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Hydrodynamic performance of an artificial aortic valve implant 1975

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HYDRODYNAMIC PERFORMANCE OF AN ARTIFICIAL AORTIC VALVE IMPLANT by MOHAMMAD AMINZADEH B.A.Sc., University of Windsor, Windsor, Ontario, 1969 A THESIS.SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA ' January, 1975 In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s under- stood that p u b l i c a t i o n , i n part or i n whole, or the copying of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. MOHAMMAD AMINZADEH Department of Mechanical Engineering The University of B r i t i s h Columbia, Vancouver V6T 1W5, Canada Date /tffifyl> £ , /9T/^ ABSTRACT The project aims at studying f l u i d dynamics of a prosthetic a o r t i c heart valve through an organized experimental programme with a view to obtain better appre- c i a t i o n of factors a f f e c t i n g i t s performance and eventual f a i l u r e . The problem i s approached i n stages representing an increasing order of d i f f i c u l t i e s i n terms of experimental set-up and i n t e r p r e t a t i o n of data. Design, construction and c a l i b r a t i o n of the g l y c e r o l tunnel, which formed the fundamental te s t f a c i l i t y for the project, are described b r i e f l y i n the beginning followed by an introduction to the test models and monitor- ing instrumentation. A simple theory for wedge shaped hot f i l m anemometer i s developed and v e l o c i t y p r o f i l e s with and without models i n the t e s t section presented. E s s e n t i a l l y the test programme consists of three stages: (i) a single sphere by i t s e l f during stationary and o s c i l l a t i n g conditions, R n = 74 - 5838; ( i i ) a stationary spherical poppet occupying d i f f e r e n t positions i n the cage representing quasi-steady opening and c l o s i n g of the valve, R = 220 - 1200; n ' ( i i i ) poppet pulsating according to the recorded displacement-time history when implanted i n a patient, R = 290 - 650. n i i i A c a r e f u l choice of reference pressure and v e l o c i t y r e s u l t s i n presentation of data i n a manner that would permit reproduction and comparison of r e s u l t s by investigators using d i f f e r e n t test f a c i l i t i e s . The s t a t i c pressure d i s t r i b u t i o n data and r e s u l t i n g information on the average base pressure, separating shear layers, i t s movement, etc. are obtained for closed and open bypass conditions of the tunnel, the l a t t e r representative of regurgitation. An extensive flow v i s u a l i z a t i o n programme using the dye i n j e c t i o n procedure i n conjunction with s t i l l and high speed movie photography complemented the experimental data. I t resulted i n better appreciation of the physical character of the flow. Studies with a single sphere v i v i d l y showed the progress of formation, elongation and i n s t a b i l i t y of the vortex r i n g with the Reynolds number. Of p a r t i c u l a r i n t e r e s t was the d i s t i n c t r i s e i n the minimum and base pressures i n the Reynolds number range of 240 - 475, which was found to be associated with the onset of i n s t a b i l i t y of the vortex ri n g leading to i t s periodic shedding. In general, the Reynolds number e f f e c t s were confined to the region near and downstream of the minimum pressure point. The experi- mental r e s u l t s c l e a r l y suggested inherent l i m i t a t i o n s i n the a n a l y t i c a l procedures for predicting pressure d i s t r i - bution as proposed by several investigators. On the other hand, the pressure integrated drag r e s u l t s compared favour- ably with the d i r e c t l y measured drag data reported i n the l i t e r a t u r e thus substantiating the r e l i a b i l i t y of the pressure measurements. Coming to the pulsating sphere, e f f e c t s of the o s c i l l a t i o n frequency, Reynolds number and sphere p o s i t i o n are investigated during both forward and reverse strokes of a cycle. Influence of the Beta number i s confined to the l o c a l changes i n the character of the base pressure plots without s u b s t a n t i a l l y a f f e c t i n g t h e i r average magnitudes. A decrease i n y leads to increase i n the minimum and s average wake pressures together with a forward movement of the separation point during the forward stroke. The general character of the pressure p r o f i l e s remains the same during the reverse stroke, however, the magnitudes involved are markedly d i f f e r e n t . Next, the thesis discusses a more r e a l i s t i c s i t u a t i o n , from the configuration consideration, of a stationary poppet occupying d i f f e r e n t positions i n the cage. The Reynolds number influence, which i s more s i g n i f i c a n t for y^ < 0.2, i s e s s e n t i a l l y r e f l e c t e d i n the increase of the wake pressure. A s i g n i f i c a n t r i s e i n the negative pressure c o e f f i c i e n t at y^ _< 0.1 would suggest large shear- ing stresses leading to possible deformation and destruction of the red blood c e l l s . The presence of d i s t r i b u t e d v o r t i c i t y and the associated c e n t r i f u g a l f i e l d i n the wake V are suspected to be the fundamental factors promoting d i s s o c i a t i o n of the blood into i t s constituents and f i n a l l y t h e i r deposition on the body of the valve. F i n a l l y , the case of poppet o s c i l l a t i n g inside the valve i s considered which shows the e f f e c t of valve opening on C to be far greater than that observed during P the stationary case. The high negative pressure, at the pulsation frequency, i n the wake region and the associated large periodic shear stresses are l i k e l y to cause not only coagulation of the c e l l s leading to c l o t t i n g but may be responsible for rupturing of the suture l i n e s i n the implanted prosthetic valve, as often encountered i n practice. v i TABLE OF CONTENTS Chapter Page 1 . INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 A B r i e f Review of the Relevant L i t e r a - ture on the Flow Past a Sphere . . . 2.3 1.3 Purpose and Scope of the Investigation 19 2. EXPERIMENTAL APPARATUS AND CALIBRATION . . 22 2.1 Glycerol Tunnel . . . . . 22 2.2 Hot Film Anemometry 32 2.3 Pressure Transducer . . . . . . . . . 44 2.4 A o r t i c Heart Valve Model . . . . . . . 45 2.5 Pulsating Mechanism . . . . . . . . . 51 2.6 Linear Displacement Transducer . . . 57 3. TEST PROCEDURES 62 3.1 V e l o c i t y P r o f i l e s Along the Test Section of the Tunnel 62 3.2 S t a t i c Pressure D i s t r i b u t i o n on Stationary and O s c i l l a t i n g Spheres 64 3.3 Pressure D i s t r i b u t i o n on the Stationary Poppet Representing Various Valve Openings 73 3.4 Mean Pressure Measurements on the Poppet While O s c i l l a t i n g Inside the Valve 73 3.5 Flow V i s u a l i z a t i o n . . . . . . . . . 75 v i i Chapter Page 4. RESULTS AND DISCUSSION 82 4.1 Tunnel V e l o c i t y P r o f i l e s . . . . . . . 83 4.2 Choice of Reference V e l o c i t y and Pressure . 91 4.3 S t a t i c Pressure D i s t r i b u t i o n . . . . . 98 4.3.1 Stationary sphere . 98 4.3.2 O s c i l l a t i n g sphere . . . . . . . 120 4.3.3 Poppet occupying d i f f e r e n t positions i n the valve . . . . . 131 4.3.4 Poppet o s c i l l a t i o n s and Beta number 153 4.4 Conclusions 167 5. CLOSING COMMENTS 175 5.1 Concluding Remarks 175 5.2 Recommendations for Future Work . . . . 178 BIBLIOGRAPHY . . . . . . . . . . 182 APPENDIX I - SIMILARITY PARAMETERS . . . . 198 1.1 Preliminary Comments . . . . . . . . . 198 1.2 S i m i l a r i t y C r i t e r i a for A r t i f i c i a l A o r t i c Valves 200 1.3 Range of Values of Relevant Parameters i n the L i v i n g System 203 1.3.1 Average blood v e l o c i t y i n the a o r t i c root based on the cardiac index 203 1.3.2 Heart rate 205 1.3.3 Properties of blood 205 1.3.4 Physical dimensions of the Starr-Edwards a o r t i c b a l l valves 213 v i i i Chapter ' Page 1.3.5 Travel and r e s t times for the poppet 214 1.4 Range of Varia t i o n of the Dimensionless Parameters . . . . . 216 APPENDIX II - A THEORETICAL APPROACH TO THE EVALUATION OF A HOT FILM PROBE PERFORMANCE 217 II. 1 Introduction 217 11.2 Heat Transfer from a Hot Film Probe . . . . 220 11.3 Experimental V e r i f i c a t i o n and „„ f i Discussion . - i x LIST OF TABLES Table Page 1-1 Independent Variables of the Problem . . . . 201 1-2 A Selection of Dimensionless Number for the Problem 202 1-3 S i g n i f i c a n t Time Parameters for an A o r t i c Valve Prosthesis . . . . 215 1-4 Observed Values of the Dimensionless Numbers and those Used i n Experiments . . . 216 I I - l Pertinent Data for the Hot Film Probes . . . 227 II-2 Heat Transfer Parameters for Hot Film Probes . . . . . . . . . . . . 227 X LIST OF FIGURES Figure Page 1-1 Several models of prosthetic a o r t i c heart valves: (a) E a r l i e r models: 3 (i) Hufengal b a l l valve; ( i i ) Bahnson t r i l e a f l e t . v a l v e ; ( i i i ) Starr and Edwards b a l l valve; (iv) d i s c valve; (v) Hufengal b u t t e r f l y hinged l e a f l e t valve. (b) Some of the newer designs: 4 (i) s i l a s t i c t r i l e a f l e t a o r t i c valve; ( i i ) t i l t i n g disc valve; ( i i i ) p ivoting d i s c valve. 1- 2 A schematic diagram of the plan of study . . 2 1 2- 1 A schematic diagram of the g l y c e r o l tunnel. . . . . . . 24 2-2 Pump c h a r a c t e r i s t i c curves for water and glycerol-water s o l u t i o n . . . . . . . . . . . 2 8 2-3 The o r i f i c e c a l i b r a t i o n p l o t . . . . . . . . . 3 0 2-4 A photograph of the g l y c e r o l tunnel . . . . . 3 1 2-5 A photograph of the probe towing mechanism and associated flume . . . . . . . 36 2-6 A photograph showing the rot a t i n g d i s h arrangement for c a l i b r a t i o n of hot f i l m probes 38 2-7 C a l i b r a t i o n data for the hot f i l m probe i n : (a) tap water; 39 (b) d i s t i l l e d water; . . . . . . . . . . . . 40 (c) water-glycerol solutions of various concentrations . . . . . 4 1 2-8 C a l i b r a t i o n p l o t s showing the e f f e c t of temperature on probe's cold resistance when immersed i n glycerol-water s o l u t i o n of d i f f e r e n t concentrations . . . . 43 x i Figure Page 2-9 Schematic diagram of a Barocel pressure transducer 45 2-10 Exploded view of the Starr and Edwards ao r t i c b a l l valve prosthesis . . . . . . . . 47 2-11 A schematic diagram showing the l o c a t i o n of pressure ports on the surface of the poppet 4 9 2-12 An exploded view of the heart valve model: (a) schematic drawing; . . . . . 52 (b) photograph of the components 53 2-13 A comparison of the t y p i c a l displacement- time h i s t o r i e s for an a o r t i c b a l l valve implants and the model 55 2-14 Details of the c i r c u i t used i n pulsating the poppet of the heart valve model . . . . 56 2-15 C i r c u i t diagram for the pulse d u p l i c a t o r : (a) D.C. power supply for the solenoid valve and the t r i g g e r i n g mechanism; 58 (b) t r i g g e r i n g mechanism . . . . . 58 2-16 A schematic diagram of the displacement transducer and associated e l e c t r o n i c c i r c u i t r y 59 2- 17 C a l i b r a t i o n p l o t for the displacement transducer 61 3- 1 Instrumentation layout used during v e l o c i t y p r o f i l e measurements . . . . . . 65 3-2 A l i n e drawing of the instrumentation setup used for pressure measurements . . . . 69 3-3 A photograph showing the instrumentation set-up used during measurements of the pulsating pressure . . . . 72 x i i Figure Page 3-4 A photograph of the dye i n j e c t i n g probes: (a) e a r l i e r model; 78 (b) f i n a l streamlined probe 78 3- 5 A sketch showing the equipment layout during flow v i s u a l i z a t i o n 79 4- 1 Typical tunnel v e l o c i t y p r o f i l e s i n d i c a t i n g l o c a l i z e d j e t type flow . . . . . . . . . . . . 84 4-2 Representative v e l o c i t y p r o f i l e s showing e f f e c t of the introduction of nylon wool: (a) i n water; 85 (b) i n glycerol-water solution, Cn=54 . . . . 86 4-3 E f f e c t of spherical model (d = 2.5 in.) on v e l o c i t y p r o f i l e s : (a) i n water; 88 (b) i n glycerol-water solution, Cn=54 . . . . 89 4-4 Ve l o c i t y p r o f i l e s as affected by the valve model 90 4-5 Blockage e f f e c t on the mean flow rate as indicated by the variac s e t t i n g 94 4-6 I l l u s t r a t i o n showing possible errors introduced by non-uniformity of the v e l o c i t y p r o f i l e . . . . . . . . 96 4-7 E f f e c t of the supporting stem upon C p r o f i l e s for a 2.5 i n . sphere 99 4-8 A schematic drawing showing the spherical model and i t s support during the pressure measurements 101 4-9 Typical Cp p r o f i l e s around the meridional section of a sphere: (a) data reduced as suggested by Grove e t a L ; 1 2 4 . . . . ? \ . 102 (b) data reduced according to the present technique . 103 x i i i Figure Page 4-10 Reynolds number dependency of the pressure d i s t r i b u t i o n around sphere: (a) Rn=74-475; (b) Rn=370-1045 . 104 4-11 E f f e c t of Reynolds number on base pressure c o e f f i c i e n t : (a) pressure c o e f f i c i e n t at 8=180° 107 (b) average wake pressure c o e f f i c i e n t . . . 108 4-12 A comparison of the t h e o r e t i c a l and experimental pressure d i s t r i b u t i o n on the surface of sphere 110 4-13 Var i a t i o n of the sphere drag c o e f f i c i e n t with Reynolds number 112 4-14 A t y p i c a l photograph i l l u s t r a t i n g formation of a vortex r i n g behind sphere 114 4-15 A flow v i s u a l i z a t i o n study showing develop- ment and i n s t a b i l i t y of vortex r i n g with Reynolds number: (a) Rn=55; 115 (b) R =92; H 5 n (c) R =176; 115 n (d) R =221; 115 n ' (e) R =265; 116 n (f) Rn=289; . . 116 (g) Rn=331; . . . . . . 116 (h) Rn=395 . 116 4-16 E f f e c t of R n on separation angle 6 g . . . . 119 4-17 Displacement and pressure time his t o r y for the pulsating sphere 123 4-18 Typical time dependent pressure p r o f i l e s for a sphere showing the e f f e c t of Reynolds number and pulsation frequency: (a) forward stroke; 125 (b) reverse stroke . . . . . . . . 126 Figure x i v Page 4-19 A flow v i s u a l i z a t i o n study showing move- ment of separation point during pulsating motion of the sphere 128 4-20 Pressure d i s t r i b u t i o n on the poppet occupy- ing d i f f e r e n t positions i n the valve during the open bypass condition: (a) y b = 0.05, 0.10, 0.15; . . . . . . . . . 133 (b) y b = 0.20, 0. 60, 1. 0 ; 134 4-21 Schematic diagram showing passage of f l u i d i n the immediate v i c i n i t y of the valve . . . 136 4-22 V i s u a l study of flow patterns past the poppet and through the bypass as effected by the valve opening at Rn= 450. 137 4-23 Va r i a t i o n of pressure d i s t r i b u t i o n with the poppet p o s i t i o n for the case of the open bypass: (a^) very small openings, Rn=620; . . . . . . 139 (a~) intermediate and large openings, ^ R =620; . . . . . . 140 n (b^) very small openings, Rn=926; . . . . . . 141 (b~) intermediate and large openings, R =926; . . . . . . 142 n (c-^) very small openings, Rn=1201; 143 (c~) intermediate and large openings, R =1201 . . . . 144 n 4-24 Pressure d i s t r i b u t i o n on the poppet occupy- ing d i f f e r e n t p o s itions i n the valve during the closed bypass condition . . . 146 4-25 Comparison of pressure d i s t r i b u t i o n on the poppet occupying various positions i n the valve for the closed and open bypass conditions . . . . . 147 4-26 Reynolds number e f f e c t on C p r o f i l e s for sphere . . . . . . . . 149 XV Figure Page 4-27 A flow v i s u a l i z a t i o n study showing the e f f e c t of valve opening on the flow past a spherical poppet during the closed bypass condition: (a) R n = 450; 150 (b) R n = 600; 151 (c) R n = 900 152 4-28 Dependence of the surface pressure p r o f i l e s on the Beta number, Reynolds number, and valve opening: (a) B n = 19, R n = 290; 155 (b) B n = 19, R n = 650; 156 (c) B n = 62, R n = 290; 157 (d) B n = 62, R n = 650 158 4-29 E f f e c t of the Beta number on the surface pressure p r o f i l e for a given R n: (a) R n = 290: 160 - (i) forward stroke; ( i i ) reverse stroke; (b) R n = 652: 162 (i) forward stroke; ( i i ) reverse stroke. 4-30 Typical photographs i l l u s t r a t i n g r o t a t i o n a l motion of streamlines about the h o r i z o n t a l axis as captured by the 16 mm. movie . . . 166 4-31 A flow v i s u a l i z a t i o n study i l l u s t r a t i n g several important characters (time dependent separation, contraction through the i n l e t o r i f i c e , j e t i n g of f l u i d i n the e x i t b e l l , turbulent wake, etc.) of the flow during p u l s a t i l e motion of the poppet inside the valve 168 5-1 A schematic diagram showing several con- figurations for momentum i n j e c t i o n . . . . 181 x v i LIST OF APPENDIX FIGURES Figure Page I I - l A schematic drawing of a hot f i l m probe „ . 218 II-2 Theoretical and experimental c a l i b r a t i o n p l o t s for the hot f i l m probes used 229 x v i i ACKNOWLEDGEMENT I would l i k e to take t h i s opportunity to express my gratitude and sincere thanks to Professor V.J. Modi for the enthusiastic guidance given throughout the research program and he l p f u l suggestions during the preparation of the t h e s i s . His help and encouragement have been invaluable. The cheerful assistance of the techn i c a l s t a f f i s g r a t e f u l l y acknowledged. Their s k i l l f u l assistance greatly accelerated the research program. Mention should be made of the occasional discussions with Dr. C.E. Rotem, C a r d i o l o g i s t at the Shaughnessy Hospital, Vancouver, B r i t i s h Columbia. The i n v e s t i g a t i o n was supported by the Medical Research Council of Canada, Grant No. MA-2637 and the National Research Council of Canada, Grants No. A-2181 and A-2772. F i n a l l y , s p e c i a l appreciation i s extended to my wife, Shaheen, for her encouragement and for her patience and understanding during d i f f i c u l t times. XV111 LIST OF SYMBOLS overheat r a t i o , (R^-R )/R 0 c ' c hot f i l m width Beta number,  R n ^ 2 S ^ 2 = D ( f / v ) 1 / 2 speed of sound i n a i r at the reference temperature s p e c i f i c heat sectional pressure drag c o e f f i c i e n t based on 1 2 sphere diameter (D) , F/(^-pU ) (diameteral cross-sectional area) percentage concentration of glycerol-water sol u t i o n by weight s t a t i c pressure c o e f f i c i e n t , ( P Q - P g 0 O ) / ( p 0 ~ P 6 o ° s t a t i c pressure c o e f f i c i e n t as suggested by Grove et a l . 1 2 4 , ( P e - P r ) / (ipU. 2) base pressure c o e f f i c i e n t , average s t a t i c pressure c o e f f i c i e n t over the portion of the body extending into the wake (6 = 120 o-180°) o r i f i c e diameter diameter of the stem supporting the sphere b a l l diameter ej e c t i o n time pulsation frequency, cpm sectio n a l pressure drag g r a v i t a t i o n a l acceleration b a l l acceleration Grashof number, [ggb (T w~T f)cos6]/v head across the pump thermal conductivity of the test f l u i d and glass support, respectively hot f i l m length a x i a l distance i n a tube viscometer torque Mach number, U/c Prandtl number, Cp/K^ Nusselt number, qb/K f(T w~T f) s t a t i c pressure s t a t i c pressure on the surface of the sphere at an angle of 6 from the front stagnation point s t a t i c pressure on the surface of the sphere at the front stagnation point s t a t i c pressure on the surface of the sphere at 9 = 60° reference s t a t i c pressure as suggested by Grove et a l . heat flux per unit area t o t a l heat flux volume flow rate radius of the ro t a t i n g c y l i n d e r i n a v i s - cometer radius of the inner and outer c y l i n d e r , respectively e l e c t r i c a l resistance cold and operating resistance of the hot f i l m probe, respectively Reynolds number, UD/v Reynolds number based on U"r as suggested by 124 Grove et a l . , U D/v r stroke length Strouhal number, fD/U time opening time, c l o s i n g time, c l o s i n g duration and opening duration of an a r t i f i c i a l a o r t i c valve, r e s p e c t i v e l y temperature bulk and mean temperature of the. f l u i d wall temperature average v e l o c i t y i n the tes t section based on a flow rate as given by the o r i f i c e meter poppet v e l o c i t y during the reverse stroke (opening of the valve) b a l l v e l o c i t y during i t s p u l s a t i l e motion poppet v e l o c i t y during the forward stroke (closing of the valve) reference v e l o c i t y , taken as the ce n t e r l i n e v e l o c i t y l o c a l v e l o c i t y as measured by a hot f i l m probe D.C. voltage output of a constant temperature anemometer piston displacement of an a i r c y l i n d e r dimensionless valve opening, y^/s valve opening probe's distance from the tunnel i n l e t dimensionless sphere displacement sphere displacement horizontal coordinate p a r a l l e l to the tunnel axis with o r i g i n at the tunnel entrance, p o s i t i v e i n the d i r e c t i o n of the flow v e r t i c a l coordinate with o r i g i n at the bottom of the t e s t section angle between the cone and the plate i n a cone-plate viscometer c o e f f i c i e n t of thermal expansion shear rate shear rate at the wall wedge angle of the hot f i l m probe angular l o c a t i o n of a pressure tap with reference to the front stagnation point angular l o c a t i o n of the separating shear layer with respect -to the rear stagnation point c o e f f i c i e n t of thermal r e s i s t i v i t y of the hot f i l m probe dynamic v i s c o s i t y Casson v i s c o s i t y kinematic v i s c o s i t y , u/p density y i e l d stress of blood shear stress shear stress at the wall d i f f e r e n t i a l pressure across the o r i f i c e plate downstream coordinate along the probe face angular v e l o c i t y of a r o t a t i o n a l viscometer x x i i i A ortic incompetence, i n s u f f i c i e n c y Aortic leakage Aortic valve Art e r i e s Blood GLOSSARY Diseases of the a o r t i c cusps and rin g causing the reverse flow of blood into the l e f t v e n t r i c l e during d i a s t o l e due to incomplete closure of the a o r t i c valve Seepage of blood from aorta to the l e f t v e n t r i c u l a r chamber during d i a s t o l e because of improper closure of the prosthetic a o r t i c valve A valve (composed of three cusps of equal size attached symmetric- a l l y around the circumference) between l e f t v e n t r i c l e and aorta Vessels carrying blood from the l e f t or the r i g h t v e n t r i c l e s to the tissues A suspension of red c e l l s (erythro- cytes) , white c e l l s (leucocytes) and the p l a t e l e t s (thrombocytes) i n the f l u i d plasma, the average xxiv 3 number of each per mm of blood 6 4 being about 5 x 10 , 10 , and 5 3 x 10 , respectively C a p i l l a r i e s The f i n e s t blood vessels, normally about 5-6 j a m . i n diameter, and about 0.5 mm. long i n the systemic c i r c u l a t i o n Cardiac Index The cardiac output as determined by the product of the heart rate 3 -1 and stroke volume, L T Cavitation The formation of c a v i t i e s f i l l e d with vapour or gas within a moving l i q u i d due to l o c a l pressure being reduced to below the vapour pressure for the l i q u i d C l o t t i n g Occurrence of c e r t a i n i r r e v e r s i b l e chemical and mechanical reactions leading to coagulation of blood Diastole The normal rhythmical d i a l a t i o n of the heart during which the chambers are f i l l i n g with blood (the r e s t i n g phase of the cardiac cycle) Ejection Time Electrocardiogram Haematocrit Heart xxv Duration of the eje c t i o n phase which l a s t s about 0.25 s e c ; approximately two thir d s of the ve n t r i c u l a r stroke volume i s pumped into the great vessels during the f i r s t h a l f of t h i s period A record of the e l e c t r i c a l a c t i v - i t y of the heart, normally obtained with electrodes on skin The r e l a t i v e volume of the c e l l i n blood, normally about 45% The mammalian heart consists of four chambers, r i g h t and l e f t a t r i a , r i g h t and l e f t v e n t r i c l e s . The a t r i a are t h i n walled chambers into which blood flows at low pressure from the veins. Between a t r i a and v e n t r i c l e s are the t r i c u s p i d (right) and m i t r a l ( l e f t ) valves. Blood at high pressure leaves the v e n t r i c l e s by the pulmonary artery (right) and aorta ( l e f t ) ; there are valves at the o r i g i n of each of these. The Hemolysis In V i t r o In Vivo Lesion Ligature Low Output Syndrome xxvi blood supply to the heart i t s e l f comes from the coronary a r t e r i e s which spring from the aorta at i t s o r i g i n . Normally both a t r i a beat together a short time before the synchronous beat of the v e n t r i c l e s The breaking down of the red blood c e l l s with l i b e r a t i o n of hemoglobin Within an a r t i f i c i a l environment, as a te s t tube Within a l i v i n g organism An injury or other change i n an organ or tissu e of the body tending to r e s u l t i n impairment or loss of function A thread or wire used to t i e up a blood vessel Heart condition, with features of simultaneous f a i l u r e of v e n t r i c l e s , r e s u l t i n g i n a low cardiac output Mean S y s t o l i c Ejection Rate Morbid Complications Regurgitation Scar Stenosis Systole Suture Ring Thrombosis Thromboembolism x x v i i The amount of blood ejected per second per square meter of body surface area Impending f a i l u r e caused by diseased condition A backward flow of blood due to imperfect closure of a heart valve A mark l e f t on the skin or other tissue a f t e r a wound A narrowing of a blood vessel or of a valve The normal rhythmical contration of the heart during which the v e n t r i c l e s are ej e c t i n g the blood A r i n g used for sewing or j o i n i n g together a severed vein Coagulation of the blood i n the heart or a blood vessel forming a c l o t The blocking of a blood vessel by an embalus (blood cl o t ) that has x x v i i i broken away from a thrombus at i t s s i t e of formation Thrombus Fibrinous c l o t attached at the s i t e of thrombosis xzix "[his thesis is dedicated to the memory of the l a t e Professor Ze'ev Rot em . 1 1. INTRODUCTION 1.1 Preliminary Remarks The topic of the present thesis i s the in v e s t i g a t i o n of the performance of a r t i f i c i a l a o r t i c valve implants. A few h i s t o r i c a l remarks about the o r i g i n of these a r t i f i c i a l non-organic devices would be i n order here. Between 1944 and 1952 the f i r s t extensive experi- mental studies concerning permanent replacement of diseased 1-3 natural valves were undertaken . The repair of damaged valves i n s i t u had, of course, been c a r r i e d out for many years, but i n some cases the complete valve replacement seemed the only way to restore proper function. The f i r s t c l i n i c a l c o r r e c t i o n of a o r t i c i n s u f f i c - iency was performed i n 1952, employing a b a l l type valve 4 prosthesis . The p l a s t i c valve, evolved following the success of the permanent intubation of the aorta, i s e s s e n t i a l l y a modification of the tube. Primarily i t consists of an i n l e t , a chamber containing a b a l l , and an o u t l e t . The e n t i r e valve, made of polyethylene, i s molded into a single piece to provide extremely smooth and seamless inner surface. The hollow b a l l with no outside seam i s normally made of methyl methacrylate with the s p e c i f i c gravity s l i g h t l y less than one. At each end of the valve, there i s a groove on the outer surface to hold the aorta by means of a f i x a t i o n ring 2 of s o l i d semiflexible nylon. The outside surface of the f i x a t i o n r i n g i s grooved so as to maintain i n p o s i t i o n the heavy braided s i l k l i g a t u r e . The arrangement i s schematic- a l l y shown i n Figure l - l a ( i ) . I t was well recognized at that time that complete control of a o r t i c leakage was not possible because of technical d i f f i c u l t i e s involved i n reaching the a o r t i c root, but t h i s introduced a new concept of s u r g i c a l treatment of a l e s i o n for which previously complete correction had been impossible. The successful demonstration that such a prosthesis could function s a t i s f a c t o r i l y for prolonged periods of time stimulated other e f f o r t s to develop prostheses which could be placed i n the normal valvular p o s i t i o n and be useful i n c o r r e c t i o n of both a o r t i c stenosis and i n s u f f i c - 5-7 lency . These e f f o r t s took three major d i r e c t i o n s : (i) The development of t r i l e a f l e t f l e x i b l e valves or i n d i v i d u a l Teflon cusp which tend to approach 8 9 quite c l o s e l y the natural valve configuration ' . Among these prostheses i n d i v i d u a l cusps, designed by Bahnson, were f i r s t applied c l i n i c a l l y i n 1959 8. ( i i ) A modification of the b a l l valve f o r the use as an a o r t i c or m i t r a l valve. One of these pros- theses i n current use i s the Starr-Edwards b a l l v a l v e ^ . I t consists of a s i l i c o n e rubber b a l l 3 Figure 1-1 Several models of prosthetic a o r t i c heart valves (a) e a r l i e r models: (i) Hufengal b a l l valve; ( i i ) Bahnson t r i l e a f l e t valve; ( i i i ) Starr and Edwards b a l l valve; (iv) d i s c valve; (v) Hufengal b u t t e r f l y hinged l e a f l e t valve 4 Figure 1-1 Several models of prosthetic a o r t i c heart valves (b) some of the newer designs: (i) s i l a s t i c t r i l e a f l e t a o r t i c valve; ( i i ) t i l t i n g disc valve; ( i i i ) pivoting disc valve 5 enclosed i n a highly polished cage of S t e l l i t e 21. A bloodtight seal i s produced by a c i r c u l a r seat at the bottom of the cage. Fix a t i o n i n c l i n i c a l implantation i s afforded by a t i g h t l y knitted short Teflon c l o t h sleeve which projects upward from the prosthesis. ( i i i ) The use of hinged d i s c or other free moving 11-13 valvular mechanisms . The important features of these valves are the low p r o f i l e housing, a r e l a t i v e l y large o r i f i c e , and modest stress on sutureline. Some of these configurations are also sketched i n Figure 1-1 (a). Many a l t e r n a t i v e valve configurations have been devised since the introduction of the b a l l valve prosthesis. Among the newer designs are s i l a s t i c t r i l e a f l e t a o r t i c 9 12 14 valve , t i l t i n g disc valve , and pivoting d i s c valve (Figure 1-lb). In s p i t e of great achievements with prosthetic heart valves there has been continued i n t e r e s t i n improving i t s performance. The l e a f l e t valve models used frequently i n the past are now knov/n to develop, almost i n v a r i a b l y , stenosis or incompetence a f t e r prolonged use due to scarring with r e t r a c t i o n of the surrounding aorta or, on the other 15-17 hand, incompetence due to fatigue fracture . Bjork 6 and associates have also concluded that s i g n i f i c a n t thicken- ing and decreased mobility of the Bahnson Cusps can r e s u l t from f i b r i n ingrowth into the Teflon f a b r i c . Recent c l i n i c a l reports on the performance of the d i s c valves reveal that none of these valves i s free from the complications of thromboembolism, s t r u c t u r a l f a i l u r e , i n f e c t i o n , and blood 18—22 destruction . On the other hand, the b a l l valve designs have shown excellent d u r a b i l i t y and r e l a t i v e freedom from c l o t t i n g d i f f i c u l t i e s . But they have been found, quite often i n postoperative studies, to produce residual stenosis, c .., . 