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The impaction of spherical particles on circular cylinders Griffin, Frank Owen 1972

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c l THE IMPACTION OF SPHERICAL PARTICLES ON CIRCULAR CYLINDERS by FRANK OWEN GRIFFIN Sc. (Eng.), Queen's U n i v e r s i t y , K i n g s t o n , 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of CHEMICAL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1972 In present ing th i s thes is in pa r t i a l f u l f i lmen t o f the requirements for an advanced degree at the Un ive rs i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make it f r ee l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying o f th i s thes i s for s cho la r l y purposes may be granted by the Head of my Department or by h is representa t i ves . It is understood that copying or pub l i c a t i on o f th is thes i s fo r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Depa rtment The Un ivers i t y of B r i t i s h Columbia Vancouver 8, Canada "ate ^V4^ *V\¥p^ / ABSTRACT I n e r t i a l and interceptive .impaction of spherical p a r t i c l e s on c i r c u l a r cylinders was investigated t h e o r e t i -c a l l y . The p a r t i c l e s were considered to be suspended in a f l u i d moving st e a d i l y through a random array of p a r a l l e l c y l i n d e r s . Fluid f l o w f i e l d s around the cylinders were obtained by numerically solving the Navier-Stokes Equation subject to Kuwabara's zero v o r t i c i t y boundary condition. These solutions were subsequently u t i l i z e d in c a l c u l a t i n g p a r t i c l e t r a j e c t o r i e s and impaction e f f i c i e n c i e s . The l a t t e r are presented as functions of Reynolds number (0.2 < Rec < 40), p a r t i c l e i n e r t i a l parameter (0 < P < 1000), p a r t i c l e to cylinder size r a t i o (0.001 < K < 1.) and cylinder concentra-tion (10-" < c < 0.111). The impaction e f f i c i e n c i e s and c r i t i c a l i n e r t i a l parameters d i f f e r s i g n i f i c a n t l y from e a r l i e r theoretical p r e d i c t i o n s . The discrepancies are primarily attributable to the inaccurate f l o w f i e l d representations used by previous authors. The agreement between Subramanyam and Kuloor's experimental work and present theory is s a t i s f a c t o r y . i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES ix ACKNOWLEDGEMENTS xv Chapter 1 INTRODUCTION 1 The Fluid Flowfield 3 The I n e r t i a l Impaction Mechanism 5 The Interceptive Mechanism 8 Combined I n e r t i a l and Interceptive Mechanisms 10 Scope of the Present Work 12 2 LITERATURE REVIEW 13 Par t i c l e - C y l i n d e r C o l l i s i o n in Vi scous Fl ow. . . 13 Par t i c l e - C y l i n d e r C o l l i s i o n in Potential Flow. . 21 Numerical Solution of Navier-Stokes Equati on 23 i i i Chapter Page 3 THEORY 27 Navier-Stokes Equation for Flow Around a Cylinder 27 Boundary Conditions 31 Numerical Solution of Navier-Stokes Equati on 33 F i n i t e Difference Formulation of the Navier-Stokes Equation 36 F i n i t e Difference Formulation of Boundary Conditions 38 Relaxation Method 39 I n i t i a l Values for Relaxation Method . . . 40 Convergence C r i t e r i a 46 Calculation of P a r t i c l e T r a j e c t o r i e s . . . . . 53 Equation of P a r t i c l e Motion 53 Drag Coefficients 55 Integration of P a r t i c l e Equations 56 Fibonacci Search for C r i t i c a l T r a j e c t o r y . . . 59 4 RESULTS AND DISCUSSION 63 Typical P a r t i c l e Trajectories 64 PARTICLE IMPACTION EFFICIENCIES 65 Effect of I n e r t i a l Parameter on E 67 Comparison with Davies 69 Experimental Impaction E f f i c i e n c i e s . . . . 71 i v Page Potential E f f i c i e n c i e s 71 C r i t i c a l P a r t i c l e I n e r t i a l Parameter . . . 73 Effect of Reynolds Number on E 74 Comparison with Householder and Goldschmidt 75 Contour Plots of E f f i c i e n c y 75 Effect of Solids Concentration. . „ 77 Sieving 78 In e r t i a l and Interceptive Mechanisms 79 Comparison with Wong et al 80 Selection of Numerical Grid 81 Angular and Radial Divisions 81 Steps Per Cell 82 Happel's Model 82 FLOWFIELDS 83 Effect of R 84 00 Characteristic Parameters 84 Davies' Flowfield at Rec = 0.2 85 Davies' Flowfield at Rec = 1.0 86 CONVERGENCE CRITERIA 87 Selection of Standard C r i t e r i o n 91 RECOMMENDATIONS FOR FUTURE WORK 9 2 v Page NOMENCLATURE 208 REFERENCES 215 APPENDICES I Rectangular Co-ordinates . . . 219 II Stagnation Pressures and Drag Coeff i c i e n t s 224 III Happel 's Model 234 IV Relaxation Factors used in this Work 237 V Computer Programmes and Sample Output. . .• 238 vi LIST OF TABLES Table Page 4-1 Impaction E f f i c i e n c i e s at = 100 153 4-2 Impaction E f f i c i e n c i e s at Rro = 3 159 4-3 Impaction E f f i c i e n c i e s at Re = 0.2, R^  = 200 . . 