e n t i Minimum drag c o e f f i c i e n t Absolute value o f the minimum pr e s s u r e c o e f f i c i e n t Minimum p r e s s u r e c o e f f i c i e n t w i t h i n a bandwidth Average root-mean square p r e s s u r e c o e f f i c i e n t Average mean pr e s s u r e c o e f f i c i e n t P r e s s u r e l o a d i n g Parent p r o b a b i l i t y d e n s i t y d i s t r i b u t i o n o f the pre s s u r e l o a d i n g Parent cumulative p r o b a b i l i t y d i s t r i b u t i o n (c.p.d.) o f L Number o f independent pressure l o a d i n g s Extreme Value (E.V.) cumulative p r o b a b i l i t y d i s t r i -b u t i o n f o r the l a r g e s t v a l u e s o f L Extreme Value p r o b a b i l i t y d e n s i t y o f the l a r g e s t v a l u e s o f L F T •  (L) I, max max mean min T R x' X b Extreme Value Type I Asymptote model f o r the p r o b a b i l i t y d i s t r i b u t i o n o f the l a r g e s t v a l u e s o f L Maximum pre s s u r e Mean p r e s s u r e - Minimum pressure - D i g i t i z i n g : o r sampling r a t e , Hz. Sampling time l e n g t h Numerical value o f r i s k , % - (Cp-Cp ) r rmean - (Cp -Cp) Mtiax • - C h a r a c t e r i s t i c model dimension o r panel s i z e XXV f N ( x ) - Normal p r o b a b i l i t y d e n s i t y f u n c t i o n o f the v a r i a b l e x y N - Mean value o f x f o r a Normal d i s t r i b u t i o n a N - Standard d e v i a t i o n o f x f o r a Normal d i s t r i b u t i o n y N - G r a p h i c a l estimate o f o" N - G r a p h i c a l estimate o f F (x;y ,a ) - Cumulative p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n f o r N N N a Normally d i s t r i b u t e d v a r i a b l e x a e - Extreme Value s c a l e parameter V& - Extreme Value l o c a t i o n parameter C h a r a c t e r i s t i c l a r g e s t v alue o f x i n a sample o f s i z e u n n H - Hazard f u n c t i o n u{ - F i r s t moment o f the Extreme Value p r o b a b i l i t y d e n s i t y f u n c t i o n Cpmin - Average minimum p r e s s u r e c o e f f i c i e n t a,b - E m p i r i c a l s t a t i s t i c a l parameters f o r the E x p o n e n t i a l D i s t r i b u t i o n o f the c.p.d. o f the low pr e s s u r e amplitudes b - Slope o f the E x p o n e n t i a l curve x^ - Design value o f x v S a f e t y F a c t o r = (Design Value of Cp . )/(Observed average value o f Cp . ) m i n • min XV ACKNOWLEDGEMENTS The author s i n c e r e l y thanks Dr. I.S. Gartshore f o r h i s unending words of encouragement and whose amiable a t t i t u d e made t h i s work a p l e a s a n t e x p e r i e n c e . S i n c e r e thanks must a l s o go t o Dr. K.V. Bury f o r h i s expert o p i n i o n s d u r i n g the s t a t i s t i c a l a n a l y s i s o f the data. The work i n Chapter 4 has been e x t e n s i v e l y drawn from h i s t e x t " S t a t i s t i c a l Models i n A p p l i e d S c i e n c e " , p u b l i s h e d by John Wiley and Sons. The author i s most g r a t e f u l t o a l l the o t h e r members of the Ae r o - A c o u s t i c s group i n the Department of Mechanical E n g i n e e r i n g f o r t h e i r i n s p i r i n g i n t e r e s t and a s s i s t a n c e . L a s t l y , t o my w i f e , my g r a t i t u d e f o r her moral support. 1 FLUCTUATING PRESSURES ON A 2-D SQUARE CYLINDER  IN TURBULENT FLOWS CHAPTER 1 INTRODUCTION 1.1. Background The i n c r e a s i n g number o f i n c i d e n t s i n v o l v i n g windstorm induced damage to b u i l d i n g s has r a i s e d much concern i n the En g i n e e r i n g communities around the wor l d . A s i g n i f i c a n t f e a t u r e of b u i l d i n g f a i l u r e t h a t has emerged from damage documentation and e v a l u a t i o n concerns c l a d d i n g and window g l a s s . Minor and * Beason (1) r e p o r t e d t h a t many o f the g l a s s f a i l u r e s o c c u r r e d when the maximum recorded windspeed was near, o r l e s s than, the d e s i g n v e l o c i t y , and gave some examples o f windstorm r e l a t e d g l a s s damage to i l l u s t r a t e the magnitude o f the problem. During the Lubbock, Texas, storm of May 19 70, the Great P l a i n s L i f e B u i l d i n g and the F i r s t N a t i o n a l Bank Pioneer N a t u r a l Gas Company B u i l d i n g each s u s t a i n e d window i n s t a l l a t i o n f a i l u r e s t h a t r e -s u l t e d i n damage t o the b u i l d i n g contents exceeding $1M. The 30-storey Guaranty Bank P l a z a b u i l d i n g , the D r i s c o l l H o t e l and the Wilson Tower were e x t e n s i v e l y damaged as a r e s u l t o f window fail u r e s v w h e n H u r r i c a n e C e l i a s t r u c k Corpus C h r i s t i , Texas. "Winds i n t h i s h u r r i c a n e gusted t o 161 mph (260 km/hr) and as * Numbers i n b r a c k e t s r e f e r t o a r t i c l e s l i s t e d i n the B i b l i o g r a p h y . 2 such, d i d not exceed wind speeds t h a t should have been a n t i c i -pated f o r a G u l f Coast exposure" (1) . The John Hancock B u i l d i n g i n Boston experienced g l a s s damage problems f o r s e v e r a l months i n e a r l y 1973. In t h i s i n c i d e n t , g l a s s damage was recorded on more than one o c c a s i o n when windspeeds reached 50 mph (80 km/hr), i . e . lower than the d esign windspeed. The Sears Tower i n Chicago, 111., s u f f e r e d g l a s s damage which was c o n f i n e d t o h e i g h t s between the n i n t h and t h i r t i e t h f l o o r s when wind gusts of 60 mph were r e p o r t e d i n February 1974 (2). Surry and Tryggvason (3), d u r i n g wind t u n n e l t e s t i n g o f a b u i l d i n g model, found t h a t f o r a p a r t i c u l a r l o c a t i o n on the b u i l d i n g , f o r a narrow range of wind d i r e c t i o n s , the peak s u c t i o n p r e s s u r e s were abnormally l a r g e and a s s o c i a t e d w i t h h i g h f r e q u e n c i e s . Some o f the l o c a t i o n s were a l s o a t low l e v e l s . Such low l e v e l g l a s s and c l a d d i n g f a i l u r e s on f u l l s c a l e s t r u c t u r e s have a l s o been observed by Eaton (4) i n the U.K. These low l e v e l f a i l u r e s have o n l y been r e c e n t l y documented and the p h y s i c a l mechanism i n v o l v e d i s not c l e a r . The d e s i g n o f g l a s s and c l a d d i n g on s t r u c t u r e s r e q u i r e s a knowledge of the peak pr e s s u r e expected to a c t on the s t r u c -t u r e d u r i n g i t s l i f e t i m e . The instantaneous pressure a c t i n g a t a p o i n t on the s t r u c t u r e i s the r e s u l t of t u r b u l e n c e i n the approach flow due to the roughness c h a r a c t e r i s t i c s o f the area upwind o f the s t r u c t u r e i n c l u d i n g nearby b u i l d i n g s . However, the s t r u c t u r e i t s e l f i n t e r a c t s w i t h the wind, m o d i f y i n g the 3 flow p a t t e r n i n i t s own neighbourhood and g e n e r a t i n g a new flow regime over i t s own s u r f a c e which c o n t r i b u t e s f u r t h e r l a r g e s c a l e unsteadiness and s m a l l s c a l e t u r b u l e n c e t o the flow. I t i s t h i s c o m p l icated flow over the s u r f a c e s o f a b u i l d i n g which determines the wind loads on the b u i l d i n g , and i n p a r t i c u l a r the l o c a l p r e s s u r e v a r i a t i o n s on the g l a s s and c l a d d i n g o f the b u i l d i n g . The wind on encountering a b u i l d i n g i s d e f l e c t e d over the r o o f and around the s i d e s . T h i s r e s u l t s i n a c c e l e r a t e d flows i n these r e g i o n s , which i n t u r n produce n e g a t i v e (suction) p r e s s u r e s on the adjacent e x t e r n a l s u r f a c e o f b u i l d i n g s . The g r e a t e r the a c c e l e r a t i o n o f flow, the g r e a t e r the s u c t i o n s experienced. T h i s i s i l l u s t r a t e d i n F i g . 1. Deflected, accelerated flow upstream positive highest point surface generally i n the wake, negative pressures Surface under reattaching Shear layer, negative pressures 4 F l u c t u a t i n g Shear Layer 4 Because o f the random nature o f the t u r b u l e n t flow around a b u i l d i n g , the p r e s s u r e on a s t r u c t u r e a l s o f l u c t u a t e s i n a random manner. Consequently, knowledge o f the s t a t i s t i c a l c h a r a c t e r i s t i c s o f these pressure f l u c t u a t i o n s i s r e q u i r e d t o p r e d i c t the peak v a l u e s l i k e l y t o occur i n a g i v e n time p e r i o d . Peterka and Cermak (6) r e p o r t e d t h a t the p r o b a b i l i t y d i s t r i b u t i o n s o f the p r e s s u r e f l u c t u a t i o n s f e l l i n t o two c a t e g o r i e s : one f o r Cp ^ > -0.1, which were n e a r l y Gaussian, and another f o r mean Cp „ < -0.25, which were non-Gaussian. These r e s u l t s were rmean from a wind-tunnel study o f a p r e l i m i n a r y d e s i g n o f the F e d e r a l Reserve Bank i n Richmond, V i r g i n i a . D a l g l i e s h (7) r e p o r t e d p r o b a b i l i t y d i s t r i b u t i o n s o f peak pre s s u r e s measured on a 45 s t o r e y o f f i c e b u i l d i n g i n Montreal. H i s p r e s s u r e s were o b t a i n e d mostly on the p o s i t i v e - p r e s s u r e upwind face o f the s t r u c t u r e , and were n e a r l y Gaussian. He r e p o r t e d o n l y one d i s t r i b u t i o n f o r a ne g a t i v e pressure on the leeward s i d e o f the s t r u c t u r e and t h i s was non-Gaussian. During a study o f pr e s s u r e f l u c t u a t i o n s on a 1:384 s c a l e model o f an i d e n t i c a l p a i r o f square Towers, Cermak and Sadeh (8) observed s i m i l a r i t i e s between the oncoming t u r b u l e n t energy s p e c t r a and the pre s s u r e f l u c t a t i o n s p e c t r a s u g g e s t i n g t h a t the upstream t u r b u l e n c e p l a y e d an important r o l e i n producing the s u r f a c e p r e s s u r e f l u c t u a t i o n s , a t l e a s t on the upwind face o f the b u i l d i n g . Large s u c t i o n p r e s s u r e s on b u i l d i n g s have been observed as " s p i k e s " on a p r e s s u r e r e c o r d (3). The d u r a t i o n and amplitude 5 of these s p i k e s have s t r o n g i m p l i c a t i o n s f o r the a b i l i t y o f the g l a s s and c l a d d i n g t o w i t h s t a n d p r e s s u r e l o a d s . These s p i k e s a l s o suggest the presence o f two flow p r o c e s s e s : one producing the broad band r e l a t i v e l y low l e v e l f l u c t u a t i o n s and a second connected to the r e l a t i v e l y r a r e , h i g h amplitude, h i g h frequency s p i k e s . P e terka and Cermak (5) e s t i m a t e d t h a t the d u r a t i o n o f these l a r g e peak gusts i s between two and f i v e seconds f o r f u l l s c a l e s t r u c t u r e s . These p r e s s u r e f l u c t a t i o n s have o n l y been r e c e n t l y r e c o g n i s e d and t h e i r a c t u a l amplitudes are not c e r -t a i n due to the poor frequency response o f the t r a n s d u c e r systems used i n the p a s t . 