23-25 varying from mild to severe Requirements of acceptable designs for an a r t i f i c - i a l replacement would be: (i) compatibility of material with body f l u i d s ; ( i i ) n e g l i g i b l e tendency to cause either thrombus growth or adjacent tissue damage; ( i i i ) good hydrodynamic performance and mechanical e f f i c i e n c y maintained over the l i f e t i m e of the valve; (iv) long fatigue l i f e ; (v) ease of i n s t a l l a t i o n ; and i d e a l l y (vi) low cost. Obviously, not a l l of these requirements can be met, hence a compromise i s i n e v i t a b l e so as to a r r i v e at an optimum configuration under imposed constraints. One of the designs evolved i n t h i s manner i s represented by the b a l l valve. 7 So far none of the e x i s t i n g b a l l valves has been e n t i r e l y successful i n operation from a l l aspects enumerated above, due to, among others, c e r t a i n inadequacies with respect to the hemodynamic performance ( p a r t i a l stenosis and/or incompetence as mentioned e a r l i e r ) , the morbid compli- 2 6 - 3 2 cations such as thromboembolism , l e f t v e n t r i c u l a r out- flow obstruction when used i n m i t r a l p o s i t i o n , and "low output syndrome" caused by obstruction to free outflow i n patients with wide a o r t i c annulus and narrow ascending 4. 33,34 aorta . Successful operation of a r t i f i c i a l heart valves i n general and b a l l type valves i n p a r t i c u l a r depend upon r e l i a b l e i n v i t r o and i n vivo evaluation. Considering that moving parts are c h a r a c t e r i s t i c of p r a c t i c a l l y a l l designs, the demand on prosthetic materials used are considerable. Therefore i t i s imperative to simulate the operating condition of prosthetic valves i n order to e s t a b l i s h whether a p a r t i c u l a r combination of design, f a b r i c , and implantation i s adequate, and compatible with requirements. In t e s t i n g prosthetic valves, i n v i t r o studies should l o g i c a l l y precede i n vivo t e s t i n g . The disadvantage of choosing the reverse sequence i s underlined by r e f l e c t - ing upon the very large number of design parameters which have to be evaluated: available prosthetic materials; the variety of mechanical valve designs; various methods of f a b r i c a t i o n and i n s t a l l a t i o n . The number of parameters 8 i s such that the evaluation of each one of them by i n vivo te s t i n g i s impossible, more p a r t i c u l a r l y so as not a l l of these are independent of one another. L a s t l y , even i f i n vivo studies were c a r r i e d out using t h e o r e t i c a l l y s a t i s f a c - tory valves, with the heart rate of common laboratory animals s i m i l a r to that of the human rate, the time for complete evaluation would be excessively long. This would mean that only a day-by-day comparison of the valve and i t s components could be made. In other words, i t would take f i v e year tests a f t e r implantation to s a t i s f y oneself that a p a r t i c u l a r valve would be s a t i s f a c t o r y i n humans over the same period. Two main objects should be achieved i n preliminary t e s t i n g . The f i r s t concerns the mechanical strength of the valve and therefore i t s a b i l i t y to withstand for many years unsteady pressures s i m i l a r to those that i t would experience i n the human body. The second problem i s to determine whether the flow c h a r a c t e r i s t i c s of the prosthetic valve are suitable for c l i n i c a l use. Many pulse duplicator and fatigue machines have been designed mainly to answer the f i r s t question; however, l i t t l e attention has been paid 35—38 to the l a t t e r aspect Experience with prosthetic heart valves has shown that a basic knowledge of p u l s a t i l e flow through the valve i s necessary i f both c l o t t i n g and hemolysis are to be eliminated. Theoretical approaches to t h i s problem are 9 limited because of the complex geometries and the time 39 dependent character of the non-Newtonian f l u i d involved Current studies of i d e a l i z e d laminar and turbulent flows i n e l a s t i c tubes are not d i r e c t l y applicable to flow i n 40 41 a o r t i c and m i t r a l valvular areas ' . Consequently, one i s forced to resort to experimental techniques for better appreciation of flow c h a r a c t e r i s t i c s associated with a r t i f i c i a l heart valves. Because of d i f f i c u l t i e s of duplicating the p u l s a t i l e pressure and flow conditions present i n the heart, quasi-steady tests have been used quite f r e q u e n t l y ^ 2 ~ ^ . The f i r s t successful attempts at v i s u a l observa- 46 t i o n of heart valve movements were those by Smith and 47 . 48 49 Kantrowitz and t h e i r associates, but i t was McMillan ' who developed post mortem cinematography as a p r a c t i c a l 50 51 technique for studying valve action. Leyse , Meisnere and others have investigated the flow pattern associated with heart valves using two dimensional models i n b i r e - fringent s o l u t i o n of bentonite clay i n aqueous-glycerol. 52 53 Davila ' employed the bentonite clay v i s u a l i z a t i o n technique to observe turbulence generated by a r t i f i c i a l heart valves, and to evaluate i t s r o l e i n the production 54 55 of thrombosis. Dye i n j e c t i o n ' as well as suspension of aluminum p a r t i c l e s i n conjunction with s l i t i l l u m i n - ation"^ ^ have also been attempted to t h i s end. Surpris- ingly Davey et a l . ~ ^ found no stagnant regions for any p o s i t i o n of the b a l l and absence of separation (with a s s o c i - ated turbulence downstream) for f u l l y open p o s i t i o n of the valve. An alternate approach to the problem and probably a more r e a l i s t i c one, would be to explore hydrodynamic character of prosthetic devices using pulse simulators. The object here i s to reproduce p u l s a t i l e character of the flow 35 i n the a o r t i c region using a p o s i t i v e displacement pump , 36 37 a cam drive mechanism , a rotary valve , and a r e c i p r o - 55 eating piston pump between constant head tanks , etc. Unfortunately, most of the e a r l i e r studies using t h i s pro- cedure are confined only to the pressure measurement across the valve for d i f f e r e n t flow rates and/or d i f f e r e n t o r i f i c e d e s i g n s ^ ' ̂ . Smeloff and a s s o c i a t e s ^ compared the flow c h a r a c t e r i s t i c s for f i v e d i f f e r e n t prostheses (Roe molded l e a f l e t valve, Gott l e a f l e t valve, Kay-Suzuki d i s c o i d valve, Starr-Edwards b a l l valve, and Smeloff-Cutter b a l l valve) and concluded the Smeloff-Cutter b a l l valve to have several favorable features which are h e l p f u l i n reducing the i n c i - dence of thromboembolism. On the other hand, using the re s u l t s of t h e i r i n v e s t i g a t i o n with six valves (Gott l e a f l e t , Teardrop d i s c o i d , Pin teardrop, Starr-Edwards, T r i l e a f l e t , 61 Heavy Teflon, and Hammersmith) Kelvin et a l . found (based on o r i f i c e s i z e , time of actuation and regurgitation) Starr- Edwards b a l l valve and Gott hinged l e a f l e t valve to be most suitable for the a o r t i c and m i t r a l p o s i t i o n , r e s p e c t i v e l y . 11 62 63 Kaster et a l . ' conducted comparative tests on eleven prostheses (Meniscus d i s c , Smeloff-Cutter b a l l #2,3, 4,5, Starr-Edwards b a l l , pivoting d i s c , Kay-Shiley disc #4,5, Toroidal d i s c , Gott-Dagget l e a f l e t ) . Their findings indicate that the b a l l , pivoting d i s c , and b u t t e r f l y l e a f l e t valves generally show good flow volumes with minimal pressure gradients. They also observed that the l e a f l e t valve requires less mechanical energy for a c t i v a t i o n than the disc or b a l l valve and allows minimal retrogation of flow during d i a s t o l e . F i n a l l y , i t was concluded that the Smeloff- Cutter f u l l flow o r i f i c e b a l l valves have good flow charac- t e r i s t i c s together with a modest pressure gradient. Wieting 59 and h i s associates investigated flow c h a r a c t e r i s t i c s of f i v e d i f f e r e n t a o r t i c valves (Starr-Edwards 12A, Kay-Shiley disc #5, Benson Roe f l e x i b l e cusp 27 mm, Gott hinged l e a f l e t 29 mm, and Barnard Poppet LICT A06). A l l of them were found to have sim i l a r a o r t i c , l e f t v e n t r i c u l a r , and mean l e f t a r t e r i a l pressure contours, but none could match that of the human a o r t i c valve. Furthermore,it was observed that the f l e x i b l e cusp, hinged l e a f l e t and caged disc valves have less regurgitation compared to the b a l l valve. This, to some extent, contradicts the conclusion of Kelvin et a l . ^ mentioned e a r l i e r . Several important parameters of the prosthetic valve c h a r a c t e r i s t i c s such as mean d i a s t o l i c pressure, incompetence, mechanical movements, and flow 64 disturbances were studied by Wright . He found: 12 (i) many of the prostheses to produce mild to moderate stenosis; ( i i ) only valves with large o r i f i c e diameter (22 mm or larger) to be free of s i g n i f i c a n t stenosis; ( i i i ) disc valves to produce a steep v e l o c i t y p r o f i l e near the wall of the v e n t r i c l e ; (iv) the Starr-Edwards 6120/3M to cause more high frequency turbulence than the other valves. Vigger^ 5 has emphasized the importance of pressure drop and associated work load on the heart i n any prosthetic valve design. Based on the measured pressure drop, an empirical hydraulic e f f i c i e n c y parameter i s suggested to assess effectiveness of an a r t i f i c i a l device i n r e l a t i o n to that of the natural valve. I t i s concluded that, despite the wear and variance that has occurred with a r e l a t i v e l y few prosthesis b a l l occluders, the b a l l valve concept i s sound, should be studied further and improved. Any systematic approach to the problem should i n - clude i n v e s t i g a t i o n of the behavior of the spherical poppet moving through a f l u i d . I t i s only through studies of such a fundamental configuration that one can hope to gain better understanding of rather complex f l u i d mechanics associated with the prosthetic heart valve. 1 • 2 A B r i e f Review of the Relevant Li t e r a t u r e on the Flow Past a Sphere Interest i n the behavior of a sphere moving through a f l u i d goes back many years with the f i r s t recorded measure- ments related to sphere drag a t t r i b u t e d to S i r Isaac Newton. Following t h i s and p r i o r to 1930, numerous measurements on the drag of sphere f a l l i n g through various f l u i d s were made, and a body of information was generated for 10 "^<Rn<10^. However, these data, which usually show a s i g n i f i c a n t degree of s c a t t e r , e t a^"' and hence are approximated by a 69 single l i n e c a l l e d the "standard" drag curve , represent only a rough estimate of the drag c o e f f i c i e n t . Ever since, t h e o r e t i c a l and p r a c t i c a l i n t e r e s t i n the subject has resulted i n a large volume of l i t e r a t u r e , and the contribu- tions up to 1960 have been s i t e d by Torobin and Gauvin i n 70 t h e i r admirable and comprehensive review of the f i e l d 71 In 1963, Hemrich et a l . c a r r i e d out sphere drag 3 4 measurements i n a wind tunnel f o r 2 x 1 0 <R < 2 x 10 n and 0.078 < M n < 0.39. Their data, however, are s i g n i f i - cantly higher than the standard values. The discrepancy 72 was attributed to the free stream turbulence. S i v i e r has measured the drag of magnetically supported spheres i n a wind tunnel with a free stream turbulence i n t e n s i t y up to 8 % . His r e s u l t s are also considerably higher than the 73 standard drag values. Zarin refi n e d the magnetic balance 72 system used by S i v i e r and varied the free stream turbulent i n t e n s i t y l e v e l . Even at the turbulence l e v e l less than 3 1%,he found, for Rfi > 10 , drag to be markedly greater than 3 the standard values. However, for R < 10 , the r e s u l t s ' n were i n good agreement with the standard values. From t h i s study, Zarin concluded that, i n the higher Reynolds number 3 range (R R > 10 ), a small degree of free stream turbulence r e s u l t s i n increased drag values. 74 Ross and Willmarth conducted drag measurements for sphere moving r e c t i l i n e a r l y through the glycerine-water 5 mixture for 5 < R n < 10 . Their r e s u l t s agree f a i r l y well 3 with the standard data f o r R < 2 x 10 but are somewhat n greater for the Reynolds number exceeding t h i s value. The study revealed that the drag on a sphere i s not s i g n i f i c a n t l y affected by the vortex shedding (5% v a r i a t i o n ) . I t was also shown that drag on a sphere accelerated from r e s t to a constant v e l o c i t y exceeds the steady state drag by as much as 30% at high Rn, u n t i l the f i n a l quasi-steady state wake configuration becomes established. Furthermore, they con- cluded that the 'potential flow apparent mass' concept i s v a l i d for the f i r s t diameter of motion of a sphere under- going constant acceleration such that R n = 30,000 when sphere has moved one diameter; beyond t h i s point, the drag i s reasonably well approximated by the steady state drag corresponding to the instantaneous sphere v e l o c i t y . B a i l y 75 and Hiatt c a r r i e d out sphere drag measurements i n a b a l l i s t i c range for 0.1 < M < 6.0 and 20 < R < 10 5 for ^ n n Ty/Too = ^ ^ T W = temperature at the wall) . There i s a reasonable agreement between t h e i r low speed data and the c l a s s i c a l standard drag curve. Based on t h e i r own r e s u l t s and other published data they were able to p r e d i c t the e f f e c t of wall temperature on C, when T /T 5* 1.0. Experimental i n v e s t i g a t i o n involving flow v i s u a l - i z a t i o n and photographing of the wake behind a sphere i n the low Reynolds number range of 5 < R R - 300 was c a r r i e d 7 6 out by Taneda using a water tank. The r e s u l t s showed that the c r i t i c a l R n at which the permanent "vortex-ring" begins to form i n the rear of a sphere i s about 24, si z e of the ring i s nearly proportional to the logarithm of the Rn, and the wake behind the r i n g begins to o s c i l l a t e for R n - 130. 77 Magarvey and Bishop studied the t r a n s i t i o n ranges for three dimensional wakes produced by the motion of a drop of an immiscible l i q u i d i n the Reynolds number range 0 < R n < 2500. They distinguished the observed wakes as steady or periodic with several s u b c l a s s i f i c a t i o n s i n each of the categories, and concluded (as can be anticipated) that the wake pattern depends e n t i r e l y on the Reynolds number regardless of the l i q u i d - l i q u i d system employed. Furthermore, i t was observed that the general values of the t r a n s i t i o n Reynolds numbers cannot be obtained as they depend on the drop deformation. However, for a l l the cases considered the t r a n s i t i o n i n the wake patterns were lim i t e d to Reynolds number spread of less than 20. A q u a l i t a t i v e i n t e r p r e t a t i o n of heat and mass transfer mechanisms i n the wake region of a sphere i n 7 8 low speed flow (R < 410) i s presented by Lee and Barrow who employ measurements of the v e l o c i t y f i e l d i n the wake through flow v i s u a l i z a t i o n by dye i n j e c t i o n . The observed flow patterns generally confirmed Taneda's r e s u l t s . An important c h a r a c t e r i s t i c of the near-wake i s the reversed flow, at the v e l o c i t y much smaller than the free stream veloc i t y , a l o n g the axis of the sphere towards the rear stagna- t i o n point. The wake t r a n s i t i o n and Strouhal number fo r the incompressible wake of various bodies was studied by 79 Goldburg and Florsheim . Based on the experimental r e s u l t s i t was suggested that the t r a n s i t i o n could be approximately correlated for a range of spheres and cones by a Reynolds number based on the t o t a l wake momentum thickness. Further- more, i t was found that for regular vortex shedding the data for spheres and cones could be correlated with Rayleigh- Strouhal formula based on the same c r i t e r i o n . Measurements of vortex shedding frequency, wake dimensions and s t a t i c pressure i n the wake for a sphere 8 0 have been reported by Calvert . Based on these data, the Strouhal number has a value of 0.188, with a scatter of 4 4 ±0.0008 for 2 x 10 < R < 6 x 10 . The r e s u l t s showed n the base pressure c o e f f i c i e n t to be s u b s t a n t i a l l y dependent on R with the v a r i a t i o n of -0.270 to -0.356 over R =1.5 n n 4 4 x 10 - 6 x 10 . The e f f e c t of a t r i p wire was to s h i f t the o r i g i n of the wake, leaving the scale unchanged. Eddy shedding from a sphere i n turbulent free streams was i n - 81 vestigated by Mujumdar and Douglas . Test data showed that for 0.5% turbulence and for 5.6 x 10 3 < R < 11.6 x 10 3 n the Strouhal number for sphere i s 0.20, a value t y p i c a l of c i r c u l a r cylinders i n cross-flow i n the same Reynolds number range. It was also suggested that i n a turbulent flow there i s no regular, well defined eddy-shedding. Theoretical i n v e s t i g a t i o n of even a steady viscous incompressible flow past a sphere i s very complex. I t 8 2 was f i r s t considered by Stokes (1851) , and has been d i s - cussed by many authors since then. A large portion of these studies have been concerned with the solutions for vanish- ing ly small R n« Stokes solved the problem by neglecting the 8 3 i n e r t i a of the f l u i d . Later, Whitehead t r i e d to improve upon t h i s s o l u t i o n by introducing higher approximations to the flow when the Reynolds number i s not n e g l i g i b l e . But as i s now well known, his solution i s not v a l i d i n problems 84 85 of uniform streaming . Oseen solved Whitehead's paradox by assuming that the sphere caused a small perturbation i n the uniform p a r a l l e l flow and neglected second order pertur- bation v e l o c i t i e s , thus taking the i n e r t i a terms into account to a l i m i t e d extent. Oseen's solution for l i n e a r i z e d 86 87 equation has been improved by Goldstein , Tomotika et a l . , 8 8 and Pearcey et a l . However, as can be anticipated, l i n e a r i z a t i o n renders these analysis inadequate for R_ > 2. 18 Of considerable i n t e r e s t are two independent 8 9 solutions: one by Kawaguti who s a t i s f i e d an integrated form of the Navier-Stokes equation for f i r s t and second- order terms when expanded by Legendre Polynomials and the 84 other by Proudman and Pearson who l i n e a r i z e d the Navier- Stokes equation by two approximations, one v a l i d at a distance from the sphere, and the other v a l i d near the 90 surface of the sphere. Kawaguti has also developed an a l t e r n a t i v e procedure to solve the Navier-Stokes equation using the f i n i t e difference method. Unfortunately, the technique, v a l i d for flow around spheres up to Rfi = 20, 91 92 proves to be extremely laborious. Fox et a l . ' and 93 A l l e n et a l . have p a r t i a l l y a l l e v i a t e d t h i s d i f f i c u l t y by t r a n s f e r r i n g the technique into the rela x a t i o n procedures. 94 On the other hand, Jenson applied the relax a t i o n method d i r e c t l y to the governing equations for v o r t i c i t y and stream function i n modified spherical coordinates to obtain solutions for flow around spheres at = 5, 10, 20, 40. Hamielec 95-97 et a l . have also used a s i m i l a r method, but with f i n e r g r i d s i z e to obtain numerical solutions of the Navier- Stokes equations for slow viscous flow around spheres. Fourier expansions for the flow variables were used to solve the problem over a wide range of the Reynolds number 98 99 by Dennis and Walker . Rimon and Cheng derived steady state solutions f o r 1 < R n < 1000 by impulsively s t a r t i n g a sphere from re s t with uniform v e l o c i t y and used a time dependent integration to carry the solut i o n to the steady state. More recently, Dennis and Walker"^^ have presented a series truncation method, f i r s t proposed by Van Dyke^^"' employing a family of Legendre functions to solve the Navier- Stokes equations for flow around spheres i n the Reynolds number range of 1-40. 1.3 Purpose and Scope of the Investigation The development of a r t i f i c i a l heart valves depends upon r e l i a b l e knowledge of the hemodynamic performance and physiology of the human cardiovascular system, i n addition to a f a i r understanding of the associated f l u i d mechanics. It i s evident from the l i t e r a t u r e survey that i n spite of extensive tests (in v i t r o as well as i n vivo) of the a o r t i c b a l l valves, c r i t i c a l information bearing on valve perform- ance i s quite unavailable. Pressure d i s t r i b u t i o n on the surface of the b a l l , exact l o c a t i o n of separation points, regions of stagnant flow, and the i n t e n s i t y of turbulence generated by the b a l l : a l l these have not received proper attention. The present thesis aims at taking a modest step i n exploring some of these fundamental parameters. In the beginning, design, construction and c a l i - bration of a g l y c e r o l tunnel i s presented. This i s followed by a b r i e f d escription of the pump drive system, hot f i l m 20 anemometry, pressure transducing set-up, a o r t i c heart valve model, and pulsating drive mechanism. Next, the tes t procedures adopted i n measuring: (i) v e l o c i t y p r o f i l e s i n the tes t length of the tunnel; ( i i ) mean pressure d i s t r i b u t i o n on stationary spheres and o s c i l l a t i n g poppet; ( i i i ) mean pressure on a stationary and o s c i l l a t i n g spherical poppet located inside the valve are described i n some d e t a i l , r e s u l t s presented and discussed. Related information from l i t e r a t u r e i s also included where appropriate for comparison and to help e s t a b l i s h trends. The res u l t s are given for both the cases — valve bypass closed as well as open — the l a t t e r condition representative of regurgitation. F i n a l l y , an extensive programme of flow v i s u a l i z - ation using dye i n j e c t i o n technique i n conjunction with s t i l l and 16 mm movie photography i s introduced. The objective i s to obtain q u a l i t a t i v e v i s u a l substantiation of the conclus- ions based on measured parameters. Figure 1-2 schematically summarizes the proposed plan of study. HYDRODYNAMIC PERFORMANCE OF AN ARTIFICIAL AORTIC VALVE IMPLANT Experimental Apparatus- Design and Construction of Anemometery Theory of Hot Film Probe Mean Pressure Measurement C a l i b r a t i o n of the Hot Film Probe at Low Reynolds Number, R n = 2-250 (based on f i l m width) C a l i b r a t i o n of the Glycerol Tunnel A Glycerol Tunnel Scaled Prosthetic Valve Model Flow V i s u a l i z a t i o n Water Glycerol Stationary Sphere by I t s e l f Pulsating Sphere Stationary Sphere Occupying D i f f e r - ent Position in an Enlarged Valve Cage Sphere O s c i l l a - t i n g i n the Cage Simulating Prosthetic Valve Motion An O s c i l l a t i n g Drive System Simulating Prosthetic Valve Motion A Linear Displacement Transducer to Figure 1-2 A schematic diagram of the plan of study 22 2. EXPERIMENTAL APPARATUS AND CALIBRATION This chapter attempts to introduce the t e s t f a c i l i t y used i n .the experimental program. Some of the instrumentation constitutes the standard equipment i n any well equipped f l u i d mechanics laboratory and hence needs no elaboration. On the other hand, design and construction- a l d e t a i l s involved i n the development of s p e c i f i c equipments are often numerous and hence, though important and relevant, cannot be covered i n t h e i r e n t i r e t y . One i s therefore, forced to confine attention to more s a l i e n t features. Un- doubtedly, the most demanding equipment i n terms of time and e f f o r t was the g l y c e r o l tunnel, which i s described f i r s t . This i s followed by b r i e f descriptions of hot f i l m anemometry, model design, pulsating mechanism, pressure and displacement transducer, etc. Wherever appropriate, c a l i b r a t i o n pro- cedures are explained and corresponding charts included. 2.1 Glycerol Tunnel The fundamental f a c i l i t y for the t e s t program i s the g l y c e r o l tunnel designed and fabricated e n t i r e l y i n the department. The main c r i t e r i o n governing the design was the Reynolds number, which for the anticipated model s i z e , was around 4 00-3000 based on the spherical poppet diameter and average flow v e l o c i t y . The choice of concentration of the working f l u i d provided a degree of f l e x i b i l i t y , but only to a c e r t a i n extent as governed by the c h a r a c t e r i s t i c s of the power unit. The tunnel i s shown schematically i n Figure 2-1. Primarily i t consists of three subassemblies: the t e s t section; the f l u i d return system; and the power unit consisting of a pump and a drive motor. The t e s t section i s b u i l t of four p l e x i g l a s walls 8 f t . l o n g , 0.7 50 i n . thick and wide enough to produce an inside cross-section of 8 x 8 i n . The long length was purposely chosen to ensure s u f f i c i e n t room fo r the i n s t a l l a - t i o n of flow d i s t r i b u t i n g and straightening devices, and to leave a r e l a t i v e l y large region f o r model p o s i t i o n i n g . A vent, 4 i n . i n diameter and one foot high, located on the downstream end of the "box" (5 i n . from the o u t l e t end) provided for f l u i d expansion as well as an escape route for the a i r bubbles. I t also serves as an e f f e c t i v e check against the overpressurization of the t e s t section, p a r t i c - u l a r l y , near the model l o c a t i o n , i r r e s p e c t i v e of the pump's operating condition. There are three accesses to the inside of the section, through each end and v i a a porthole at the top of the "box". The porthole, 5 i n . i n diameter, i s j u d i c i o u s l y , located 33 i n . from the entrance to admit an arm to reach, p o s i t i o n and adjust models. During operation vent pipe radiator hose honeycombs portholes A " t od ra in "tfp =@=T 8x8 in. box filter •wvw HI heat exchange orifice plate cooling water in Figure 2 - 1 A schematic diagram of the gl y c e r o l tunnel 25 the porthole i s sealed watertight employing an "0"-ring. In addition, several smaller portholes which could take 5/8 i n . N-C plugs were d r i l l e d and tapped on the top of the p l a s t i c "box". These openings were used to mount models, take out pressure conducting l i n e s and to support a hot f i l m probe i n the t e s t section. When not i n use they were sealed o f f employing the plugs with "0"-rings. Two glass plates, 25 x 5 1/2 x 1/2 i n . , recess-mounted i n the sides of the t e s t section provided o p t i c a l l y f l a t , homogeneous and thermally stable walls for inspection and photography. A drain positioned at the bottom of the "box" f a c i l i t a t e d complete draining and flu s h cleaning of the tunnel. Of c r i t i c a l importance, from v e l o c i t y p r o f i l e consideration, was the t r a n s i t i o n of the flow from the pump outlet (3 i n . diameter) to the 8 x 8 i n . "box". The j e t l i k e flow has to be diff u s e d and spread evenly across the test cross-section. To achieve t h i s the following arrange- ments were used: (i) d e f l e c t i o n annular vanes were positioned i n the incoming stream to force some of the f l u i d away from the center of the stream; ( i i ) several sections of honeycombs followed the annular vanes to straighten the flow through turbulent exchange and laminar damping; ( i i i ) brass screens of d i f f e r e n t pore size with or without f i b e r g l a s s wool i n between. 