165 4-4 Comparison of Impaction E f f i c i e n c i e s for Potential Flow 166 4-5 Impaction E f f i c i e n c i e s for Potential Flow . . . . 167 4-6 Impaction E f f i c i e n c i e s at Re = 0.2 168 4-7 Impaction E f f i c i e n c i e s at Rec = 40 171 4-8 Interception E f f i c i e n c i e s 174 4-9 Impaction E f f i c i e n c i e s Calculated at Re = 40, R = 100 under Various Conditions 175 CO 4-10 Stagnation Pressures and Drag C o e f f i c i e n t s . . . . 178 4-11 Fluid v e l o c i t y , v x , according to Davies and Peetz [7] at Rec = 0.2 179 4-12 Fluid v e l o c i t y , v , according to Davies and Peetz [7] at Rec = 0.2 182 vi i Table Page 4-13 Fluid v e l o c i t y , v , as interpolated from the Numerical Flowfield at Re„ = 0.2, R = 200. . . . 185 C C O • w 4-14 Fluid V e l o c i t y , v , as Interpolated from the Numerical Flowfield at Re = 0.2, R = 200. . . . 188 p CO 4-15 Fluid V e l o c i t i e s According to Davies and Peetz [7] at Rec =10 191 4 -16 Fluid v e l o c i t y , v , as Interpolated from Numerical Flowfield at Re = 10, R = 100 C ' oo the • • • . 192 4 -17 Fluid V e l o c i t y , v , as Interpolated from Numerical Flowfield at Rec = 10, R^  = 100 the . 195 4 -18 Convergence C r i t e r i a for Rec = 0.2, R = ' 00 100. . • 198 4 -19 Convergence C r i t e r i a for Rec = 0.2, R = ' 00 3. . . . 200 4 -20 Convergence C r i t e r i a for Rec = 0.5, R = ' CO 100. . . 202 4 -21 Convergence C r i t e r i a for Rec = 40, R = 100 . . oo . 204 4 -22 Convergence C r i t e r i a for Rec = 40, R =3 ' oo . . . . 206 IV -1 . 237 v i i i LIST OF FIGURES Fi gure Page 1-1 Typical flowfield around a circular cylinder. . . 4 1-2 Trajectories of three very small particles (p p > p) 6 1-3 Trajectories of three particles having f i n i t e size (p = p) 9 3-1 Co-ordinate system for circular cylinder and surrounding fluid 30 3-2 Grid system in the 6-z plane 34 3-3 Masliyah's [43] relaxation factors for vorticity 42 3- 4 Collision function for Fibonacci search 61 4- 1 Particle trajectories for K = 0.1, P = 10, Re„ = 0.2, R =100 93 c 00 4-2 Particle trajectories for K = 0.1, P = 10, Re^ = 40, R = 100 93 C co 4-3 Particle Reynolds number as a function o f x and starting position, y p (K = 0.1, P = 10, Re„ = 40, R = 100. 94 C ' C O 4-4 Impaction efficiency as a function of inertial parameter for Rec = 0.2, Rro = 100 95 4-5 Impaction efficiency as a function of inertial parameter for Rec = 1, R^  = 100 96 ix Fi gure Page 4-6 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 5, R^  = 100 97 4-7 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 10, R^  = 100 . . 98 4-8 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 20, Rro = 100 99 4-9 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 40, R^  = 100 100 4-10 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 0.2, R^  = 3 101 4-11 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 1 , Rro = 3 102 4-12 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 5, Rro = 3 103 4-13 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 10, R^  = 3 104 4-14 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 20, Rro = 3 105 4-15 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for Rec = 40, Rro = 3 106 4-16 Comparison of impaction e f f i c i e n c i e s for Rec = 0.2, R^  = 200 as predicted by: ( i ) Davies and Peetz [ 7 ] , ( i i ) this work 107 4-17 Comparison of impaction e f f i c i e n c i e s for Rec = 10, R = 100 as predicted by: ( i ) Davies and Peetz [ 7 ] , ( i i ) this work 108 4-18 Comparison of impaction e f f i c i e n c i e s given by: ( i ) Subramanyam and Kuloor [12] (experi-mental), ( i i ) this work 109 x Fi gure Page 4-19 Impaction e f f i c i e n c y as a function of i n e r t i a l parameter for potential flow 110 4-20 C r i t i c a l i n e r t i a l parameter as a function of Reynolds number I l l 4-21 Impaction e f f i c i e n c y as a function of Reynolds number for K = 0.001, R = 100 112 ' CO 4-22 Impaction e f f i c i e n c y as a function of Reynolds number for K = 0.1, R =100 113 oo 4-23 Impaction e f f i c i e n c y as a function of Reynolds number for K = 1.0, R = 100 114 ' CO 4-24 Impaction e f f i c i e n c y as a function of Reynolds number for K = 0.001, R =3 115 ' C O 4-25 Impaction e f f i c i e n c y as a function of Reynolds number for K = 0.1 , R = 3 116 CO 4-26 Impaction e f f i c i e n c y as a function of Reynolds number for K = 1.0, R =3 117 ' oo 4-27 Comparison of impaction e f f i c i e n c i e s for K = 0.1, Rro = 100 as predicted by: ( i ) Householder and Goldschmidt [13], ( i i ) this work 118 4-28 Comparison of impaction e f f i c i e n c i e s for K = 1.0, R e = 100 as predicted by: ( i ) Householder and Goldschmidt [13], ( i i ) thi s work 119 4-29 Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter for K = 0.001, R = 100 120 00 4-30 Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter for K = 0.5, R = 100 121 CO xi Fi gure Page 4-31 Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter for K = 1.0, R = 100 122 00 4-32 Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter for K = 0.001, R = 3 . 