1.2. O b j e c t i v e In o r d e r to b e t t e r understand the nature o f the p r e s -sure f l u c t a t i o n s on b u i l d i n g s , the p r e s e n t work i n v e s t i g a t e s the f l u c t u a t i o n s on a two dimensional square p r i s m a t i c c y l i n d e r r e p r e s e n t i n g a s i m p l i f i e d square b u i l d i n g b l o c k ; p a r t i c u l a r upstream e f f e c t s are used v i z . , smooth flow, g r i d produced t u r b u l e n c e and a wake g e n e r a t i n g r o d . L o c a t i o n s on the b u i l d -i n g f o r v a r i o u s wind d i r e c t i o n s where p o s s i b l e g l a s s f a i l u r e c o u l d occur are i d e n t i f i e d , and a t these l o c a t i o n s , p r o b a b i l i t y d i s t r i b u t i o n s and s p e c t r a of the f l u c t u a t i o n s are o b t a i n e d to develop a technique f o r p r e d i c t i n g peak n e g a t i v e v a l u e s o f Cp, the non-dimensional p r e s s u r e c o e f f i c i e n t . 6 CHAPTER 2 DESCRIPTION OF THE APPARATUS AND EXPERIMENTS 2.1. Wind-Tunnel A l l the experiments were conducted i n the U.B.C. low speed, low t u r b u l e n c e , c l o s e d r e t u r n type wind t u n n e l i n which the v e l o c i t y can be v a r i e d from 3 f . p . s . (0.91 m/s) to 150 f . p . s . (45.7 m/s) with an i n h e r e n t u n d i s t u r b e d t u r b u l e n c e l e v e l o f l e s s than 0.1%. Three screens smooth the flow a t the entrance o f the s e t t l i n g chamber and a 7:1 c o n t r a c t i o n a c c e l e r a t e s the flow, improving i t s u n i f o r m i t y as i t reaches the t e s t s e c t i o n . The t e s t s e c t i o n i s 9 f e e t (2.74 m) lo n g w i t h a c r o s s - s e c t i o n o f 36 i n . (0.91 m) by 27 i n . (0.69 m). Four 45° f i l l e t s d e c r e a s i n g from 6 i n . (0.15 m) a t the upstream to 4.75 i n . (0.121 m) a t the downstream end o f f s e t the e f f e c t o f boundary l a y e r growth i n the t e s t s e c t i o n . The t u n n e l i s powered by a 15 HP d i r e c t - c u r r e n t motor, d r i v i n g a commercial a x i a l flow fan wit h T h y r i s t o r speed c o n t r o l . F i g . 2 shows the o u t l i n e o f the t u n n e l . 2.2. The Model The model used i n t h i s study was a square p r i s m a t i c c y l i n d e r , measuring 3 i n . by 3 i n . (0.076 m) , which v e r t i c a l l y spanned the 27 i n . t e s t s e c t i o n o f the wind t u n n e l . The model.then 7 represents the cross-section of a square b u i l d i n g of moderate dimensions on a scale of about 1:400. This model was mounted on an Aerolab pyramidal s t r a i n gauge balance, which i s designed to support the model i n the wind tunnel and vary i t s yaw angle over a 360° range with a pr e c i s i o n of 0.1°. The model orient a t i o n was such that, at an angle of incidence, a, of zero degrees, face A was the windward face normal to the flow, faces B and D were p a r a l l e l to the flow d i r e c t i o n , and face C was the leeward face. F i g . 3 defines a and shows the pressure tappings provided at the mid-span of the model: seven equidistant pressure tappings on face A, one each at the mid-sections of faces B and C, and eleven on face D. 2.3. Turbulence Generating Grid The square mesh g r i d employed i n generating turbulence was the larger of the two square mesh grids designed and used by Laneville (9). The g r i d i s made of wood with rectangular bars of dimensions -2.25 i n . (0.057 m) by 0.75 i n . (0.019 m) and mesh size of 9 i n . (0.23 m) . The properties of the turbu-lence generated downstream of t h i s g r i d were thoroughly i n v e s t i -gated by L a n e v i l l e (9) and F i g . 4a and 4b show his d i s t r i b u t i o n U u 1 of |j and — respectively across the tunnel dowstream of the mesh g r i d . As a check, the turbulence l e v e l was measured i n the tunnel without the model but at the p o s i t i o n occupied by 8 the pressure tappings on face A of the model. Values were found to be 14% with the g r i d and 0.02% without the g r i d . The p o s i -tion of the pressure taps on face A was measured to be 54.9 i n . (1.39m) downstream of the g r i d . The r a t i o of t h i s distance to the g r i d bar si z e i s 24.40 and i n t e r p o l a t i n g Laneville's data the turbulence i n t e n s i t y at t h i s p o s i t i o n i s shown as 15.7%. i n moderate aareement with presently measured values. For use with the a r i d , a c a l i b r a t i o n of the dvnamic head. q. was performed.usincr two p i t o t s t a t i c tubes i n the tunnel without the model. One Ditot tube was positioned up-stream of the gri d i n the contraction chamber and the second p i t o t s t a t i c tube was mounted at the position- of the pressure taps on the model face A. The c a l i b r a t i o n curve i s shown i n F i g . 5. In subsequent measurements, the second tube was re-moved and the e f f e c t i v e dynamic: pressure i n the test section inferred from the curve of F i g . 5 and the reading of the up-stream probe. 2.4. Localised Turbulence I t has been suggested that only the turbulence close to the upstream stagnation l i n e of a b l u f f body i s e f f e c t i v e i n modifying i t s flow pattern and t h i s suggests the use of a thi n wake produced by a narrow rod located on the upstream centre l i n e . The concept was suggested and v e r i f i e d by Gartshore (10). This experimental technique, which emphasises 9 the importance o f nearby upstream s t r u c t u r e s when c o n s i d e r i n g the flow around b u i l d i n g s i n the n a t u r a l wind, c o u l d i n d i c a t e whether such upstream s t r u c t u r e s are r e s p o n s i b l e f o r p r o d u c i n g h i g h frequency n e g a t i v e p r e s s u r e s . Since v o r t e x s t r e t c h i n g around the b u i l d i n g may be o f p a r t i c u l a r importance here, a wake was used which c o n t a i n e d v o r t i c e s w i t h axes normal to the square c y l i n d e r a x i s . T h i s was accomplished by mounting a rod d e s c r i b e d below which shed a Karman v o r t e x s t r e e t t o p r o -v i d e v o r t i c e s . A wake g e n e r a t i n g rod o f 0.25 i n . (0.635 cm.) diameter was mounted h o r i z o n t a l l y 10 rod diameters upstream o f model face A at a = 0°. Wind t u n n e l speeds, of, approximately 35 f . p . s , (10.7 m/s) were used f o r a l l t e s t s , so t h a t the rod Reynolds number (Re) was 5200. The v e l o c i t y and t u r b u l e n c e i n t e n s i t y p r o f i l e s 10 and 16 rod diameters downstream o f the rod, without the square model i n p o s i t i o n , were measured and are shown i n F i g s . 6 and 7. From the former p l o t , the v e l o c i t y d e f i c i t i n the wake i s observed and a l s o the s i m i l a r i t i e s i n the v e l o c i t y p r o f i l e s 10 and 16 rod diameters-downstream are q u i t e v i s i b l e . As would be expected, the v e l o c i t y p r o f i l e s are symmetrical. The l a t t e r p l o t shows the s i m i l a r i t i e s f o r the d i s t r i b u t i o n of the r e l a t i v e l o n g i t u d i n a l t u r b u l e n t i n t e n s i t i e s downstream o f the r o d . F i g . 7(a) shows the l o n g i t u d i n a l t u r b u l e n c e s p e c t r a a t 10 and 16 r o d diameters downstream o f the rod, measured a t the rod c e n t r e l i n e . Two 10 peaks are observed, the second peak occurs a t twice the f r e -quency o f the f i r s t peak, y i e l d i n g S t r o u h a l numbers e q u a l t o 0 . 2 0 and 0 . 3 8 r e s p e c t i v e l y . At s i x t e e n diameters downstream o f the rod, the two peaks are not so obvious, s u g g e s t i n g t h a t f u r t h e r downstream, the e f f e c t o f the v o r t e x shedding frequency d i m i n i s h e s . 2 . 5 . P r e s s u r e Measuring System In the p a s t , a t y p i c a l t e s t procedure has been the i n s t r u m e n t a t i o n o f a model w i t h p r e s s u r e t a p s . Each p r e s s u r e tap was connected v i a a l e n g t h o f p l a s t i c t u b i n g t o a s c a n i v a l v e c o n t a i n i n g a pr e s s u r e t r a n s d u c e r . Such a system i s convenient f o r measuring the mean and the root-mean square p r e s s u r e s . However, such a system cannot be used to measure the i n s t a n -taneous p r e s s u r e s d i r e c t l y because of the t u b i n g frequency response. A t c e r t a i n f r e q u e n c i e s the t u b i n g w i l l a t t e n u a t e the magnitude of the instantaneous p r e s s u r e and may a m p l i f y i t a t o t h e r f r e q u e n c i e s . Consequently, to enable a more accurate measurement o f the f l u c t a t i n g p r e s s u r e s , a p r e s s u r e measuring system w i t h a broader range o f a c c e p t a b l e frequency response was developed. For the mean pr e s s u r e s and the r o o t -mean square p r e s s u r e v a r i a t i o n s on the model f a c e s , a s c a n i -v a l v e system was used however as d e s c r i b e d i n s e c t i o n 2 . 7 . A f t e r i n v e s t i g a t i n g the frequency response o f v a r i o u s l e n g t h s o f t u b i n g o f d i f f e r e n t diameters, the f i n a l system 11 employed here c o n s i s t e d o f a Model 237 S e t r a systems p r e s s u r e transducer connected v i a a p l a s t i c cap to a p i e c e o f p l a s t i c t u b i n g 1.50 i n . (0.038 m) l o n g , w i t h i n t e r n a l diameter 0.045 i n . (0.00114 m) and w i t h e x t e r n a l diameter o f 0.062 i n . (0.00157 m) . The o t h e r end o f the t u b i n g was connected t o the i n s i d e o f a p r e s s u r e tap on the model. The arrangement i s shown i n F i g . 8. To o b t a i n the frequency c a l i b r a t i o n o f t h i s system, the f o l l o w i n g equipment was used: 1. 