26 In view of the a v a i l a b l e information on v e l o c i t y 102 d i s t r i b u t i o n at the entrance of the a o r t i c passage , i t was desirable to obtain uniform v e l o c i t y p r o f i l e i n the t e s t section. A preliminary c a l c u l a t i o n of the boundary layer growth on a f l a t plate indicated adequate region of uniform flow one foot downstream of the honeycomb. Later on, experi- mental measurements confirmed t h i s observation. An i n c i d e n t a l advantage of the t e s t i n a uniform flow i s , of course, the ease of comparison of data obtained i n d i f f e r e n t tunnels. Aluminum honeycombs used o r i g i n a l l y were susceptible to p i t t i n g due to electrogalvanic action and fungus growth on i t s surface. A coating of epoxy r e s i n decelerated the process but f a i l e d to solve the problem permanently. F i n a l l y p l a s t i c honeycomb sections, 6 i n . thick, were employed with success. Ethylene Diamine Tetraacetate (EDTA) i n small quantity (1:120) helped suppress b i o l o g i c a l growth. Located between the end of the p l e x i g l a s "box" and the power drive system i s the return section e s s e n t i a l l y comprising of heat exchanger, Poly V i n y l Chloride (PVC) pipes and elbows with connecting flanges and radiator hose. A copper pipe, 10 f t . x 3 i n . , i n conjunction with 8 f t . x 6 i n . PVC p l a s t i c sewer pipe formed an annular s i n g l e pass heat exchanger. With the coolant supplied by a water main i t was possible to maintain temperature of the working f l u i d within ±.0.2°C. PVC elbows and sections of the radiator hose provided r e l a t i v e l y easy, anti-corrosion and v i b r a t i o n free connections between the t e s t section and heat exchanger. 27 The power unit consists of a c e n t r i f u g a l pump (Auora type GAPB, 200 gal/min, 25 f t . head, 1750 rpm) driven by a three horsepower variable speed D.C. motor. The pump impeller and housing are of cast brass to guard against possible corrosion. The motor i s energized by a three phase g r i d , the voltage being adjusted through an autotransformer (Variac model 4T11) and r e c t i f i e d by selenium diodes. No further smoothing of the D.C. output was required. To minimize f l u c t u a t i o n i n pump performance, i t was desirable to operate on the stable (descending) leg of i t s c h a r a c t e r i s t i c s . Unfortunately, performance charts as supplied by the manufacturer are only for water. Hence, t e s t r e s u l t s with o i l of d i f f e r e n t v i s c o s i t y as obtained by Abdurashitov 103 et a l . were used to estimate pump behaviour i n g l y c e r o l . Later on, when the system was operational, i t was possible to evaluate the pump performance with the s p e c i f i c concen- t r a t i o n of g l y c e r o l used to obtain the desired Reynolds number. This p l o t i s shown i n Figure 2-2. I t i s of i n t e r e s t to recognize that the pump performance remains v i r t u a l l y unaffected by an increase i n v i s c o s i t y as high as 7-10 times that of water. This confirms the t e s t data presented by 103 Abdurashitov et a l . as well as the American Hydraulic T 4.-4_ a. 104 Instit u t e To promote damping and i s o l a t i o n of vib r a t i o n s generated by the pump motor assembly, a concrete foundation of 6 x 2 x 1 f t . was used with the unit mounted on a s t e e l Q. filter 28 o n ? 005 010 0,15 0.20 0.25 0. i i r~ i 1 o water • water-glycerol solution , Cn=54 pump / J L 0 50 .' 100 150 Q.U.S.gpm Figure 2r-2 Pump c h a r a c t e r i s t i c curves for water and glycerol-water solution 29 channel base and shafts aligned within ±0.002 i n . A f l e x - i b l e coupling with a damping i n s e r t was used to connect the drive shafts. Radiator hose were used to attach the pump i n l e t and o u t l e t to the tunnel. Flow rate i n the tunnel was monitored using a sharp edge o r i f i c e plate mounted two feet upstream of the pump i n l e t . The plate l o c a t i o n was selected so as to make i t s reading r e l a t i v e l y independent of upstream and downstream disturbances i n the form of elbows, change i n section at the pump i n l e t , pump suction, etc. Before f i n a l assembly, the o r i f i c e plate and associated plumbing were c a l i b r a t e d , under simulated t e s t conditions, by pumping water from a large sump into a weighing tank. For various flow rates, adjusted by means of a gate valve, time taken to c o l l e c t 1000 l b . of water was recorded along with pressure difference across the o r i f i c e p l ate. For each s e t t i n g of the valve s u f f i c i e n t time was allowed for the flow to reach the steady state condition In a l l , ten valve settings were used to give the flow rate v a r i a t i o n from 2.5-15.0 cm/sec. For a given s e t t i n g the procedure was repeated three times and t h e i r arithmetic mean was used to obtain the f i n a l c a l i b r a t i o n chart (Figure 2-3). It was important to minimize d i r t contamination of f l u i d . This was achieved by incorporating a f i l t e r (10 u) in a bypass c i r c u i t across the pump. The system f i l t e r s the e n t i r e volume at le a s t once i n twenty four hours of operation (Figure 2-2). A photograph of the g l y c e r o l tunnel i s shown i n Figure 2-4. 1.0i 0.4 0.2h 9' Of 0 10 • 20 30 40 Q,U.S.gpm Figure 2-3 The o r i f i c e c a l i b r a t i o n p l o t A photograph of the g l y c e r o l tunnel: E, honeycombs; F, f l e x i b l e couplings; 1-1 H, heat exchanger; M, manifold; 0 , o r i f i c e plate; P, pump; S, power switch; V, variac; W, plate-glass window 32 2.2 Hot Film Anemometry The next l o g i c a l step would be to c a l i b r a t e the g l y c e r o l tunnel. In the Reynolds number range of i n t e r e s t t h i s presents several problems and hence requires c a r e f u l evaluation of the avai l a b l e procedures. During the course of the experimental work i t was required to measure flow v e l o c i t i e s of the order 2.5-15 cm/sec. with a f a i r degree of accuracy, i n order to achieve dynamic s i m i l a r i t y between the model and the prototype (Appendix I ) . Measurement of f l u i d v e l o c i t i e s at low values of Reynolds number (R n) has long been known to be exceptionally d i f f i c u l t . Boundary layer theory i s a v a i l a b l e for s u f f i c i e n t l y large R R ; the dynamic head may then be measured, e.g. with a P i t o t tube, and as the pressure of the "outer" e s s e n t i a l l y i n v i s c i d flow i s impressed upon the boundary ( i . e . , the outer envelope of the P i t o t tube) i t can be converted d i r e c t l y to a reading of v e l o c i t y . Hot f i l m probes i n conjunction with the theory of 1am- 105 inar flow forced convection can also y i e l d predictable c h a r a c t e r i s t i c s (Appendix I I ) . Should R be very low n 1 (up to 1.5), Stokes 1 c a l c u l a t i o n for drag on a spherical body 84 with Oseen's (subsequently Proudman and Pearson's ) improve- ment w i l l enable the i n t e r p r e t a t i o n of a pressure read by an impact tube. The r e s u l t s depend, however, on probe shape, and pressures tend to be so low as to be v i r t u a l l y unrecord- able. In p r i n c i p l e , a theory for hot f i l m probes may also be obtained for t h i s range (e.g. Reference 106 for a spherical surface) but free convective heat transfer from the probe 33 107 renders the a n a l y t i c a l r e s u l t of no value . In the range of R n = 2-150 there i s at present no theory avai l a b l e which w i l l r e l a t e measured values of the t o t a l head, as of the convective heat d i s s i p a t i o n , to flow v e l o c i t i e s . It i s therefore necessary to resort to c a r e f u l s e l e c t i o n and c a l i - bration of a probe which could perform successfully i n t h i s range. Apart from laser-doppler anemometer, which was s t i l l i n the stage of development when c a l i b r a t i o n of the tunnel was undertaken (1970), a hot f i l m probe appeared to be the only instrument to meet the requirements of high resolution i n time and space of flow v e l o c i t i e s . Hence a quartz coated wedge shaped platinum f i l m probe (DISA 55A83) was used i n conjunction with standard constant temperature anemometric equipment (DISA model 55A01). Despite the existence of a comprehensive l i t e r a t u r e on measurement i n gases, very few papers deal with the use of hot f i l m anemome- try for i n v e s t i g a t i o n of slow l i q u i d flow. I t i s mainly because of several d i f f i c u l t i e s involved i n adapting the anemometer to use i n water or other l i q u i d s : (i) E l e c t r o l y s i s i s by far the worst source of trouble, causing corrosion of the probe, generation of gases and i n s t a b i l i t y i n the e l e c t r o n i c control c i r c u i t r y . This p a r t i c - ular problem does not a r i s e i n non-conducting liquids,such as d i s t i l l e d water or kerosene. 34 Another way of avoiding serious corrosion could be the use of high frequency a l t e r n a t i n g current to heat the probe, and/or coating the probe to provide e l e c t r i c a l i n s u l a t i o n from the l i q u i d . ( i i ) Often the formation of bubbles on the sensor causes inc o r r e c t and unstable operation of the 108 probe . Bubble formation can be reduced by cleaning the probe i n a solvent, e.g. methyl alcohol, with the anemometer i n "stand-by" condition, and/or by adding some surface reac- tants to reduce f l u i d surface tension, thus preventing the formation of bubbles and t h e i r attachment to the sensor. In the case of water a "wetting agent" (Kodak Photo-Flo 200.) can be used. ( i i i ) Contamination of the probe by dust p a r t i c l e s or other deposits reduces and modifies i t s s e n s i t i v - 109 l t y . To eliminate d i r t contamination the surface of the l i q u i d should be shielded. Con- tinuous f i l t r a t i o n of a part of the c i r c u l a t i n g f l u i d should also help i n minimizing the problem. Both these methods and frequent cleaning of the probe were found necessary i n the present set of experiments. 35 For the reasons c i t e d above each probe had to be ca l i b r a t e d independently before i t was used. There are many ways to c a l i b r a t e hot f i l m p r o b e s ' * " ^ ' . The choice of method depends on the a v a i l a b i l i t y of a sui t a b l e standard of comparison, the ease of measurement, and the desired degree of accuracy. In most cases, the v e l o c i t y measured by mechanical means at a s p e c i f i c point i n the f l u i d f i e l d is compared with the e l e c t r i c a l s i g n a l of the anemometer. The degree of accuracy then depends mainly on the accuracy with which the reference v e l o c i t y i s known. Two c a l i b r a t i o n procedures were used mainly to substantiate the methods adopted. i n the f i r s t case,a probe was mounted on the t o o l holder of a lathe and towed at c o n t r o l l e d speeds i n a s l o t t e d flume, employing the feed mechanism. The multispeed gear box allows s a t i s f a c t o r y determination of the tow v e l o c i t i e s . The probe was immersed i n the f l u i d to a depth of at l e a s t ten times i t s diameter. The arrangement worked s a t i s f a c - t o r i l y up to a v e l o c i t y of about 15.0 cm/sec. beyond which noise caused by spurious v i b r a t i o n s u b s t a n t i a l l y affected the s i g n a l . Furthermore, at higher v e l o c i t i e s the towing time became so short, due to l i m i t e d length of the flume (5 f t . ) , t h a t the output signal from the probe could not reach the steady state condition. A photograph of the arrangement is given i n Figure 2-5. In the a l t e r n a t i v e arrangement the probe was held stationary i n a rotating dish, 1 f t . diameter and 10 i n . high, Figure 2-5 A photograph of the probe towing mechanism and associated flume: F, feed worm; P, probe; S, flume; V, speed control gear box mounted h o r i z o n t a l l y on a turntable with i n f i n i t e l y variable speed drive (Figure 2-6). This arrangement was found s a t i s f a c t o r y over the v e l o c i t y range of i n t e r e s t . S u f f i c i e n t time had to be allowed for a quasi steady state of motion 112 to be set-up . The motion obtained was very c l o s e l y s o l i d body r o t a t i o n when the probe was not too f a r from the c y l i n - d r i c a l or bottom walls of the dish. The c i r c u l a r dish had to be s u f f i c i e n t l y large to allow for the d i s s i p a t i o n of v o r t i c i t y generated by the probe between successive passes; obviously the time constant of t h i s e f f e c t i s of the order 2 113 v/r (r = distance from the probe to the axis of r o t a t i o n ) . Absolute cleanliness was found to be e s s e n t i a l i n these t e s t s . The complete r i g was kept i n a glass enclosure which greatly reduced the frequency of probe cleaning required. In general, i f properly designed, either of the procedures can be expected to y i e l d r e l i a b l e r e s u l t s . This was indeed the case with the c a l i b r a t i o n p l o t s obtained, which correspond p r e c i s e l y . However, the second arrangement, because of i t s rather l i m i t e d volume, was anticipated to be more susceptible to temperature v a r i a t i o n s , and at high speeds to mechanical v i b r a t i o n s . Hence the f i r s t procedure was used when a r e c a l i b r a t i o n was desired. C a l i b r a t i o n tests were c a r r i e d out i n tap water, d i s t i l l e d water and water-glycerol solutions of several concentrations (Figure 2-7). As anticipated, the experimental points clustered around a st r a i g h t l i n e down to quite low Figure 2-6 A photograph showing the rota t i n g dish arrangement for c a l i b r a t i o n of hot-film probe: C, constant temperature anemometer; D, drive wheel; co M, drive motor; P, probe; R, rota t i n g disn; V, D.C. d i g i t a l voltmeter 0 0 39 800 600H 2 21 V, (volts) 40 (H 200H a=0.121 0.097 0.047 — e — 9— B s— Tap water T= 23.20 °C 0 Figure 2-7 I U°f ( c m / s e c ) 0 ' 5 C a l i b r a t i o n data for the hot f i l m probe i n : (a) tap water Figure 2-7 C a l i b r a t i o n data for the hot f i l m probe i n : (b) d i s t i l l e d water 41 800 600H 2 \ V, (volts) 400H 200H Cn=60 - © — o - 65 •* * 70 a _ a=0.10 0-50 Glycerol-water solution 0 Figure 2-7 I U , (cm/sec) C a l i b r a t i o n data for the hot f i l m probe in; (c) water-glycerol solutions of various concentrations values of the true v e l o c i t y . The indicated s t r a i g h t l i n e i s a le a s t mean square f i t through the measured data. The maximum percentage deviation from the f i t i s 8.8% for tap water and the minimum i s 0.40% for water-glycerol s o l u t i o n of 60% concentration. The scatter of experimental r e s u l t s i s of the order of that which i s expected from the a n c i l l a r y equipment alone. Though the v i s c o s i t y of g l y c e r o l i s rather sensitive to changes i n temperature, the e f f e c t of rates of heating i n standard f i l m equipment appears low enough to be ne g l i g i b l e . Since the hot f i l m temperature T m i s kept con- stant by v i r t u e of the overheat r a t i o , a change of R c (probe's cold resistance) during the measurements would imply a change of T . It i s , therefore, useful to investigate d r i f t i n the overheat r a t i o induced by v a r i a t i o n s i n R . J c This would give some appreciation as to the changes i n the ambient temperature that can be tolerated during a given test. To t h i s end, dependence of probe cold resistance on f l u i d temperature was measured using a constant temperature bath. Figure 2-8 shows these r e s u l t s for various concen- trations of g l y c e r o l s o l u t i o n . A l l the curves have almost the same slope suggesting the constant c o e f f i c i e n t of r e s i s t i v i t y . In the worst case,the maximum deviation was observed to be about 1.2%. The arrangement was also used to locate the intersection of the v e l o c i t y c a l i b r a t i o n curves with axis Glycerol-water solution U = 0 13 Figure 2-8 3b T, °C 50 70 C a l i b r a t i o n plots showing the e f f e c t of temperature on probe's cold resistance when immersed i n glycerol-water solution of different•concentrations 44 2.3 Pressure Transducer The mean pressure component, being extremely small -4 (of the order of 10 p s i ) , demanded a highly s e n s i t i v e i n s t r u - mentation for i t s measurement. This was accomplished using a "Barocel Modular Pressure Transducing System" developed by Datametric Inc. of Waltham, Massachusetts. The type 550-5 Barocel sensor i s designed to operate with f l u i d s over the pressure range of 0-10 p s i a . The unit i s a high p r e c i s i o n , stable capacitive voltage d i v i d e r , the v a r i a b l e element of which i s a t h i n prestressed s t e e l diaphragm positioned between fixed capacitor plates. The diaphragm d e f l e c t s p r o p o r t i o n a l l y to the magnitude of the applied pressure. To i s o l a t e the external pressure medium from the sensor diaphragm-capacitance system, the unit uses highly s e n s i t i v e m e t a l l i c bellows. The volume between the bellows, i s o l a t o r and sensor diaphragm i s f i l l e d with degassed s i l i c o n e o i l which serves both as pressure transmitting f l u i d and as a d i e l e c t r i c . The pressure signal from the external l i q u i d medium i s transmitted by the bellows to the s i l i c o n e o i l which i n turn d e f l e c t s the diaphragm to produce the required change i n capacitance. An A.c. c a r r i e r voltage at 10 Kc/sec. i s applied to the stationary capacitor plates, and a bridge c i r c u i t determines an output voltage dependent on the r a t i o of the capacitance of the diaphragm to each of the stationary plates. The c a r r i e r voltage i s therefore modulated according to the input pressure. The unit s e n s i t i v i t y i s 10 p s i provided the pressure sensor i s f u l l y i s o l a t e d from external sources of v i b r a t i o n and noise. I t was imperative to ensure removal of a l l traces of a i r pockets from the pressure ducting for s a t i s f a c t o r y operation. Barocel i s accurately c a l i b r a t e d for steady pressures. Figure 2-9 presents a schematic diagram of the pressure transducer. Figure 2-9 Schematic diagram of a Barocel pressure transducer 46 It was important to minimize the e f f e c t of ambient temperature excursions on the Barocel's performance. This was achieved by mounting the transducer on a heat sink, a large aluminum block with working f l u i d c i r c u l a t i n g i n s i d e . The arrangement v i r t u a l l y eliminated the influence of temperature transients. 2.4 A o r t i c Heart Valve Model A o r t i c replacement b a l l values presently i n use consist of the following parts (Figure 2-10): (i) a metal cage of a highly polished uncoated casting of S t e l l i t e 21, a cobalt a l l o y noted for high strength and excellent corrosion resistance; ( i i ) a spherical b a l l of s i l i c o n e rubber with diameter ranging from 0.4 82 - 0.868 i n . ; ( i i i ) a metal seat, normally c a l l e d o r i f i c e , carry- ing a sewing margin of knitted Teflon c l o t h . Unfortunately, the actual a o r t i c valve i s f a r too small for the purpose of pressure measurement on the b a l l surface and i t s wake. Furthermore, v e l o c i t y data upstream and downstream of the b a l l would be affected because of the flow disturbances produced by the presence of the hot f i l m probe and i t s stem. Hence i t was decided to use an enlarged (iv) suture ring Figure 2-10 Exploded view of the Starr and Edwards a o r t i c b a l l valve prosthesis: (i) case; ( i i ) b a l l ; ( i i i ) o r i f i c e r i n g ; (iv) suture ring 48 model of the a o r t i c b a l l valve for t e s t evaluation. The enlargement r a t i o was governed by the following f a c t o r s : (i) a t t a i n a b i l i t y of the dynamic s i m i l a r i t y ; ( i i ) size of the tunnel t e s t section; ( i i i ) s u f f i c i e n t l y large sphere to provide adequate number of pressure ports and room for pressure conducting tubings; (iv) p r a c t i c a l d i f f i c u l t i e s involved i n pulsating the b a l l . Above considerations i n conjunction with a few compromises led to a 5:1 enlargement of an actual p r o s t h e t i c valve for experimental i n v e s t i g a t i o n . I t consists of: (i) A p l e x i g l a s spherical b a l l of 2.5 i n . diameter with fourteen pressure taps d r i l l e d r a d i a l l y to accommodate "Intramedic" polyethylene tubing of 0.038 i n . O.D., 0.023 i n . I.D. and 3 f t . i n length. The exact l o c a t i o n of the pressure ports are indicated i n Figure 2-11. (i i ) A s t a i n l e s s s t e e l tube of 0.375 O.D., 0.325 I.D., and 6 i n . long i s inserted into the b a l l , i n the plane of pressure taps, by means of a p l e x i g l a s plug with 0.750 N.C. threads. The tube acts both as a stem to p o s i t i o n the b a l l c e n t r a l l y with respect to the seat as well as a conduit (b) u (c) ^ stem to the pu I sating mechanism 2 3 2 n e° 1 130 2 110 3 90 4 70 5 50 6 30 7 10 8 0 9 20 10 40 11 60 12 80 13 100 14 120 15 140 16 150 Figure 2-11 A schematic diagram showing the lo c a t i o n of pressure ports on the surface of the poppet: (a) used i n water, 6 = 0 - 130° (b) employed i n glycerol-water s o l u t i o n , 0 = 0 - 110° (c) used i n glycerol-water solution, 6 = 90 150' 50 accoiranodating polyethylene tubings conveying pressure signals to the externally located transducer. ( i i i ) A three prong cage, made of 0.184 i n . O.D., 0.125 i n . I.D. brass tubing, encloses the b a l l and r e s t r i c t s i t s movement. A port, 0.061 i n . diameter, i s d r i l l e d i n each of the prongs to help estimate pressure v a r i a t i o n s i n the wake of the b a l l . The cage i s fixed to the base r i n g , acting as a seat, by a bayonet f i x t u r e arrangement. The junction of the prongs forming the downstream end of the cage c a r r i e s a brass bushing which acts as an external support for the stem. The bushing i s l i n e d with a Teflon cylinder to reduce f r i c t i o n wear, (iv) A p l e x i g l a s seat, 6 i n . O.D., 2.250 i n . I.D., 1 i n . thick, consisting of three l i p s spaced c i r c u m f e r e n t i a l l y at 120°, i s so machined as to produce watertight f i t through matching curvatures when the b a l l i s pressed against i t . The arrangement i s e s s e n t i a l i n actual valves to reduce contact stresses through increased impact area; (v) Two concentric p l e x i g l a s c y l i n d e r s , held i n pos i t i o n by means of four rectangular p l e x i g l a s r i b s , r e s u l t i n an annular space to serve as a 51 bypass when the valve i s shut. This i s e s s e n t i a l to prevent possible water hammer damage to the tunnel. Connected at the out- l e t end i s a brass f i x t u r e which supports a double acting a i r c y l i n d e r , an i n t e g r a l part of the pulsating mechanism described l a t e r . Two p l e x i g l a s plates, attached to the top and bottom of the outer c y l i n d e r by means of two rectangular p l e x i g l a s r i b s , help mount the model inside the tunnel. Figure 2-12(a) shows an exploded view of the model with a photograph of the components presented i n Figure 2-12(b). The assembled model was s l i d into p o s i t i o n through t2se o u t l e t end of the t e s t section and was locked i n place bf a s t a i n l e s s s t e e l pin extending about 0.500 i n . i n the r i b . A porthole (0.500 i n . diameter), d r i l l e d through the tunnel wall and concentric c y l i n d e r s f a c i l i t a t e d v e l o c i t y measurements i n the wake of the poppet. 2.5 Pulsating Mechanism Poppet of the prosthetic valve exhibits a rather complex time-displacement his t o r y . However, for dynamic s i m i l a r i t y between the prototype and the model, i t i s essen- t i a l to simulate important c h a r a c t e r i s t i c s of t h i s motion. The time history used i n the program was based on a frame p r e s s u r e convey ing t ub i n g s su ppo r t i ng c y l i nder P l ex i g l a s seat p l e x i g l a s c y l i n d e r re p r e s e n t i n g a o r t a p o p p e t s tem connected to o s c i l l a t i n g mechanism Figure 2-12 An exploded view of the heart valve model: (a) schematic drawing Figure 2 -12 An exploded view of the heart valve model: (b) photograph of the components: C, cage; D, concentric c i r c u l a r cylinder; M, mounting plate; R, support r i b s ; S, o r i f i c e seat OJ by frame analysis of the x-ray movies (taken at Shaughnessy hospital) of patients with prosthetic valve employing s p e c i a l radio opaque poppet.Furthermore, there has been some i n f o r - mation on t h i s aspect recorded i n l i t e r a t u r e . As can be expected, the displacement function varied from patient to patient. A t y p i c a l p l o t i s presented i n Figure 2-13. In the t e s t described i n d e t a i l s l a t e r , a representative p l o t was selected for simulation. The desired o s c i l l a t i n g motion of the b a l l i s produced by means of a double acting a i r cylinder (A) i n conjunction with an e l e c t r o n i c pulse duplicator and a s p e c i a l l y designed pneumatic c i r c u i t (Figure 2-14). The supply of compressed a i r , held constant at 80 psig through a needle valve (N), i s fed into two pressure regulating valves (R^,I^) adjusted to produce o u t l e t pressures of 4 0 and 5 psig, respectively. The 40 psig a i r supply, c o n t r o l l e d by a solenoid valve (s), i s used to drive the p i l o t of a four way a i r valve (p). On the other side, the a i r supply at 5 psig i s connected to the intake of the four way valve with i t s outlets connected to the a i r c y l i n d e r . As the solenoid opens the port, the p i l o t a i r activates the four way valve, which i n turn causes the b a l l to move upstream u n t i l i t rests against the o r i f i c e seat. When the solenoid shuts the valve, the b a l l returns to i t s o r i g i n a l p o s i t i o n under the reversed piston actuation. The speed of t h i s fore and a f t motion of the b a l l can be controlled by means of two needle valves (v) placed i n l i n e 1 : 1 1 1 _ 0 0.5 1.0 1.5 time»sec Ln Figure 2-13 A comparison of the t y p i c a l displacement-time h i s t o r i e s f o r an a o r t i c b a l l valve implant and the model 56 to the heart valve vie 3 to the, displacement transducer L - J R . .40 PSi from pulse duplicator Figure 2-14 Details of the c i r c u i t used i n pulsating the poppet of the heart valve model: C, compressor; F, adjustable muffler; M, motor; N, needle valve; P, four way a i r valve; R,, fi n e control pressure regulating valve; R2, pressure regulat-ing valve with coarse s e t t i n g ; S, three way solenoid valve; V, one way fine control needle valve 57 at the intake of the a i r c y l i n d e r . Duration of each cycle and the t r a v e l time are adjusted through a pulse duplicator which controls the solenoid (Figure 2-15). 2.6 Linear Displacement Transducer The time h i s t o r y of the pulsating poppet was moni- tored by a l i n e a r displacement transducer s p e c i f i c a l l y designed for underwater a p p l i c a t i o n (Figure 2-16). The transducer employs the p r i n c i p l e of d i f f e r e n t i a l transformer and consists of three main components: (i) a sof t iron core; ( i i ) a non-conducting spool upon which the primary and secondary windings are wound; ( i i i ) a brass tubular casing. A 10 Kc/sec., 10 v o l t s rms si g n a l from a function generator (Interstate E l e c t r o n i c Corporation) supplied to the primary i s modulated by the core o s c i l l a t i o n . This modulated sig n a l from the secondary i s r e c t i f i e d to 2.4 - 4.4 v o l t s D.C. i n l i n e a r proportion to the core di s p l a c e - ments. The transducer i s c a l i b r a t e d to the accuracy of ±.001 i n . employing a mechanical d i a l gauge i n conjunction with a traversing mechanism. The transducer's body was held i n a vise with the actuating rod connected to the 15n JlOOOjif ,1000fi 110v A.C (a) 6 8 0 a r V W 2N4923 L+ -^trigger inpuli-r- 28v D.C.supply to external trigger 58 solenoid valve o/p (b) 10kn Figure 2-15 C i r c u i t diagram for the pulse duplicator: (a) D.C. power supply for the solenoid valve and the (b) tr i g g e r i n g mechanism 59 secondary winding 1 movable core case primary winding .electrical connections •3 c e C c out i T in 9 O out 1 V i X,inch _ i_t 2 V.volts in Figure 2-16 A schematic diagram of the displacement transducer and associated e l e c t r o n i c c i r c u i t r y 1 60 traversing gear. For each p o s i t i o n of the rod, set by the traversing device and indicated by the d i a l gauge, the output voltage from the r e c t i f i e r was recorded. The c a l i - bration p l o t thus obtained i s shown in Figure 2-17. The st r a i g h t l i n e represents the l e a s t square f i t to the t e s t data. 4-5r 4-0 3-5h V,(volts) 3-01 25 • exp. data ....fitted line 2.01 0 0-2 04 X, inch Figure 2-17 C a l i b r a t i o n p l o t for the displacement transducer 6 2 3. TEST PROCEDURES Before proceeding to present the t e s t r e s u l t s and th e i r discussion i n d e t a i l , i t would be appropriate to b r i e f l y describe some of the important t e s t procedures. Mostly, the techniques employed are conceptually well known but t h e i r implementation a t t a i n complexity of a higher order, mainly because of the character of the working f l u i d (water or g l y c e r o l ) . Often p e c u l i a r i t i e s of s p e c i f i c experiments make certain measurements quite d i f f i c u l t . Throughout, the emphasis i s on p r a c t i c a l considerations involved i n executing the experimental programme. At times the factors involved are, seemingly, so t r i v i a l that one would seldom give them a second look. However, a common experience of most experi- menters i s that r e s o l u t i o n of apparently simple problems occasionally takes days, i f not weeks or months. This chapter aims at touching upon such relevant points encountered during the present t e s t programme. 