123 oo 4-33 Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter for K = 0.5, R =3 124 00 4-34 Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter for K = 1.0, R =3 125 oo 4-35 Impaction e f f i c i e n c y as a function of sol i d s concentration for Rec = 0.2, K = 0.001 126 4-36 Impaction e f f i c i e n c y as a function of so l i d s concentration for Rec = 0.2, K = 0.1 127 4-37 Impaction e f f i c i e n c y as a function of s o l i d s concentration for Rec = 0.2, K = 1.0 128 4-38 Impaction e f f i c i e n c y as a function of s o l i d s concentration for Rec = 40, K = 0.001 129 4-39 Impaction e f f i c i e n c y as a function of s o l i d s concentration for Re„ = 40, K = 0.1 130 c 4-40 Impaction e f f i c i e n c y as a function of sol i d s concentration for Rec = 40, K = 1.0 . 131 4-41 Sieving mechanism at high solids concentration 132 xi i Fi gure Page 4 - 4 2 Impaction e f f i c i e n c i e s (Ex, E K , E X + E R , E) as a function of Reynolds number for K = 1 . 0 , P = 5 , R = 1 0 0 1 3 3 4 - 4 3 Impaction e f f i c i e n c i e s (Ex, E K , E x + E K , E) as a function of Reynolds number for K = 0 . 5 , P = 2 , R = 3 1 3 4 ' C O 4 - 4 4 Comparison of impaction e f f i c i e n c i e s due to interception for K = 1 . 0 , Roo = 1 0 0 as pre-dicted by: ( i ) Wong et al. [ 1 0 ] , ( i i ) this work 1 3 5 4 - 4 5 Streamlines and e q u i - v o r t i c i t y lines at R^ = 1 0 0 f o r : (a) Rec = 0 . 2 , (b) Re = 1 . . . . 1 3 6 4 - 4 6 Streamlines and e q u i - v o r t i c i t y lines at R = 1 0 0 f o r : (a) Re„ = 5 , (b) Re„ = 1 0 1 3 7 C O c c 4 - 4 7 Streamlines and e q u i - v o r t i c i t y lines at R = 1 0 0 f o r : (a) Re„ = 2 0 , (b) Re = 4 0 . . . . 1 3 8 oo C C 4 - 4 8 Streamlines and e q u i - v o r t i c i t y lines at R^ = 3 f o r : (a) Rec = 0 . 2 , (b) Re = 1 . . . . 1 3 9 4 - 4 9 Streamlines and e q u i - v o r t i c i t y lines at R = 3 f o r : (a) Rer = 5 , (b) Rer = 1 0 1 4 0 4 - 5 0 Streamlines and e q u i - v o r t i c i t y lines at R = 3 f o r : (a) Re_ = 2 0 , (b) Re = 4 0 . . . . 1 4 1 4 - 5 1 Streamlines and e q u i - v o r t i c i t y lines at Re = 0 . 2 f o r : (a) R = 5 0 , (b) R = 2 5 1 4 2 4 - 5 2 Streamlines and e q u i - v o r t i c i t y lines at Re„ = 0 . 2 for R = 1 0 1 4 3 c °° 4 - 5 3 Streamlines and e q u i - v o r t i c i t y lines at Re„ = 4 0 f o r : (a) R = 5 0 , (b) R = 2 5 . . . . . 1 4 4 C CO ? v * 0 0 x i i i Fi gure Page 4-54 Streamlines and e q u i - v o r t i c i t y lines at Re = 40 for R =10 145 c 00 4-55 Comparison of drag c o e f f i c i e n t s as given by T r i t t o n [31], Takami and Keller [39], Hamielec and Raal [40], Dennis and Chang [41], and this work 146 4-56 Streamlines at Rec = 0.2, Rro = 200 f o r : (a) Davies and Peetz [ 7 ] , (b) this work 147 4-57 Streamlines in the v i c i n i t y of the cylinder f o r : (a) Davies and Peetz [ 7 ] , (b) this work 148 4-58 Fluid v e l o c i t y , v , as a function of x at Re = 0.2, R = 200 f o r : ( i ) Davies and C ' oo x ' Peetz [ 7 ] , ( i i ) this work 149 4-59 Fluid v e l o c i t y , v^, as a function of x at Re = 0.2, R = 200 f o r : ( i ) Davies and C 00 1 ' Peetz [ 7 ] , ( i i ) this work 150 4-60 Fluid v e l o c i t y , v , as a function of x at Re„ = 10, R = 100 f o r : ( i ) Davies and Peetz [ 7 ] , ( i i ) this work 151 4-61 Fluid v e l o c i t y , v^, as a function of x at Rec = 10, Rro = 100 f o r : ( i ) Davies and Peetz [ 7 ] , ( i i ) this work 152 1-1 Typical grid c e l l 222 xiv ACKNOWLEDGEMENTS My sincere thanks to Dr. Axel Meisen for his va l u -able guidance and encouragement throughout the course of this work. In addition I am indebted to Dr. J.H. Masliyah for his assistance in explaining numerical techniques. I am very grateful for the f i n a n c i a l support pro-vided by the National Research Council of Canada and the University of B r i t i s h Columbia. I also wish to acknowledge the co-operation and suggestions from the members of the U.B.C. Computing Centre. xv Chapter 1 INTRODUCTION The removal of p a r t i c u l a t e matter from l i q u i d s and gases has been an important operation since the incep-tion of Chemical Engineering. Although the subject of p a r t i c l e separation has in the past received intensive study, present concern about a i r and water p o l l u t i o n is stimulating further research. Special e f f o r t s are being made to improve or develop techniques suitable for the re-moval of very fine p a r t i c l e s . Several methods are available for removing very small (i . e . micron-s i ze) p a r t i c l e s , the p r i n c i p l e ones being e l e c t r o s t a t i c p r e c i p i t a t i o n , scrubbing, centrifugal cleaning and f i l t r a t i o n . Since the former three techniques are costly and e l e c t r o s t a t i c p r e c i p i t a t i o n and scrubbing are only s u i t -able for gases, f i l t r a t i o n is frequently the preferred method. There are b a s i c a l l y two d i f f e r e n t ways in which f i l t e r s separate p a r t i c l e s and they may therefore be c l a s s i -f i e d as surface or deep bed f i l t e r s . 