1" - 4146 B&K Microphone 2. V i n y l E a r p i e c e adaptor (B&K) 3. Earphone - Donavox 4640-57 (B&K) 4. S e t r a Transducer - Model 237 5. Wavetek F u n c t i o n Generator 6. Measuring TAmplifier 7. D i g i t a l Voltmeter: S o l a r t r o n Time Domain A n a l y s e r 8. O s c i l l o s c o p e The p r e s s u r e measuring system was c a l i b r a t e d i n i t i a l l y a t v a r i o u s f r e q u e n c i e s a t c o n s t a n t p r e s s u r e through a l a r g e ex-t e r n a l volume glued over a p r e s s u r e tap. The experimental s e t -up i s shown i n F i g . 9. V a r i o u s l e n g t h s o f c o n n e c t i n g tube were c a l i b r a t e d , as shown i n F i g . 10. The c a l i b r a t i o n w i t h the 1.50 i n . long t u b i n g i s shown i n F i g . 11. To reduce the peak which occurs a t 740 Hz. a simple e l e c t r i c a l f i l t e r was used, t h a t c ontained i n the DISA A u x i l i a r y u n i t Type 55D25. The 12 output from the t r a n s d u c e r was passed through the Low Pass f i l t e r s e t a t v a r i o u s f r e q u e n c i e s and the High Pass f i l t e r shut out. The optimum s e t t i n g o f the LP f i l t e r was chosen as 1 KHz. T h i s r e s u l t e d i n an a c c e p t a b l e frequency response to a p p r o x i -mately 900 Hz. w i t h i n a ±3 ab. range. I f the v e l o c i t i e s used i n the t e s t are r e p r e s e n t a t i v e of those p r e s e n t i n a f u l l s c a l e wind, and the l e n g t h s c a l e o f the model i s 1/400 o f the f u l l s c a l e case, the frequency response o f 900 Hz. i s e q u i -v a l e n t t o f u l l s c a l e measurements o f about 2 Hz., some fou r times the frequency o f peak gusts d e s c r i b e d by Cermak and Sadeh (5) . The s t a t i c c a l i b r a t i o n o f the p r e s s u r e measuring system was accomplished w i t h a Lambrecht manometer. The c a l i b r a t i o n curve i s shown i n F i g . 12, and the system s e n s i t i v i t y i s 5.97 p s f / v o l t . Care was taken not to exceed the l i n e a r range o f the t r a n s d u c e r (±2.5 v o l t s ) . The t u n n e l s t a t i c p r e s s u r e , p 0 , was measured a t about 12 i n . (0.31 m) upstream from the model a t the t u n n e l s i d e w a l l . T h i s same s t a t i c p r e s s u r e was connected to the t r a n s d u c e r r e f e r -ence p r e s s u r e tap, so t h a t the t r a n s d u c e r always sensed the r e l a t i v e p r e s s u r e , ( p - p 0 ) . 2.6. Data A c q u i s i t i o n For the measurement o f f l u c t u a t i n g p r e s s u r e s , the s i g n a l from the p r e s s u r e measuring system was d i g i t i s e d by a PDP 11/10 minicomputer. The sampling r a t e was s e t a t 2000 Hz 13 and three one second samples were taken a t each s e t t i n g o f the angle o f i n c i d e n c e f o r p r e s s u r e taps 1 and 4. T h i s t h r e e i second sample c o n s i s t i n g o f 6000 readings would correspond t o a 20 minute sample i n f u l l s c a l e on a s c a l e o f 1:400. The s t a t i s t i c s developed f o r each sample were as f o l l o w s : 1. Cp = mean pre s s u r e c o e f f i c i e n t mean 2. c P r m s = r o o t mean square p r e s s u r e c o e f f i c i e n t 3. Cp . = maximum pre s s u r e c o e f f i c i e n t observed maximum c P - Po rmax HP u 2 4 * C pminimum = m i n i m u m p r e s s u r e c o e f f i c i e n t observed pmin " p ° hp u 2 5. G(+): the l a r g e s t gust f a c t o r = (Cp - Cp ) / 3 • 3 rmax rmean Cp ^rms 6. G ( - ) : the lowest gust f a c t o r = (Cp . - Cp ) / 3 ^min rmean ' Cp ^rms At p o s i t i o n s where the Cp was r e l a t i v e l y h i g h , and/or the Cp . . was r e l a t i v e l y low, and/or the gust f a c t o r s were minimum 3 l a r g e and p o s i t i v e or n e g a t i v e , a f u r t h e r s e t of s t a t i s t i c s were developed. T h i s s e t was e s s e n t i a l l y a p r o b a b i l i t y d i s t r i -b u t i o n histogram (or frequency d i s t r i b u t i o n ) where three one-second samples were d i g i t i z e d and then analysed t o y i e l d the number o f readings whose amplitudes were w i t h i n a s p e c i f i e d 14 bandwidth o f 2 or 4 "computer" v o l t s depending on the s i g n a l a n alyzed. The concept o f "computer" v o l t s i s used f o r con-venience. The computer samples i n the b i - p o l a r mode w i t h i n a range o f ±2.5 v o l t s . T h i s s i g n a l when d i g i t i z e d g i v e s a computer output i n the range from 0 t o 1023 "computer" v o l t s . Hence 1 "computer" v o l t corresponds t o 4.89 mV. (see F i g . 13) or about 0.029 p s f ( 2.7% o f q ) . From the histograms so o b t a i n e d , f o r the sum o f the 3 samples, the p r o b a b i l i t y d e n s i t i e s were c a l c u l a t e d , and the data t e s t e d t o i d e n t i f y the type o f d i s t r i b u t i o n (e.g. Gaussian, W e i b u l l , Log^Normal, e t c ) . At these same p o s i t i o n s where the p r o b a b i l i t y d e n s i t i e s were e v a l u a t e d , pressure s p e c t r a were ob t a i n e d u s i n g a B&K Type 1614, 1/3 octave band f i l t e r and the S o l a r t r o n Time-Domain A n a l y z e r . 2.7. S c a n i v a l v e Measurements Pr e s s u r e measurements were made u s i n g a d i f f e r e n t i a l p r e s -sure t r a n s d u c e r , a Statham PDCR 22, i n the s c a n i v a l v e . The p r e s s u taps were connected v i a 24 i n . (0.61 m) tubes o f i n t e r n a l diameter 0.045" -(0.0011 m) and e x t e r n a l diameter 0.0625" (0.0016 m) to the scar v a l v e . The mean and the r o o t mean square p r e s s u r e s were ob t a i n e d u s i n g the S o l a r t r o n Time-Domain A n a l y s e r . S i n c e the model was a l r e a d y mounted on the Aerolab pyramidal s t r a i n gauge b a l a n c e , drag f o r c e measurements were a l s o made. The mean ,1 15 pressure c o e f f i c i e n t s were c o r r e c t e d f o r blockage u s i n g the method suggested by M a s k e l l (11). 16 CHAPTER 3  RESULTS AND DISCUSSION I Test s were conducted on the square p r i s m i n uniform 4 smooth flow and i n homogenous t u r b u l e n t flow a t Re = 7 x 10 4 and Re = 8.9 x 10 r e s p e c t i v e l y . In the case o f the wake genera-t i n g r od, the Re = 5200 i s based on the r o d diameter. 3.1. Mean P r e s s u r e s , Forces The mean p r e s s u r e , Cp , d i s t r i b u t i o n s f o r angles o f ^ 'mean y i n c i d e n c e a = 0 ° , 10°, 14° and 20° were o b t a i n e d i n smooth flow, and f o r a = 0 ° , 5°, 9° and 15° i n g r i d t u r b u l e n c e . The data i s shown i n F i g s . 14 and 15. A l s o o b t a i n e d i s the base pr e s s u r e v a r i a t i o n f o r 0° < a < 90°, which i s shown i n F i g . 16. Since the model was a l r e a d y mounted on the Aerolob S t r a i n Gauge Balance, the v a r i a t i o n o f C Q w i t h a was o b t a i n e d and i s shown i n F i g . 17. The C Q i s based.on a constant area and not the f r o n t a l a r ea. A l l these r e s u l t s have been c o r r e c t e d f o r wind t u n n e l blockage u s i n g M a s k e l l ' s c o r r e c t i o n s . The above r e s u l t s i n d i c a t e the e f f e c t s o f angle of i n c i d e n c e and the tu r b u l e n c e i n t e n s i t y on the mean p r e s s u r e d i s -t r i b u t i o n and the drag on the prism. When the prism i s r o t a t e d w i t h r e s p e c t t o the flow, the base p r e s s u r e r i s e s i n smooth flow and the drag decreases. The drag continues t o f a l l o r , i n t u r -b u l e n t flow s t a y s constant u n t i l the minimum drag p o s i t i o n i s reached, and then g r a d u a l l y r i s e s . A l s o , an i n c r e a s e i n the angle o f i n c i d e n c e causes a b e t t e r p r e s s u r e r e c o v e r y on the s i d e f a c e s , l e a d i n g t o the r e d u c t i o n i n the mean drag. The e f f e c t o f i n c r e a s i n g the t u r b u l e n c e i n t e n s i t y i s very much l i k e t h a t o f i n c r e a s i n g the angle o f i n c i d e n c e a t s m a l l a. I t causes more p o s i t i v e p r e s s u r e s t o occur on the s i d e f a c e s , produces a r i s e i n the base p r e s s u r e , l e a d i n g t o a r e d u c t i o n i n the drag. The tu r b u l e n c e a l s o causes the C p o s i t i o n t o min occur a t s m a l l e r angles o f a t t a c k . The e f f e c t o f t u r b u l e n c e on the drag i s more pronounced a t s m a l l e r a. For 20° < a < 76°, the C D i s almost the same i n smooth and t u r b u l e n t flows, as i s e v i d e n t from F i g . 17. T h i s i s due to the f a c t t h a t i n the range 20° <..:ot < 76°, the shear l a y e r s s e p a r a t i n g from the sharp corner have such a l a r g e r a d i u s o f c u r v a t u r e t h a t they cannot impinge on the leeward model fa c e s and r e a t t a c h , even when the on-coming flow i s t u r b u l e n t . A t a = 0° i n smooth flow, the c o r r e c t e d mean p r e s s u r e v a r i a t i o n on the windward face compared w e l l w i t h r e s u l t s from Parkinson's, wake source model (12) as shown i n F i g . 18. A l s o at a = 0 ° , the measured value o f the drag f o r c e was compared wit h the va l u e obtained by i n t e g r a t i n g the mean pre s s u r e d i s t r i -b u t i o n on the model and agreement was found t o be w i t h i n 5%. The p h y s i c a l mechanism i n v o l v e d i n the e f f e c t s o f t u r -bulence and the angle o f i n c i d e n c e on the mean pre s s u r e f i e l d o f the model have been p r e v i o u s l y e x p l a i n e d by Gartshore (10) and Lee (13) . 18 3.2. Fluctuating Pressure Measurements Using pressure taps #9 and #6 i n the three types of upstream flows, an i n i t i a l survey was made to i d e n t i f y positions of a where large values of I c p .1 and Cp occur. These 1 rmin 1 rms positions are l i s t e d i n Table I and Table II for taps #9 and #6 respectively. The data from the survey revealed that, with a few exceptions, the l i k e l y positions of a where possible glass and cladding f a i l u r e could occur l i e i n the wake region, i . e . 90° < a < 270°. The Cp v a r i a t i o n with a obtained rmean from the survey runs i s shown i n F i g . 19 for tap #9 and F i g . 20 for tap #6. When the pressure tap i s i n the wake zone, i . e . 80° < a < 300°, the Cp o s c i l l a t e s with a. This phenomenon ' 'mean occurs i n a l l the three flow regimes, and for both pressure taps. The c P m e a n observed i n the presence of the wake generating rod i s s l i g h t l y lower than that observed i n smooth flow. This i s expected and can be attributed to the v e l o c i t y d e f i c i t i n the wake of the rod. The o s c i l l a t i o n of Cp with a appears mean to "damp" out i n gr i d turbulence flow, i . e . the number of "cycles", as i t were, of the o s c i l l a t i o n with a are reduced. These plots again indicate the e f f e c t of turbulence on the Cp v a r i a t i o n , i . e . the negative Cp observed i n smooth ^raean 3 means flow are more negative than the Cp at the same a i n g r i d means turbulent flow, or i n other words, turbulence produces a better pressure recovery. 