3.1 V e l o c i t y P r o f i l e s Along the Test Section of the Tunnel The f i r s t step i n the test programme was to c a l i - brate the tunnel, i . e . , to obtain information about the boundary layer growth as r e f l e c t e d i n the v e l o c i t y p r o f i l e s along the t e s t section. To t h i s end, the tunnel (without the test model) was f i l l e d with the working l i q u i d of a fi x e d concentration. A l l a i r pockets and bubbles were removed from the tunnel by c i r c u l a t i n g the t e s t f l u i d , with the wetting agent, for at l e a s t eight hours at around 30°C. V e l o c i t y p r o f i l e s at several t e s t sections were then obtained using the c a l i b r a t e d hot f i l m probe i n conjunction with a travers- ing gear, which can p o s i t i o n the probe with an accuracy of around ±0.01 i n . I t should be pointed out that the probe movement i s confined to the v e r t i c a l d i r e c t i o n i n the c e n t r a l plane of the tunnel. Step si z e for the probe movement was regulated according to the v e l o c i t y gradient so as to provide an accurate p r o f i l e near the w a l l . I n i t i a l tests revealed the regions where the v e l o c i t y d i s t r i b u t i o n deviated from the desired uniform value. Logical a p p l i c a t i o n of the a v a i l a b l e information"''^ -'*"^ together with considerable amount of experimentation led to a suitable combination of honeycombs, brass screens and f i b e r g l a s wool, which produced a f a i r l y uniform flow over the c e n t r a l portion of the t e s t section i n the desired range of the Reynolds number. Deposition of dust p a r t i c l e s and/or a i r bubbles on the probe, leading to a sudden change i n the probe re- sponse, represented an occasional source of annoyance. Addition of the surface reactant and introduction of the f i l t e r i n g c i r c u i t reduced the problem but did not completely 64 eliminate i t because of the gradual wear of the pump sea l . To avoid d i r t contamination the probe was washed i n methyl alcohol a f t e r each run. Figure 3-1 shows the instrumentation layout used during these measurements. 3.2 S t a t i c Pressure D i s t r i b u t i o n on Stationary and O s c i l l a t i n g Spheres The main element of the valve being the spherical poppet, i t seemed appropriate to conduct pressure measure- ments on an i s o l a t e d sphere during both stationary and o s c i l l a t i n g conditions. This would n a t u r a l l y serve as a useful reference i n assessing e f f e c t s of presence of the cage, seat and the surrounding c y l i n d e r s approximately repre- senting the a o r t i c vein. Furthermore, an extensive review of the l i t e r a t u r e showed that most investigations on sphere are confined to the o v e r a l l drag measurements. Detailed studies of the pressure d i s t r i b u t i o n i n the Reynolds number range of i n t e r e s t (100-5000) are rather scarce. Of course, with i t s symmetrical geometry together with the a v a i l a b l e information on the drag data, i t i s convenient to check the performance of the t e s t set-up. To cover a wide range of Reynolds number, seven di f f e r e n t spheres (1/4, 3/8, 1/2, 1, 1 1/2, 2, 2 1/2 i n . diameter) were employed as models with water constant temperature annemometer d.c. digital voltmeter . oscilloscope honeycombs DISA hot.film probe,55A83, supported by a traversing brass screens 2 Figure 3-1 Instrumentation layout used during v e l o c i t y p r o f i l e measurement CTl 66 -5 2 (v = 1.05 x 10 f t . /sec.) and glycerol-water solution (54% -5 2 glycerin by weight, v = 7 x 10 f t . /sec.) as working f l u i d s . After a complete removal of a i r bubbles from the f l u i d , a model was positioned i n the t e s t section with i t s center 18 i n . downstream of the l a s t screen. Next, the pressure ducting was f i l l e d with the t e s t f l u i d and was connected to a Barocel pressure transducer v i a a manifold, which f a c i l i t a t e d removal of the a i r pockets from the l i n e and provided a central station for connection of the pressure tubings. The pressure sensing unit was balanced to read zero output i n the no-flow condition. With the pump operating at a preselected speed to give a desired Reynolds number and the t e s t f l u i d held at a constant temperature, the mean pressure d i s t r i b u t i o n around the horizontal meridional cross-section was measured. For each run the v e l o c i t y p r o f i l e upstream of the sphere was also recorded. The hot f i l m probe, mounted on a traversing gear, was positioned 10 i n . upstream of the sphere. The procedure was repeated over a range of mean flow rates. A point concerning an appropriate choice of the si z e of the pressure tubings must be emphasized here. A systematic study with tubes of d i f f e r e n t size and associated time to reach steady state pressure showed the tubes with i n t e r n a l diameter less than 0.062 i n . to have an excessively large time constant (>20 min.) Of course, as suggested by several theoretical and experimental studies on the dynamic response 121-123 of f l u i d l i n e s , the time constant would depend on a number of parameters including the diameter and length of the tubings, v i s c o s i t y of the f l u i d , i n l i n e volume including the transducer's cavity, character of pressure signals, etc. For the mean pressure measurements under consideration, i t was convenient to use f l u i d l i n e s of 0.023 - 0.187 i n . I.D., depending on the size of the sphere and the f l u i d , r e s u l t i n g i n the time constant of around f i v e minutes. To insure accuracy as well as r e p e a t a b i l i t y of the measured data, i t was of utmost importance to minimize and compensate for any d r i f t of the pressure transducer and associated e l e c t r o n i c c i r c u i t r y . Minute character of -4 the pressure signals (10 psi) together with the r e l a t i v e l y long time involved i n reaching the steady state made t h i s a l l the more necessary. Chart recordings of the d r i f t over periods of 24-48 hours showed them to be quite s i g n i f i c a n t , at times as large as 50% of the actual s i g n a l , but of no well defined pattern. The d r i f t compensation procedure involved c a r e f u l measurements of pressures at a reference step (say 6 = 60°) before and a f t e r a given t e s t and a t t r i - buting the difference to the d r i f t . The procedure yielded data that can be reproduced within the accuracy of ±2.0%. Furthermore, any f l u c t u a t i o n i n the l i n e voltage would be r e f l e c t e d on the pump speed and hence on the pressure signals from the model. The speed fluctuations were monitored through v a r i a t i o n s i n the o r i f i c e meter data. The output voltage from the pressure transducer was damped 68 using a DISA type 550 d i g i t a l D.C. voltmeter equipped with a R-C damping c i r c u i t to provide an adjustable time constant of up to 100 seconds. Schematic diagram of the instrumen- tation layout i s shown i n Figure 3-2. The pressure measurements were also conducted during controlled o s c i l l a t i o n s of the spherical poppet of the a o r t i c heart valve model. For this,the poppet was connected to the end of a double acting a i r cylinder and the displacement transducer to the other. The entire assembly was placed inside the t e s t section and locked i n p o s i t i o n . The power cables for the transducer and pressure conducting intramedic tubings were brought outside the tes t section through the portholes on the top of the tunnel. Compressed a i r l i n e was connected to the cylind e r ports and the actuating mechan- ism checked f o r any misalignment before introducing the working f l u i d . A f t e r s a t i s f a c t o r y operation of the pulsating system the tunnel was f i l l e d with the tes t f l u i d and a i r bubbles removed. Few t r i a l runs were c a r r i e d out to check the performance of the pressure measuring unit and the d i s - placement transducer. Upon assurance of successful operation of a l l the instruments involved, tests were c a r r i e d out for two conditions. In the f i r s t , the poppet was o s c i l l a t e d at various frequencies (6-80 cpm) with a constant mean flow rate through the tunnel. In the alternate case, the pulse rate was held constant but the mean flow rate was varied. As before, for both the cases, the mean pressure measurements \rssssssssssss/sss/ssssssss/sv rsss/ss\ 69 u rssssss/rssssssssssssssssssssss\ manifold -0- reference pressure Barocel air supply to flush liquid in Ii power supply signal conditioner d.c. digital voltmeter oscil loscope u-v. recorder Figure 3-2 A line drawing of the instrumentation set-up used for pressure measurements were confined to a meridional section of the poppet. Above tests were c a r r i e d out with a constant amplitude of 1 i n . in the range 300 < R < 1100. 3 n An aspect of considerable importance i s the a c c e l - eration of the l i q u i d body contained inside the pressure conveying l i n e s leading to an a d d i t i o n a l s i g n a l . I t was d i f f i c u l t to i s o l a t e e n t i r e l y t h i s i n e r t i a e f f e c t from the pressure signals due to the movement of the sphere and f l u i d surrounding i t . To overcome t h i s problem several procedures were attempted, including: (i) o s c i l l a t i o n of a conveying l i n e by i t s e l f and when connected to a pressure tap, the di f f e r e n c e leading to the desired information; ( i i ) d i f f e r e n t i a l pressure measurements with two l i n e s connected to two d i f f e r e n t pressure taps, one of which used as a reference; ( i i i ) pulsating the poppet with no l i q u i d inside the conveying tubes thus minimizing (though not eliminating) the l a t t e r ' s acceleration e f f e c t . As the l a s t method was found to be, r e l a t i v e l y , more e f f e c t i v e , i t was decided to conduct pulsating sphere t e s t s without any l i q u i d (water or glycerol) inside the tubes. Accomplishment of t h i s demanded a degree of ingenuity. With the sphere stationary and having established a desired flow 71 v e l o c i t y , compressed a i r at 5 psig was pumped into a given tube to force the l i q u i d column down to a minimum value (=0.10 i n . ) . The o u t l e t end of the tube was now connected to a shut o f f valve leading to Barocel v i a a manifold. In- side of the tube at the entrance was coated with a low surface tension compound (shoe polish) to avoid breaking up of the l i q u i d column r e s u l t i n g i n trapped a i r and hence, erroneous signals. The poppet was now set into motion at a desired frequency and pressure measurements conducted, opening i n d i v i d u a l valves one at a time. The time h i s t o r y of the s t a t i c pressure was logged continuously, a f t e r f i l t e r i n g the extraneous high frequency noise, using a U.V, recorder (S.E. Laboratories, Model 3006). The mean value of the signal was also obtained using a DISA TYPE 550 D.C. d i g i t a l voltmeter employing a damping c i r c u i t with the maximum time constant of 100 sec. A l l pressure measurements were made with respect to a constant head provided by an external l i q u i d column. To co r r e l a t e pressure signals with the poppet t r a v e l during a cycle, the sphere displacement was also recorded on the same chart. Care was taken to f l u s h pressure conveying l i n e s before each set of t e s t s . Figure 3-3 shows the instrumentation set-up used during the measurements of the pulsating pressures. A photograph showing the instrumentation set-up used during measurements of the pulsating pressure: A, double acting a i r cylinder; C, signal conditioner; D, d i g i t a l D.C. voltmeter; F, f i l t e r ; G, e l e c t r i c a l pulse duplicator; M, modelof the a o r t i c heart valve; N, function generator; 0, osciloscope; P, pressure transducer; Q, manifold; R, recorder 73 3.3 Pressure D i s t r i b u t i o n on the Stationary Poppet Representing Various Valve Openings Here the complete heart valve model was placed inside the tunnel test section and locked i n p o s i t i o n . Employing a gear and rack system, the poppet was positioned so as to produce a preselected valve opening. The poppet p o s i t i o n was indicated within ±0.001 i n . using the l i n e a r displacement transducer. The t e s t l i q u i d was c i r c u l a t e d u n t i l a l l a i r bubbles were removed (=6 h r s . ) . For various openings of the valve and over a range of the Reynolds number, mean pressures around the meridional section of the poppet and three stations on the cage were measured. At the same time,velocity p r o f i l e s at several stations upstream of the model were also recorded. The tests were conducted for both open and close bypass conditions. In the case of the open bypass, the valve opening ranged from 0-1 i n . , i . e . from f u l l y closed to f u l l y open positions of the valve, i n the range 300 < R n < 3100. However, for the closed bypass, the minimum valve opening was r e s t r i c t e d to 0.050 i n . and the maximum Reynolds number to 1100 to avoid over s t r e s - sing of the test section. 3.4 Mean Pressure Measurements on the Poppet while O s c i l l a - ting Inside the Valve For t h i s study, i t was necessary to o s c i l l a t e the poppet in the manner so as to simulate actual motion of the 7 4 prosthetic valve i n a patient. Thus the model arrangement in the tunnel remained e s s e n t i a l l y the same as before except for the pulsating mechanism and monitoring device. After mounting the double acting a i r c y l i n d e r and the displacement transducer, the entire assembly was s l i d into the t e s t section and locked i n p o s i t i o n . Two sets of pressure conducting tubings were involved: one supplying the necessary compressed ai r for operation of the a i r c y l i n d e r and the other set of polyethylene tubes transmitting s t a t i c pressures from the surface of the poppet to the externally located Barocel. These tubes were brought out of the t e s t section through the downstream portholes at the top of the tunnel. Now the tunnel was f i l l e d with the t e s t l i q u i d and a i r bubbles removed as before. With zero mean flow through the tunnel ( i . e . no discharge from the pump), the poppet was forced to o s c i l l a t e and i t s time displacement history was recorded using the displacement transducer ( d i f f e r e n t i a l transformer) and a Honeywell V i s i c o r d e r . After a few t r i a l runs to check alignment of the a i r cylinder and proper functioning of the four way .air valve, the pulse duplicator was set so as to simulate the actual motion of the prosthetic heart valve. The tests were conducted for two sets of conditions. In the f i r s t , the frequency of o s c i l l a t i o n was varied with no net flow through the tunnel. The second case involved a systematic v a r i a t i o n of flow rate through the valve but 75 at a constant frequency. For both the cases, time h i s t o r y of pressures around the poppet and v e l o c i t y p r o f i l e s upstream of the valve were measured. As the pressure at the i n d i v i d r ual tap was recorded one at a time (only one unit of Barocel being available) i t was quite important to maintain the precise character of the o s c i l l a t i n g function. This was ascertained by a continuous recording of the poppet displace- ment and maintaining the constant actuating pressure. To reduce impact stresses r e s u l t i n g from the c o l l i s i o n of the b a l l with the seat, a i r cushions were provided during the terminal portions of the forward and reverse strokes. The a x i a l r o t a t i o n of the poppet was prevented by a s l o t t e d sleeve, forced around i t s stem, and locked i n p o s i t i o n by a key on the a i r cylinder shaft. Care was taken to p o s i t i o n the poppet so that the meridional section carrying the pressure taps was h o r i z o n t a l . As before, the tests were c a r r i e d out for both open and closed bypass conditions, and i n the Reynolds number range of 300-1100. The pressure measuring technique was similar to the one described i n Section 3.2. 3.5 Flow V i s u a l i z a t i o n To better appreciate the character of the flow through the a o r t i c heart valve model and trends indicated by the pressure measurements, flow v i s u a l i z a t i o n was under- 76 taken. Among the several a v a i l a b l e techniques Schlieren, shadowgraph, and dye i n j e c t i o n procedures were considered for t h e i r s u i t a b i l i t y . A preliminary study suggested that the Schlieren technique would present obvious problems of alignment and s t a b i l i t y due to the f l e x i b l e character of the wooden f l o o r . Furthermore, there was also a p o s s i b i l i t y of excessive o p t i c a l noise from the tunnel, l i q u i d and model materials. The shadowgraph technique was discarded because of the l i m i t a t i o n s i n marking the flow (heating or cooling the l i q u i d upstream of the model). Here three d i f f e r e n t procedures were attempted: (i) introduction of a heating element i n the form of a tungsten wire (6 i n . long) to mark the flow; ( i i ) i n j e c t i o n of hot or cold t e s t f l u i d up- stream of the model; ( i i i ) introduction of glycerol-water sol u t i o n of higher density than the t e s t f l u i d upstream of the model. Unfortunately, none of the above attempts proved s a t i s f a c t o r y i n terms of high r e s o l u t i o n flow patterns that can be photographed with c l a r i t y and interpreted. Conse- quently, dye i n j e c t i o n method was used. The dyed glycerol-water solution of the same con- centration as that of the t e s t f l u i d was injected 10 i n . upstream of the model. The dye employed was an im i t a t i o n Cochineal food colour. Appropriate volumes of the dye and pure glycerin were mixed to produce a glycerol-water solu- t i o n of the same density as that of the t e s t f l u i d . At f i r s t the dyed solution was injected through a set of f i v e syringe needles (#16) as shown i n Figure 3-4 (a). Although the arrangement worked f a i r l y w e l l , i t was f e l t that the diameter of the i n j e c t i n g needles was a l i t t l e too large (0.052 in.) and could be reduced. With t h i s i n mind, a new i n j e c t i n g probe consisting of seven #23 syringe needles (0.015 in.) placed 0.25-0.50 i n . apart on a streamlined support, was constructed (Figure 3-4b). "Intramedic" tubings (0.023 i n . I.D.) were used to connect the needles to a manifold. The rate of i n j e c t i o n was c o n t r o l l e d with brass needle valves. To ensure adequate flow through each needle, i . e . to provide s u f f i c i e n t head, the supply b o t t l e was suspended from the c e i l i n g 10 f t . above the i n j e c t i o n l e v e l . A schematic diagram of the complete set-up i s shown i n Figure 3-5. After ascertaining a successful operation of the entire assembly, flow patterns around the stationary and pulsating sphere, by i t s e l f and when located within the a o r t i c heart valve model, were studied. The Reynolds number and o s c i l l a t i n g frequency [Beta number = ( R n S n ) ^ 2 = D t f / v ) ^ were varied systematically and flow patterns recorded. It would be appropriate to point out here the type of l i g h t i n g system used as i t played a c r i t i c a l r o l e i n the 7 8 (a) (b) Figure 3-4 A photograph of dye i n j e c t i n g probes: (a) e a r l i e r model (b) f i n a l streamlined probe camera A sketch showing the equipment layout during flow v i s u a l i z a t i o n 80 photographing process. A combination of three variable i n t e n s i t y photo floods (maximum 500 watts, 3400°K) back- illuminated the subject through the tunnel glass window. To eliminate hot spots, the l i g h t beam was evenly d i f f u s e d by masking the t e s t section wall with a tracing paper. A set of t r i a l runs helped a r r i v e at the appropriate aperture setting and exposure time for the type of f i l m used ( s t i l l - Kodak high speed Ektachrome type EHB-135, tungsten, 3200°K, ASA 125, f i l t e r 81A; movie-Kodak Ektachrome EF7242, tungsten, 3200°K, ASA 125, f i l t e r 81A). During the course of v i s u a l i z a t i o n study, i t was discovered that i n spite of the large volume of the t e s t f l u i d (40 U.S. g a l l o n s ) , a r e l a t i v e l y small amount of dye (8 f l u i d oz.) was s u f f i c i e n t to p o l l u t e the working f l u i d to the point that no c l e a r photographs could be taken. This presented a rather serious problem i n terms of time, e f f o r t and cost involved i n replenishing the working f l u i d . C l early, i t was necessary to f i n d an agent that would neutral i z e the dye without attacking the tunnel material or i t s c i r c u - l a t i n g system and which does not a l t e r the physical properties of the t e s t f l u i d . Unfortunately, no such agent has been reported i n the l i t e r a t u r e . A considerable amount of patient t e s t i n g with numerous o x i d i z i n g agents led to sodium hypochlorite which has a l l the desirable a t t r i b u t e s . Only 300 cc of the agent was s u f f i c i e n t to completely n e u t r a l i z e 81 the dye. To keep the concentration of the test f l u i d constant, s u f f i c i e n t amounts of g l y c e r i n were p e r i o d i c a l l y added thus o f f s e t t i n g the d i l u t i n g e f f e c t of the dye removing agent. 82 4 . RESULTS AND DISCUSSION With some appreciation of the background to the problem, instrumentation used and the experimental procedures adopted, we are ready to look into the t e s t r e s u l t s and t h e i r i n t e r p r e t a t i o n . The amount of experimental data obtained i s rather enormous, thus d i c t a t i n g a compromise between concise- ness and comprehensibility. The guiding p r i n c i p l e has been to include only those r e s u l t s which have immediate relevance to the study i n hand, and help i n e s t a b l i s h i n g d e f i n i t e trends. In general, the sequence i n which the r e s u l t s are presented also denote the chronological order of the t e s t s . To begin with, v e l o c i t y p r o f i l e s along the t e s t section are measured which give some i n d i c a t i o n as to the performance of the designed t e s t f a c i l i t y . This i s followed by the steady and instantaneous mean pressure on stationary and o s c i l l a - ting spheres, respectively. The study not only concerns one of the basic elements i n the more complex configuration undertaken l a t e r , but also provides information of fundamen- t a l importance not recorded i n l i t e r a t u r e . The next l o g i c a l step was to study the hydrodynamics of the poppet i n the simulated b i o l o g i c a l environment systematically — f i r s t with the spherical poppet occupying d i f f e r e n t positions i n the valve followed by i t s o s c i l l a t i o n i n the p r e c i s e l y controlled fashion representing the time hi s t o r y of a 83 prosthetic valve i n patient. The r e s u l t s of both the bypass open and closed si t u a t i o n s are presented to have at l e a s t a q u a l i t a t i v e appreciation of the influence of regurgitation and leaky valve. The flow v i s u a l i z a t i o n data are also i n - cluded which generally substantiate the flow behavior predic- ted by pressure r e s u l t s . When av a i l a b l e , published r e s u l t s from l i t e r a t u r e are included to aid comparison and help e s t a b l i s h trends. 4.1 Tunnel V e l o c i t y P r o f i l e s V e l o c i t y p r o f i l e s were measured at several locations i n the t e s t section i n the Reynolds number range 300-20,000 based on the hydraulic diameter of the t e s t section and the average v e l o c i t y as deduced from the flowmeter data. Both with water and g l y c e r o l as working f l u i d s , a j e t type of flow was observed over several l o c a l i z e d regions t y p i c a l l y i l l u s - trated by the plots i n Figure 4-1. This suggested that the d i f f u s e r r i n g , honeycombs and screens introduced to uniformly d i s t r i b u t e the flow were not completely e f f e c t i v e . The undesirable behavior was s u b s t a n t i a l l y reduced (Figure 4-2) through the introduction of nylon wool i n conjunction with e x i s t i n g straighteners i n the entrance portion of the te s t section. Note that now the v e l o c i t y p r o f i l e i s e s s e n t i a l l y f l a t at l e a s t over c e n t r a l 10 cm. of the tunnel 20 16 h 12 Z,cm 8 4 • • • • 4* I • +A • » +A 0 O • A o o a O D a CD a o a o D O A a o A a o A • A a o A o o A • • «D O A • D O A • • § A O A • a o A • • • 6 O A • a o A yp=100 cm • • 4 o o A • O O A • • 4* a O A • O A • • A O O A D O A • • A • O O A . . * • " a • ^ CD A • • A • OD • a • • A4 O D >• A , o D , , O O o o o o z U,cm /sec Y^cm • 3-10 A 4.40 • 3.10 \ • 440 i' A 9.65 a 6.73 o 12.49 o o 6.73 A A I 12 U 2,cm /sec 15 18 21 0-0 2.5 0.0 2.5 24 water water glycerol solution Figure 4-1 Typical tunnel v e l o c i t y p r o f i l e s i n d i c a t i n g l o c a l i z e d jet type flow 00 20.0 17.5 15.0 12.5 10.0 Z,cm 7.5 5.0 2.5 cx a <o < o < o < o < o < o < o < o < o < o < o o < o < o o < o < o < o < 6 ° e«—B • • • • U,cm /sec • 5.60 o '2.66 8 5 < 1.27 y p=100 cm 0.0 4.0 8.0 12.0 U z,cm /sec 16.0 Figure 4-2 Representative v e l o c i t y p r o f i l e s showing e f f e c t of the introduction of nylon wool: (a) i n water 20.0 ._ 17.5 L 15.0 125 10.0 Z, cm 7.5 5.0 2.5 < < < < < < < < < < < < < < < < < < . < < < < < < < < < < o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o • • • • • • • q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • o • • • 8 6 U,cm /sec D12.70 .11.30 o 9.80 < 4.70 yp=100 cm o 0.0 4.0 Figure 4-2 8.0 12.0 16.0 20.0 U z ,cm /sec Representative v e l o c i t y p r o f i l e s showing e f f e c t of the introduction of nylon wool: (b) i n glycerol- water solution, C = 5 4 87 height. Of course, t h i s i s desirable for reproduction and comparison of the t e s t r e s u l t s . In general, the uniform portion of the p r o f i l e s decreased with increase i n distance from the tunnel i n l e t and v e l o c i t y . This i s to be expected i n the l i g h t of the flow development downstream and stren- gthening of the j e t e f f e c t with an increase i n the flow rate. Another point of i n t e r e s t was the character of the v e l o c i t y p r o f i l e s i n presence of the models as shown i n Figures 4-3, 4-4. In both the cases, the v e l o c i t y r e s u l t s are for the plane 12.5 cm. ahead of the model (distance measured from the stem center l i n e for the spherical model and from the o r i f i c e plane i n the case of the valve model). Note that with the spherical model (Figure 4-3) the increase i n resistance and i r r e g u l a r i t y of the p r o f i l e s are r e l a t i v e l y small compared to the valve model representing higher r e s i s - tance (Figure 4-4). Apparently t h i s i s an outcome of rather complex int e r a c t i o n s between in t e r r u p t i o n of the boundary layer growth, blockage, changed t o t a l c i r c u i t re- sistance and associated s h i f t i n the operating point of the pump. Prosthetic valve openings also had a d i r e c t e f f e c t on v e l o c i t y d i s t r i b u t i o n s . Smaller valve openings led to f l a t t e r p r o f i l e s i n water; however, no such discernable trend was observed i n g l y c e r o l . The worst s i t u a t i o n was represented by the valve model, f u l l y open, i n the g l y c e r o l 20.0 I— A O 17.5 15.0 12.5 10.01 Z.cm 7.5! o A A A A A O O • 88 O O • • 5.0 2.5 A A A o o o • o • o • o a • U, cm/sec a 6-03 o 4.32 A 3.05 vp=100cm _J L_ 0.0 2.5 5.0 75 U z,cm/sec 10.0 12.5 Figure 4-3 E f f e c t of spherical model (d = 2.5 in.) on v e l o c i t y p r o f i l e s : (a) i n water 20-0 17.5 < o a < o • <j o a < o • < o • 89 15.0 12.5 10.0 Z.cm 7.5 5.0 2.5 U,cm /sec a 12.70 o 11.30 < 9.80 v p = 100 cm o • • • 0.0 -G-B- 4.0 Figure 4-3 8.0 12.0 16.0 U z,cm /sec 20.0 E f f e c t of spherical model (d = 2.5 in.) on v e l o c i t y p r o f i l e s : (b) i n glycerol-water s o l u t i o n , C = 54 n ' 20 16 12 ,cm 8 U,cm /sec A 7.28 D 5.71 o 8.25* • 4.24 A 7.28 4r- 0 • A a A V e n n 147 135 122 • • • • • A a • • A • A • A • A • A • A • A • A • A • A • • A O A O A O A O A O A O A O A O A . O A O A O A O A O A O A O A O A • A O O 8 1 0 U 2 > cm/sec 12 14 16 Figure 4-4 V e l o c i t y p r o f i l e s as affected by the valve model 91 solution at the Reynolds number of 3000 (the highest f o r the g l y c e r o l solution i n the tes t program). Even here, the maximum deviation from the mean over the valve entrance amounted to only 2.5%. 