1 2 Surface f i l t e r s achieve p a r t i c l e separation by a s t r a i n i n g or sieving action because the pores of the f i l t e r cake are smaller than the p a r t i c l e s . The p a r t i c l e s are retained at the surface of the cake and the cake t h i c k -ness increases as the f i l t r a t i o n proceeds. Since micron size p a r t i c l e s form very compact cakes with concomitant high pressure drops, surface f i l t e r s are only suitable for coarser p a r t i c l e s (generally having diameters in excess of 10y) or under circumstances where high pressure drops are i nconsequenti a l . Deep bed f i l t e r s , on the other hand, are loosely packed assemblages of granules or c y l i n d r i c a l f i b r e s . Their solids fraction is generally less than 0.1 and the pressure drops are therefore s i g n i f i c a n t l y lower than those of surface f i l t e r s . The main disadvantages of deep bed f i l t e r s arise from the fact that they separate and retain the p a r t i c l e s inside the f i l t e r medium which makes regeneration d i f f i c u l t . Consequently they are mainly used for cleaning d i l u t e suspensions and the f i l t e r s are discarded once they have become loaded. Very l i t t l e sieving occurs in deep bed f i l t e r s and p a r t i c l e s are removed mainly by i n e r t i a l impaction and i n t e r c e p t i o n . Other separation mechanisms such as e l e c t r o -s t a t i c , gravitational and Brownian motion effects are frequently less important. 3 In the present study the l a t t e r effects were omitted and the work was r e s t r i c t e d to deep bed f i l t e r s composed of randomly spaced, p a r a l l e l cylinders lying at right angles to the main dir e c t i o n of flow. In order to gain a better understanding of the i n e r t i a l and interceptive mechanisms in such a f i l t e r i t is i n s t r u c t i v e to consider a very d i l u t e bed. Under these conditions the mechanisms can be described in reference to a single cylinder situated in a very large amount of moving f l u i d . After providing a b r i e f account of the f l u i d f l o w f i e l d , i n e r t i a l impaction' and interception are discussed separately. F i n a l l y i t w i l l be shown that in r e a l i t y the effects always occur together and should therefore be considered j o i n t l y . The Fluid Flowfield A t y p i c a l , steady state f l o w f i e l d around a stationary cylinder is shown in Figure 1-1. The ve l o c i t y of the f l u i d far away from the cylinder is r e c t i l i n e a r and of magnitude U 0. In the v i c i n i t y of the cylinder the stream lines are curved and closely spaced. A wake may also be present. It may be pointed out that the flow in deep bed f i l t e r s is generally steady and laminar. The following arguments apply however equally well to turbulent flows Figure 1-1. Typical flowfield around a c i r c u l a r c y l i n d e r . 5 provided the p a r t i c l e motion is not affected by the micro-structure of the turbulence. The I n e r t i a l Impaction Mechanism The mechanism of i n e r t i a l impaction w i l l be i l l u s t r a t e d by considering three i d e n t i c a l spherical p a r t i c l e s having radius, Rp, and density, p p. The r a t i o of p a r t i c l e to cylinder radius, i . e . K = p£ (1-1) is taken to be very small so that the p a r t i c l e s may be re-garded as points. The p a r t i c l e density is sti p u l a t e d to exceed the f l u i d density, i . e . pp > p . When the three p a r t i c l e s are started far upstream of the cylinder with velocity U0 at x' = - R' and at p <» three d i f f e r e n t heights, y p , above the centre l i n e , they travel along d i f f e r e n t t r a j e c t o r i e s as shown in Figure 1-2. Since the i n e r t i a of the p a r t i c l e s are higher than those of equivalent volumes of f l u i d , t h e i r t r a j e c t o r i e s deviate from the streamlines and approach the front of the c y l i n d e r . In the case of p a r t i c l e 1 the deviation is s u f f i c i e n t for the trajectory to i n t e r s e c t the cylinder surface. 