19 TABLE I : P o s i t i o n s o f a f o r Pr e s s u r e Tap #9 where p o s s i b l e c l a d d i n g damage c o u l d o c c u r . a 0 Type o f Flow Cp Cp Cp . G(+) G(-) mean rms min. 90 SMOOTH -1.9 7 0.90 -4.5 110 -1.69 -3.4 -4.5 130 -1.91 -5.0 160 -5.3 290 -1.81 3.4 90 GRID -1.46 -3.8 -4.4 110 -1.55 -3.8 -4.3 160 -1.26 -5.5 200 -1.22 -5.6 290 0.52 300 -5.9 15 ROD 6.2 -5.0 90 -2.00 130 -1.99 -6.0 180 -1.75 -5.6 290 4.5 TABLE I I : P o s i t i o n s o f a f o r Pressure Tap #i S where p o s s i b l e c l a d d i n g damage c o u l d o c c u r . Type o f Flow Cp ^mean Cp rms Cp . min. G(+) G(-) 90 SMOOTH -1.87 0.81 -4.2 130 -1.76 -4.5 140 -1.67 -7.2 160 -1.45 -5.4 180 -1.81 0 GRID 4.6 90 -1.64 0.51 -4.0 -4.5 160 -5.8 180 -5.1 0 ROD 4.4 -4.9 70 -4.2 80 -1.95 -3.1 90 -2.13 0.74 -4.8 130 -4.2 -6.7 Note: 1) values o f the v a r i o u s parameters l i s t e d are a t value s o f a f o r which r e l a t i v e l y l a r g e v a l u e s were observed as compared w i t h v a l u e s o b t a i n e d f o r o t h e r a i n the range 0 ° < a < 360°. 2) ho blockage c o r r e c t i o n s have been made t o data i n the above Tables I and I I . 20 The data i n Tables I and I I show the range o f the mag-nitudes f o r the Gust f a c t o r s a t l o c a t i o n s on the b u i l d i n g f o r v a r i o u s d i r e c t i o n s o f the i n c i d e n t wind where g l a s s on c l a d d i n g damage i s most l i k e l y t o o c c u r . The maximum p o s i t i v e Gust f a c -t o r l i e s i n the range 3 < G(+) < 7, the maximum observed b e i n g 6.2. The minimum ne g a t i v e gust fiactor l i e s i n the range -8 < G(-) < -4, the minimum observed value b e i n g -7-2. For the same angle o f i n c i d e n c e a and l o c a t i o n on the model, the data i n the t a b l e s i l l u s t r a t e s the e f f e c t o f the G r i d and rod p r o -duced t u r b u l e n c e p r o f l i e s . For i n s t a n c e , f o r p r e s s u r e tap #9 a t a = 130° i n smooth flow, the observed v a l u e o f Cp i s 'mean -1.91 and G(-) i s -5.0. However, i n presence of the wake-rod, the c P m e a n becomes s l i g h t l y more ne g a t i v e to -1.99 and G(-) becomes -6.0. For the same case o f i n c i d e n t flow f o r tap #6, the G(-) decreases to -6.7 i n presence of the rod from -4.5 i n i smooth flow. T h i s i l l u s t r a t e s the f a c t t h a t the t u r b u l e n c e produced on a b u i l d i n g by s m a l l upstream s t r u c t u r e s can exag-gerate the p r e s s u r e f l u c t u a t i o n s on the b u i l d i n g i n q u e s t i o n . At the same p o s i t i o n s of a t a b u l a t e d i n Tables I and I I , f o r p r e s s u r e taps #9 and #6 r e s p e c t i v e l y , p r e s s u r e s p e c t r a were o b t a i n e d . At a few o f these p o s i t i o n s , the s p e c t r a r e -v e a l e d h i g h values o f p r e s s u r e energy a t h i g h f r e q u e n c i e s , o f t e n p a s t the S t r o u h a l number o f 5 (here St = Reduced f cl Frequency•= — , where f = frequency, Hz., d = c h a r a c t e r i s t i c dimension of the model, i n t h i s case, the width; and U = free-stream wind speed). For tap #9: i n g r i d t u r b u l e n c e a t a = 15° and a = 2 9 0 ° , F i g . 21 shows the h i g h energy l e v e l s p a s t S t = 5.0. S i m i l a r o b s e r v a t i o n s were made f o r Tap #6 i n g r i d turbulence a t a = 9 0 ° and i n rod t u r b u l e n c e a t a = 70° which are shown i n F i g . 22. T h i s should be borne i n mind when d e s i g n i n g a pr e s s u r e measuring system w i t h a broad frequency response. F i g . 23 shows the s p e c t r a o b t a i n e d a t ot h e r p o s i -t i o n s ; energy peaks were found a t around S t = 0.12 and are very t y p i c a l . The p r o b a b i l i t y d i s t r i b u t i o n s o f the Cp amplitudes o b t a i n e d a t v a r i o u s p o s i t i o n s o f a f o r taps #9 and #6 are p r e -sented and d i s c u s s e d i n the next chapter. 3.3. Bandwidth A n a l y s i s An experiment was d e v i s e d t o i n v e s t i g a t e the v a r i a t i o n o f Cp . w i t h frequency, the o b j e c t i v e b e i n g t o determine mm. whether a p a r t i c u l a r frequency was a s s o c i a t e d w i t h the occurrence o f the minimum value of Cp . . T h i s was accomplished by min. p a s s i n g the e l e c t r i c a l s i g n a l from the pressure t r a n s d u c e r through the B&K 1/3 Octave Bandwidth A n a l y z e r b e f o r e d i g i t i z a t i o n i n the PDP 11/10. At each Bandwidth s e t t i n g : the sampling r a t e was a d j u s t e d to be f i v e times the centre frequency f o r a t o t a l o f 2000 samples. As b e f o r e , the PDP 11/10 p r i n t e d out the C prms'• C pmax' C p m i n . ' G ^ + * ' G ^ - ) and the power s p e c t r a l d e n s i t y w i t h i n the bandwidth. The r e s u l t i n g p r e s s u r e s p e c t r a and the C p m i n v a r : L a t : i - o n were then p l o t t e d on the same graph a g a i n s t the S t r o u h a l number. T y p i c a l r e s u l t s are shown i n F i g s . 24 t o 28. A l l the p l o t s r e v e a l t h a t the Cp . v a r i a t i o n w i t h r min. frequency f o l l o w s the pr e s s u r e spectrum, w i t h the minimum values o f Cp . o c c u r i n g near the S t r o u h a l number where the min. spectrum a t t a i n s the maxima. However, the magnitude o f j ^ ) ^ o c c u r i n g a t spectrum peaks i s lower than the c P m ^ n measured fo r the t o t a l s i g n a l , and t h i s i s shown on the p l o t s . T h i s i m p l i e s t h a t the o c c u r r e n c e o f these f l u c t u a t i n g p r e s s u r e s i s not simply due to a vo r t e x shedding phenomena s i n c e i t i s a f f e c t e d by f r e q u e n c i e s not c l o s e t o the shedding frequency. The o t h e r l i k e l y p o s s i b i l i t y i s t h a t superimposed onto the v o r -tex shedding p r o c e s s , i s a "broad-band" i n t e r m i t t e n t process i n v o l v i n g the motion o f the reattachment p o i n t along the s i d e w a l l s o f the model. As the unsteady shear l a y e r s e p arates from the corners and r e a t t a c h e s , the r a d i u s o f c u r v a t u r e of the shear l a y e r changes and the c a v i t y under the shear l a y e r can become vented. T h i s , or a s i m i l a r process must be pr e s e n t i n the wake r e g i o n , i . e . 80° < a < 300°, and must produce a changing " o f f s e t " i n the p r e s s u r e f i e l d , hence r e s u l t i n g i n the Cp . being more n e g a t i v e f o r the t o t a l s i g n a l than the (Cp . ) • 3 3 r rmm. bw observed a t , or near, the v o r t e x shedding frequency. T h i s s e t o f experiments again produced the pr e s s u r e s p e c t r a , and as had been observed b e f o r e , r e v e a l e d the p o s i t i o n s of a f o r taps #9 and #6where h i g h p r e s s u r e energy l e v e l s o c c u r r e d h i g h e r f r e q u e n c i e s . T h i s c o n f i r m s , t h a t t o i n v e s t i -gate h i g h frequency p r e s s u r e f l u c t u a t i o n s , the p r e s s u r e measuring system s h o u l d have a broad response, p a s t a t l e a s t the St = 6.0. F i g . 28 shows a spectrum f o r tap #6. f o r a = 70° i n r o d tu r b u l e n c e , i n t e r e s t i n g from the energy content p o i n t o f view: the energy spectrum continues t o r i s e up t o the c a p a b i l i t y o f the p r e s e n t p r e s s u r e measuring system. A r e s u l t o f having measured the Cp and the Cp mean rrms u s i n g the two d i f f e r e n t systems v i z . , the s c a n i v a l v e system and the high-frequency response system,is t o be able t o compare the r e s u l t s . Table I I I l i s t s the values o f Cp measured u s i n g rmean ^ the two d i f f e r e n t systems. (Note: both s e t s o f Cp have been c o r r e c t e d here f o r blockage u s i n g M a s k e l l ' s c o r r e c t i o n s . ) One can conclude t h a t the Cp can be measured by e i t h e r system w i t h i n mean a f i v e percentage d i f f e r e n c e o v e r a l l . Table IV l i s t s the v a l u e s of C p r m s measured u s i n g the two systems. (Note: no blockage c o r r e c t i o n s are made t o the f l u c t u a t i n g p r e s s u r e measurements.) R e c a l l , the Cp' g measured u s i n g the hig h frequency response system i s c a l c u l a t e d f o r a one-second sample c o n s i s t i n g o f 2000 r e a d i n g s . The value's, o f C p r m g l i s t e d i n Table IV are averages o f at l e a s t t h r e e such one-second samples. The averaging time f o r the Cp measures with the s c a n i v a l v e system ranged between rms J t h i r t y t o s i x t y seconds. Comparing the r e s u l t s i n Table IV, the percentage d i f f e r e n c e r e l a t i v e t o the s c a n i v a l v e r e a d i n g i s s c a t t e r e d : from +25% t o -17%. I t i s s u r p r i s i n g t o n o t i c e t h a t the high-frequency response system measurements o f C p r m g are f r e q u e n t l y lower than the s c a n i v a l v e system. A d i f f e r e n t o f ±10% i n the measurement o f f l u c t u a t i n g q u a n t i t i e s would appear reasonable, and as such, 70% o f the readings f a l l w e l l w i t h i n t h i s e r r o r -band. The s l i g h t d i f f e r e n c e s i n the value s measured w i t h the high frequency system c o u l d be a t t r i b u t e d t o the d i p (of 1.2 db) i n the frequency response c a l i b r a t i o n as shown i n F i g . 11. TABLE I I I : Comparison o f f e r e n t systems CPmean values measured w i t h two d i f -P.T.# a° TYPE OF FLOW PRESSURE MEASURING SYSTEM s c a n i v a l v e S e t r a H i - F r PERCENT* DIFF. 9 6 0° SMOOTH 0.8771 1.000 0.925 1.026 +5.5 +2.6 9 6 3 90° SMOOTH -1.541 -1.506 -1.472 -1.667 -1.571 -1.522 + 8 .2 +4.3 + 3.4 9 6 3 180° SMOOTH -1.541 -1.636 -1.567 -1.617 -1.523 -1.429 +4.9 -6.9 -8.8 9 6 3 270° SMOOTH -1.532 -1.576 -1.593 -1.597 -1.57 -1.667 +4.2 -0.3 +4.7 * % d i f f e r e n c e as compared t o s c a n i v a l v e r e a d i n g . Note: The values o f Cp f o r both systems have been c o r r e c t e d mean f o r wind-tunnel blockage. 