4.2 Choice of Reference V e l o c i t y and Pressure The v e l o c i t y p r o f i l e s bring to l i g h t an important question of reference or c h a r a c t e r i s t i c v e l o c i t y and pressure that can be used i n presenting data. C l e a r l y , with model immersed i n an unbounded uniform stream there i s no ambiguity i n the d e f i n i t i o n of these parameters: i t i s the constant velocity and pressure of the stream far away from the model. For low Reynolds number flow i n a tunnel, however, the f l u i d v elocity and pressure vary s i g n i f i c a n t l y along the axis of the test section, even i n the absence of the model due to the boundary layer growth along the walls. The presence of model and the associated wake would only accentuate t h i s problem. Obviously some compromise i s indicated i n selec- t i o n of these parameters. 124 125 Grove et a l . ' have suggested the use of the pressure d i r e c t l y below the cente r l i n e of t h e i r model as the reference s t a t i c pressure and the cen t e r l i n e v e l o c i t y , with the model absent but at the same s e t t i n g of the pump, as the c h a r a c t e r i s t i c v e l o c i t y . For models with small blockage t h i s choice of reference pressure may prove to be adequate but at a larger blockage, due to acceleration of the flow at the model loc a t i o n , the reference pressure i s indeed affected and becomes a function of wall confinement (besides other parameters). To put i t d i f f e r e n t l y , the choice of reference pressure as suggested above has a degree of optimism i m p l i c i t i n i t . I t assumes that the e f f e c t s of the upstream adverse pressure gradient created by the presence of the model exactly cancels the influence of the acceleration i n the gaps at the model lo c a t i o n thus giving the desired P^. One possible improvement i n the choice of P^ would be to take i t as the pressure at the model l o c a t i o n (but without the model) with operating condition of the tunnel kept the same as that used with the model i n p o s i t i o n . How- ever, t h i s s t i l l leaves us with the problem of s e l e c t i n g a suitable reference v e l o c i t y . Usefulness of the cen t e r l i n e v e l o c i t y as a charac- t e r i s t i c v e l o c i t y also poses several questions. As indicated by Figures 4-1 to 4-4, the v e l o c i t y p r o f i l e s are s u b s t a n t i a l l y affected by l o c a t i o n , boundary layer growth, screen's mesh size, blockage, pump speed and the t o t a l c i r c u i t resistance. Hence the U^ proposed by Grove et a l . can hardly be considered a suitable reference. Another possible compromise would be to take uniform portion of the v e l o c i t y p r o f i l e far upstream and use i t as a c h a r a c t e r i s t i c v e l o c i t y . However, the distance 93 involved to account for boundary layer e f f e c t s would, i n general, depend upon the tunnel used, model and i t s l o c a t i o n . A rather s i g n i f i c a n t point to keep i n mind i n pre- senting data i s to ensure i t s r e p e a t a b i l i t y by other i n v e s t i - gators, using d i f f e r e n t t e s t f a c i l i t i e s , to permit comparison. With t h i s i n mind and a f t e r c a r e f u l consideration of the a l t e r n a t i v e methods discussed above i n conjunction with the pressure and v e l o c i t y data i n hand, a compromise ch a r a c t e r i s - t i c v e l o c i t y , average v e l o c i t y i n the t e s t section based on the mean flow rate, was adopted. This approach has several obvious advantages. I t eliminates most of the problems mentioned above. A series of tests conducted with and without models (but at the same pump se t t i n g as with the model) showed the reference v e l o c i t y to vary by l e s s than 8% (Figure 4-5). Thus, not only does i t eliminate the question of model l o c a t i o n , type of tunnel, flow straighteners used and size of the t e s t section but also overcomes problems of pressure gradient and blockage. The choice would f a c i l i t a t e the d u p l i c a t i o n of Rn, reference v e l o c i t y being more p r e c i s e l y defined. Further- more, i t s measurement i s quite simple and involves only conventional instrumentation. However, i t must be emphasized that t h i s does not correct for changes i n v e l o c i t y p r o f i l e with distance and hence the l o c a t i o n of the model. We are s t i l l faced with that e l l u s i v e task of se l e c t i n g P . As discussed e a r l i e r , the P advocated by •* 00 ' OO -* 94 40r 30 variac setting 20 10 0 J' pi' / * .A/O • model inserted o empty test sec t i on 1 4 Figure 4-5 8 10 U,cm/sec 1 2 1 4 Blockage e f f e c t on the mean flow rate as indicated by the variac s e t t i n g Grove et a l . has l i t t l e meaning here i n view of the large blockage presented by the model. From the point of view of r e p e a t a b i l i t y and comparison of data, the use of pressure at a spe c i f i e d tap on the surface of the model as reference appears quite a t t r a c t i v e . Although t h i s cannot account for the l o c a l v a r i a t i o n of blockage e f f e c t s (from point to point at the surface of the model) i t could compensate for i t i n an average fashion. Thus one way to present pressure data i n c o e f f i c - ient form would be as C = (P. - P )/(pU /2) where P p v 6 r 0 0 r corresponds to the pressure at the s p e c i f i e d tap on the surface of the poppet and as calculated from the average flow rate (average flow r a t e / t e s t - s e c t i o n area of 8 i n . x 8 i n . ) . However, t h i s d e f i n i t i o n i s s t i l l susceptible to errors introduced by non-uniformity of the v e l o c i t y p r o f i l e (at a pressure tap and the reference l o c a t i o n ) . One way to v i r t u a l l y eliminate i t i s to express the pressure c o e f f i c - ient as explained below (Figure 4-6). Let errors i n pressure due to non-uniformity of the v e l o c i t y p r o f i l e be e n at P n, e. at P Q and e at P . J 0 0 6 6 r r Expressing pressure c o e f f i c i e n t as the r a t i o of the d i f f e r - e n t i a l pressures, between that at a tap i n question and the stagnation point with respect to the reference pressure, gives 96 * U 0 1 V Figure 4-6 An i l l u s t r a t i o n showing possible errors introduced by non-uniformity of the v e l o c i t y p r o f i l e (pe + e e ) - ( p r + £r> ( P0 + e0>- ( Pr + £r> where Pg, P̂ ., Pg correspond to pressures with uniform v e l o c i t y p r o f i l e . Thus e. - e 1 + - J £ P a - P P 0 - P c = ( _§ £ ) ( 6 r ) r O r 1 + 0 r P n - P 0 r Note that - and Eg - are l i k e l y to be very small. On the other hand, Pg - P and P Q - P represent r e l a t i v e l y large quantities compared to the respective error d i f f e r e n - t i a l s . Therefore, e = = =- and e n = are l i k e l y to be vanishingly small. Consequently, the term 1 + £ 9 r P9 " P r - 1 and C - =, =— . This d e f i n i t i o n of the l + e 0 r P P 0 " P r pressure c o e f f i c i e n t promises to provide adequate compensation for errors introduced by non-uniformity of the v e l o c i t y p r o f i l e . The reference l o c a t i o n was taken to be at 9 = 6 0 ° . The choice was prompted by the t e s t data which showed C^ to reach zero i n the general v i c i n i t y of 9 = 6 0 ° , i . e . Pgg - Pm. Of course, i n general, l o c a t i o n of the reference pressure i s e n t i r e l y a r b i t r a r y . The pressure data presented i n t h i s chapter use the d e f i n i t i o n of pressure c o e f f i c i e n t as P - P C = 6 6 0 P P 0 " P 6 0 I t i s easy to recognize the term P Q - P ^ Q as an approximation 2 of 0.50 pU^. However, now we are l i k e l y to account for the errors introduced by non-uniformity of the v e l o c i t y p r o f i l e . Thus, i n summary, t h i s c o e f f i c i e n t has several advantages: i t tends to compensate for blockage e f f e c t s , i r r e g u l a r i t y of the v e l o c i t y p r o f i l e and possible errors i n pressure measure- ments caused by e l e c t r i c a l d r i f t s of the pressure measuring 98 system (the e l e c t r i c a l d r i f t was discussed i n Chapter 3 ) . Furthermore, i n conjunction with the Reynolds number (based on average flow v e l o c i t y and the poppet diameter), i t promises to a s s i s t i n du p l i c a t i o n and comparison of s i m i l a r data by other investigators using d i f f e r e n t t e s t f a c i l i t i e s . 4.3 S t a t i c Pressure D i s t r i b u t i o n 4.3.1 Stationary sphere The tests were conducted on a set of spheres ranging i n diameter from 0.5 - 2.5 i n . i n g l y c e r o l as well as water in the Reynolds number range of 74 - 5848. In a l l the cases, the model was supported by a v e r t i c a l stem, a s t a i n l e s s s t e e l tubing, which also served as a pressure conducting l i n e . Its outside diameter was d i c t a t e d by the r e l a t i v e s i z e of the sphere and the stem influence on the pressure f i e l d . On the other hand, the inside diameter was governed by the time constant to reach the steady state pressure as discussed before. A series of tests conducted with 2.5 i n . diameter sphere supported either by a v e r t i c a l or horizontal stem showed that the sphere to stem diameter r a t i o must be l e a s t seven to make stem interference n e g l i g i b l e (Figure 4-7). The pressure measurements were confined to the h o r i z o n t a l meridional section of the model. A 1/16 i n . pressure tap 99 1.0 0.8 0.6 C p 0-4 0.2 0 -0.2 -0-4 I * D/d s = 1 0 * 7 • l 5 R n = 917 • • • • • A 5 • • • • • • ± 9 J 1 1 I I I 0 30 6 0 9 0 o 120 150 180 9 Figure 4-7 E f f e c t of the supporting stem on the p r o f i l e s for a 2.5 i n . sphere 100 connected the stem through a groove (1/16 i n . dia.) d r i l l e d i n the body of the sphere (Figure 4-8). The entire horizon- t a l plane was covered by a c o n t r o l l e d r o t a t i o n of the stem i n a step s i z e of 10°. Due to sectional symmetry, the measurements i n general were confined to only one side of the sphere, except for occasional checks to confirm flow symmetry. The heat exchanger described e a r l i e r held tempera- ture of the working f l u i d constant within 0.2°C during the test period. Before proceeding to present and analyze the r e s u l t s i n d e t a i l , i t would be of i n t e r e s t to compare the data reduction procedure adopted here with that discussed by Grove et a l . The comparison for a representative set of data i s shown i n Figure 4-9. Note the use of R r i n Figure 4-9(a) due to a d i f f e r e n t d e f i n i t i o n of the Reynolds number used by these authors. Several points of i n t e r e s t become appar- ent. E s s e n t i a l l y l i n e a r drop from the front stagnation to zero pressure c o e f f i c i e n t s i s exhibited by both the systems of data presentation. Note that C = 0 at 0 - 50°, thus P our choice of P r at = 60° i s indeed going to be close to P^. However, the most important departure, and here the advantage of using becomes evident, i s i n and around the separated region. The large spread i n Cp data, which i s due to the combined e f f e c t s of the Reynolds number and blockage, i s d e f i n i t e l y reduced by the use of as shown i n Figure 4-9(b). This suggests at l e a s t p a r t i a l compensation of the blockage e f f e c t s as anticipated before i n Section 4.2. 101 to pressure transducer Figure 4-8 A schematic drawing showing the sp h e r i c a l model and i t s support during the pressure measurements Reynolds number e f f e c t s , over i t s d i s c r e t e ranges, on the surface pressure d i s t r i b u t i o n are presented i n Figure 4-10. I t i s apparent that the region bounded by the front stagnation and zero pressure points i s v i r t u a l l y independent of the Reynolds number (Figure 4-9 (b) also leads to the same conclusion). Note that l o c a t i o n of the zero pressure point (C n =0) i s fix e d by the choice of the r e f e r - 1.4 1.1 0.8 0-5 0.2 C p -0.1 -0.4 -0.7 -1.0 102 • R n = 1116 D = 2.500in I o 983 2.500 758 1-495 594 1.495 397 1-495 A 9 A o 250 0.500 173 0.3 75 S . 103 0.3 75 2 8 o \ ] i : ? 6 A * ° a v • ^ o ° p. R . ̂ <>• - * • 8 8 ° 8 8 A - O A ^ 2 $ o 8 • ° • _L 0.0 30 60 0 o 90 120 150 180 Figure 4-9 Typical C p r o f i l e s around the meridional section of a sphere: (a) data reduced as suggested by Grove et a l . 1 ^ 4 • A • R n =1045 o 918 A 7 0 5 A 5 5 3 • 3 7 0 o 2 3 6 V 161 • 93 1 8 * 3 O v • 8 A O v 0 A 8 v • • 4 * o « 103 D=2.50 0 2.50 0 1.495 1.495 1.495 0.50 0 0.375 0.37 5 A * 2 S t -8 o I 4 - 8 8 8 8 ° o • $ n fi 0.0 30 60 90 120 150 180 6° Figure 4-9 Typical C p r o f i l e s around the meridional section of a sphere: (b) data reduced according to the proposed technique 1.00 0.75 0.50 0.25 C p 0.00 -0.25 -0.50 "o 1 1 > 0 1 I " - Q 0 D=1.0 In. D=0.5 in. D= 0.375 in. a 6 o 6 ' 0 — Q R n =243-475 O A O " R n=124-236 0 R n=74-161 O A A o o A A 2 ° * A : A » 9 O A A 0 o A o - 0 O A i O A ft O A A O * <° o A • • 1 1 0 60 Figure 4-10 120 0 60 0 O 120 60 120 180 Reynolds number dependency of the pressure d i s t r i b u t i o n around a sphere: t-> (a) R n = 74-475 . g i j O O r 0.75 D==2.5 in. 0.50 0.25 C p 0.00 -0.25 - 0 5 0 o A fi R n=9l 7-1045 0 ° c> 6 fi 6 © D=2.1 in. R n=1045-5848 _ o o o o fi °4 fi x Dr1.5 in, 6 A O 8 'n R,=370-705 2 9 0.0 60 120 0.0 60 a 120 0.0 60 120 180 F i g u r e 4-10 Reynolds number dependency of the pressure d i s t r i b u t i o n around a sphere: R n = 370-1045 o 106 ence pressure. However, as indicated by Figure 4-9(a), A. even lo c a t i o n of the zero pressure point C p = 0 remains e s s e n t i a l l y fixed i n the Reynolds number range investigated. Thus influence of the Reynolds number i s primarily confined to the region downstream of t h i s l o c a t i o n . Figure 4-10 also shows an i n t e r e s t i n g trend of forward s h i f t of both, l o c a t i o n of the minimum pressure point and the separation point (approximately represented by the beginning of a f l a t p r o f i l e i n the wake) with increase i n the Reynolds number. This can also be deduced from Figure 4-9(b). Magnitude of the minimum pressure remains e s s e n t i a l l y constant for R R < 240 (Figure 4-10a). However, i n the Reynolds number range of 240-475 i t shows a sudden r i s e . Beyond t h i s the minimum pressure again at t a i n s an almost constant value but at a s l i g h t l y lower l e v e l (Figure 4-10b). In f a c t , t h i s form of behavior i s also exhibited by the pressure plots i n separation and wake regions. To better v i s u a l i z e t h i s sudden r i s e i n pressure, both pressure at 8 = 180° and average base pressure (average over the region 0 = 120° - 180°) are plotted i n Figure 4-11. The r i s e i n pressure i s indeed quite spectacular. No d e f i n i t e explanation was r e a d i l y a v a i l a b l e , however, i t was suspected to be associated with a fundamental change i n the character of the vortex r i n g and the wake downstream. In any case, i t c l e a r l y suggested a need for flow v i s u a l i z a t i o n study to confirm t h i s explanation. 0.12 0.08 0.04| W 0-001 • • • • V V • o D,in. 2.5 2.1 1.5 1.0 0.5 0.375 • o -0.04 -0.08 •4 < < -0.1-2' 100 200 4 00 R 600 800 1000 n Figure 4-11 E f f e c t of Reynolds number on base pressure c o e f f i c i e n t : (a) pressure c o e f f i c i e n t at 0 = 180° 1200 o 0.12i 0.08 0.04 • • • • D,in. o 2.5 o 2.1 v 1.5 • 1.0 . 05 <• 0.375 0.00 . b -0.04 V o • a • a o - o . o 8 ; -0.12 4 1 100 200 4 0 0 R 600 800 1000 1200 n Figure 4-11 E f f e c t of Reynolds number on base pressure c o e f f i c i e n t : (b) average wake pressure o 00 109 It must be emphasized here that such d e t a i l e d measurements of pressure d i s t r i b u t i o n on the surface of a sphere i n the indicated low Reynolds number range (R n = 90 - 5000) have not been recorded i n l i t e r a t u r e . On 82-101 the other hand, numerous a n a l y t i c a l attempts do e x i s t Probably the most relevant to the present range of Reynolds 99 number i s the numerical analysis by Rimon and Cheng who solved the complete Navier-Stokes equations using a f i n i t e difference approximation to v o r t i c i t y transport i n conjunc- t i o n with spherical polar g r i d system. Large v e l o c i t y gradients encountered i n the analysis r a i s e d some questions concerning the s t a b i l i t y of the integration procedure. Rimon and Cheng's r e s u l t s (R n = 1 - 1000) are compared with present experimental data i n Figure 4-12. Discrepancies between the two sets of r e s u l t s i n the v i c i n i t y of the separated region are rather s t r i k i n g and tend to increase with an increase i n the Reynolds number. The numerical r e s u l t s consistently predict higher pressure i n t h i s region. The discrepancy may be a t t r i b u t e d to the inherent l i m i t a t i o n s i n the numerical procedure used and would lead to lower drag predict i o n . Furthermore, as pointed out by the authors themselves, t h e i r t h e o r e t i c a l development i s v a l i d f o r a x i - symmetric flows. However, for R n - 250 - 300, the flow becomes .asymmetric due to the onset of wake o s c i l l a t i o n as explained i n the following section. Thus v a l i d i t y of the theory beyond R n > 300 i s indeed questionable. 110 • Rn=1045 370 239 117 D= 2.090 in. 1.495 0.500 0.375 o • A 2 • o X V A O 1000 300 200 100 t h e o r y 99 i v • V D T v v - X A • T • 0 cm„ ° o o • A V g ID I 8 X • V • 60 90 120 150 180 A comparison of the theoretical.and experimental pressure d i s t r i b u t i o n s on the surface of a sphere The pressure d i s t r i b u t i o n data were integrated to obtain v a r i a t i o n of with the Reynolds number as shown i n Figure 4-13. Important r e s u l t s of experiments and semi- 126 empirical theory as reported i n the l i t e r a t u r e are also included for comparison. The c l a s s i c a l drop i n drag with the Reynolds number represents resultant influence of the minimum pressure, i t s l o c a t i o n and the pressure d i s t r i b u t i o n downstream of i t . Considering the f a c t that the experi- 75 mental data themselves show wide deviations (often as much as 20%), the agreement may be considered rather good. This i s indeed comforting and tends to substantiate r e l i a b i l i t y of the presented pressure data. To provide better appreciation as well as substan- t i a t i o n of the c e r t a i n behavior exhibited by the measured data, i t was decided to undertake extensive flow v i s u a l i z - ation program. A set of spheres ranging i n diameter from 0.5 - 2.0 i n . were used i n g l y c e r o l s o l u t i o n medium to provide lower end of the Reynolds number range of i n t e r e s t . The main objective was to observe the formation, development and i n s t a b i l i t y of the vortex r i n g and the associated influence on the measured pressure data. I t was also hoped that t h i s would provide some i n d i c a t i o n concerning l o c a t i o n of the separation point and i t s movement. The use of dye i n j e c t i o n procedure, explained i n d e t a i l e a r l i e r , proved to be quite e f f e c t i v e i n achieving these objectives. I t showed the formation of vortex r i n g i n a rather spectacular 1.4, 1.2 1.0h ft * 0.6 A d h • • - A 0.8 A • • - A A 67 • Liebster 7 4 • Ross & Willmarth • Present work A T h e o r y 1 2 6 A • • • A • • • ^ 0.4 L u _ 1 , | | | | i i i I • A 8 1 0 2 2 3 4 5 1 0 3 2 3 4 Rn Figure 4-13 V a r i a t i o n of the sphere drag c o e f f i c i e n t with Reynolds number 113 fashion as presented i n Figure 4-14. Numerous photographs were taken at systematic increments of the Reynolds number. Only a few of the t y p i c a l pictures i l l u s t r a t i n g formation, symmetric elongation, onset of asymmetry and i n s t a b i l i t y followed by turbulent shedding are presented i n Figure 4-15. The existence of an axisymmetric, stable vortex rin g for low Reynolds number i n a stream, e s s e n t i a l l y free of macroscopic turbulence i s shown i n Figures 4-15 (a) and (b). For the Reynolds number above a c r i t i c a l value (correspond- ing to the f i r s t formation of a stable r i n g , 10 < R n < 25), the streamlines separate from the surface and form a closed region immediately behind the sphere. A sing l e stream emerges from the vortex of the closed region extending to a long distance behind the sphere. The si z e of the r i n g i s such as to maintain an equilibrium between the rate at which v o r t i c i t y i s generated and di s s i p a t e d into the main stream. As the Reynolds number i s increased the vortex r i n g becomes elongated i n the flow d i r e c t i o n to maintain t h i s equilibrium, and the separation points move upstream towards the front stagnation point (Figure 4-15a-d). This forward movement of the separation points was also suggested by the pressure plots presented e a r l i e r (Figure 4-9). For Reynolds number between 170 - 230 an asymmetry i n the c i r c u l a t o r y motion within the vortex sheet produces a corresponding asymmetry i n the c i r c u l a t o r y motion i n the sheet i t s e l f and a resultant s h i f t from the center l i n e . Figure 4-14 A t y p i c a l photograph i l l u s t r a t i n g formation of a vortex ri n g behind sphere N H Figure 4-15 A flow v i s u a l i z a t i o n study showing development and i n s t a b i l i t y of vortex H ri n g with Reynolds number: (a) Rn=55; (b) Rn=92; (c) Rn=176; (d) Rn=221 0 1  117 This wake i s followed by two d i s t i n c t l y i n c l i n e d streamlines which maintain an equilibrium between rates of generation and d i f f u s i o n of v o r t i c i t y . As no r o t a t i o n i s observed, i t i s assumed that the lack of symmetry i s responsible for the 7 6 sidewise force component (Figure 4-15 e - f ) . Taneda rela t e d the s l i g h t asymmetry of the vortex r i n g to the support e f f e c t . However, t h i s could not be so as the asymmetry 1 s i prevailed regardless of the support diameter d g (j^o < ~fy < X 6 ^ ' The state of unsymmetrical but steady wake i s dis= turbed by further increase i n the Reynolds number. The rate at which v o r t i c i t y i s d i f f u s e d from the sheet into the main body of the f l u i d remains p r a c t i c a l l y constant, but the increased rate at which i t i s transferred to the vortex r i n g creates unstable condition within the vortex sheet. Basic- a l l y , the process i s one of build-up and release* but no sizeable portion of the r i n g escapes through an opening i n the end of the vortex sheet during the c y c l e . This i n turn causes the o s c i l l a t i o n of the asymmetrical wake about the axis of symmetry. When the vortex strength of the r i n g reaches a c r i t i c a l value, a sudden motion of the r i n g d i s ^ turbs the sheet, which i n turn i s responsible for a release of v o r t i c i t y and a consequent return of the r i n g to i t s o r i g i n a l p o s i t i o n and shape. This phenomenon appears to happen i n the Reynolds number range of about 250 - 300 (Figures 4-15 e - f ) . Thus we have a c l e a r explanation as to the sudden r i s e i n wake pressure observed e a r l i e r (Figure 118 4 - l l b ) . I t can now be a t t r i b u t e d , with a measure of con- fidence, to the o s c i l l a t i n g character of the asymmetrical r i n g and the attached filaments. With further increase i n the Reynolds number, the o s c i l l a t o r y motion of the vortex r i n g assumes higher frequency and the c i r c u l a t i o n within the sheet ceases to be symmetrical. In the cycle of build-up and release, the v o r t i c i t y generated i n the boundary layer becomes concentrated on d i a m e t r i c a l l y opposite sides of the flow axis within the vortex sheet.' The sections i n which the vortex strength i s greatest are a l t e r n a t e l y discharged into the main body of f l u i d . With each ejection a portion of the sheet i s c a r r i e d away. The vortex element discharged into the stream i n t e r a c t s with the disperse l i q u i d to form a regular wake pattern. Figures 4-15 (g) and (h) represent the i n i t i a t i o n and development of the unsteady wake which promotes regular vortex shedding. The flow v i s u a l i z a t i o n r e s u l t s also provide useful substantiation as to the forward movement of the separation point as suggested by the pressure plots (Figure 4-9b). For t h i s the photographs were analyzed systematically and the separation p o s i t i o n plotted as a function of R n as shown i n Figure 4-16. Note that the separation point moves forward by as much as 17° over the R n range of 55 - 315. For com- 99 parison, the a n a l y t i c a l r e s u l t s of Rimon and Cheng are also included. The r e s u l t s show increasing discrepancy at lower d=0.375 in. 0.50 0 1.500 t h e o r y " 40 50 10 0 p 2 0 0 3 0 0 Figure 4-16 E f f e c t of R on the separation angle 6 H R n suggesting inconsistency of some of the assumptions i n - herent i n Rimon and Cheng's analysis. On the other hand, i t must be emphasized that the v i s u a l determination of separation point i s , at best, approximate. Considering t h i s and the unstable character of the process, scatter i n the experimental r e s u l t s i s s u r p r i s i n g l y small. 4.3.2 Pulsating sphere The problem of dynamic forces exerted by r e a l f l u i d on a submerged object when the r e l a t i v e v e l o c i t i e s between the two change with time i s c l a s s i c a l and complex. 127 As early as i n 1851, Stokes investigated simple harmonic and other r e c t i l i n e a r motions of several objects including spheres, but h i s approximate analysis neglected convective acceleration terms i n the Navier-Stokes equations. Later 128 129 130 Basset , Boussinesq , and Oseen studied the same problem using e s s e n t i a l l y the same s i m p l i f i c a t i o n concerning the convective terms, but with one useful conclusion. They observed that the force on the sphere depends not only on i t s instantaneous v e l o c i t y and acceleration but also on an int e g r a l representing the e f f e c t of i t s e n t i r e h i s t o r y of acceleration. The force expression thus obtained i s v a l i d only for slowly moving but r a p i d l y accelerating sphere. A l u l l of nearly t h i r t y years prevailed before any other important contribution i n the f i e l d appeared. 121 131 Lin's method for periodic external flows past a body of revolution r e l i e s on forming appropriate averages of the quantities under i n v e s t i g a t i o n and on a l i n e a r i z a t i o n of the equation describing o s c i l l a t i o n s of the laminar separation. 