7 Hence when the c e n t r e of a p a r t i c l e coincides with the cylinder surface as the result of the aforementioned cause, i n e r t i a l impaction is said to occur. Since in r e a l i t y the p a r t i c l e centre cannot touch the cylinder surface, this is c l e a r l y an i d e a l i z e d s i t u a t i o n . However, for small radius r a t i o s , K, the i d e a l i z a t i o n does not lead to s i g n i f i c a n t e r r o r s . As may be seen from Figure 1-2 p a r t i c l e 3 misses the cylinder whereas the trajectory of the centre of p a r t i c l e 2 just grazes the surface. The l a t t e r is c a l l e d the c r i t i c a l t rajectory since p a r t i c l e s s t a r t i n g at x' = - R' with y' < P oo J p yp , c r i t c o l l i d e with the cylinder I whereas p a r t i c l e s s t a r t i n g with y^ >|^p C r i t ] m i s s ' J t 1 S therefore possible to define a dimensionless i n e r t i a l impac-tion c o e f f i c i e n t , , as follows y Jp , c r i t - 1 1 (1-2) It may be noted that e increases with p a r t i c l e i n e r t i a and ranges from 0 to 1. The upper l i m i t is attained by p a r t i c l e s with i n f i n i t e i n e r t i a whose t r a j e c t o r i e s are therefore straight l i n e s . 8 The I n t e r c e p t i v e Mechanism As i n the previous s e c t i o n three i d e n t i c a l s p h e r i c a l p a r t i c l e s are c o n s i d e r e d to approach the c y l i n d e r with v e l o c i t y U 0• However, i n the present case the p a r t i c l e d e n s i t y equals the f l u i d d e n s i t y and the r a d i u s r a t i o K i s s i g n i f i c a n t l y g r e a t e r than z e r o . I f i t i s assumed that the p a r t i c l e s do not a p p r e c i -ably d i s t u r b the f l u i d f l o w f i e l d , the t r a j e c t o r i e s of t h e i r centres c o i n c i d e with the stream l i n e s . Hence, when a p a r t i c l e of radius R p t r a v e l s along a s t r e a m l i n e , which approaches the c y l i n d e r w i t h i n a d i s t a n c e Rp , i t i s i n t e r -cepted by the c y l i n d e r . The c o l l i s i o n i s s o l e l y due to the s i z e of the p a r t i c l e and i s c a l l e d the i n t e r c e p t i v e mechani sm. As seen from Figure 1-3 there i s a c r i t i c a l s t a r t i n g , which r e s u l t s i n the p a r t i c l e s u r f a c e K j u s t g r a z i n g the c y l i n d e r s u r f a c e . A dimension 1 ess i n t e r -ception c o e f f i c i e n t , zv , may t h e r e f o r e be d e f i n e d analogous to E q u a t i o n (1-2): pos i t i on , p , c r i t c r i t 'K K (1-3) 10 This c o e f f i c i e n t may however exceed unity for large p a r t i c l e s i . e . K >> 0, and i t is therefore convenient to normalize i t by introducing the impaction e f f i c i e n c y , Ev , where K The impaction e f f i c i e n c y therefore ranges from 0 to 1. Combined I n e r t i a l and Interceptive Mechanisms In most p r a c t i c a l situations Pp > P a n c* K > 0 so that both the i n e r t i a l and interceptive mechanisms are operative. It is possible to define an overall impaction c o e f f i c i e n t and e f f i c i e n c y : e = y P > c r i t ( 1 _ 5 ) RC 1 + K (1-6) The c r i t i c a l trajectory s t a r t i n g at x^ = - and y^ = y^ c r i t is determined by selecting the trajectory 11 which causes the p a r t i c l e surface to just touch the cylinder surface. In p r i n c i p l e this trajectory can be determined by a t r i a l and error procedure. The above discussion has dealt s o l e l y with the mechanisms of p a r t i c l e transport to the surface of the cyl i n d e r . No mention has been made of the int e r a c t i o n which occurs between the p a r t i c l e and the cylinder at the instance of impaction. The impaction e f f i c i e n c y indicates how many of the p a r t i c l e s approaching a cylinder w i l l actually h i t under given flow conditions. Since p a r t i c l e s c o l l i d i n g with a cylinder do not necessarily adhere, the actual c o l l e c t i o n e f f i c i e n c y of the fibres is less than or equal to the impaction e f f i c i e n c y . In practice i t is known that micron-size p a r t i c l e s are permanently retained in f i b r e f i l t e r s by virtue of van der Waals f o r c e s , surface tension and e l e c t r o s t a t i c e f f e c t s . In the case where a l l p a r t i c l e s contacting the cylinder are thus retained without rebound from the surface, the c o l l e c t i o n e f f i c i e n c y is numerically equal to the impaction e f f i c i e n c y . 12 Scope of the Present Work The primary objective of this thesis was to calculate impaction e f f i c i e n c i e s for spherical p a r t i c l e s impinging on c i r c u l a r c y l i n d e r s . The following mechanisms and range of variables were considered: - i n e r t i a l and i n t e r c e p t i v e e f f e c t s - Re y n o l d s number based on c y l i n d e r d i a m e t e r . 