25 TABLE IV: Comparison o f C p r m s v a l u e s measured by two d i f f e r e n t systems. TYPE OF PRESSURE MEASURING PERCENT"1" FLOW SYSTEM DIFF. s c a n i v a l v e S e t r a 9 0° SMOOTH .0740 .0723 -2.3 6 .0155 .0207 +25 9 0° GRID .2345 .2321 -1.0 6 .2278 .2147 -5.75 9 90° GRID .4665 .4831 + 3.5 6 .5792 .5143 -11.2 9 110° SMOOTH .3816 .3581 -6.2 6 .3314 .2748 -17.1 9 290° SMOOTH .2251 .2021 -10.2 6 .1903 .1770 -7.0 * The values l i s t e d are averages o f a t l e a s t 3,1-second samples. + % d i f f e r e n c e as compared wi t h the s c a n i v a l v e system. 26 CHAPTER 4 RESULTS AND DISCUSSION I I ;  PROBABILITY CONCEPTS AND ANALYSIS (Note: M a t e r i a l i n t h i s Chapter has been e x t e n s i v e l y drawn from Ref. 14) 4.1. A p p l i e d Pressures The windows and c l a d d i n g on a b u i l d i n g are s u b j e c t e d to p r e ssure l o a d i n g t h a t v a r i e s i n magnitude. From the d e s i g n p o i n t o f view, instantaneous v a l u e s o f the p r e s s u r e and times of these o c c u r r e n c e s a r e u s u a l l y u n p r e d i c t a b l e , i . e . not d e t e r -m i n i s t i c . T h e r e f o r e , as a r u l e , the p r e s s u r e f l u c t u a t i o n s are c o n s i d e r e d to obey s t a t i s t i c a l r e l a t i o n s o n l y . A complete knowledge o f the p r e s s u r e s i g n a l as i t v a r i e s w i t h time i s not wanted and cannot be o b t a i n e d anyway. The g l a s s or c l a d d i n g f a i l u r e w i l l occur when the p r e s -sure l o a d i n g exceeds the v a l u e o f the a p p l i c a b l e f a i l u r e c r i -t e r i o n . I f d i s t i n c t l o a d i n g occurs a t a frequency t h a t i s the same order o f magnitude as the n a t u r a l frequency of the c l a d d i n g element, the dynamic response of the element to the l o a d e x c i -t a t i o n must be c o n s i d e r e d . Representing the p r e s s u r e f l u c t u a t i o n amplitude, L, i n terms o f i t s frequency of occurrence r e s u l t s i n a s t a t i s t i c a l p r o b a b i l i t y d e n s i t y f ( L ) such as t h a t sketched o n ; t h e f o l l o w i n g page. f(L) 1 27 L S i n c e the f a i l u r e s are caused by l a r g e l o a d s , i t i s most im-p o r t a n t t h a t the r i g h t hand t a i l o f the s t a t i s t i c a l model f ( L ) be an ac c u r a t e r e p r e s e n t a t i o n o f the a c t u a l l o a d f r e q u e n c i e s . For t h i s reason, the Log-Normal, Gamma and W e i b u l l d i s t r i b u -t i o n s , w i t h t h e i r i n h e r e n t f l e x i b i l i t y i n the t a i l r e g i o n s are prominently employed i n r e p r e s e n t i n g the random behaviour of L. I f the c l a d d i n g s t r e n g t h , S c, i s known, the area <£ under f(L) i n the above diagram, i n d i c a t e s the p r o p o r t i o n o f loads L with values t h a t are above the c l a d d i n g s t r e n g t h S c . To the d e s i g n e r , who knows t h a t the c l a d d i n g w i l l f a i l a t the occurrence o f the f i r s t exceedance o f S c, t h i s p i e c e o f i n f o r m a t i o n alone i s o f no v a l u e . The p r o b a b i l i t y t h a t the s t r e n g t h S c w i l l not be ex-ceeded i n n c o n s e c u t i v e independent l o a d a p p l i c a t i o n s i s giv e n as (see r e f . 14): *n ( S c } = { F < L > I L = S } n 4.1 where 28 F(L) = cumulative p r o b a b i l i t y d i s t r i b u t i o n o f the l o a d i n g , L $ (S_) = extreme value p r o b a b i l i t y f o r the l a r g e s t v alues of L — S_ I f the i n i t i a l l o a d i n g d i s t r i b u t i o n i s Normal, Log-Normal, Gamma or W e i b u l l , the Type I Extreme Value Asymptote model, F „,_„(D , can be used i n p l a c e o f equation 4.1. I t f o l l o w s t h a t a knowledge o f the i n i t i a l l o a d d i s t r i b u t i o n F(L) or f ( L ) , together w i t h the number, n, o f a p p l i e d loads L, leads t o the extreme value model, which i n t u r n p r o v i d e s the d e s i g n process w i t h the a p p r o p r i a t e i n p u t i n f o r m a t i o n on l o a d s . 4.2. Extreme Value S t a t i s t i c s as A p p l i e d to Pre s s u r e Loading  ("Theory") The diagram below shows a t y p i c a l p r e s s u r e r e c o r d as a f u n c t i o n o f time. I t i s a t o t a l l y random (not t o be confused w i t h Gaussian) s i g n a l . maximum t, sees 29 Such a pr e s s u r e s i g n a l i s d i g i t i z e d a t a d i g i t i z i n g r a t e (or a where ( t ^ - tg) i s the sampling i n t e r v a l . In the p r e s e n t study f Q i s s e t a t 2 KHz., about twice the frequency response o f the pr e s s u r e measuring system, ( t ^ - t^) i s s e t a t 1 second, and consequently n = 2000 r e a d i n g s . The d i g i t i z e d v a l u e s of the p r e s s u r e are converted i n t o the non-dimensional p r e s s u r e c o e f f i c i e n t s , Cp, and these Cp v a l u e s are s o r t e d i n t o a p p r o p r i a t e b i n s to y i e l d a frequency d i s t r i b u t i o n and a cumulative pro-b a b i l i t y d i s t r i b u t i o n , i . e . f(Cp) and F(Cp) r e s p e c t i v e l y . Now, Eq. 4.1 can be w r i t t e n as f o r any value Cp = Cp.,, as: sampling rate) f D Hz., which g i v e s a l l the s i g n i f i c a n t i n f o r -mation. We assume t h a t the number o f independent l o a d i n g s , n, i s g i v e n by $ n ( C p l ) = { F ( c P i ) }  r C p i n (4.2) where n extreme value cumulative p r o b a b i l i t y d i s t r i b u t i o n n extreme value p r o b a b i l i t y d e n s i t y F p a rent cumulative p r o b a b i l i t y d i s t r i b u t i o n I f f parent p r o b a b i l i t y d e n s i t y f u n c t i o n ' ; 30 . " . '.•(Cp) = n [ F ( C p ) ] n - 1 ^ £ E L = n t F / C p ) ] 1 1 - 1 f(Cp) Then, a t any value Cp^ a t which f a i l u r e occurs p i $ n ( C p l ) E \ ^ ( C p J d C p i s the f r a c t i o n a l p r o b a b i l i t y t h a t the maximum from n samples w i l l be a t or lower than the value Cp^. The numerical r i s k t h a t the g l a s s w i l l break i s then g i v e n by R = 1 - $ n ( C p x ) 4.3 . R = 1 - { F ( C p 1 ) } n A l t e r n a t i v e l y , F f C p ^ = (1 - R ) 1 / n 4 .4(a) I f R i s s m a l l , F(Cp,) = (1 - -) 4 .4(b) 1 n Th e r e f o r e , i f F(Cp) i s known or can be approximated by a known curve, and i f the maximum v a l u e o f the p r e s s u r e l o a d i n g Cp^ and n are known, the r i s k can be e v a l u a t e d from Eq. 4.3. A l t e r n a t i v e l y , i f the r i s k i s chosen, w i t h F(Cp) and n b e i n g known, the maximum pr e s s u r e l o a d i n g Cp^ f o r d e s i g n purposes can be e v a l u a t e d from Eq. 4 . 4 . : 4.3. A p p l i c a t i o n s T e s t s o f f i t ( i . e . curve f i t t i n g ) o f the observed cumu-l a t i v e p r o b a b i l i t y d i s t r i b u t i o n s o f the p r e s s u r e f l u c t u a t i o n s 31 are performed u s i n g the commercially a v a i l a b l e Normal and W e i b u l l p r o b a b i l i t y paper. B e s i d e s the Normal and W e i b u l l f i t s , an attempt was made to t e s t f o r Log-Normal f i t s . Data ex-h i b i t i n g a s t r a i g h t - l i n e f o r any o f . t h e s e f i t s admits the p o s s i b l i t y o f t h a t p a r t i c u l a r d i s t r i b u t i o n f o r f(Cp) (14). Such p l o t s on the p r o b a b i l i t y paper would a l s o h e l p i d e n t i f y i f one or more flow processes were a t work (14). D e f i n i n g a new v a r i a b l e as x' = (Cp - C p ^ ) mean then, the pressure f l u c t u a t i o n s are d e s c r i b e d s t a t i s t i c a l l y by F ( x ' ) . However, s i n c e (Cp - Cp ) i s both p o s i t i v e and nega-^ 'mean ^ r t i v e t h i s i s not a u s e f u l form. The v a r i a b l e x = (Cp - Cp) i s always p o s i t i v e , and as such i s a more u s e f u l v a r i a b l e , and was used here. Consequently, the p l o t s on the p r o b a b i l i t y paper show the data s h i f t e d by the observed value of Cp i n ^ ^ . 'max or d e r t o o b t a i n completely p o s i t i v e s t a t i s t i c s . The p r e s e n t measurements f i n d the F(x) from a 1 second sample d i g i t i z e d a t 2 KHz., i . e . a sample of l e n g t h T = 1 s e c , f D = 2000 Hz., y i e l d i n g ^ = 1 2 0 b and 32 f b where b = c h a r a c t e r i s t i c model dimension = 3 i n . U = t u n n e l wind speed 30 f p s . n NOTE: the product (^-) and (-^—) i s equal to n. 4.3. (a) P o s i t i v e - G o i n g Cp R e s u l t s . «! mean For 270° < a £ 360° and 0 £ a < 90°, f o r both taps #9 and #6, the cumulative p r o b a b i l i t y d i s t r i b u t i o n p l o t s on Normal paper e x h i b i t e d s t r o n g Gaussian t e n d e n c i e s , f o r a l l three k i n d s of flows. A few t y p i c a l r e s u l t s are shown i n F i g s . 29-35 f o r taps #9 and #6. i n a l l the three flow s i t u a t i o n s . These p l o t s are f o r a three second accumulative sample. However, F i g . 36 shows three i n d i v i d u a l one-second samples f o r tap #6 i n g r i d generated t u r b u l e n t flow. The data s t i l l e x h i b i t s s t r o n g Gaussian tendencies i m p l y i n g t h a t the sample s i z e a p p a r e n t l y does not i n f l u e n c e the shape. However, there appear t o be d i s -c o n t i n u i t i e s i n the upper t a i l r e g i o n . These c o u l d be generated by s i n g l e readings i n the extreme ( c P m a x ~ Cp) b i n v a l u e s , o r c o u l d r e a l l y be the "second flow p r o c e s s " mentioned i n Chapter 1. Because these d i s c o n t i n u i t i e s are o n l y p r e s e n t i n the d i s t r i -b u t i o n e x t r e m i t i e s , no c o n c l u s i o n s can be drawn from them. Note t h a t they disappear i n the 3-second accumulated sample. Having i d e n t i f i e d the k i n d of d i s t r i b u t i o n f o r the regions i n which the Cp i s p o s i t i v e going, the extreme value mean s t a t i s t i c s can now be e a s i l y generated. Such Gaussian tendencies o f the p r e s s u r e amplitudes have been observed i n s i m u l a t e d and f u l l s c a l e atmospheric flows p a s t b u i l d i n g models by Peterka and Cermak (5), D a l g l i e s h (6) and Davenport (15). The Normal (Gaussian) p a r e n t d i s t r i b u t i o n f N i x ) has a shape which depends on i t s mean v a l u e , and the standard d e v i a t i o n , a . The model a l s o f e a t u r e s an "unbounded" t a i l a t N the upper and: lower extremes, t h e r e f o r e , the Type I Extreme Value asymptote can be used to evaluate