132 In the early s i x t i e s , Odar and Hamilton under- took to eliminate the assumption of n e g l i g i b l e convective acceleration terms and proposed a new force expression, which showed excellent agreement with t h e i r experimental data up to around the R R of 60, f o r a sphere executing simple harmonic motion. The expression was also successful i n p r e d i c t i n g 133 v e l o c i t i e s of spheres during free f a l l i n viscous f l u i d 134 With appropriate modifications, Odar was also able to use the expression for p r e d i c t i n g forces on sphere accelerating along a c i r c u l a r t r a j e c t o r y . As pointed out before, a numerical study of impulsively started sphere i n the R n range of 1 - 1000 was 99 presented by Rimon and Cheng . Using the "complete" Navier-Stokes equations i n conjunction with time dependent stream f u n c t i o n - v o r t i c i t y formulation, the authors obtained pressure d i s t r i b u t i o n and hence drag. Unfortunately, t h e i r steady state pressure data do not agree with the present experimental r e s u l t s (Figure 4-12) while t h e i r unsteady drag r e s u l t s showed considerable discrepancy with the experimental 74 values of Ross and Willmarth Several items of i n t e r e s t become apparent at t h i s stage: 122 (i) No attempt i s recorded i n l i t e r a t u r e to measure time history of the mean pressure on an o s c i l l a t i n g sphere, even when the surrounding f l u i d i s stationary ( i . e . zero free stream v e l o c i t y ) . ( i i ) Any e f f o r t at t h e o r e t i c a l and experimental c o r r e l a t i o n between the time dependent drag of o s c i l l a t o r y sphere i s confined to the R n of around 60. ( i i i ) Even here the comparison shows considerable 74 discrepancy during the f i r s t 1.5 seconds This section studies time h i s t o r y of the mean pressure d i s t r i b u t i o n s i n the R n range of 600 - 1000, and complements the r e s u l t s with the flow v i s u a l i z a t i o n i n f o r - mation to confirm several physical c h a r a c t e r i s t i c s of the flow. I t should be emphasized that here the time dependent displacement corresponds to a prosthetic valve i n a patient (Figure 4-17). Furthermore, the free stream v e l o c i t y i s no longer zero as has been the case i n the past. Time history of the pressure at d i f f e r e n t surface taps on the poppet was obtained using a Barocel pressure transducer i n conjunction with the U.V. recorder (Section 3.2). The displacement-time h i s t o r y of the poppet was recorded simultaneously along with the pressure signals. (Figure 4-17). With these i n hand, C for a given 6 can P b e calculated for any p a r t i c u l a r valve opening. Obviously, 2.5 2.0 1.5 y b,cm. 1.0 0.5 Oh-r 0 / / forward open dwel l p r e s s u re on the p o p p e t / poppet d IsPlacment c l o s e d reve rse •dwell- per iod- i 0-5 1.0 0.3 0.2 0.1 0 P(mbar) -0.1 -02 -0.3 •0.4 1.5 time ,sec. Figure 4-17 Displacement and pressure time h i s t o r i e s for the pulsating sphere 124 depending on the number of values of selected, many such C p r o f i l e s can be constructed. However, for the sake of P conciseness only a few of the representative p r o f i l e s are given here. Pressure d i s t r i b u t i o n bn the surface of an o s c i l l a - ting sphere placed i n a uniform stream i s shown i n Figure 4-18. The p l o t s correspond to four d i f f e r e n t instantaneous locations of the poppet corresponding to the valve being f u l l y open, p a r t i a l l y open and f u l l y closed. Both forward and reverse portions of the cycle are considered together with two d i f f e r e n t values of the o s c i l l a t i o n frequency. For comparison, the pressure d i s t r i b u t i o n on a stationary sphere i s also included. (a) Forward stroke During forward motion of the b a l l ( i . e . moving upstream against the flow), i t i s apparent that the pressure d i s t r i b u t i o n over the f r o n t a l region of the sphere ( 9 - 0-60°) i s not s i g n i f i c a n t l y a l t e r e d ; however, beyond t h i s the pressure data are markedly affected (Figure 4-18 (i)) . The mean pressure i n the wake displayed a d i r e c t dependence upon the time hi s t o r y of the poppet p o s i t i o n . Of p a r t i c u l a r interest i s the increase i n minimum and average wake pres- sures together with forward movement of the separation point with decreasing y . Although the movement of the separation point i s not d i s t i n c t l y demonstrated by the p r o f i l e s -0.25 (i) 40 Figure 4-18 80 (ii) 0 40 e 80 / (iii) 0 40 80 120 Typical time dependent pressure p r o f i l e s for a sphere showing the e f f e c t of Reynolds number and pulsation frequency: (a) forward stroke to Figure 4-18 Typical time dependent pressure p r o f i l e s for a sphere showing the e f f e c t of Reynolds number and pulsation frequency: (b) reverse stroke (note that the avail a b l e pressure d i s t r i b u t i o n do not cover the e n t i r e wake region), i t was substantiated by the flow v i s u a l i z a t i o n (Figure 4-19). The large changes i n the wake pressure, i n the beginning of the cycle, can be antic i p a t e d as here the b a l l undergoes a large acceleration before reaching a constant speed. Impact of the surrounding f l u i d , rushing i n to f i l l the void created by the accelerating sphere, r e s u l t s i n an increase i n the wake pressure as i n - dicated. With the attainment of a nearly uniform v e l o c i t y , the rate of change of momentum i n the wake i s reduced lead- ing to a r e l a t i v e l y smaller pressure. Of course, a motion of the b a l l against the free stream d i r e c t i o n leads to an increase i n the e f f e c t i v e R and an associated decrease i n n drag as depicted by the large p o s i t i v e pressure i n the wake (y = 1/4). As the b a l l continues i t s forward movement at a constant v e l o c i t y , the acceleration e f f e c t i n the wake decays. Consequently, the drag increases compared to the previous case as indicated by smaller pressure i n the wake region (y = 3/4). Further journey of the b a l l i s at decelerating rates r e s u l t i n g i n reduction of the b a l l v e l o c i t y , i . e . lowering the e f f e c t i v e Rn, leading to further reduction of the wake pressure. Thus r e c a l l i n g the e f f e c t of R n upon C p i n the range of i n t e r e s t (Figure 4-10b), the observed time his t o r y of the pressure d i s t r i b u t i o n i s consistent with what one would expect. It i s noteworthy that at the end of the stroke, y =1, the pressure d i s t r i - bution i s cl o s e s t to the steady state condition. 128 Figure 4-19 A flow visualization study showing movement of separation point during pulsating motion of the sphere Figure 4-18a(ii) shows the e f f e c t of increasing the free stream R^ on the time hi s t o r y of the pressure. The important aspect to notice i s the c l u s t e r i n g of pressure plots i n the wake, suggesting t h e i r l i t t l e dependence on the poppet p o s i t i o n . Recognizing that the e f f e c t i v e R n v a r i a t i o n for t h i s case i s i n the range 935 <_ R^ <_ 3000, where i t has very l i t t l e influence on (Figure 4-9b), the behavior i s i n conformity with the expected trend. Figure 4-18a(iii) shows the e f f e c t of Beta number [B n = D ( w / v )®"^], which i s a measure of the pulsation f r e - quency, on Cp. In the R n range considered and for the given v a r i a t i o n i n B n, the pressure d i s t r i b u t i o n plots are affected only by a small amount. In general, the e f f e c t of frequency i s confined to l o c a l changes i n the character of the base pressure plots without s u b s t a n t i a l l y a f f e c t i n g the average magnitude of the pressure. This behavior i s better under- stood through recognition of the f a c t that an increase i n B n i s accomplished by changing the dwell without a f f e c t i n g time history of the forward (and reverse) motion. At higher frequencies, obtained by further reduction of the dwell, the poppet would be e s s e n t i a l l y quasi-stationary suggesting the pressure p l o t to s h i f t towards the stationary case. (b) Reverse stroke The time dependent pressure d i s t r i b u t i o n on the surface of the poppet during the reverse stroke i s shown i n Figure 4-18 (b). In general, the plots are s i m i l a r to those 130 obtained for the forward stroke but with s i g n i f i c a n t d i f f e r - ences i n d e t a i l . Although the pressure p r o f i l e s are s i m i l a r i n shape, t h e i r magnitudes and dependence on the poppet position are markedly d i f f e r e n t . Note that for the values of R n and o s c i l l a t i n g frequency considered, the wake pressure peaks at around y = 1/2. The pressure i n the wake i s quite s e n s i t i v e to the p o s i t i o n parameter, c e r t a i n l y more than that observed during the forward stroke. These d i f f e r - ences i n pressure c h a r a c t e r i s t i c s are understandable i n l i g h t of the reduction i n r e l a t i v e v e l o c i t y and hence the ef f e c t i v e Rn. Discontinuous reduction i n the r e l a t i v e l y small R n range leads to d i s t i n c t pressure p r o f i l e s as observed e a r l i e r (Figure 4-9b). The minimum r e l a t i v e v e l o c i t y occurs i n the v i c i n i t y of the midstroke (Figure 4—17) causing the wake to a t t a i n i t s largest p o s i t i v e pressure. However, t h i s region of near minimum v e l o c i t y i s rather short and the b a l l enters a period of high deceleration j u s t before completing the cyc l e . This i n turn s h i f t s the corresponding C p r o f i l e s towards the steady state condition. P E f f e c t of R on C plo t s i s quite s i m i l a r to that n p ^ ^ observed during the case of the forward motion. V a r i a t i o n of Cp with B n (Figure 4-18b(iii)) i s also as expected: time constant of the pressure transducer being f i x e d , i t eliminates l o c a l i r r e g u l a r i t i e s without s u b s t a n t i a l l y a f f e c t i n g the l e v e l of the base pressure. As before, the flow v i s u a l i z - ation study showed movement of the separation point and associated changes i n the size of the wake. 131 During the past ten years, several investigators have conducted measurements of the f l u c t u a t i n g force on 132 133 spheres ' . However, there appears to be no published experimental or t h e o r e t i c a l r e s u l t s on the pressure d i s t r i - bution i n the R range of i n t e r e s t . The lack of information n 3 may be attr i b u t e d to the int r a c t a b l e nature of the system, which i s not amenable to known a n a l y t i c a l procedures, and several challenging problems of instrumentation. 4.3.3 Poppet occupying d i f f e r e n t p o s itions i n the valve Having obtained some appreciation as to the f l u i d dynamics of the poppet by i t s e l f , i n a r e l a t i v e l y unconfined condition, the next l o g i c a l step would be to consider a more r e a l i s t i c s i t u a t i o n of the b a l l operating i n the cage. Two cases are of i n t e r e s t : (i) Bypass open, i . e . when a part of the f l u i d flows through the valve while the remaining f l u i d i s conveyed through the outside passages. Thus the arrangement provides for possible backflow (re- gurgitation) as encountered i n a leaky valve. From the point of view of the tunnel's s t r u c t u r a l safety, t h i s i s a desirable condition, ( i i ) Closed bypass condition corresponds to the normal operation of the heart valve but pre- sents a danger of high pressure b u i l d up, p a r t i c - u l a r l y at a small opening of the valve, due to the increased resistance. 132 The t e s t were conducted for both the cases over the desired range of Rn, however, for the l a t t e r case the highest R n was purposely l i m i t e d to 1200 from the considera- t i o n of the g l y c e r o l tunnel's s t r u c t u r a l i n t e g r i t y . (a) Bypass open Figure 4-20 gives a set of representative pl o t s showing the Reynolds number e f f e c t on the pressure d i s t r i b u - tions at various valve openings. At the outset i t i s apparent that the Reynolds number e f f e c t i s e s s e n t i a l l y confined to the negative pressure region. For r e l a t i v e l y smaller valve openings, 0 < y^ < 0.2, when t h i s influence i s more s i g n i f i - cant, the wake pressure increases with R R (Figure 4-20a). This has a degree of s i m i l a r i t y with the observed behavior of the single sphere i n the range 90 < R n < 300 discussed e a r l i e r (Figure 4-10). Recognizing the f a c t that with a + gradual c l o s i n g of the valve more and more f l u i d i s diverted into the bypass, the apparent discrepancy i n R n i s r e a d i l y explained. Thus although a free stream Reynolds number for smaller valve openings be the same, t h e i r e f f e c t i v e values at the poppet w i l l be v a s t l y d i f f e r e n t due to the presence of the bypass. Another factor of importance which promotes R n dependency at smaller openings i s related to the c o n t r o l l i n g e f f e c t produced by the poppet i n the wake of the e x i t b e l l . As the poppet trav e l s forward to reduce the outflow o r i f i c e 1.5 1.0 0-5 0.0 C P -0.5 -1.0 -1.5 yb=o.i5 0.10 < 4 0 120 Figure 4-20 Pressure d i s t r i b u t i o n on the poppet occupying d i f f e r e n t positions i n the valve during the open bypass condition: (a) y f c = 0.05,0.10,0.15 C O 11 I l_ I I I I I I I 0 40 80 0 40 80 0 40 80 120 Fxgure 4-20 Pressure d i s t r i b u t i o n on the poppet occupying d i f f e r e n t positions i n the M valve during the open bypass condition: (b) y = 0 . 2,0. 6 ,1. 0 0 0 (Figure 4-21),the separation from the e x i t b e l l i s delayed or even suppressed. Consequently, the v o r t i c i t y d i s t r i b u - t i o n immediately upstream of the poppet i s c o n t r o l l e d . This i n turn produces a more uniform flow of low e f f e c t i v e R n which r e s u l t s i n the delay of separation on the poppet. On the other hand, for valve openings i n the range 0 . 2 < y^ < 1.0, the pressure d i s t r i b u t i o n i n the wake i s v i r t u a l l y unaffected as shown i n Figure 4-20(b). Recalling the r e s u l t s on the v e l o c i t y p r o f i l e s i n the t e s t section with the model present (Figure 4-4) and the relevant data for a single sphere i n the corresponding range (Figure 4-9b) the above behavior appears consistent with what one would expect. Figure 4-22 presents t y p i c a l flow v i s u a l i z a t i o n photographs to i l l u s t r a t e d i s t r i b u t i o n of flow past the poppet and through the bypass as affected by the valve open- ing. For an extremely small opening (y^ = 0.050, Figure 4-22a) most f l u i d takes the bypass route. However, flow past the sphere gradually increases with opening (y^ = 0.50, Figure 4-22b) and with the valve f u l l y open (y^ = 1.00, Figure 4-22c) considerable amounts of f l u i d flows through the i n l e t o r i f i c e . Variations i n the e f f e c t i v e R caused n by the valve openings are thus apparent. Dependence of the mean pressure d i s t r i b u t i o n on the valve opening for a set of R r i s presented i n Figure 4-23. In contrast to R r , the valve opening s u b s t a n t i a l l y a f f e c t s Figure 4-21 A schematic diagram showing passage of f l u i d i n the immediate v i c i n i t y M of the valve w  138 Figure 4-22 Visual study of flow patterns past the poppet and through the bypass as effected by the valve openings at R - 290 Figure 4-23 Vari a t i o n of pressure d i s t r i b u t i o n with the poppet p o s i t i o n for the case of the open by- pass: (al) very small openings, R = 620 1.00 R n =620 yb=0.150 0.200 0.75 0.50 0.25 C P 0.00I 0.25 - 0 5 0 140 -0.75 1.001 0.0 20 40 9 ( 60 80 100 120 Figure 4-23 Va r i a t i o n of pressure d i s t r i b u t i o n with the poppet p o s i t i o n for the case of the open by- pass: (a2) intermediate and large openings, R = 620 n 1.2 0-6 0 0 -0-6 -1.2 -o- 141 R n = 9 2 6 y b =o.o5 0.1 o • o Or -©-- -1-8 2.4 3.0 \ 3.6 0 20 40 60 *t3- — O — -E> -a - L 80 100 120 Figure 4-23 Varia t i o n of pressure d i s t r i b u t i o n with the poppet posxtion f o r the case of the open bypass: (bl) R n = 926, y = 0.05, 0.10 1.25 142 1 .0 0 0-7 5 0.50 0.25 0.00 -0-25 -0-50 R n =926 o 0.150 A 0.200 A 0.25 0 . 0.30 0 o 0.40 0 v 0.60 0 - 1.0 0 0 -0.75 0.0 20 40 60 80 100 120 9' Figure 4-23 Var i a t i o n of pressure d i s t r i b u t i o n with the poppet p o s i t i o n for the case of the open bypass: (b2) R n = 926, y b = 0.15 , 1.0 • R n=1201 y b =o.05 0.10 D 143 o — — o — — 0 —o - 0 20 Figure 4-23 40 9' 60 80 100 120 V a r i a t i o n of pressure d i s t r i b u t i o n with the poppet p o s i t i o n for the case of the open bypass: (cl) R n = 1201, y b = 0.05, 0.1 1.25 1.00 0.75 0.50 0.25 C P 0.00 - 0 2 5 -0.50 -0.75 144 0.0 Rn=1201 s o 0.150 A 0.200 A 0.250 . 0.30 0 o 0.40 0 v 0.60 0 . 1.000 • « f . v - - A ,o o 1 20 40 60 8.0 100 120 Figure 4-23 Variat i o n of pressure d i s t r i b u t i o n with the poppet p o s i t i o n for the case of the open bypass: (c2) R r = 1201, y b = 0.15, 1.0 the pressure d i s t r i b u t i o n over the e n t i r e surface of the poppet. In general, with progressive reduction i n (i.e. valve c l o s i n g ) , the pressure on the f r o n t a l section of the sphere 8 - 0-60°) increases while that on the downstream section (0 - 60-180°) diminishes. I t i s t h i s increase i n pressure on the f r o n t a l section that i s responsible for the reversal of flow observed e a r l i e r (Figure 4-22a). Of p a r t i c u l a r s i g n i f i c a n c e i s the dramatic r i s e i n the negative pressure c o e f f i c i e n t at smaller openings (y^ £ 0.1) i n the near wake region. This would suggest large shearing stresses leading to possible deformations and 135 destruction of the blood c e l l s . The presence of v o r t i c i t y d i s t r i b u t i o n characterizing the wake, generates a c e n t r i f u g a l f i e l d which may cause d i s s o c i a t i o n of the blood i n t o i t s constituents and f i n a l l y t h e i r deposition on the body of the valve. (b) Closed bypass As pointed out before, i n t h i s study the range of R n (R n = 221-900) and valve openings (y b = 0.125-1.0) were c a r e f u l l y selected to obtain meaningful data without excess- i v e l y stressing the t e s t section. In general, the pressure results (Figure 4-24) exhibited the same trends as those observed during the open bypass case with one s i g n i f i c a n t difference: c l o s i n g of the bypass tends to increase the pressure on the downstream region of the poppet ( i . e . 0 > 60°, Figure 4-25). This increase i n pressure may be .1.00" ' - 1 1 1 1 r 1 „ 0 25 50 75 100 125 150 9° Figure 4-24 Pressure d i s t r i b u t i o n on the poppet occupying d i f f e r e n t positions i n the valve during the closed bypass condition . - - -.-.̂-"-"-"-V.V.-- . 0 20 R n =592 147 • 1.00 bypass \ 0 0-50 .open A 0.25 A o 0.1 25 • bypass closed 3—. * <• ,o o 40 60 80 100 120 Figure 4-25 Comparison of pressure d i s t r i b u t i o n s on the poppet occupying various positions i n the valve for the closed and open bypass conditions 148 attributed to a large difference i n the l o c a l R n for the two cases even when the free stream value i s the same. With the bypass blocked the f l u i d must pass through the valve with the i n l e t much smaller than that of the t e s t section. The j e t - l i k e flow issuing from the e x i t b e l l i s accelerated past the poppet leading to a higher e f f e c t i v e Rn. A preliminary c a l c u l a t i o n based on the area r a t i o showed the e f f e c t i v e R to be several times the mean free stream value for the n closed bypass case (3700-15300). To ascertain the e f f e c t of such a dramatic increase i n R n a separate t e s t program was undertaken using a set of spheres with g l y c e r o l as well as water as working f l u i d s . The r e s u l t s are presented i n Figure 4-26. Recognizing the f a c t that an increase i n R R i n the range of i n t e r e s t leads to a corresponding increase i n pressure on the downstream region of the sphere, the observed pressure d i s t r i b u t i o n f o r the closed bypass case follows the l o g i c a l trend. As before, the flow v i s u a l i z a t i o n study proved to be quite e f f e c t i v e i n providing better appreciation of the phenomenon. Tests with dye i n j e c t i o n were conducted at three Reynolds numbers: R n = 450, 600, 900, and photographs of the flow pattern taken as shown i n Figure 4-27. Several aspects of i n t e r e s t become apparent. Acceleration of the f l u i d through and downstream of the i n l e t o r i f i c e i s v i v i d l y demonstrated by stretching of the dye filaments which become 149 1.0P a 0-8 0.6 0.2 H o Rn=5848 A 121 3 • 7 37 o 117 * 0.4 C p 0.2 0.0 h • A© a A A A A A A Q O • D O o o o o o o o _04 ! L_ 1 1—t I ! I 0 30 60 90 120 150 180 6 o Figure 4-26 Reynolds number e f f e c t on C p r o f i l e s for a sphere p yb=o.75 V 1 , 0 I—1 Figure 4-27 A flow visualization study showing the effect of valve openings on g the flow past a spherical poppet during the closed bypass condition: (a) R • 450 n V y b=o.75 y b = i . o Figure 4-27 A flow visualization study showing the effect of valve openings on the flow past a spherical poppet during the closed bypass condition; (b) Rn = 600 Figure 4-27 A flow v i s u a l i z a t i o n study 'showing the e f f e c t of valve openings on the flow past a spherical poppet during the closed bypass condition: (C) R n = 900 d r a s t i c a l l y thin and eventually break up i n the wake. Of pa r t i c u l a r importance i s the loc a t i o n of the separating shear layer and i t s movement with poppet p o s i t i o n . Although i t would be somewhat fortuitous to assign numerical values, downstream movement of the separation point with progressive closing of the valve i s quite apparent at a l l the three Reynolds numbers. This i s quite s i g n i f i c a n t as now the poppet of the prosthetic device has a p e r i o d i c a l l y o s c i l l a t i n g boundary layer with associated wake that also grows p e r i o d i c - a l l y . Obviously, t h i s adds to the turbulent character of the wake which appears to be the main cause of concern i n successful operation of the device. 4.3.4 Poppet o s c i l l a t i o n and Beta number Having obtained some understanding of the hydro- dynamics of the poppet during stationary condition, the next l o g i c a l step was to explore the e f f e c t of pulsation frequency as represented by the Beta number. Using the dimensional analysis of Appendix I and the r e s u l t s of a si n g l e sphere described e a r l i e r , an experimental program was organized to obtain appreciation of the f l u i d dynamics of an o s c i l l a t i n g poppet. This section presents r e s u l t s on the time h i s t o r y of the s t a t i c pressure d i s t r i b u t i o n i n the Reynolds number range of 300-650. Both the forward and reverse stroke of the poppet are considered with the o s c i l l a t i o n frequency 154 ranging over 6 - 6 0 cpm (B = 19-61) . One of the important requirements i n th i s set of experiments was a c a r e f u l simula- t i o n of the displacement-time hi s t o r y of the poppet as observed i n a patient with such a prosthetic device (Figure 4-7). The r e s u l t s are complemented by the flow v i s u a l i z a t i o n which provides better understanding as to the physical character of the flow. Figure 4-28 studies dependence of the surface pressure p r o f i l e s on the Beta number, Reynolds number and valve opening, thus summarizing a rather extensive amount of information i n a concise form. At the outset, one recon- izes a degree of s i m i l a r i t y among a l l the p l o t s . In general, during the forward stroke, the pressure decreases from the front stagnation value reaching a minimum around 80° followed by a s l i g h t increase. I t appears that, i r r e s p e c t i v e of the pulsation frequency, the Reynolds number e f f e c t i s rather i n s i g n i f i c a n t for larger valve openings (y^ > 0.20). This i s consistent with the trend observed e a r l i e r f o r the stationary poppet occupying d i f f e r e n t positions i n s i d e the valve (Figure 4-20b). However, for narrow valve settings (y^ < 0.2), Rfi influence i s rather s u b s t a n t i a l , p a r t i c u l a r l y i n the negative pressure region. In general, an increase i n R leads to a decrease i n the wake pressure. Based on n the data f o r stationary poppet inside the valve (Figure 4-24) , t h i s again i s an expected behavior. 155 3 r B n= 1 9 R n =290 f o r w a r d s t r o k e 0 -1 -2 -3 -4 -5 0 \v^«;;:;-. V V\>C"« ^ V * » A y b=i.o 0-8 0-6 0-5 0-4 0-3 0.2 0.1 i * • « ! \ i « • i i \ %"B5- A.'"" *' ^ A i. 1.25 I.OOHKS««ES5 0.75 0.50 0.25 Cp 0.00 -.0.25 k -0.50H 20 40 9( 60 80 Figure 4-28 100 120 -0.75 0 r e v e r s e s t r o k e * \ » \ -4 20 40 60 80 100 120 Dependence of the surface pressure p r o f i l e s on the Beta number, Reynolds number, and valve opening: (a) B n = 19, R n =290 ,̂156 2r- o _1 .2 -3 -4 -5 -6 0 v y b= t o o 0.8 0.6 o 0-5 • 0.4 0.3 0.2 0.1 Bn= 19 Rn=650 forward stroke .or Ti- tS 20 40 60 80 100 120 1.25 0-75 0-50 0-25 0-00 -0-25 -0.50 r- -0-75 u 0 yb= 1.0 0.8 0.6 0.5 0.4 0.3 0.2 0.1 reverse stroke + I'AW ^ 'AW \ * • * * . » * . » » » * < x>—s 20 40 60 80 100 120 Figure 4-28 Dependence of the :surface pressure p r o f i l e s on the Beta number, Reynolds number, arid valve opening: (b) B = 19, R„ = 650 2 r U 0 -5 1.00 B n = 6 2 R n = 2 9 0 f o r w a r d stroke 0.75 ^:::f^;;:- O- -0--— — : 3 "O 'TD 0.50 !.• -1 V V 1 0 -2 O 0.8 A 0.6 o 0.5 -3 • • 0.4 A 0.3 -4 0.2 • 0.1 Q25 0 -0.25 -0.50 h 20 40 6 0 80 100 120 -0.75 »v O- 157 reverse stroke 0 20 40 to I I I 60 - n - - A - • 80 100 120 Figure 4-28 Dependence of the surface pressure p r o f i l e s on the Beta number, Reynolds number, and valve opening: (c) B = 62, R_ = 290 n 1,25 B n = 62 R n=650 forward stroke 0.75 0.50 yb= io v 0-8 o 0.6 ̂ 0.5 o 0.4 • 0-3 * 0-2 • 0.1 o I n * % • » « » •-o < • « 0-25 C p 0.00 -0.25 • J -0-50 V 20 40 60 80 100 120 -0.75 u 158 reverse stroke 0 20 40 0V 60 80 100 120 F i g u r e 4-28 Dependence o f " t h e s u r f a c e p r e s s u r e p r o f i l e s on the Beta number, Reynolds number, and v a l v e opening: (d) B = 62, R n n 650 1 Moving to the e f f e c t of pulsation frequency on the pressure p r o f i l e , for a given Rn, i t appears to be prim a r i l y confined to the negative pressure region and for smaller valve openings (0 <_ y^ _< 0.20). This i s better appreciated through Figure 4-29 where pressure v a r i a t i o n s are plotted as functions of four d i f f e r e n t values of the Beta number. A degree of s i m i l a r i t y with the singl e pulsat ing sphere studied e a r l i e r (Figure 4-18a) i s quite apparent Of p a r t i c u l a r i n t e r e s t i s the time hi s t o r y of the pressure as a function of the poppet p o s i t i o n for a given value of the Reynolds number and Beta number. As the poppet moves closer to the seat, narrowing the valve opening, the back pressure decreases quite sharply. We may r e c a l l s i m i l a r behavior f o r the stationary poppet (Figure 4-23). However, here due to a r e l a t i v e l y higher v e l o c i t y (poppet v e l o c i t y plus mean flow v e l o c i t y during forward stroke) e f f e c t of the valve opening on C i s much greater than that i n the stationary case. Note the high negative pressure i n the wake region for y^ < 0.50. The phenomenon, which i s a di r e c t consequence of the main flow being blocked by the poppet, i s c y c l i c and has the same frequency as that of the poppet. The r e s u l t i n g large p e r i o d i c shear stresses may cause not only d i s t r u c t i o n and coagulation of the c e l l s leading to c l o t t i n g as pointed out before,but may also account for rupturing of the suture l i n e s i n the implanted prosthetic valve, as occasionally observed. Figure 4-29 E f f e c t of the Beta number on the surface pressure p r o f i l e for a given R n : ( a ) R n = 2 9 0 : ( i ) f°rward stroke; M O 1-6i 1.2 0-8 0.4| C p 0.01 -0.4 -0.8 1 1 R n= 290 yb= 0.05 U y b =o.50 reverse stroke Vb=1.00 Bn=19 . 36 . 49 A 61 A 0 40 80 120 40 0 o 80 120 40 80 120 Figure 4-29 E f f e c t of the Beta number on the surface pressure p r o f i l e for a given 2 R : (a) R̂  = 290: ( i i ) reverse stroke n n Figure 4-29 E f f e c t of the Beta number on the surface pressure p r o f i l e f o r a given Rn: (b) R n = 652: (i) forward stroke M to Rn=652 y b=0.05 y 5 = o . 5 0 \ Bn=19 • u reverse stroke =1.00 40 80 120 40 0 o 80 120 40 Figure 4-29 E f f e c t of the Beta number on the surface pressure p r o f i l e given R n: (b) R R = 652: ( i i ) reverse stroke 80 for a 164 Di s c u s s i o n so f a r was concerned w i t h the forward p o r t i o n of the p u l s a t i o n c y c l e . Coming t o the re v e r s e stroke, i t i s apparent t h a t the general c h a r a c t e r of the p l o t s remains e s s e n t i a l l y the same. However, there are several s i g n i f i c a n t d i f f e r e n c e s . As a g a i n s t the forward case, now the e f f e c t s o f the Reynolds number and Beta number on the pressure p r o f i l e s are n e g l i g i b l e even f o r s m a l l open- ing s . Furthermore., n o t i c e the d r a s t i c r e d u c t i o n s i n negative pressures f o r small openings (y^ < 0.3). During the forward c y c l e t h i s p o r t i o n of narrow v a l v e openings corresponded to t h e d e c e l e r a t i o n w h i l e d u r i n g the reverse stroke the same p o r t i o n represents an a c c e l e r a t i n g poppet. This a c c e l e r a t i n g motion w i t h i n c r e a s e i n opening does not r e s u l t i n a s t a b l e j}:et and consequently the lower negative pressures. On t h e other hand, f o r l a r g e r openings dur i n g the reverse s t r o k e p r e s s u r e c o e f f i c i e n t s are l e s s p o s i t i v e compared t o those f o r the forward case, thus emphasizing the R Q e f f e c t discussed before ( i . e . higher Reynolds number i n the range 100 < iR^ <^ 1200 l e a d s to higher p o s i t i v e pressures but f o r 1200 < R < 5800 the R e f f e c t s seems t o n n be n e g l i g i b l e ; S e c t i o n -4. 3.3, F i g u r e 4-26). No attempt i s made here t o e x p l a i n the observed dependence of surface pressures on v a r i o u s systems parameters, as any such e f f o r t would be, a t b e s t , mere s p e c u l a t i o n . We are dealing w i t h a Hiighly complex system c o n s i s t i n g o f i n t r i c a t e geometry and (numerous v a r i a b l e s . One i s thus forced to be s a t i s f i e d with the desc r i p t i o n of the observed behavior and correlate i t s possible influence i n l i g h t of the recorded prosthetic valve performance. On the other hand, the r e s u l t s c l e a r l y emphasize the usefulness of studies of a single sphere by i t s e l f and when occupying various stationary positions i n the valve. Although d i f f e r e n t i n d e t a i l s , a general s i m i l a r i t y i n the plots i s quite evident. Hence the discussion given e a r l i e r i n Sections 4.3.1 - 4.3.3 may have some relevance here. A flow v i s u a l i z a t i o n study helped gain better appreciation of t h i s rather complex character of the f l u i d dynamics problem. As explained before, the flow pattern was vi s u a l i z e d using the dye i n j e c t i o n procedure and recorded through a 16 mm high speed photography. An analysis of the movie c l e a r l y showed accelera- t i o n and deceleration of flow during the cycle through ex- tension and compression of the dye filaments. Formation of nodal points on the streamlines suggested pressure b u i l d - up upstream during the valve closure. The streamlines on the sphere were, i n general, i n c l i n e d to and executed r o t a t i o n a l motion about the horizontal axis i n d i c a t i n g h e l i c a l character of the separating vortex r i n g (Figure 4-30). By f a r the most s t r i k i n g contribution of the flow v i s u a l i z a t i o n was a v i v i d confirmation as to the time dependent character of the separation p o s i t i o n during the 166 Figure 4-30 Typical photographs illustrating rotational motion of streamlines about the horizontal axis as captured by the 16mm movie 167 poppet o s c i l l a t i o n . Although i t would be imprecise to assign numerical values to data obtained through such a crude v i s u a l - i z a t i o n procedure, the trends were quite d i s t i n c t ^ They substantiated general behavior of the separation movement as 131 predicted a n a l y t i c a l l y by L i n Contraction of the flow past the i n l e t o r i f i c e was evident through the crowding of streamlines. This was followed by j e t t i n g of the f l u i d i n the e x i t b e l l as indicated by stretching of the dye filament, which f i n a l l y disintegrated i n the turbulent wake (Figure 4-31). Of considerable p r a c t i c a l i n t e r e s t was the flow r e v e r s a l , i l l u s t r a t e d by the back flow of the dye filaments during the valve closure, suggesting the p o s s i b i l i t y of regurgitation. Surprising as i t may seem, the movie showed periods of almost stagnant condition i n the wake, p a r t i c u l a r l y at a higher pulsation frequency and when the poppet was moving forward to close the valve, which may stimulate deposition of the dissociated blood constituents and thus promote c l o t t i n g . 