0.2 < Re < 40 3 — c - p a r t i c l e i n e r t i a l p a r a m e t e r , 0 1 P < 1000 - r a t i o o f p a r t i c l e r a d i u s t o c y l i n d e r r a d i u s , 0.001 < K < 1.0 - c o n c e n t r a t i o n o f c y l i n d e r s r a n g i n g from 10-Ho 0.111 The f l u i d f lowfields around the cylinder were obtained by solving the Navier-Stokes Equation numerically with a relaxation technique. Various c r i t e r i a for monitor-ing the convergence of the solution were developed. Chapter 2 LITERATURE REVIEW This section is primarily r e s t r i c t e d to a review of the l i t e r a t u r e on p a r t i c l e and f l u i d motion around cylinders in the viscous flow regime. However, a b r i e f account of the key papers describing p a r t i c l e c o l l i s i o n s with cylinders in the potential flow regime is also pro-vided. The l a t t e r were used to v e r i f y the computer pro-grammes ca l c u l a t i n g the p a r t i c l e t r a j e c t o r i e s (Appendix V). Fuchs [ 1 ] , Dorman [2 ] , Pich [3 ] and L o f f l e r [ 4 ] have written general, review-s on p a r t i c l e deposition on cylinders and other objects. P a r t i c l e - C y l i n d e r C o l l i s i o n in Viscous Flow Landahl and Herrmann [ 5 ] were the f i r s t who t h e o r e t i c a l l y calculated the impaction c o e f f i c i e n t due to i n e r t i a l and interceptive mechanisms in the viscous flow regime. Their work was r e s t r i c t e d to a c i r c u l a r 13 14 cylinder with Rec = 10 , and Thorn's [ 6 ] approximate flow-f i e l d was used. The p a r t i c l e s were assumed to obey Stokes1 Law, and t h e i r motion was calculated by an i t e r a t i v e pro-cedure involving piecewise l i n e a r i z a t i o n of the f l o w f i e l d and t r a j e c t o r i e s . The p a r t i c l e i n e r t i a l parameter was varied from 0 < P < 3 , and t h e i r predictions have been summarized [3 ] by the following equation: e = P3 + 1 .54P2 + 1.76 + K (2-1) Equation (2-1) indicates that e is an increasing function of P and K. However, since i t is based on Thorn's data i t is somewhat inaccurate. Furthermore, the expression implies that the i n e r t i a l and interceptive contributions to the impaction c o e f f i c i e n t are a d d i t i v e . This is not s t r i c t l y correct because the f l u i d streamlines are not equispaced at Re£ = 10 . Davies and Peetz [ 7] also used Thorn's results at Re£ = 10 . By means of i n t e r p o l a t i o n and extrapolation polynomials they were able to obtain a f i n e r g r i d , and to extend the f l o w f i e l d to 5 r a d i i from the centre of the c y l i n d e r . The authors i n i t i a t e d t h e i r p a r t i c l e trajec tories on the outer boundary of the i r f l o w f i e l d . Stokes' Law was used to calculate the drag on the p a r t i c l e s and numerical integration with variable step size was employed to calculate the t r a j e c t o r i e s (see Davies and Aylward [ 8 ] for a description of this technique). Both i n e r t i a l and interceptive effects were studied, plots being given for e vs P in the ranges 0 < P < 30, 0 < K < 1 . Contrary to Landahl and Herrmann, Davies and Peetz found that the intercep t i v e and i n e r t i a l mechanisms of impaction were not s t r i c t l y a d d i t i v e . Davies and Peetz also determined a so-called " c r i t i c a l i n e r t i a l parameter," P c of 0.417. P c i s defined such that an i n f i n i t e l y small p a r t i c l e (K 0) having i n e r t i a l parameter P < Pc would f a i l to c o l l i d e with the c y l i n d e r . These authors also studied p a r t i c l e c o l l e c t i o n at a Reynold's number of Rec = 0.2 , by u t i l i z i n g a Bessel function representation of the f l o w f i e l d [ 9 ] . The following equations were used: 16 Re v x = 1 + rRe ^ ( c 1 + X 2r^ r Ki Re. "16 K° ^ J 1 - — 5 r" Re 64 r2 Re fRe - exp Kc Re„ • r c + 7 Kx fRe K( fRe + 0.5 Ki Re 16 Re 1 -Re 64 1 - 2 + J F2" r + x^ x^_ . Re exp Re„ • x c L K fRe c • r K c Re + 0.5 (2-2) where «Q A N D Ki a r e modified Bessel functions of the second kind. 17 The p a r t i c l e t r a j e c t o r i e s v/ere calculated as before at Re = 10 , except that the s t a r t i n g position was 200 cylinder r a d i i upstream from the cylinder centre. The results were presented in graphical form by pl o t t i n g e versus P with K as a parameter. The ranges investigated were 0 < P < 80 , and 0 < K < 1 , and P c in this instance was estimated to be 0.899. Davies and Peetz' results are the most accurate to date in the viscous regime because they are based on good approximations of the f l u i d f l o w f i e l d s . The Bessel function expression at Rec = 0.2 s a t i s f i e s the Navier-Stokes Equation and the boundary condition far away from the c y l i n d e r . However, as is shown l a t e r (Chapter 4 ) , i t does not predict zero f l u i d v e l o c i t i e s at the cylinder