the maximum and the m i n i -mum extremes of the Normal sample. Table V(a) compares the measured v a l u e s of the Cp ^ rraean and Cp _ w i t h the g r a p h i c a l estimates o f u„ and o\,, denoted rrms N . N by $ N and , o b t a i n e d from the p l o t s of the cumulative p r o -b a b i l i t y d e n s i t y f o r those p o s i t i o n s o f a f o r tap #9 e x h i b i t i n g a Normal d i s t r i b u t i o n . The v a l u e s are i n good agreement. The Normal cumulative d i s t r i b u t i o n f u n c t i o n F._(x;y„,a ) N N N cannot be expressed i n a l g e b r a i c form. Hence, the Extreme Value (E.V.) s c a l e parameter, a g , and the l o c a t i o n parameter, y g , cannot be o b t a i n e d e x p l i c i t l y . I f the sample s i z e , n, and the p a r e n t d i s t r i b u t i o n parameters y N and o"N are known however, the E.V. parameters can be e v a l u a t e d from t a b l e s . The s c a l e parameter, a g , f o r the type I asymptote model can be expressed i n terms o f i t s "Hazard-function" (14) as a e = [ H ( u n ) ] " 1 (4.5) where u i s d e f i n e d as the c h a r a c t e r i s t i c v alue i n a sample o f n c s i z e n, and u n works out to be the same as the E.V. l o c a t i o n 34 parameter, y (14) . y^ i s d e f i n e d as the q u a n t i l e X and t h e r e f o r e 1-1/n' a e = [ H C y ^ ] " 1 = [ H C X ^ ) ] - 1 (4.6) TABLE V(a) : A Comparison o f t h e Gra p h i c a l R s h i m a f P s and q N w i t h the Observed Values f o r Tap o f #9 > a° TYPE OF NORMAL FIT OBSERVED FLOW . f t 0 M Cp N N rmean VALUES Cp rms 0° 290° SMOOTH 0.87 0.06 0.9 3 -1.74 0.18 -1.75 0.073 0.18 0° 160° 290° GRID 0.98 0.22 0.93 -1.34 0.25 -1.26 -1.20 0.50 -1.24 0.23 0.25 0.50 15° 290° ROD 0.70 0.06 0.70 -1.61 0.34 -1.61 0.054 0.31 Note: y N and o"N are l i s t e d here i n terms o f Cp and the Hazard f u n c t i o n e v a l u a t e d from (14) „ , v _ f N ( X l - l / n ; y ' a )  H ( X l - l / n > " 1 - F N ( X l _ 1 / n ; y,a) 4.7 The E.V. d i s t r i b u t i o n then, i s o f the form f I , m a x ( x e ) = fI,max ( x ? W 4.8 35 I t can be shewn (14) t h a t the expected v a l u e o f the E.V. random v a r i a b l e x g i n a sample o f s i z e n i s g i v e n by the f i r s t moment o f fI,max ( x e } ' i ' e * U' (x .) = U + 0.57722 cr 4.9 l e e e A sample c a l c u l a t i o n f o r pressure tap #9 a t a = 0° i n smooth flow i s shown i n Appendix I. Table V(b) compares the observed v a l u e s o f Cp . w i t h min the expected values o f Cp . c a l c u l a t e d as d i s c u s s e d above. I t should be emphasised here t h a t the observed v a l u e o f Cp . min i s the average minimum observed i n a c o l l e c t i o n of 6000 l o a d i n g s , i . e ^ i n three samples:of 2000 l o a d i n g s each. The c a l c u l a t e d value i s the expected average minimum seen i n many s u c h . c o l l e c t i o n s of n samples. I t can be seen t h a t i n Table V(b) t h a t , w i t h the e x c e p t i o n o f two cases, the magnitude o f the C p m ^ n c a l c u -l a t e d i s always l e s s than the observed value of Cp . . In o t h e r min words, the p r e d i c t e d v a l u e would l e a d , t o a c o n s e r v a t i v e design, v a l u e . One o f the exceptions i s the case of p r e s s u r e tap #6 a t a = 7 0 ° i n the wake o f the t u r b u l e n c e g e n e r a t i n g rod. Although F(Cp) i s Normal, the c P m e a n = -0.57, and consequently, may not be d e f i n e d as p o s i t i v e - g o i n g , i n which case the a n a l y s i s should be as v. d i s c u s s e d i n S e c t i o n 4.2. The a n a l y s i s d i s c u s s e d above u s i n g the Type I asymptote model i s s i m i l a r to the one put forward by Davenport (16). Davenport performs h i s a n a l y s i s i n the "frequency domain", and 36 as such r e q u i r e s s p e c t r a l i n f o r m a t i o n . However, the pr e s e j i t a n a l y s i s r e q u i r e s o n l y the Cp ( u ) and the Cp (Q.) , .• •• /mean N rms N TU v a l u e s which are u s u a l l y p r o v i d e d t o the wind-engineer, — f b b and — f j — / t o c a l c u l a t e the expected minimum or maximum v a l u e of the pre s s u r e l o a d i n g . TABLE V(b) A Comparison o f the Observed Cp . and the C a l c u l a t e d Expected Values m i n P.T.# a FLOW TYPE V a. °N V a e y l .Pre-d i c t e d Ob-served 9 0° SMOOTH 0.21 .06 0 .420 0 .036 .423 0 .617 0.780 290° 0 .53 .18 1.16 .0464 1.19 -2.39 -2.28 0° GRID .79 .22 1.56 .0763 1.60 0 .27 0.132 290° 1.27 .49 2.98 .324 3.27 -3.20 -2.92 15° ROD .20 .06 0.41 .0546 .413 0.49 0.48 290° 1.23 .34 2 .42 .183 2 .53 -2.9 -2 .49 6 0° SMOOTH .14 .019 .206 .0001 .207 1.10 0 .994 0° GRID .82 .23 1.62 .0829 1.67 0.11 0 .388 0° ROD .15 .042 .297 .0028 .299 0.83 . 842 70° ROD .52 .16 1.08 .0383 .542 -0 .57 -1.22 CP. mm Note: 1) the expected v a l u e s have been c a l c u l a t e d f o r n = 6000 2) the values o f l l and o l i s t e d are i n terms o f (Cp - Cp) . N N  rmax ^ 4.3. (b) Negative Cp R e s u l t s — 2 — — - % e a n For o t h e r p o s i t i o n s o f a, i n the range 90° < a < 270°, the cumulative p r o b a b i l i t y d i s t r i b u t i o n o f the Cp value s p l o t t e d on Normal, W e i b u l l and Log-Normal paper f a i l e d t o r e v e a l a p a r t i c u l a r type o f d i s t r i b u t i o n . P l o t s on W e i b u l l paper f o r Taps #9, #6 and #3 i n three flows a t a = 90° and a = 130° are 37 shown i n F i g s . 37-42. A prominent f e a t u r e o f these p l o t s i s the "S" shape which might imply t h a t some k i n d o f a c e n s o r i n g process i s t a k i n g p l a c e (14). However, t h i s was r e c o g n i z e d i n the e a r l y stages o f the i n v e s t i g a t i o n , and the sampling r a t e was h a l v e d to 1 KHz. and a two-second sample ( i . e . 2000 read-ings) analysed. However, the "S" shape s t i l l p e r s i s t e d , l e a d -i n g t o the c o n c l u s i o n t h a t i f there i s a c e n s o r i n g p r o c e s s , i t must be i n h e r e n t i n the system and not due to the sampling r a t e . The attempts on Log-Normal s c a l e f o r the c.p.d. are shown i n F i g s . 43-50. Instead o f e x h i b i t i n g a s t r a i g h t l i n e , the p l o t s a l l r e v e a l one smooth curve, i n d i c a t i n g the non-Log Normal f e a t u r e s o f the d i s t r i b u t i o n . Both" P e t e r k a and Cermak (5) and Davenport (15) have suggested t h a t the t a i l s o f the d i s t r i b u t i o n of the s t r o n g n e g a t i v e p r e s s u r e s are roughly e x p o n e n t i a l i n form. Consequently, the t a i l o f the (Cp - Cp) c.p.d. was curve f i t t e d w i t h an max e x p o n e n t i a l o f the form: b F(x) = 1 - e " a x 4.10 where x i s the random v a r i a b l e (Cp - Cp). rmax c T h i s s a t i s f i e s the c o n d i t i o n t h a t the c.p.d. should aporoach u n i t y a s y m p t o t i c a l l y . Then, r e a r r a n g i n g and t a k i n g logarithms to the base 10 twice: L o g 1 0 ^~ L o g 1 0 f 1 - FJ^ = L °9 1 0 a + bL°9iox + Loaio(Loaioe* 4 , 1 1 38 The e q u a t i o n i s of the form: y = c + bLog^gX 4.12 where y = L o g 1 Q { - [ L o g 1 Q 1-F(x)]} c = L o g 1 Q a + L o g 1 0 ( L o g 1 ( ) e ) When p l o t t e d on s e m i - l o g a r i t h m i c paper, the s t r a i g h t l i n e f i t to the " t a i l " data y i e l d s (c) as the i n t e r c e p t and (b) as the s l o p e , and consequently, the equation to the low p r e s s u r e t a i l o f the d i s t r i b u t i o n i s o b t a i n e d . get From Eq. 4.4(b), r e p l a c i n g F f C p ^ with F ( x ) , we F(x) = l - e " a x b = 1 - -n l e a d i n g to e - a x b = R 4.13 n T h e r e f o r e , f o r a chosen value o f the r i s k R, g i v e n the number o f samples of independent l o a d i n g s , n, and having e v a l u a t e d the parameters "a" and "b" f o r the e x p o n e n t i a l f i t of the parent t a i l - d a t a , the d e s i g n v a l u e of x, denoted by x,, can be e v a l u a t e d from d 1/b x d = [ - | l n | ] - 4.14 39 For the p l o t s o f the parent t a i l - d a t a on semi-l o g a r i t h m i c paper, the Cp was a r b i t r a r i l y s e t by d e f i n i n g a constant Gust f a c t o r , G(+), equal to t h r e e . However, any value c o u l d have been s e t , and although the numerical v a l u e s of "a" and "b" would be d i f f e r e n t , i t was found t h a t there would be l i t t l e or no d i f f e r e n c e i n the value of x,. d F i g s . 51-53 show the p l o t s o f the t a i l data on semi-l o g a r i t h m i c paper f o r the three flow c o n d i t i o n s f o r tap #9 at v a r i o u s s e t t i n g s o f a. In almost every case, the data was w e l l - f i t t e d by t h i s form of e q u a t i o n , f o r l a r g e v a l u e s o f x. S i m i l a r r e s u l t s f o r tap #6 are shown i n F i g s . 54-56. From these p l o t s the v a l u e s o f "a" and "b" were e x t r a c t e d to o b t a i n the d e s i g n value f o r the Cp. However, two other v a l u e s are needed b e f o r e such a c a l c u l a t i o n can be made, i . e . R and n. The r i s k was chosen a t 10% and n as equal to 6000, corresponding to the number of readings i n a 3-second sample, or h equal t o 14400, corresponding to the number o f one-second p e r i o d s i n an average, say 4 hours, storm. Table VI l i s t s the values o f "a", "b", x. and • . . A l s o l i s t e d a6000 d14400 are the,corresponding values of Cp . e v a l u a t e d u s i n g the .... mm. • observed Cp and Cp v a l u e s . The v a l i d i t y o f t h i s t e c h -rmean crms . nique can be estimated by c r o s s - c h e c k i n g the c a l c u l a t e d value of the d e s i g n Cp . ... w ith the observed value of Cp . , a l s o min. m m . l i s t e d i n Table VI. N o t i c e the c a l c u l a t e d of (Cp . ) , i s r m i n d always l e s s than the observed C p m ^ n , wi t h o n l y one e x c e p t i o n , v i z . , tap #9 i n g r i d t u r b u l e n t flow a t a = 110°. In perf o r m i n g the c a l c u l a t i o n s , there was an assumed r i s k , R, o f 10% o f ex-ceeding the c a l c u l a t e d v a l u e , and t h e r e f o r e e x c e p t i o n s w i l l occur T h i s technique can e l i m i n a t e o v e r d e s i g n o f the g l a s s and c l a d d i n g f i x t u r e s . I f we d e f i n e the s a f e t y f a c t o r as V ' = ^ W d K n n n / ( C p m i n ) n = 6 0 0 0 n=6000 then Table VI shows t h a t V v a r i e s from 1.02 to 1.70. I t sh o u l d be borne i n mind t h a t t h i s depends on the e x a c t curve draw, a s u b j e c t i v e process I Having o b t a i n e d the v a r i a b l e s (a) and (b), the obvious t h i n g t o do was t o i n v e s t i g a t e i f 'a' o r *b' v a r i e d w i t h x, i . e . (Cp • „ - Cp) . F i g . 