4.4 Conclusion I t would be useful to review some of the more s i g n i f i c a n t r e s u l t s of the experimental i n v e s t i g a t i o n : (i) The designed g l y c e r o l tunnel performs s a t i s - f a c t o r i l y and produces e s s e n t i a l l y f l a t v e l o c i t y p r o f i l e s , at l e a s t over the ce n t r a l 10 cm. of 168 Figure 4-31 A flow v i s u a l i z a t i o n study i l l u s t r a t i n g s e v e r a l important characters (time-dependent s e p a r a t i o n , c o n t r a c t i o n through the i n l e t o r i f i c e , j e t i n g of f l u i d i n the e x i t b e l l , t u r b u l e n t wake, etc.) of the flow during p u l s a t i l e motion of the poppet i n s i d e the valve 169 the tunnel section, which are acceptable f o r the test program. In general, the uniform portion of the p r o f i l e decreases with an i n - crease i n v e l o c i t y or distance from the tunnel i n l e t . ( i i ) Introduction of a model i n the tunnel t e s t section s u b s t a n t i a l l y modifies the v e l o c i t y p r o f i l e s . The spherical model, with a small blockage r a t i o , tends to improve the v e l o c i t y p r o f i l e , i . e . i t extends the region of nearly uniform flow. On the other hand, the a o r t i c valve model a f f e c t s the p r o f i l e s adversely. The valve opening also has d i r e c t influence on the v e l o c i t y p r o f i l e s and hence on d i s t r i - butions on the surface of the poppet, ( i i i ) Use of the average v e l o c i t y i n the t e s t section (based on the mean flow rate) as a reference v e l o c i t y promises to promote r e p e a t a b i l i t y and comparison of data regardless of the t e s t f a c i l i t y used. Furthermore, defining as: c P - P 0 ref tends to compensate for blockage e f f e c t s , i r r e g u l a r i t y of the v e l o c i t y p r o f i l e s and possible errors i n pressure measurements caused 170 by e l e c t r i c a l d r i f t s of the pressure measuring system. Furthermore, i n conjunction with the Reynolds number (based on the suggested average v e l o c i t y ) , i t promises to a s s i s t i n dupl i c a t i o n and comparison of s i m i l a r data by other i n v e s t i - gators using d i f f e r e n t t e s t f a c i l i t i e s , (iv) A v e r t i c a l stem supporting the spherical model has n e g l i g i b l e influence on the pressure d i s t r i - bution i f the sphere to stem diameter r a t i o i s greater than seven, (v) For pressure d i s t r i b u t i o n on the surface of a sphere, the Reynolds number e f f e c t s are primarily confined to the region downstream of the zero pressure point. Locations of the minimum pressure and the separation points tend to s h i f t forward with an increase i n R . Of p a r t i c u l a r i n t e r e s t n i s the sudden r i s e i n the magnitude of the minimum pressure and the average base pressure downstream of i t around R = 240 - 475. As confirmed by n J the flow v i s u a l i z a t i o n , t h i s i s associated with the onset of i n s t a b i l i t y of the vortex r i n g , (vi) Drag c o e f f i c i e n t obtained by integrating pressure agrees rather well with the available experimen- t a l and t h e o r e t i c a l r e s u l t s and thus tends to substantiate r e l i a b i l i t y of the pressure data. 171 (vii) Pressure d i s t r i b u t i o n on the downstream section of a pulsating sphere (60° <_ 9 <_ 110°) displays a d i r e c t dependence upon the time h i s t o r y of the poppet p o s i t i o n . A decrease i n y causes increase' i n the minimum and average wake pressures to- gether with a forward movement of the separation point during the forward stroke. However, Cp i s e s s e n t i a l l y independent of the Reynolds number. E f f e c t of the Beta number i s confined to l o c a l changes i n the character of the base pressure plo t s without s u b s t a n t i a l l y a f f e c t i n g t h e i r average magnitudes. Although during the reverse stroke the general character of the pressure p r o f i l e s remains e s s e n t i a l l y the same, t h e i r magnitudes and s e n s i t i v i t y to the p o s i t i o n para- meters are markedly d i f f e r e n t , ( v i i i ) As i n the case of a single sphere by i t s e l f , R n e f f e c t s are confined to the wake region for a poppet occupying d i f f e r e n t positions i n the valve (bypass open). The R n influence, which i s more s i g n i f i c a n t for y^ < 0.20, i s e s s e n t i a l l y r e f l e c t e d i n increase of the wake pressure. On the other hand, valve opening for a given Reynolds number su b s t a n t i a l l y a f f e c t s the pressure d i s t r i b u t i o n over the en t i r e surface of the poppet. With a progressive reduction i n y b ( i . e . valve c l o s i n g ) , 172 the pressure on the f r o n t a l section of the sphere increases while that on the downstream section diminishes, leading to a reversal of the flow. A dramatic r i s e i n the negative pressure c o e f f i c i e n t at < 0.1 i n the near wake region has far-reaching s i g n i f i c a n c e . This would suggest large shearing stresses leading to possible deformation and destruc- t i o n of the red blood c e l l s . The presence of v o r t i c i t y d i s t r i b u t i o n and the associated c e n t r i f u g a l f i e l d i n the wake are the prime suspects promoting d i s s o c i a t i o n of the blood into i t s constituents and f i n a l l y t h e i r deposition on the body of the valve, (ix) In general, pressure d i s t r i b u t i o n on the poppet occupying d i f f e r e n t positions inside the valve, with bypass closed, exhibits the same trends as those observed during the open bypass case with one s i g n i f i c a n t d i f f e r e n c e : c l o s i n g of the by- pass tends to increase the pressure on the down- stream region of the poppet ( i . e . 9 ? 60° ). This i s attr i b u t e d to the j e t - l i k e flow issuing from the e x i t b e l l leading to a higher e f f e c t i v e RR. Of p a r t i c u l a r importance i s the lo c a t i o n of the separating shear layer and i t s movement 173 with the poppet p o s i t i o n . Now the poppet of the prosthetic device has a p e r i o d i c a l l y o s c i l l a t i n g boundary layer with associated wake that also grows p e r i o d i c a l l y . Obviously, t h i s adds to the turbulent character of the wake, which appears to be the main cause of concern i n successful operation of the device. (x) For the poppet o s c i l l a t i n g inside the valve, i r r e s p e c t i v e of the pulsation frequency, the Reynolds number e f f e c t i s rather i n s i g n i f i c a n t for larger valve openings (y^ > 0.2). However, for narrow valve settings (y^ < 0.2), R n influence i s s u b s t a n t i a l : an increase i n R leads to a n decrease i n the wake pressure. E f f e c t of the pulsation frequency i s primarily confined to the negative pressure region at smaller valve openings ( 0 £ y b £ 0.20). On the other hand, influence of the valve opening on Cp i s quite s t r i k i n g and much greater than that i n the stationary case. High negative pressure i n the wake region for y^ < 0.50, which i s a d i r e c t consequence of the main flow being obstructed by the poppet, i s c y c l i c and has the same frequency as that of the poppet. The r e s u l t i n g large periodic shear stresses may cause not only d i s t o r t i o n and coagulation of the c e l l s leading 174 to c l o t t i n g as pointed out e a r l i e r but may also account for rupturing of the suture l i n e s i n the implanted prosthetic valve, as occasion- a l l y observed, (xi) Flow v i s u a l i z a t i o n proved to be quite useful i n providing better appreciation of t h i s rather complex f l u i d dynamics problem. Formation, symmetric elongation, onset of asymmetry and i n s t a b i l i t y followed by turbulent shedding of the vortex r i n g were c l e a r l y i l l u s t r a t e d by the dye i n j e c t i o n procedure. Forward movement of the separation point with increasing R n (single sphere) and i t s rearward movement with a pro- gressive closure of the valve were also v i v i d l y observable. A v i s u a l study of the c y c l i c a cceleration and deceleration of the flow, h e l i c a l character of the separating vortex r i n g , flow re v e r s a l , stagnant condition of f l u i d i n the wake followed by a period of high turbulent motion, etc., not only complemented the t e s t data but also helped understand the phenomenon better. 175 5. CLOSING COMMENTS 5.1 Concluding Remarks The primary objectives of the research program have been twofold: (i) to design, construct, c a l i b r a t e and instrument a l i q u i d tunnel f a c i l i t y p a r t i c u l a r l y suited for low Reynolds number research i n the range 75 < R < 5000; — n — ( i i ) to u t i l i z e the f a c i l i t y to study hydrodynamic performance of an a r t i f i c i a l a o r t i c b a l l valve under simulated condition with a hope to gain some in s i g h t into the problems encountered i n i t s operation. One can say with a measure of s a t i s f a c t i o n that both the objectives have been r e a l i z e d to an extent. However, I must hasten to add that i n a very r e a l sense, the thesis represents only a beginning i n exploration of the challenging and equally i n t e r e s t i n g problem. I have barely touched upon the fringes of an extremely complex problem which only now I am able to understand a l i t t l e better. Probably the progress of knowledge rests on such modest steps. The l i q u i d tunnel with co n t r o l l e d v a r i a t i o n of the Reynolds number should represent a welcome addition to the departmental f a c i l i t i e s . It would undoubtedly be u t i l i z e d i n further i n v e s t i g a t i o n of a v a r i e t y of aspects associated with t h i s and other bioengineering oriented programs. However, what i s more important to recognize i s the f a c t that with i t s precise temperature control of the working f l u i d , t e s t section with o p t i c a l l y f l a t glass sides ( i d e a l l y suited for Schlieren photography and lase r doppler anemometry), c a r e f u l l y designed hydraulic and e l e c t r o n i c c i r c u i t r y for pulse d u p l i c a t i o n , r e c i r c u l a t i o n and f i l t e r i n g loop, etc. the tunnel represents an i d e a l t o o l for studying a v a r i e t y of fundemental problems i n f l u i d mechanics. The pressure d i s t r i b u t i o n data on the surface of a sphere i n the Reynolds number range 74 £ R R _< 5838 represents a s i g n i f i c a n t contribution i n the area of fundamen- t a l f l u i d mechanics. More importantly, the flow v i s u a l i z a - t i o n of the formation, growth and i n s t a b i l i t y of the vortex rin g and i t s c o r r e l a t i o n with the accompanying pressure v a r i a t i o n provides better appreciation of the physics of the phenomenon, and remain unrecorded so f a r . Coming to the hydrodynamics of the prosthetic heart valve proper, the project appears to be the f i r s t systematic attempt at e s t a b l i s h i n g f l u i d dynamic char a c t e r i s - t i c s of the widely used configuration. This i s very important as now we have a reference that can be used to evaluate merits of any future inovations i n the design of such prosthesis. 177 In obtaining pressure r e s u l t s over the enti r e range of v a r i a t i o n of nine dimensionless II numbers during the c a r e f u l l y simulated displacement-time hi s t o r y of the valve l i e s the strength of the project. It throws some welcome l i g h t on several fundamental questions concerning blood c l o t t i n g and hemolysis. The dramatic r i s e i n the negative pressure for small valve openings suggests large shearing stresses leading to possible deformation and destruction of the red blood c e l l s . The wake v o r t i c i t y and the associated . c e n t r i f u g a l f i e l d are probably the two fundamental factors promoting d i s s o c i a t i o n of the blood and deposition of i t s constituents on the valve. Periodic character of the phenomenon only accentuates the problem and may also be responsible f o r ruptured suture l i n e s . Contribution of the flow v i s u a l i z a t i o n i n such a study can hardly be overemphasized. A v i s u a l study of the c y c l i c acceleration and deceleration of the flow, h e l i c a l character of the separating vortex r i n g , flow reversal, stagnant condition of the f l u i d i n the wake, periods of high turbulent motion, etc. not only complemen- ted the t e s t data but also helped to understand the phenomenon better. 178 5.2 Recommendation for Future Work The i n v e s t i g a t i o n reported here suggests several topics for future exploration. Much can be learned about the problem through a well-organized experimental program. However, the number of system variables involved are rather enormous hence, any attempts i n achieving precise dynamic s i m i l a r i t y i s l i k e l y to demand considerable time and patience. Although i d e a l l y one would l i k e to employ a model that reproduces the actual prosthesis i n a l l i t s d e t a i l s — s t r u c t u r a l , geometrical and f l u i d dynamic —• p r a c t i c a l considerations would require us to e s t a b l i s h r e l a t i v e s i g n i f i c a n c e of the parameters involved and d i c t a t e some j u d i c i o u s l y thought out compromises. Only a few of the more s i g n i f i c a n t avenues of future e f f o r t s , which are l i k e l y to be rewarding, are b r i e f l y indicated here, (i) The obvious l o g i c a l extension of the present work would be to aim at precise d u p l i c a t i o n of the; (a) f l e x i b l e character of the surrounding? (b) p u l s a t i l e character of the flow; (c) tappered character of the aorta at the implantation s i t e ; (d) non-Newtonian character of the flow using, 136 may be, blood compatible synthetic polymers ( i i ) Character of the f l u i d i n the near wake region plays an important r o l e i n governing the valve 179 performance. Hence, a study leading to i n f o r - mation concerning periodic movement of the separation points, mean and unsteady v e l o c i t y p r o f i l e s of the separating shear layers, trans- i t i o n to turbulence, mechanism of the f i r s t vortex formation together with generation, d i s s i p a t i o n , d i f f u s i o n and retention of v o r t i c i t y , etc. would go a long way i n providing better appreciation of the process. Of p a r t i c u l a r i n t e r e s t would be a d e t a i l e d study of the turbulent character of the flow, both near the poppet surface as well as i n the wake. Attention may also be directed towards the measurement of the surface shear stress using f l u s h mounted hot films. These have f a r reaching implications, as the data would i d e n t i f y regions i n the f l u i d f i e l d most responsible for deformation and destruction of the red blood c e l l s , ( i i i ) Serious e f f o r t s should be made to assess blockage e f f e c t s due to the poppet operating i n the con- fined aorta and under adverse pressure gradient. Such a study as a function of the Reynolds and Beta numbers should provide useful information concerning the wake-body i n t e r a c t i o n . A pre- liminary study of t h i s aspect i s already i n progress. 180 (iv) With the test f a c i l i t y and instrumentation having been well organized, i t would be useful to undertake a study concerning the f l u i d dynamics of an accelerating sphere with d i f f e r e n t time- displacement h i s t o r i e s . The experiments may be conducted under stationary and streaming conditions of the f l u i d with the desired pressure gradient. This would help assess v a l i d i t y of the e x i s t i n g theories" 1* 2^ and contribute towards evolution of a better model for an a n a l y t i c a l approach to such a complex problem, (v) A systematic study of the drag force acting on the sphere during opening and c l o s i n g conditions would be of considerable i n t e r e s t . This would a s s i s t i n reshaping the e x i t b e l l such that the i n t e n s i t y of the impact between the cage and poppet i s minimized. Obviously, t h i s would reduce the wear of the components involved. (vi) One approach that seems promising i n improving the valve design would be to study the e f f e c t i v e - ness of momentum i n j e c t i o n i n delaying separation. Several a t t r a c t i v e configurations are possible (Figure 5-1). 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Grove, A.S., Shair, F.H., Petersen, F.E., and Acrivos, A., "An Experimental Investigation of the Steady Separated Flow Past a C i r c u l a r Cylinder," J. F l u i d Mechanics, Vol. 19, Pt. 1, May 1964, pp. 60-80. 125. Acrivos, A., Leal, L.G., Snowden, D.D., and Pan, F., "Further Experiments on Steady Separated Flows Past B l u f f Objects," J. F l u i d Mechanics, Vol. 34, Pt. 1, 1968, pp. 25-48. 126. Abraham, F.F., "Functional Dependence of Drag Co- e f f i c i e n t of a Sphere on Reynolds Number," Physics of F l u i d s , Vol. 13, No. 8, August 1970, pp. 2194- 2195. 127. Stoke, G.G., Mathematical and Physical Tapers, Vol. 3, 1851, p. 1. 128. Basset, A.B., A Treatise on Hydrodynamics, Vol. 2, Ch. 21, Deighton, B e l l and Co., Cambridge, 1888 (also Dover Publications Inc., New York, 1961). 129. Boussinesq, M.J., "Sur l a resistance qu-oppose un l i q u i d e i n d e f i n i en repos, sans pesanteur, au mouve- ment varie d'une sphere solide q u ' i l mouille sur toute sa surface, quand les vitesses restent bien continues et assez f a i b l e s pour que leurs carres et produits soient negligeables," Academic des sciences, Paris Comptes Renaus, Vol. 100, 1885, pp. 935-937. 130. Oseen, C.W., Hydrodynamik, Akademische Verlagsgesell- schaft, Leipzig, 1927. 131. L i n , C.C., "Motion i n the Boundary Layer with a Rapidly O s c i l l a t i n g External Flow," Proc. 9th Intern. Con. Appl. Mech., Brussels, Vol. 4, 1957, pp. 155-167. 132. Odar, F., and Hamilton, W.S., "Forces on a Sphere Accelerating i n a Viscous F l u i d , " J . F l u i d Mechanics, Vol. 18, Pt. 2, February 1964, pp. 302-314. 195 133. Odar, F., " V e r i f i c a t i o n of the Proposed Equation for Calculation of the Forces on a Sphere Accelerating i n a Viscous F l u i d , " J . F l u i d Mechanics, Vol. 25, 1966, pp. 591-592. 134. Odar, F., "Unsteady Motion of a Sphere Along a C i r c u l a r Path i n a Viscous F l u i d , " Transaction of the ASME, Series E, Journal of Appl. Mech., Vol. 35, No. 4 , December 1968, pp. 652-654. 135. Fung, Y.C., Perrone, N., and Anliker, M., Biomechanics: Its Foundations and Objectives, Prentice-Hall Inc., Englewood C l i f f s , 1972, pp. 507-523. 136. Bruck, S.D., Blood Compatible Synthetic Polymers, Charles C. Thomas Publisher, S p r i n g f i e l d , 111., 1974. 137. Kline, S.J., Similitude and Approximation Theory, McGraw-Hill, New York, 1965. 138. Buckingham, E., "Model Experiments and the Form of Empirical Equations," Trans. ASME, Vol. 37, No. 1487, June 1915, pp. 263-296. 139. Gimenez, J.L., Winters, W.L., Davila, J.C., Connell, J . , and K l e i n , K.S., "Dynamics of the Starr-Edwards B a l l Valve Prosthesis: A Cine-Fluorographic and Ultrasonic Study i n Humans," The American Journal of the Medical Sciences, Vol. 250, No. 6, December 1965, pp. 652-657. 140. Dayem, M.K.A., and Raftery, E.B., "Phonocardiogram of the Ball-and-Cage A o r t i c Valve Prosthesis," B r i t i s h Heart Journal, Vol. 29, No. 3, 1967, pp. 446-452. 141. Winters, W.L., Gimenez, J.L., and So l o f f , L.A., " C l i n i c a l Application of Ultrasound i n the Analysis of Prosthetic B a l l Valve Function," The American Journal of Cardiology, Vol. 19, No. 1, January 1967, pp. 97-107. 142. Bjork, V.O., Grenvik, A., Hergog, P., and Holmdhahl, M.H., "Aortic B a l l Valve Resistance," Thorax, Vol. 21, No. 2, March 1966, pp. 118-120. 143. Benchimol, A., and Matsuo, S., "Ejection Time Before and After A o r t i c Valve Replacement," The American Journal of Cardiology, Vol. 27, No. 3, March 1971, pp. 244-249. 196 144. Bristow, J.D., McCord, C.W.D., Starr, A., Ritzmann, L.W., and Griswold, H.E., " C l i n i c a l and Hemodynamic Results of Ao r t i c Valvular Replacement with a B a l l - Valve Prosthesis, Supplement to C i r c u l a t i o n , Vol. 29, No. 4, A p r i l 1964, pp. 36-46. 145. Brandfonbrener, M., Landowne, M., and Shock, N.W., "Changes i n Cardiac Output with Age," C i r c u l a t i o n , Vol. 12, No. 4, October 1955, pp. 557-566. 146. Levine, H.J., N e i l , W.A., Wagman, R.J., Krasnow, N., and G o r l i n , R., "The E f f e c t of Exercise on Mean Left Ventricular Ejection Rate i n Man," Journal of C l i n i c a l Investigation, Vol. 41, No. 5, 1962, pp. 1050-1058. 147. Yu, N.P., Pulmonary Blood Volume i n Health and Disease, Lea and Febiger Publishers, Philadelphia, 1969, p. 142. 148. M e r r i l , E.W., "Rheology of Blood," Ph y s i o l o g i c a l Reviews, Vol. 49, No. 4, October 1969, pp. 863-888. 149. A l b r i t t o n , E.C., Standard Valves i n Blood, W.B. Saunders Company, Philadelphia and London, 1952, p. 5. 150. Van Wazer, J.R., Lyons, J.W., Kim, K.Y., and Colwell, R.E., V i s c o s i t y and Flow Measurement, Interscience Publishers, a d i v i s i o n of John Wiley and Sons, New York, London, 196 3. 151. Charm, S., and Kurland, "Viscometry of Human Blood for Shear Rates of 0-1000,000 Sec." 1," Nature, Vol. 206, No. 4984, May 8, 1965, pp. 617-618. 152. Cassen, W., "A Flow Equation for Pigment-Oil Suspensions of the Pr i n t i n g Ink Type," Rheology of Disperse Systems, edit o r : C C . M i l l , Pergmon Press, 1959, pp. 84-104. 153. M e t r i l l , E.W., Benis, A.M., G i l l i l a n d , E.R., Sherwood, T.K., and Salzman, E.W., "Pressure-Flow Relations of Human Blood i n Hollow Fibers at Low Flow Rates, J . Applied Physiology, Vol. 20, No. 5, September 1965, pp. 954-967. 154. Dinsdale, A., and Moore, F., V i s c o s i t y and i t s Measurements, Chapman and H a l l , London, 1962. 155. Hylen, J.C., Kloster, F.E., Herr, R.H., H u l l , P.Q., Ames, A.W., Starr, A., and Griswald, H.E., "Phono- cardiographic Diagnosis of A o r t i c B a l l Valve Variance," C i r c u l a t i o n , Vol. 38, No. 1, July 1968, pp. 90-102. 197 156. Dring, R.P., "A Theoretical and Experimental I n v e s t i - gation of Disturbance Amplification i n External Laminar Natural Convection," Ph.D. the s i s , Cornell University, June 1968. 157. Ranstadler, P.W., "Stable Operation of Hot-Film Probes i n Water," ASME Symposium on Measurement i n Unsteady Flow, E d i t o r s : R.E. McNair, G.L. Mellor, and R.E. Kronauer, Worchester, Mass., May 1962, pp. 83-84. 158. Richardson, E.V., and McQuivery, R.S., "Measurements of Turbulence i n Waters," Journal of the Hydraulic D i v i s i o n , ASCE, Vol. 94, No. 2, March 1968, pp. 411- 430. 159. Richardson, E.V., McQuivery, R.S., Sandborn, V.A., and Jog, P.M., "Comparison Between Hot-Film and Hot- wire Measurements of Turbulence," Proceedings: 10th Midwestern Mechanics Conference, Edi t o r s : J.E. Cermak and J.R. Goodman, Colorado State University, Fort C o l l i n s , Colorado, August 1967, pp. 1213-1223. 160. C o l l i s , D.C, and Williams, M.J. , "Two Dimensional Convection from Heated Wires at Low Reynolds Numbers," J . of F l u i d Mechanics, V o l . 6, Pt. 3, October 1959, pp. 357-384. 161. Wood, W.W., "Calculations for Anemometry with Fine Hot Wires," J . F l u i d Mechanics, Vol. 32, Pt. 1, 1968, pp. 9-19. 162. Perry, A.E., and Morrison, G.L., "A Study of the Constant-Temperature Hot-Wire Anemometer," J . F l u i d Mechanics, Vol. 47, Pt. 3, 1971, pp. 577-599. 163. Moore, F.K., Theory of Laminar Flows, Princeton University Press, Princeton, New Jersey, 1964, p. 127. 164. Rotem, Z., and Claassen, L., "Natural Convection above Unconfined Horizontal Surfaces," J . F l u i d Mechanics, Vol. 38, Pt. 1, 1969, pp. 173-192. 165. Sokolowski, M., "Heat Flow i n a Wedge with Discontin- uous Boundary Conditions," Archiwum Mechaniki Stosowanej, Vol. 4, No. 13, 1961, pp. 433-455. 198 APPENDIX I SIMILARITY PARAMETERS I.1 Preliminary Comments There are three commonly used procedures for developing s i m i l a r i t y c r i t e r i a for a given problem: evalua- t i o n of force r a t i o s , dimensional analysis, and c r i t i c a l study of the governing equations of the system. Each method has i t s own advantages and l i m i t a t i o n s , and these have been 137 discussed i n d e t a i l by Kline . Here the method of dimensional analysis based on the well known Buckingham's 138 H-theorem i s employed to a r r i v e at the dimensionless group of numbers governing the problem. As the procedure i s rather routine, only the e s s e n t i a l ideas are touched upon. The usual problem i s one of p r e d i c t i n g some charac- t e r i s t i c function y of a system i n terms of a set of indepen- dent variables x.. , i . e . = f (x 1' 2' , x . ) . . ( I - D Application of the II-theorem leads to a new set of variables for the problem. 199 " l = f l ( I I 2 ' n 3 ' • * , ' n j ) . . . . (1-2) where the value of j i s governed by the number of variables and the independent basic dimensions required to describe these varia b l e s . The c h a r a c t e r i s t i c s of II terms are that they are dimensionless and independent. It follows that i f functional relationships for are the same for two systems, i . e . the same phenomenon i s involved i n both the systems, then i i , = n . i f n „ = n_ , n . = n . 1 lm 2 2m' ' j jm . . . . (1-3) where the subscript m r e f e r s to the model. The s c a l i n g relationships f o r the dependent variable y and the indepen- dent variables follow immediately from (1-3). Although t h i s procedure i s simple i n p r i n c i p l e , there are two inherent d i f f i c u l t i e s : (i) the v a l i d i t y of the analysis depends on the correct s e l e c t i o n of the variables x.; l ( i i ) i t i s not always possible to s a t i s f y the design conditions s p e c i f i e d by (1-3), i . e . there ex i s t s a p o s s i b i l i t y of model d i s t o r t i o n . 200 I.2 S i m i l a r i t y C r i t e r i a for A r t i f i c i a l A o r t i c Valves The dependent variable i s considered to be the pressure on the surface of the spherical poppet, P Q, and u the variables that are assumed to influence the pressure are l i s t e d i n Table 1. This l i s t , by no means complete, i s believed to include more s i g n i f i c a n t parameters l i k e l y to a f f e c t the pressure. Of course, the v e l o c i t y d i s t r i b u t i o n upstream and downstream of the valve, f l e x i b i l i t y of the surrounding aorta, time history of the poppet acceleration, etc. w i l l have some influence upon the pressure. Unfortunately, i t i s very d i f f i c u l t to account for them pr i m a r i l y because of the lack of any a v a i l a b l e measured information about these parameters. The dimensional analysis gives a set of II numbers as l i s t e d i n Table 2. 11^, correspond to the r a t i o of the b a l l v e l - o c i t y to the mean flow v e l o c i t y for opening and c l o s i n g of the valve, respectively. The importance of these two parameters are well substantiated by post-operative evalua- 139-141 tio n of prosthetic replacements . 