57 shows a p l o t 'a' v s . (Cp - Cp) . rmax . c max Although no s i n g l e curve r e s u l t s from the data, a few tendencies are r e v e a l e d . For very l a r g e ( c P m a x ~ Cp), the val u e o f 'a' i s sma'll, and the r e v e r s e i s a l s o t r u e , i . e . f o r s m a l l (Cp rmax Cp), the value o f 'a' i s r e l a t i v e l y l a r g e . One a l s o observes from F i g s . 48-53, t h a t f o r a = 9 0 ° , 'a' ranges from 0.002 t o 0.45 f o r tap #9, and from 0.02 to 0.22 f o r p r e s s u r e tap #6. However, a = 130°, 'a' v a r i e s form 0.37 to 0.58 f o r p r e s s u r e taps #9 and #6, w i t h the e x c e p t i o n o f the case i n G r i d t u r b u l e n t flow f o r tap #9 where a = 1.08. I t should be noted t h a t the values o f the parameter b l i e i n a f a i r l y narrow range, from 1.35 to 5.03. I t may be p o s s i b l e t o use the average v a l u e o f b i n de s i g n c a l c u l a t i o n s . 41 S i n c e a p a r t i c u l a r d i s t r i b u t i o n has not been i d e n t i f i e d f o r the negative p r e s s u r e s , and f u r t h e r a n a l y s i s has f a i l e d t o r e v e a l any exact r e l a t i o n s h i p between the u s u a l l y a v a i l a b l e C P ™ ^ „ and C p ^ „ and the e x p o n e n t i a l t a i l d i s t r i b u t i o n p a r a -mean rms c meters, o n l y the technique has. been e s t a b l i s h e d f o r the p r e -d i c t i o n method t o pro v i d e a design value f o r the Cp. However, f o r p o s i t i o n s o f a f o r p r e s s u r e taps w i t h p o s i t i v e going p r e s s u r e s , the p r e d i c t i o n method o n l y r e q u i r e s the knowledge of the Cp and Cp , values which are r o u t i n e l y p r o v i d e d rmean rrms . 2 * to the wind-engineer. TABLE V I : L i s t s the Values o f the Parameters o f the E x p o n e n t i a l Form for the T a i l o f the Non-Normal P r o b a b i l i t y D i s t r i b u t i o n s P r e s s u r e TYPE OF . X d X d Cp . min Cp . rmin Cp . mm Tap # FLOW - a b (n = 6000) (n = 14400) (n = 6000) (n = 14400) 9 90 SMOOTH .002 5.03 5.54 5.63 -4.81 -4.90 -3.47 1.40 130 .58 2.24 3.72 3.85 -4.57 -4.70 -3.86 1.20 90 GRID .45 2.'4 8 3.63 3.74 -3.64 -3.75 -3.58 1.02 110 .22 3.44 3.12 3.19 -3.22 -3.29 -3.42 0.94 130 1.08 1.35 5.58 5.91 -5.86 -6.19 -3.40 1.74 160 2.16 1.73 2.56 2.68 -3.06 -3.18 -2.60 1.20 90 ROD .009 5.00 4.14 4.21 -4.05 -4.12 -3.63 1.12 130 .48 2.22 4.10 4.24 -4.69 -4.83 -4.00 1.17 180 1.24 1.59 3.95 4.14 -4.56 -4.75 -3.90 1.17 6 90 SMOOTH .02 3.86 5.13 5.23 -4 .57 -4.67 -3.79 1.21 130 .52 2.72 3.07 3.16 -3.66 -3.75 -3.19 1.15 180 .73 2.53 2.92 3.01 -3.59 -3.68 -3.02 1.19 90 GRID: .22 2.69 4.28 4.41 -4.38 -4.51 -3.63 1.21 130 .51 2 .52 3.38 3.49 -3.80 -3.91 -2 .76 1.38 90 ROD .02 3.9 3 4.98 5.08 -4.88 -4.98 -4.35 1.12 130 .37 3.21 2. 88 2.95 -3.63 -3.70 -3.37 1.08 The equations used f o r the c a l c u l a t i o n s are: 1/b where R = 0.10 or 10% and n = 6000 or 14400 / I T R\ x, = {-- In -} a a n then x, = (-—) n=14400 a where A d Cp. = (Cp . - Cp . ) a n d M n a x r m i n = {G(+) x Cp } + Cp m a x r r m s ' m e a n For the dat a above G(+) i s s e t e q u a l t o 3.0. CHAPTER 5 43 CONCLUSIONS The o b j e c t i v e o f t h i s p r o j e c t was t o study the h i g h frequency p r e s s u r e f l u c t u a t i o n s on ,a two-dimensional square c y l i n d e r i n t u r b u l e n t f l o w s . The approach taken i n t i a l l y i n -v e s t i g a t e d the e f f e c t s o f t u r b u l e n c e on the man p r e s s u r e f i e l d . The c o n c l u s i o n s drawn i n t h i s study a r e : 1) The e f f e c t o f i n c r e a s i n g the t u r b u l e n c e i n t e n s i t y i n the flow normal t o the pris m i s t o produce a more complete p r e s -sure r e c o v e r y on the s i d e faces and a r i s e i n the base p r e s s u r e , l e a d i n g t o a r e d u c t i o n i n the mean drag. T h i s t r e n d i s e v i d e n t o n l y a t s m a l l angles o f the i n c i d e n t flow. 2) When the p r i s m i s r o t a t e d w i t h r e s p e c t t o the flow, at s m a l l angles o f i n c i d e n c e s , the base p r e s s u r e r i s e s i n smooth flow and stays n e a r l y constant i n t u r b u l e n t flow r e s u l t i n g i n the drag d e c r e a s i n g or remaining c o n s t a n t c o r r e s p o n d i n g l y . The drag f a l l s t o a minimum a t the C D p o s i t i o n , e.g. a = 14° i n min smooth flow, and then g r a d u a l l y r i s e s to an almost c o n s t a n t value between 40° < a < 50°. As the t u r b u l e n c e i n t e n s i t y i s i n c r e a s e d , the p o s i t i o n a t which drag r i s e s o ccurs a t s m a l l e r u 1 u 1 angles o f i n c i d e n c e : a = 1 0 ° f o r — = 7.3% and a = 9° f o r ^ j -- 14%. The next stage i n the i n v e s t i g a t i o n i n v o l v e d the measure-ment o f the f l u c t u a t i n g p r e s s u r e s u s i n g a p r e s s u r e measuring system w i t h a broad frequency response. The c o n c l u s i o n s drawn a r e : 44 3) the very h i g h " s p i k e s " c o r r e s p o n d i n g to -10 < G(-) < -9 observed by oth e r workers on t h r e e - d i m e n s i o n a l s t u d i e s were not seen i n t h i s two-dimensional study. 4) the p o s i t i o n s o f a f o r l o c a t i o n s on the b u i l d i n g near the centre or corners o f a f a c e , e.g. pre s s u r e taps #6 and #9, where p o s s i b l e g l a s s and c l a d d i n g f a i l u r e c o u l d occur are found mostly i n the wake r e g i o n . 5) the s p e c t r a l a n a l y s i s o f the pre s s u r e s i g n a l r e -v e a l e d t h a t a t p a r t i c u l a r l o c a t i o n s on the b u i l d i n g f o r s p e c i f i c angles o f the i n c i d e n t wind, h i g h l e v e l s o f p r e s s u r e energy occur a t h i g h e r f r e q u e n c i e s , o f t e n p a s t St = 5.0: e.g. f o r pressure tap #9 a t a = 290° i n both g r i d t u r b u l e n t flow and rod t u r b u l e n c e , and a = 70° f o r tap #6 i n rod t u r b u l e n c e . A t ot h e r p o s i t i o n s o f a, f o r both taps #9 and #6, most o f the energy i s c o n c e n t r a t e d a t the vo r t e x shedding frequency o f the model, i . e . near St = 0.12. The s t a t i s t i c a l analyses o f the p r e s s u r e f l u c t u a t i o n amplitudes r e v e a l e d t h a t 6) f o r p o s i t i o n s o f a i n the range 270° < a £ 360° and 0° £ a < 90°, where the mean p r e s s u r e are p o s i t i v e going, the p r o b a b i l i t y d i s t r i b u t i o n o f the Cp v a l u e s , s t r o n g l y e x h i b i t Gaussian t e n d e n c i e s , i . e . are Normally d i s t r i b u t e d i n the form F(Cp) = F N ( C p ; U J J / O ^ ) . Consequently, a t such l o c a t i o n s , the Extreme Value Type I asymptote model can be employed t o e v a l u a t e the expected v a l u e s o f the Cp . and Cp by having o n l y the c rmm 'max Cp and the Cp value s a v a i l a b l e . 7) For other p o s i t i o n s of a where the Cp v a l u e s c rmean are s t r o n g l y n e g a t i v e , the Cp v a l u e s are not d i s t r i b u t e d i n a Normal f a s h i o n , o r i n a W e i b u l l o r Log-Normal f a s h i o n s . Where a s p e c i f i c p r o b a b i l i t y d i s t r i b u t i o n model was not i d e n t i f i e d f o r the Cp amplitudes, an a l t e r n a t i v e method of a n a l y s i s was developed, r e v e a l i n g t h a t 8) the low pressure t a i l can be modelled by a decaying e x p o n e n t i a l o f the form b F(Cp) = 1 - e a x With such a model, the d e s i g n e r can e v a l u a t e the d e s i g n value o f the p r e s s u r e f o r a chosen exposure time n, the number o f independent pressure l o a d i n g s , and an a c c e p t a b l e value o f the r i s k R u s i n g the e q u a t i o n 1 x, = — i l n — i d a n The experiments to i n v e s t i g a t e the v a r i a t i o n o f c P m ^ n w i t h frequency y i e l d e d : 9) the C p m ^ n does vary w i t h frequency: i t f o l l o w s the i n v e r t e d p r e s s u r e spectrum w i t h the lowest value o f the C p m i n P e a ^ v a l u e * i . e . a t the vortex-shedding frequency of the model. However, the value of the Cp . observed f o r the t o t a l mxn s i g n a l i s more negative than the value observed a t the S t r o u h a l shedding frequency. 4 6 The r e s u l t s o b t a i n e d w i t h the wake g e n e r a t i n g rod i n p o s i t i o n l e d to the c o n c l u s i o n t h a t 10) upstream s t r u c t u r e s do i n f a c t cause f l u c t u a t i n g p r e s s u r e s to occur, o f t e n a m p l i f y i n g the negative p r e s s u r e s be-yond those observed without the upstream s t r u c t u r e . 47 BIBLIOGRAPHY 1) J.E. Minor and W.L. Beason 2) Herbert S. S a f f l r 3) D. Surry and B.V. Tryggvason 4) K.J. Eaton 5) J.A. Peterka and J.E. Cermak 6) J.A. Peterka and J.E. Cermak 7) W.A. D a l g l i e s h 8) J.E. Cermak and W.Z. Sadeh 9) A. L a n e v i l l e 10) I.S. Gartshore 11) E.C. M a s k e l l "Window Glass F a i l u r e s i n Wind-Storms", J l . o f S t r . Div., A.S.C.E., Jan. 1976. " E f f e c t s o f High Wind on G l a z i n g and C u r t a i n w a l l s , and R a t i o n a l Design Methods f o r G l a z i n g and C u r t a i n w a l l s " , Proceedings o f the 2nd. U.S.-Japan seminar on Wind E f f e c t s on S t r u c t u r e s , Sept. 19 74. " F l u c t u a t i n g P r essures on T a l l B u i l d i n g s " , P roceedings, 2nd U.S. N a t i o n a l Conference on Wind Engineer-i n g Research, 19 75. "Cladding and the Wind", B.R.E., U.K., CP 47/75. " F l u c t u a t i n g Wind Pressures on S t r u c t u r e s " , Proceedings, 2nd U.S. N a t i o n a l Conference on Wind E n g i n e e r i n g Research, 19 75. "Wind Pressues on B u i l d i n g s -P r o b a b i l i t y D e n s i t i e s " , J l . o f S t r . Div., A.S.C.E., June 1975. " S t a t i s t i c a l Treatment o f Peak Gusts on C l a d d i n g " , J l . of S t r . Div., A.S.C.E., Sept. 1971. "Pressure F l u c t u a t i o n s on B u i l d i n g s " , 19 71 Wind E f f e c t s on B u i l d i n g s Symposium. " E f f e c t s o f Turbulence on Wind-Induced V i b r a t i o n s o f B l u f f Bodies", Ph.D. T h e s i s , U.B.C, May 19 73. "The E f f e c t s o f Freestream Tur-bulence on the Drag o f a Rectangular Prism", Research Report BLWT-4-73, U n i v e r s i t y o f Western O n t a r i o . "A Theory o f Blockage E f f e c t s on B l u f f Bodies and S t a l l e d Wings In A C l o s e d Wind Tunnel", R.A.E. Report No. 3400, Nov. 1963. 48 12) G.V. Parkinson and T. J a n d a l i 13) B.E. Lee 14) K.V. Bury 15) A.G. Davenport 16) A.G. Davenport "A Wake Source Model f o r B l u f f Body P o t e n t i a l Flow", J l . o f F l . Mech. (19 70), V o l . 40, 577-594. " E f f e c t o f Turbulence on the Surface P r e s s u r e F i e l d o f a Square Prism", J l . o f F l . Mech. (1975), V61.-69, p t . 2., 26 3-282. " S t a t i s t i c a l Models i n A p p l i e d S c i e n c e " , p u b l i s h e d by John Wiley and Sons, 19 75. " D i s c u s s i o n t o (5): J l . o f S t r . Div., A.S.C.E..., Nov. 1976 . " A p p l i c a t i o n o f S t a t i s t i c a l Con-cepts t o Wind Loading o f S t r u c t u r e s " , Proceedings o f the I n s t i t u t i o n o f C i v i l E n g i n e e r s , V o l . 19, 1961, pp. 449-472. 49 APPENDIX I To i l l u s t r a t e the e v a l u a t i o n o f the extreme value type I model parameters f o r a parent Normal d i s t r i b u t i o n . From the c.p.d. p l o t s on Normal p r o b a i b l i t y paper we a l r e a d y have u N and cr . E.g. at a = 0°, f o r tap #1 i n Smooth flow. fi„ = 0.21 of = 0.06 N N For n = 6000: Z l - 1/n = Z0.9998 = 3 ' 4 9 f r o m T a b l e S X l - 1/n = K ( Z 1 - l/n> + ^N = 0 ' ° 6 ( 3 ' 4 9 ) + ° ' 2 1 = 0.4194 But u = X, . . y = 0.4194 e 1 - l/n ^e To c a l c u l a t e the s c a l e parameter a e / we have to e v a l u a t e the hazard f u n c t i o n . H(u e) = f N ( y e ; y N , a N ) / i - r ^ v v ^ where f ( y ^ J = - f . Q # 1 ) y e aN ' a N and ^ VVVV " F N ^  ' ^  •• N •'• H ( y e ) - 0.4194*0.06 (0 -0009)/(l-0.9998) = 0.04/0.0002 = 178.83 ."... a = tH(y e) ] _ 1 = 0.0056 50 T h e r e f o r e , the expected v a l u e o f X , i s g i v e n by e U" = y + 0.5 7722 a 1 ^e e = 0.423 R e c a l l , the p l o t t e d data had been s h i f t e d by Cp , . . the l a r g e s t expected value o f ( c P m a x ~ Cp) i s g i v e n by y|, i . e . the value o f the lowest Cp. . . Cp . = Cp - u' = 1.040 - 0.423 rmm 'max M l =0.617 An e n t i r e l y d i f f e r e n t approximation t o the d i s t r i b u t i o n o f Hofimal extremes can be o b t a i n e d as f o l l o w s (14 ) . Consider a &tan.da.fid Normal o r d e r s t a t i s t i c Z. . and i t s (n) t r a n s f o r m a t i o n i n t o another random v a r i a b l e Z^: Z 2 = n { l - F N ( Z ; 0 , l ) j , 0 £ Z 2 ^ n A l The l a r g e sample approximate p. d . f . o f Z^ i s o b t a i n e d as Y(n) ( Z 2 ) = e x P ( - Z 2 ) ' 0 ^ z 2 ^ n A 2 I t i s p o s s i b l e t o approximate Eq. A l by an a l g e b r a i c e x p r e s s i o n i n v o l v i n g Z and n: Z 2 = exp {- } A3 where • , _ 21n n - h In In n - In (2 /?) A 4 /2 In n LEAF 50a OMITTED IN PAGE NUMBERING. 50b and B« = 1 A5 •2 In n Transforming Eq. A2 from Z 2 to Z by Eq. A3, the approximate p.d.f. of the standard Normal order s t a t i s t i c Z, v = Z i s (n) obtained as ^(n) ( Z ; n ) = k> e x p { _ ^ g r 1 ~ e x p { ~ ^ T - } } A 6 which i s recognised as a type I Extreme Value model with u .= e A' , C e = B ' For the same case as above, for n = 6000, Eq. A4 and Eq. A5 y i e l d A' = 3.61 and B' = 0.24. Converting these to the non-6ta.ndafLd-16Q.d Normal s t a t i s t i c s , gives / e = X l - l / n = ° N ( A , ) + V = 0 ' 4 3 a « = 0 . 3 ' = 0.014 e N Hence, the expected value of X e i s given by V, = u + 0.57722a 1 e e = 0.438 This corresponds to an expected value of 0.602 for the Cp m^ n« This alternative method of evaluating Cp . i s i l l u s t r a t e d because the approximation based on Eq. A4 and Eq. A5 i s closest to the uppo.fi t a i l of the exact model. In applications of the extreme value s t a t i s t i c s , the upper t a i l of $ n i s usually of prime importance, so that the a-proximate model f with X 0 ITlclX parameters A 1 and B 1 i s preferred. F i g . 2: O u t l i n e o f the U.B.C. A e r o n a u t i c a l Wind Tunnel F i g . 3: ' "Diagram showing Model dim and Numbering of Pressure Taps and D e f i n i n g the Angle, o f Incidence .9 1 1 . 1 . 9 1 1.1 .9 1 1.1 0.9 1 1.1 0 . 4 .81.2 1.62.0 F i g . 4 ( a ) : D i s t r i b u t i o n o f = a c r o s s the tunnel downstream o f the 9" mesh g r i d . Ul to = 32.2 • • • • • • • J L ,1 .11 .12 = 26.9 x . b 24.4. .13 .14 .15 J L 17 .18 .19 = 16.2 • • J L&J ,23 J L 25 u' F i g . 4 ( b ) : D i s t r i b u t i o n o f — across the tunnel downstream of the 9" mesh £ g r i d . 55 Fig. 5: Calibration of q for Large Grid qupstream (BETZ) nunw 56 l U 0! CO s: o c • B B 10 ROD DIAMETERS DOWNSTREAM • 16 ROD DIAMETERS DOWNSTREAM Bm u Uo , i i i i i i i i i 1 I I I I I i I I I 0.8 0.9 1.0 Fig. 6: Diagram shows the velocity d e f i c i t in the wake o f the rod. a • a • m a am -(1.-0) ta _ » 0 J3 O c El 1 1 • • %17% %15% 2 i 1 _L 0.05 0.10 u'/Uo 0.15 F i g . 7: R e l a t i v e Turbulence I n t e n s i t y •16 ROD DIAMETERS °10 ROD DIAMETERS 0.20 —I 58 Fig. 7(a): Longitudinal Velocity Spectra Downstream of the Wake Generating Rod 1.0 .10 16 dia. down U * 33.58 t/s fSuu(f) - " * * " • A • A A-.10 dia. downstream U = 33.45'/s .01 I * I I k 1 1 1 ' 1 I I I I I 1 L 11 I I I I I I 1 I 10 100 100 f, frequency (Hz) 59 Reference Pressure Tap (Static pressure) E l e c t r i c Cable Model 2 37 Setra Systems Pressure Transducer Fig. 8: High Frequency Response Pressure Measuring System. 60 1" B&K Microphone Clamp to hold the transducer Plexiglass Cap i y of 0 . 0 4 5 " I . D . & 0 . 0 6 2 O . D . tubing Model Face 0 . 1 2 5 " thick Aluminum with Pressure Tap F i g . 9: Experimental Set-Up of the C a l i b r a t i o n of the Pressure Measuring System LIST OF EQUIPMENT 1 B&K 11 MICROPHONE 2 VINYL ADAPTOR FOR THE EARPHONE 3 B&K EARPHONE 4 SETRA TRANSDUCER 5 WAVETEK FUNCTION GENERATOR "6 MEASURING AMPLIFIER 7 DIGITAL VOLTMETER 8 DUAL BEAM SCOPE 9 PRESSURE TAP 1 1—I—I I I I I T 1 1 1 I I I | " t 2.4-1 Pig. 10: Frequency Response o f V a r i o u s Lengths o f 0.045 i n . I . D . and 0.062 i n . O.D. T u b i n g 2.0" 1.6H © © © © • O © afc= Transmission Coefficient Pressure Measured by the Transducer f the presssure i n the Calibrating Cavity Note: A l l calibrations at constant pressure i n the outside cavity of 0.44 psf. 1.2-1:0- <r 0.8^ 0.4 -I o.o 10 © © • * _ « m i ITT © © A A AA OA ® • J I I I l i l t FREQUENCY (Hz) I L_ I I I I 1 I 100 1000 A - 24" © - 1?" • - 6" • - 3" 1 1—|—T 1—r 63 64 1023 Real Volts Input Pig. 13: Input-Output Calibration of the PDP 11/10 F i g . 14: Mean Pressure D i s t r i b u t i o n on Model for Various a i n Smooth Flow l . O i O.SOY 8 * • 0 A fi A • A fi A o - J — I — I l l i I I I I • JO 11 12 13 l l i 15 1 6 1 7 1 8 1 9 20 -0.5 •1.0 •1.5 KEY • o ct-10' A ot-14* 3 * 5 6 7 8 9 -I I—l I I I I A A A A 6 fl «2 o A A A A • • • • • • 10 11 12 1 3 m 15 16 17 18 19 20 —I 1 I I I I I I I I I B A A A A A A A A A A So fi o £ # # • ft 3* 5 6 7 6 9 °g,A . 9 Si& cn Ul F i g . 15: Mean Pressure D i s t r i b u t i o n on Model f o r Various a i n Grid Turbulent Flow 1.0 Cp .50 hi 3 A A A * 9 A i ' i i2i'a A h ihA \h ih h K . o K . 5 KEY • oc » 0 * • 5* A 9 ' A 1 5 ' 3 <t S S 7 8 9 I I I I I I L. n fi • A A A 10 11 12 13 m 1516 17 18 19 20 B 3 » 5 6 7 8 9 i • ' • ' L_l_ * A A A • A • 6 A A P r e s s u r e Tap # F i g . 16i V a r i a t i o n of Base Pressure With Angle of Incidence -1.4 -1.2 *» a v U-1.0 • • s : i 9 t • % 2 ; ' I* 1 o • • • • • -0.8 0) m OJ u a « • 01 n) X l U ai f-o. <u M M O U -0.6 f | -0.2 XI a KEY • SMOOTH FLOW • GRID FLOW cn 16 24 32 40 48 56 64 . 72 80 88 C Q : C o r r e c t e d Drag C o e f f i c i e n t o _1_ o > 3 t—1 o| (B 0 Ml 3 O H-a ro 3 O ® 10 < 01 H H* Pi rt-r** O 3 O Hi rt-3* (D D M P> o o n Ml M) H-o H-ID 3 rt-rt 3" 4 • • CA -< z -J o i — •1 • 89 69 0.75 H 0.50 H 0.25 .500 -0.333 -0.167 J I I I I L -.25 J -.75 H Fig. 18: To Compare 2-D Theory and Measured Cp c on Faoe D at a = 0° i n Smooth Flow -1.25 4 .167 .333 A . 500 J I I I 1 L-L £ n A A Parkinson's Wake Source Model O Corrected Cp using Maskell's Corrections 70 71 72 73 F i g . 22: Pressure Tap #6: Revealing high energy l e v e l s at higher frequencies 1 0 ' a = 90 / G r i d 0 a = 70 /Rod 74 i o . F i g . 23: Some t y p i c a l pressure spectra KEY 1.0 fSpp(f) , . 2 .10 O a= 90°/P.T. #9/ SMOOTH • o= 130°/P.T. #9/ SMOOTH •o= 130°A>.T. #9/ GRID Oa= 90°/P.T. #9/ GRID .01 I I 1 I 10 Seduced V e l o c i t y ^ 76 77 78. 7 9 80 81 82 83 F i g . 32: C.P.D. P l o t on Normal Paper f o r Tap #9, a = 15°, Rod Turbulence 84 F i g . 33: C.P.D. P l o t on Normal Paper f o r Tap #9, a = 290°, Rod Turbulence 99.99, : ; — — 86 F i g . 35: C.P.D. P l o t on Normal Paper f o r Tap # 6 , a = 70°, Rod Turbulence 87 Fig. 36: C.P.D. Plots on Normal Paper for Tap #6, a = 0°, Rod Turbulence 99.99 99.95 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 > i £ 30 rH -H X) 20 o u 10 .5 .2 .1 .05 .01 a © A READS RON V Cp ^rnean o" CPrms 2000 B 1.17 1.03 0.28 0.23 2000 9 1.12 1.05 0.25 0.22 2000 A 1.04 1.03 0.21 0.21 B r- • nil i 1 11 I 1 1 1 1 .13 .27 . 40 0.54 . .67 . 80 • .94 . 1.20. 1.48 1.74 F i g . 37: C.P.D. P l o t on Weibull Paper f o r Tap #9, a = 90° In the three kinds of flow. F i g . 38: C.P.D. P l o t on Weibull Paper f o r Tap #9, a = 130° In the three types of Flow. Probability % 06 F i g . 41: C.P.D. P l o t s on Weibull Paper for Tap #3, a = 90 In the three Types of Flow 94 95 Fig. 44: C.P.D. Plot on Log-Normal Paper for Tap #9., a = 90°, Smooth Flow - 3 . 0 -2.0 - 1 . 0 In (Cpnnx-Cp) 1.0 1.25 96 F i g . 45: C.P.D. P l o t on Log-Normal Paper f o r Tap #9, a = 130°, Smooth Flow 99.99 \t 99.95 U 1" (Cp^-Cp) 97 98 F i g . 47: C . P . D . Slot on Log-Normal S c a l e F o r P r e s s u r e Tap # 6 , 0 = 90°, G r i d Flow 99.99 00 -u •H f-i •H XI £ o u i n (CP^-CP) 99 F i g . 48: C.P.D. P l o t on Log-Normal Scale For Tap #6, a = 160°, G r i d Turbulence 99.99 100 F i g . 49: C.P.D. On Log-Normal Scale For Tap #9, a = 180° Rod Turbulence 99.99 F i g . 51: Exponential Curve F i t t i n g (E.C.F.) to the Low-Pressure T a i l at Tap #9, Smooth Flow at Various a. Fig. 52: E.C.F. to Low-Pressure T a i l at Tap #9, Grid Turbulence at Various a Fiff. 53: E . C . F . t o Lower P r e s s u r e T a i l at Tao #9, Rod T u r b u l e n c e , at V a r i o u s a Fig. 56: E.C.F. to Low-Pressure T a i l at Tap #6, Rod Turbulence, at Various o 108 1.4 1.2 1.0 .6 F i g . 57: A p l o t of a vs. [Cp - Cp] max. A P.T.#1 P.T. #4 9 © • E A A J I I L ® L Aj_ © -0.75 1.00 - 1.25 1.50 1.75 2.00 2.25 2.50 

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