1^ i s a measure of the r e l a t i v e time that the valve stays open and closed. The s i g n i f i c a n c e of t h i s factor also has been established 142 143 by published medical reports ' . II^ r e l a t e s the t o t a l e j e c tion time and the heart rate, quite c r i t i c a l parameters 14 2 143 of the problem, whose importance cannot be overemphasized ' 201 TABLE 1-1 Independent Variables of the Problem Variable Symbol Dimension Comments Velo c i t y U L T - 1 Average based on the car- diac index Pulsation Frequency f T" 1 Based on average heart rates of patients with a r t i f i c i a l a o r t i c valves B a l l Diameter D L A o r t i c b a l l valve diameter, an average based on more commonly used models O r i f i c e Diameter d L O r i f i c e diameter of the heart valve Stroke s L Stroke length based on the manufacturer's s p e c i f i c a t i o n Valve Opening ? b L A o r t i c b a l l valve opening as measured from the f u l l y closed p o s i t i o n Opening Time fc0 T Time of t r a v e l from f u l l y closed to f u l l y open positions Open Time fcs0 T Time elapsed during which the valve remains f u l l y open Closing Time t c T Time of t r a v e l from f u l l y open to f u l l y closed p o s i t i o n Closed Time Density t sc p T 2 -4 FT L Elapsed time during which the valve remains f u l l y closed Assumed constant V i s c o s i t y y FTL~ 2 Corresponds to blood assumed Newtonian 202 lit-, also c a l l e d Beta number, i s an index of i n e r t i a l e f f e c t s due to the o s c i l l a t i n g flow as compared to the viscous e f f e c t s . Reynolds number 11̂ , of course, must play an im- portant r o l e . The remaining three dimensionless numbers are associated with geometry of the system. o TABLE 1-2 A Selection of Dimensionless Numbers for the Problem n l i • = ( s / t Q ) / U • v u n 5 = D ( f / V ) 1 / 2 = B n n2 = ( s / t c ) / U = uc/u n6 = UD/v = R n n 3 = t / t so sc n 7 = s/D n4 = f ( t 0 + t c + t s 0 ) n8 n 9 = d/D = y b / s 2 0 3 3 Range of Values of the Relevant Parameters i n the Livi n g System 1-3.1 Average blood v e l o c i t y i n the a o r t i c root based on the cardiac index Two sets of data on the a o r t i c valve have been re- 144 ported i n the l i t e r a t u r e , by B r i s t o l et a l . and Judson 24 et a l . , which lead to somewhat d i f f e r e n t ranges of v a r i a t i o n for t h i s parameter. For comparison, the average blood v e l o c i t y has been obtained using information from both these sources. (a) Average v e l o c i t y based on data by B r i s t o l et a l . Cardiac index = volume flow rate i n the a o r t i c passage per u n i t body surface area Rest: 2.26-3 . 3 9 L/min/M2 Mean* = 2.756 L/min/M2 Exercise: 2.85-4.73 L/min/M2 Mean = 3.717 L/min/M2 Valve si z e = 9A - 12A, model 1000, Starr and Edwards B a l l diameter = 0.655 - 0.748 i n . O r i f i c e diameter = 0.610 - 0.685 i n . Mounting diameter = 0.920 - 1.070 i n . , a o r t i c area (A) = 4.29 - 5.8 cm •Mean refer s to average based on the number of patients examined. 204 Blood density (p) = 1.055 gr/cc (based on the standard value, see Section 13.3) U = average blood v e l o c i t y based on cardiac index = (body surface area)(cardiac index)/A Rest: 10.77 - 21.84 cm/sec Mean = 15.24 cm/sec Exercise: 13.58 - 30.47 cm/sec Mean = 22.0 cm/sec Average v e l o c i t y based on data by Judson et a l . Cardiac index = volume flow rate i n the a o r t i c passage per u n i t body surface area Rest: 2.18 - 3.36 L/min/M2 Mean = 2.967 L/min/M2 2 Exercise 2.65 - 7.75 L/min/M Mean = 4.957 L/min/M2 Valve s i z e = 9A - 13A, model 1000, Starr & Edwards B a l l diameter = 0.6555 - 0.815 i n . O r i f i c e diameter = 0.610 - 0.730 i n . Mounting diameter = 0.920 - 1.160 i n . , a o r t i c 2 area = 4.29 - 6.8 cm 205 U = average blood v e l o c i t y based on cardiac index Rest: 8.86 - 23.71 cm/sec Mean = 9.33 cm/sec Exercise: 10.77 - 49.93 cm/sec Mean = 15.60 cm/sec 1.3.2 Heart rate Based on extensive tests on numerous healthy adults (without prosthetic valve) conducted by several i n v e s t i - 145-147 gators , i t can be concluded that the normal sinus rhythm ranges over 57 - 111 beats/min. at r e s t (exercise: 72 - 164 beats/min.) with the mean rhythm of 69 - 85 beats/min. (exercise: 117 beats/min.). On the other hand, f o r patients with a o r t i c replacements the beat rate has been observed to 24 vary over 56 - 138 (exercise: 90 - 150; Judson et a l . ), and 62 - 118 (exercise: 82 - 140; B r i s t o l et a l . 1 4 5 ) beats/min. at r e s t with the mean values of 74 (exercise 114) and 74.3 (exercise 103.5), re s p e c t i v e l y . In each case, the data represent an average of a large number of samples and hence accounts for observed differences due to age, weight, body surface area, sex and environment. 1.3.3 Properties of blood Human blood i s a suspension of p a r t i c l e s i n a complex aqueous solution (plasma) of organic and inorganic 206 substances. In addition, about 7%, by weight, of the blood i s macromolecules c a l l e d proteins. The volume f r a c t i o n of the p a r t i c l e s i s of the order of 50%, but the red c e l l s constitute about 97% of the t o t a l c e l l volume i n the blood with c e l l concentration of about 5 m i l l i o n per mm . The red c e l l s have a d i s c o i d shape of about 8 u maximum diameter but are f l e x i b l e enough to pass e a s i l y through c a p i l l a r i e s of about one-half t h i s dimension. In normal blood, the red c e l l s aggregate, face-to-face. Usually these aggregates contain 6 - 10 red c e l l s i n a stack, such a primary aggregate being c a l l e d rouleaux. Secondary aggregation of the rouleaux also occurs. When blood i s sheared these secondary aggregates and rouleaux break up, and at s u f f i c i e n t l y high shear rates, the c e l l s e x i s t as i n d i v i d u a l s . However, i f the shear rate reduces to about zero, these aggregate structures reform very r a p i d l y . The other c e l l s (white c e l l and p l a t e l e t s ) are normally not numerous enough to a f f e c t the f l u i d dynamic behavior of blood. Because of t h i s , and as p l a t e l e t s i n pa r t i c u l a r can cause experimental d i f f i c u l t i e s i n viscometer, most blood rheological studies are performed with suspen- sions which have had most of the white c e l l s and p l a t e l e t s removed. However, i t should be pointed out that p l a t e l e t s play an important r o l e i n the formation of blood c l o t s and, in t h i s manner may seriou s l y i n t e r f e r e with the flow. Considering these factors, the rheo l o g i c a l prop- e r t i e s of blood might be expected to be rather complex. 207 This i s indeed the case, and the complexity becomes apparent when one examines a recent review of the blood rheology 148 l i t e r a t u r e s by M e r r i l . Although numerous reports have been published on the subject, the r e s u l t s show consider- able scatter and may present some d i f f i c u l t y i n the s e l e c t i o n of appropriate values. In contrast to the r h e o l o g i c a l properties, determin- ation of blood.density i s a r e l a t i v e l y simple matter. I t s 149 s p e c i f i c g r a v i t y has been found to vary over the range 1.052 - 1.061. The reference here i s the water at 4°C. Problems associated with the measurement of blood v i s c o s i t y are quite challenging. So f a r , i t has not been possible to p r e d i c t pressure-velocity r e l a t i o n s h i p s for blood over a wide range of conditions without employing empirical functions. Consequently, many models have been proposed for blood. Employing these models i n conjunction with experimental data obtained with various viscometers, the rh e o l o g i c a l properties of blood have been compiled. Almost a l l researchers have used one or more of the three types of viscometers: (i) the c a p i l l a r y tube type; ( i i ) the concentric c y l i n d e r type; and ( i i i ) the cone and plate configuration. Each of these o f f e r s several advantages and l i m i t a t i o n s . Some of these are re l a t e d to p e c u l i a r i t i e s of the instrument while others r e l a t e to the c h a r a c t e r i s t i c s of blood. 208 (i) Tube viscometers Here a f l u i d i s allowed to flow from a r e s e r v o i r down a tube of p r e c i s e l y known dimensions and the volume conveyed i n a given time under a given pressure i s recorded. For a simple steady laminar flow i n the tube, the correspond- 150 ing shear stress and shear rate at the tube wall are given by T w = APr/2L . . . . ( 1 - 4 ) Y w = APr/2uL . . . . (1-5) where r = tube radius L = a x i a l distance AP = pressure difference between two points distance L apart u = c o e f f i c i e n t of dynamic v i s c o s i t y . Equation (1-4) i s v a l i d regardless of the nature of the f l u i d However, the shear rate v a r i a t i o n , Equation (1-5), depends on the v e l o c i t y d i s t r i b u t i o n which, i n turn, i s determined by the nature of the f l u i d . 151 Recently, Charm' and Kurland have shown that 152 blood obeys Cassen's equation at shear rate from 1 to 209 100,000 sec. and that the general expression for blood viscometry i s T V 2 . K y l / 2 + o l / 2 _ _ ( i _ s ) or ( A P r / 2 L ) 1 / 2 = u ( 4 Q / T i r 3 ) 1 / 2 + o 1 / 2 where u c = Cassen v i s c o s i t y a = y i e l d stress Q = volume flow rate "1/2 1/2 Thus a p l o t of Q ' against (AP/L) ' should be a s t r a i g h t l i n e from whose slope K can be obtained and whose intercept at AP/L = 0 gives o1^2. If the tube viscometer i s too narrow, however, the recorded v i s c o s i t y of a heterogeneous suspension such as blood may become dependent upon the diameter of the tube (Fahraeus-Lindqvist e f f e c t ) . Corrections may also be required for flow pattern changes, anomalous pressure gradient at the i n l e t and o u t l e t of the tube, and meniscus e f f e c t s . The tube viscometer has been used extensively for work on blood because i t i s inexpensive, requires only a small volume of f l u i d , i s easy to use and gives good r e p r o d u c i b i l i t y . Moreover, i t simulates extra vivum, to a degree, the flow of blood i n vivo. Unfortunately, i n the 210 case of a non-Newtonian f l u i d each v i s c o s i t y p l o t covers a wide range of shear rate for which allowance has to be made and, i f the tube i s narrow, wall e f f e c t s may influence the r e s u l t s . ( i i ) Co-axial cylinder viscometers The f l u i d i s held i n a c y l i n d r i c a l pot containing a c o n c e n t r i c a l l y mounted bob. The pot i s rotated at a constant v e l o c i t y and the viscous drag transmitted through the f l u i d i s measured by the angular d e f l e c t i o n of the inner c y l i n d e r or by the torque required to return i t to i t s o r i g i n a l p o s i t i o n . This torque i s balanced by the moment due to viscous shear so that for a Newtonian f l u i d y = M(r 2 - r 2)/(4Trr 2r 2La>) . . . .(1-7) where M = measured torque r^ = radius of inner cylinder ^ = radius of outer cylinder L = height of inner c y l i n d e r angular v e l o c i t y of outer c y l i n d e r . CO = The v i s c o s i t y of an unknown f l u i d can then be found from the torque which i t transmits at a predetermined angular v e l o c i t y . 211 For non-Newtonian f l u i d s the shearing stress i s no longer d i r e c t l y proportional to the rate of shear and thus the measured torque M i s not d i r e c t l y proportional to the angular v e l o c i t y t o . For example, for a p l a s t i c f l u i d r>- u v- J 150 or Bingham body u = M f r ^ - r ^ / ^ T f r J r ^ L w ) - (a/to) In ( r ^ r ^ . . (1 -8 ) The co-axial cylinder viscometer has the advantage of shearing a great portion of the f l u i d at something approach- ing a constant rate of shear. On the other hand, end e f f e c t s on the cylinders are sometimes d i f f i c u l t to overcome or allow f o r , p a r t i c u l a r l y i n the case of non-Newtonian f l u i d s . A well designed instrument i s expensive, p a r t i c u l a r l y i f i t i s to work on small samples of f l u i d at low rate of shear (which i s often necessary i n the case of blood). ( i i i ) Cone-plate viscometers The f l u i d i s contained i n the space between a cone of very large apex angle and a f l a t surface normal to i t s axis. One unit i s rotated and the drag transmitted by the f l u i d i s measured on the other. In order that i d e a l i z e d flow e x i s t i n the f l u i d space between the cone and plate, i t i s necessary that i n e r t i a l e f f e c t s be n e g l i g i b l e and that a, the angle between the cone and plate, be very small (less 212 than 4°). Due to the system geometry the perpendicular distance between the cone and the plate increases proportion- a l l y with the radius (z = r tan a). Moreover, the l i n e a r r e l a t i v e v e l o c i t y of the cone or the plate i s proportional n to the radius so that the rate of shear i n the f l u i d i s constant throughout. For the case where the cone rotates at speed co, the shear rate y and shear stress T are given by Y = to/a . . . . (1-9) x = 3M/27rr3. . . . . (1-10) Thus for a Newtonian f l u i d y = 3Ma/2Trr 3w . . . . . (1-11) Equations (1-9) and (1-10) are generally used without correc- 154 ti o n for p l o t t i n g flow curves of non-Newtonian f l u i d s , the viscometer f i r s t being c a l i b r a t e d with Newtonian f l u i d s of known v i s c o s i t y . The apparent v i s c o s i t y of the unknown non-Newtonian f l u i d i s then found at various rates of shear and the corresponding shear stress obtained from Equation (1-11). The flow curve can then be constructed r e a d i l y . The cone-plate viscometer possesses many advantages over the co-axial cylinder instrument and, i n addition, 213 normally requires a smaller sample. However, the distance over which shearing takes place may be so small that the discontinuous nature of f l u i d such as blood could affect the magnitude of the transmitted stress. The long length and short width of the air f l u i d interface at the perimeter of the shearing surfaces may also, in certain circumstances, affect stress transmission unless i t can be allowed for by the inclusion of a guard ring or other special design features. Furthermore, there i s the possibility of circu- lation developing between the cone and the plate during the shearing of certain non-Newtonian fluids and suspensions, leading to incorrect results. From the foregoing i t i s clear that the problems of evaluating rheological properties of normal human blood are overwhelming and in spite of extensive experimental research no precise value can be pinpointed due to the wide scatter in data and the problem of reproducibility. An average value of u = 3.196 centipoises, usually quoted in hand 149 books , thus offers a good compromise. 1.3.4 Physical dimensions of the Starr-Edwards aortic b a l l  valves Physical dimensions of the human heart and associ- ated appendages vary considerably requiring, preferably, personalized construction of prosthetic replacements. This presents a rather formidable problem of manufacture where standardization i s naturally preferred. For example, Edwards Laboratories has produced at l e a s t f i f t y types of a o r t i c valves accounting for d i f f e r e n t models and s i z e s . The range of v a r i a t i o n of important parameters and t h e i r main values, based on the information supplied by the manufacturer through private communication, are l i s t e d below. B a l l diameter = 0.482 - 0.868 i n . ; mean 0.681 i n . Stroke = 0.232 - 0.405 i n . ; mean 0.348 i n . O r i f i c e diameter = 0.420 - 0.757 i n . ; mean 0.594 i n . Mounting diameter = 0.618 - 1.207 i n . ; mean 0.938 i n . 1.3.5 Travel and r e s t times for the poppet As seen i n Figure 2-13, displacement hi s t o r y of the b a l l t r a v e l includes four important time parameters: opening, opening duration, c l o s i n g , c l o s i n g duration. 'Opening time' represents the time taken by a poppet to t r a v e l between f u l l y closed to f u l l y open positions ( i . e . from the seat to the t i p of the cage). This i s followed by a dwell period during which the valve remains f u l l y open. It i s referred to as 'opening duration'. Now the poppet s t a r t s i t s return journey which ends with f u l l c l o s i n g of the valve. This represents 'closing time'. The following dwell period before i n i t i a t i o n of the new cycle i s c a l l e d 'closing duration'. From p h y s i o l o g i c a l consideration a parameter of importance i s the 'ejection time' (Ê .) i n d i c a t i n g duration 215 over which blood i s pumped into aorta. Considering the fac t that the flow reverses during the terminal period of the impending closure of the valve, the e j e c t i o n time i s approxi- mately represented by E f c - tg + t gQ + t c . The recorded information on these parameters i s rather sketchy. Among 139 the useful reports are those by Gimenez et a l . and Hylene 155 et a l . Unfortunately, even the mean values given by them vary s i g n i f i c a n t l y as indicated below: TABLE 1-3 S i g n i f i c a n t Time Parameters for an A o r t i c Valve Prosthesis Time Parameters Gimenez et a l . Hylene et a l . Opening time, tg 50 m sec. 59 m sec. Opening duration, t 125 m sec. 110 m s e c * Closing time, t 50 m sec. 89 m s e c * Closing duration, t sc 310 m sec. 419 m sec. Ejection time, E f c 225 m sec. 258 m s e c Of i n t e r e s t would be the values of these parameters f o r patients (with prosthetic valves) with abnormal sinus rhythm. Based on the information obtained at Shaughnessey •Calculated and/or estimated from the ca r o t i d p r o f i l e s . 216 Hospital i t appears that, although tg and t remain i n the ranges given above, t ^ and t g c may show considerable deviation. I.4 Range of V a r i a t i o n of the Dimensionless Parameters The wide range of v a r i a t i o n s of the system parameters would be r e f l e c t e d i n the corresponding v a r i a t i o n of the dimensionless numbers involved i n the dynamic simulation. Obviously with nine II numbers t h i s presents a rather challeng- ing experimental project. Fortunately, the t e s t program showed some parameters to be more s i g n i f i c a n t than others. The ranges of dimensionless numbers as obtained using the information given i n Section 1.3 and those used during the actual t e s t are presented i n Table 1-4. TABLE 1-4 Observed Values of the Dimensionless Numbers and Those Used i n Experiments Dimensionless Number Observed Range Experimental Range n x = ( s / t Q ) / U = U Q / U 0.88 - 1.06 1.0 - 2.0 n2 = ( s / V / U = U c / U 0.80 - 1.06 1.0 - 2.0 0.30 - 0.35 0.255 - 0.950 n4 = f ( W t s O ) = f E t 0.298 - 0.460 0.299 - 0.530 n 5 = D ( f / v ) V 2 27.68 - 34.36 18.60 - 61.20 n g = UD/V 533 - 1257 290 - 1200 n ? = S/D 0.467 - 0.488 0.02 - 0.500 n 8 = d/D 0.871 - 0.872 0.871 n 9 = V s 0 - 1.0 0 - 1.0 217 APPENDIX II A THEORETICAL APPROACH TO EVALUATION OF THE HOT FILM PROBE PERFORMANCE II.1 Introduction Hot wire anemometer has been used for many years as a research t o o l i n f l u i d mechanics. I t s small dimensions and time constant have allowed precise measurements of steady and time dependent flow v e l o c i t i e s . The successful operation of the system has led to i t s modification for measurements i n l i q u i d s , p a r t i c u l a r l y water. Although some r e l i a b l e r e s u l t s have been obtained using the hot wire anemometer i n 136 l i q u i d s , under co n t r o l l e d conditions, recent investigations would seem to indicate that hot films are better suited f o r , • . , 157,158 use i n l i q u i d s ' A hot f i l m probe e s s e n t i a l l y consists of a platinum f i l m deposited on a glass substrate i n the form of a wedge and coated with a quartz protective layer. Gold leads convey the signa l to two concentric s i l v e r tubes with i n s u l a t i o n i n between. The outer tube, normally grounded, provides a heavy metal contact for fastening the sensor to the support (Figure I I - l ) . As compared to hot wires, the f i l m sensors have the following advantages: 218 Figure I I - l A schematic drawing of a hot f i l m probe 219 (i) A hot f i l m probe has better frequency response than a hot wire of the same se n s i t i v e area. This i s because the sensor i s d i s t r i b u t e d on the surface and hence o f f e r s very l i t t l e thermal i n e r t i a as against the c i r c u l a r cross section of the wire. ( i i ) In general, f i l m probes e x h i b i t lower heat conduction to the substrate (end losses) f o r a given power input due to the lower thermal conductivity of the base material. A shorter sensing length can thus be used. On the other hand, size of the s i l v e r supports and t h e i r proximity to the wire normally r e s u l t s i n greater loss due to conduction and free convection. ( i i i ) Film probes o f f e r more f l e x i b i l i t y i n the sensor configuration. Wedge, c o n i c a l , parabolic and f l a t surface shapes are a v a i l - able. (iv) Comparatively speaking, probes with heated films are less susceptible to f o u l i n g and easier to clean. A t h i n quartz coating on the surface r e s i s t s accumulation of foreign matter. Fouling tends to be a d i r e c t function of s i z e . On the other hand, hot wire probes oxidize r e a d i l y , are very f r a g i l e and prone to breakage. 220 Several detailed studies of r e l a t i v e behavior of hot wire and hot f i l m probes and the d i f f i c u l t i e s encountered 158 1 in t h e i r a p p l i c a t i o n i n l i q u i d mediums have been reported ' et a l . I n general, the investigations e s t a b l i s h s u p e r i o r i t y of hot films for l i q u i d a p p l i c a t i o n . II.2 Heat Transfer from a Hot Film Probe There are several well developed theories reported in l i t e r a t u r e for evaluating performance of a hot wire 160 161 anemometer ' . On the other hand, although hot f i l m probes play a rather useful r o l e i n the experimental l i q u i d flow studies, no well developed theory systematically accounting for conduction and free convection losses i s ava i l a b l e , p a r t i c u l a r l y , i n the Reynolds number range of 2-350 (based on average flow rate and f i l m width). This i s indeed s u r p r i s i n g , since the f a c t that i n t h i s range of R n viscous e f f e c t s are of the same order as those due to 107 i n e r t i a , was f i r s t recognized by Barker as early as i n 19 22. Thus i t has been necessary to re s o r t to c a r e f u l c a l i b r a t i o n for each shape of the probe. The theory presented here represents a modest step towards minimizing dependence on such tedious experimentation. The thermal energy loss from a f i l m may be due to forced and free convection, conduction, and r a d i a t i o n of heat to the f l u i d and to the glass support. At higher 221 v e l o c i t i e s the primary heat loss i s by forced convection, the contribution through other sources being negligible,and may be considered as secondary. The convective heat loss from the f i l m i s much greater i n l i q u i d than i n gas, consequently demanding add i t i o n a l heat flux to keep the f i l m at a constant tempera- ture. Hence, the a p p l i c a t i o n of hot f i l m anemometers i n l i q u i d s i s l i m i t e d to low v e l o c i t i e s . However, i n t h i s case, only r a d i a t i o n e f f e c t s tend to be n e g l i g i b l e , heat loss by the other modes being s i g n i f i c a n t . Therefore, t h e o r e t i c a l analysis of a f i l m probe, p a r t i c u l a r l y i n the low v e l o c i t y range, should account for the losses by both forced and free convection. From s i m i l a r i t y arguments i t follows that, f o r forced convective heat transfer, the Nusselt number i s a function of Prandtl and Reynolds numbers only. For heat transfer i n a laminar boundary layer and N^>>1, the function takes the form"*"^ N (x) = C(x)N 1 / 3 R 1 / 2 u P n (I I - l ) where x = x / ° / dimensionless downstream coordinate along . the probe face, C(x) = { [B(x) ] 1 / 2 / [ 9 1 / 3 r ( 4 / 3 ) ] } * [ B ( S ) ] 1 / 2 d U 1/3 (H-2) 222 B(x) = C[(l+ m ) / 2 ] 1 / 2 x ( 3 m _ 1 ) / 2 . . . . (II-3) C = f" (0) X 6 3 = 0.887 (based on data by Moore ) m = ty/ (2 - i>) . . . . (II-4) \l> = 6/2II, 6 = the wedge angle. . . . . (II-5) For the f i l m probe used (DISA type 55A83), 6 = 80°. With t h i s , equation ( I I - l ) , a f t e r i n t e g r a t i o n and s i m p l i f i c a t i o n , i s reduced to N (x) = 0.4748 N 1 / 3 R 1 / 2 x 5 / 1 4 u p n the average Nusselt number over the wedge face i s obtained as J n u N = N,(x)dx I Q = 0.7386 N 1 / / 3 R 1/ 2 P n Q, = 0.7386 (2JK.) ( T T - T . ) N y 3 R y 2 . . . . (II-6) ± r w r. p n Heat transfer by natural convection depends on the or i e n t a t i o n of the f i l m r e l a t i v e to the g r a v i t a t i o n a l f i e l d . For a f l a t plate i n c l i n e d to the horizontal (half of the 223 wedge shaped hot film) the heat loss from the heated surface to the f l u i d due to convection and conduction, assuming constant wall temperature, may be evaluated by N u = 0.7668(Gr N p ) 1 / 5 Le. Q 2 = 0. 7668 (2ZKf) (T w-T f) (G r N p) 1 / 5 . . . . (II-7) providing that N » 1 and the wedge angle obeys the r e l a t i o n P 6 = O | t g * 1 ( G r N px 3) 1 / 5 Fortunately, both the conditions are s a t i s f i e d here. F i n a l l y , the heat conduction to the quartz support can be estimated from Q 3 = 2C 3K gZb(T w-T f)/b . . . .(II-8) where K g = thermal conductivity of the quartz support I = f i l m length b = f i l m width = bulk temperature of the f l u i d T = wall temperature w C- = constant of p r o p o r t i o n a l i t y . 224 The value of can be evaluated, approximately, by using a s p e c i a l equation for heat flow i n a two dimensional wedge as given by Sokolowski (here l/h = 5). Taking the average 1 f 1 heat flow as — q(x)dx to avoid s i n g u l a r i t y at x = 0, x J l O " 7 the constant i s found to be 0.38. The t o t a l heat flux from the f i l m may now be written as the sum of the contributions represented by Equations (II-6), (II-7) and (II-8). This can then be equated to the power supplied to the f i l m . Thus, R o I2/4.2 =. [ C l ^ N l / 3 R ^ 2 + C 2 ^ (G r N ) X / 5 + c 3 ] * g g * *2ZK g(T w-T f) . . . . (H-9) = C. 2 K (T -T.) 4 g w f' where I = probe current = 0.7386 C 2 = 0.7668, As suggested by the manufacturer, the difference between temperature of the wall and the f l u i d may be expressed by 225 w - T, a X _ / 0 c. 1 (—g ) j . . . . (11-10) c With t h i s , Equation (II-9) can be rewritten as 0 8.4K IC. I ^ ( a+l ) / a = A R g 4 . . . .(II-H) c where X = c o e f f i c i e n t of thermal r e s i s t i v i t y . Now the voltage output of a DISA 55A01 constant temperature anemometer i s the bridge voltage, which i s related to the probe current by I = 1.04 V/(100.6 + R c) . . . . .(n-12) Inserting (11-12) i n (11-11), the following r e l a t i o n between bridge voltage and flow v e l o c i t y i s obtained: 2 ~ V c. c 2 + c±u 1/2 (H-13) where 226 C, = [ (a+l)/a{100.6 + R (1+a)} 2] * *1.08 A R /8.4 K I c g ^2 = C3 + C2 IT ( G r N p ) 1 / 5 ^ i = c i i r N y 3 ( b / v ) 1 / 2 . g p II.3 Experimental V e r i f i c a t i o n and Discussion To assess v a l i d i t y of the approach, c o e f f i c i e n t s ^1' ^2' a n < ^ ^3 w e r e f° u n <l experimentally. This was accomplish- ed by c a l i b r a t i n g two d i f f e r e n t probes i n a water-glycerol solution, 55% by weight, for two overheat r a t i o s (Figure II-2). Employing the l e a s t square f i t , the c a l i b r a t i o n data were f i t t e d into a s t r a i g h t l i n e of the form V 2 = F j U 1 7 2 + F 2 . . . . . (11-14) Knowing and F^, and c a l c u l a t i n g for the probe i n question, A. >\ c o e f f i c i e n t s C 2 and are determined. Pertinent information for the two hot f i l m probes are l i s t e d i n Table I I - l , while experimental and t h e o r e t i c a l values for F-̂  and F 2 are compared in Table II-2. Figure II-2 Theoretical and experimental c a l i b r a t i o n p l o t s for the hot f i l m probes used 228 TABLE I I - l P e r t i n e n t Data f o r the Hot F i l m Probes P r o b < 3 NO. 1 2 I, f i l m l e n g t h , cm 0.10 0.10 b, f i l m width, cm 0.02 0.02 A, temperature c o e f f i c i e n t o f f i l m resistance,° C" 1 4.41xl0~ 3 4 . 4 x l 0 ~ 3 Kg, heat c o n d u c t i v i t y o f the g l a s s support, [ { c a l / s e c . c m 2 ) / ( o c / c m ) ] 2 . 7 x l 0 ~ 3 2 . 7 x l 0 ~ 3 R , probe c o l d r e s i s t a n c e , Q 7.8 10.3 TABLE II-2 Heat T r a n s f e r Parameters f o r Hot F i l m Probes Probe No. Overheat R a t i o F L % E r r o r F l F2 % E r r o r F *2 Theory Experiment Theory Experiment 1 0.051 31.17 29.84 -4.45 21.19 21.79 3.14 0.103 60.24 61.20 1.26 43.19 40.18 -7. 48 2 0. 049 23.82 22.06 -7. 69 16.09 16.82 + 4.36 0.097 45.50 43.77 -3.43 32.36 31.60 -2.42 229 Figure II-2 presents t h e o r e t i c a l and experimental c a l i b r a t i o n plots for the hot f i l m probes used. The agree- ment may be considered quite s a t i s f a c t o r y p a r t i c u l a r l y when one r e a l i z e s the fac t that the r e s u l t s depend heavily on the r e s i s t i v i t y and conductivity of the glass support, accuracy of the wedge angle, and r e l a t i o n (11-12) governing the ampli- f i e r c h a r a c t e r i s t i c s , etc., which are quoted by the manufac- turer (usually as constant values) and can change from probe to probe. Any discrepancy i n the slope (F^ = C^/C^) can be minimized by better determination of the probe and f l u i d physical parameters, s p e c i f i c a l l y b, I, K^, and A . The accurate evaluation of these parameters w i l l also improve 2 r e l i a b i l i t y of the V intercept ( i . e . F^) for the probes. Furthermore, recognizing that F^ i s s u b s t a n t i a l l y affected by the value of the constant C^, i t s accuracy w i l l improve with that of C^, i . e . by eliminating the assumption of two dimensionality of the wedge and improving upon the power law character of the temperature v a r i a t i o n as used i n the Sokolowski theory. In any case, even i n the absence of any further s o p h i s t i c a t i o n , the theory as i t stands gives res u l t s of s u f f i c i e n t accuracy for p r a c t i c a l a pplications.

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