surface. This feature produces s i g n i f i c a n t errors in the calculated e f f i c i e n c i e s of small p a r t i c l e s which closely approach the cylinder surface. Wong, Ranz and Johnstone [10] used the equation given by Lamb [11] for viscous flow around a cyli n d e r : 18 V = v X ' x2+ y2 v y = Y[*2+y2-l Y = 1 + x2+y2-l ' * y2-x2 /2 x2+y2 • I J L J xy / x2+y2 2.002 - In Re. (2-3) Assuming Stokes' Law for the p a r t i c l e s they calculated the following theoretical expression for zv, i . e . , the impaction c o e f f i c i e n t due to interception alone: (1+K) In (1+K) - K(2 + K)/2(1+K)J (2-4) The authors present curves for Equation (2-4) at Re = 0.01, 0.1, 1.0 , the highest values of Re extending past the l i m i t of a p p l i c a b i l i t y of Equation (2-3) Furthermore, Equation (2-3) is v a l i d only close to the cylinder and does not s a t i s f y the boundary conditions at i n f i n i t y . 19 Subramanyam and Kuloor [12] used Davies and Peetz' extension of Thorn's calculations at Rec = 10. They c a l -culated p a r t i c l e t r a j e c t o r i e s s t a r t i n g at 5 r a d i i upstream, for 0 < P < 10 , and K = 0 . Neither the method of c a l -culating the p a r t i c l e t r a j e c t o r i e s nor the drag law for the p a r t i c l e s were c l e a r l y stated. However, t h e i r drag c o e f f i c i e n t would appear to be very close to that given by Stokes' Law. Householder and Goldschmidt [13] used Davies' [ 9 ] a n a l y t i c a l approximation to the f l o w f i e l d at Rec = 0.2. They assumed Stokes' drag law for the p a r t i c l e s and used a second-order Runge-Kutta integration technique to deter-mine p a r t i c l e t r a j e c t o r i e s . The s t a r t i n g position was 400 r a d i i upstream. Impaction e f f i c i e n c y , E, was plotted as a function of P and K for the ranges 1 < P < 101* and 0 5 K 5 7 . The authors also calculated values of E at Re ~ 1000 for s i m i l a r ranges of P and K. Using a least-squares technique they f i t t e d the following equation to t h e i r t h e o r e t i c a l predictions: 20 E = 0.5 1 + tanhjCj + CalnO+P) + C3K + C\ l n ( l + Rec) + C5 ln(l+P)K + C6 l n ( l + P) ln(l+Re r) + C7K ln(l+Rej|- (2-5) + c, ln(l+P) + C9K2 + C i o(l nO + R e c ) where -0.9955 c 6 = 0.0010 c 2 = 0.1921 c 7 = 0.0197 c 3 = 0.1426 c 8 = 0.0464 c- = 0.0066 c 9 = -0.0036 C5 = -0.0251 C i o = 0.0287 The above expression is intended to predict the impaction e f f i c i e n c y , E, by incorporating the effects of f l o w f i e l d , p a r t i c l e i n e r t i a l parameter, P, and p a r t i c l e size parameter, K. No mention is made of a c r i t i c a l i n e r t i a l parameter P . 21 Equation (2-5) cannot be regarded as well-founded because i t is based on results obtained at just two Reynolds numbers. Furthermore, the trajectory calculations assumed that the p a r t i c l e s do not influence the f l u i d f l o w f i e l d . This condition is c l e a r l y violated for diameter r a t i o s , K, greater than unity. Experimental work on p a r t i c l e c o l l e c t i o n by cylinders has been mainly concerned with potential flow situations [5,10,12-17] (Re £ 1000). Although data are available [5,10,12,16,17] at lower values of Re c, the variable flowrates make analysis d i f f i c u l t . In addi t i o n , the p a r t i c l e s used were generally so small or so polydisperse as to make i t impossible to determine the interceptive contributions to impaction e f f i c i e n c y . Subramanyam and Kuloor performed impaction e f f i c i -ency experiments in the ranges: 4 $ Rec < 240, 0 < K < 0.10 and 1 5 P 5 60 . Their results gave at least a quantita-tive i n d i c a t i o n of the effects of p a r t i c l e size on impac-tion e f f i c i e n c y . P a r t i c l e - C y l i n d e r C o l l i s i o n in Potential Flow Langmuir and Blodgett [18] presented the f i r s t comprehensive study of p a r t i c l e deposition on cylinders in 22 the potential flow regime. The work was e n t i r e l y theoretical and u t i l i z e d Lamb's [11] equations to define the f l o w f i e l d : v = 1 + y2 - x2 (x2 + y2)2 (2-6) - 2xy f y (x2 + y2) The p a r t i c l e t r a j e c t o r i e s were i n i t i a t e d four cylinder diameters upstream sof the cylinder and the cal-. culations performed on a d i f f e r e n t i a l analyzer (analog computer). The drag experienced by the p a r t i c l e s was expressed either by Stokes Law at low Reynolds numbers, or by an empirical equation at high values of Re p. Langmuir and Blodgett considered p a r t i c l e depo-s i t i o n due to i n e r t i a l effects only, and presented graphs of the impaction c o e f f i c i e n t , e , as a function of P, with <j) as a parameter. <J> was defined as: R e2 • K2 18p2 U0 Rr 4> = — ^ - p = - (2-7) P y pp 23 The authors investigated the ranges 0 < <j> < 106 and 0.1 1 P 1 100 . Using s t r i c t l y t h eoretical argu-ments based on Equation (2-6) they determined a value of 0.125 for the c r i t i c a l i n e r t i a l parameter P . Davies and Peetz [7] also reported th e o r e t i c a l results on p a r t i c l e c o l l i s i o n s with cylinders in the potential flow regime. Their work was b a s i c a l l y s i m i l a r to that of Langmuir and Blodgett, except that they also considered the interception e f f e c t . The t r a j e c t o r i e s were started 5 r a d i i upstream of the c y l i n d e r , and the p a r t i c l e s were assumed to obey Stokes' Law at a l l times. Impaction c o e f f i c i e n t s , e, were reported for the ranges 0 < K < 1 , and 0 < P < 40 . By using a method s i m i l a r to Langumuir and B'lodgett's, they also calculated the c r i t i c a l i n e r t i a l parameter, P , to be 0.125. Further theoretical and experimental re s u l t s i n the potential flow regime may be found in references [18] to [24]. Numerical Solution of Navier-Stokes Equation Thorn [ 6 ] was one of the f i r s t to use a f i n i t e difference i t e r a t i v e method to solve the steady state 24 Navier-Stokes Equation for flow around c y l i n d e r s . He c a l -culated fl o w f i e l d s at Rec = 10 and 20 and achieved good agreement with his experimental r e s u l t s . Tomotoika and Aoi [25] obtained an exact s o l u -tion to Oseen's l i n e a r i z e d form of the Navier-Stokes Equa-tion for Rec up to 10. However, they erroneously pre-dicted wake formation at Re£ = 0.05 , and Yamada [26] (quoted by Underwood [27]) indicated serious numerical errors in their work. Proudman and Pearson [28] used matched asymptotic expansions, but the results were only v a l i d up to Rec z 5 . Happel [29] presented a solution to the creeping flow equation for c y l i n d e r s , using the boundary condition that the f l u i d shear stress becomes zero at large but f i n i t e distances from the c y l i n d e r . Kawaguti [30] solved the steady state Navier-Stokes Equation in f i n i t e difference form for a c i r c u l a r cylinder at Rec = 40. The results of his laborious calcu-lations agree reasonably well with experimental data [31]. The work of Allen and Southwell [32] and Dennis and Shimshoni [33] is generally regarded as imprecise [34], since they p r e d i c t , for example, a decrease in wake size with increasing Reynolds number at Re = 40. 25 Jenson [35] used a Gauss-Seidel relaxation tech-nique to solve the f i n i t e difference form of the Navier-Stokes Equation for spheres, under non-turbulent flow con-d i t i o n s . In establishing the boundary conditions for numerical solution he used Kuwabara's [35] model, which stip u l a t e s that the f l u i d v o r t i c i t y tends to zero at large distances from the sphere. This technique of relaxation may also be applied to the solution of the Navier-Stokes Equation for c i r c u l a r c y l i n d e r s , using either Happel's or Kuwabara's model for the numerical boundary condition far from the c y l i n d e r . Jenson suggested c r i t e r i a to ensure the convergence of the numerical s o l u t i o n , but these were found to lead to excessive computation times [37]. He also introduced the transformation r = ez, for determining appropriate radial spacing of the numerical solution points. Hamielec et al. [37] discussed several of the problems associated with numerical solution of the Navier-Stokes equation for spheres. Their discussion included angular and radial spacing of grid p o i n t s , as well as the e f f e c t of the size of the outer boundary. Subsequently, Kawaguti and Jain [38] and Son and Hanratty [34] solved the unsteady state form of the Navier-Stokes equation for cylinders and extrapolated th e i r 26 values to i n f i n i t e times. Kawaguti and Jain found that t h e i r results approached the steady state solutions of Kawaguti [30] and Thorn [ 6 ] . However, the unsteady state solution predicted wakes larger than expected Takami and Keller [39], Hamielec and Raal [40] and Dennis and Chang [41] numerically solved the steady-state Navier-Stokes Equation for c y l i n d e r s , and t h e i r tabulated results afford easy comparison. Hamielec and Raal used a modified relaxation technique to study the range 1 < Re 5 500 , and obtained reasonable agreement with previous experimental and theoretical r e s u l t s . They used two relaxation f a c t o r s , one for stream function and one for v o r t i c i t y . Pruppacher et al. [42] compiled a useful summary of the best theoretical and experimental data on c i r c u l a r c y l i n d e r s . Masliyah [.43] and Masliyah and Epstein [44] solved the steady state Navi er-Stokes Equation for flow around spheroids and e l l i p t i c a l c y l i n d e r s . They introduced an additional relaxation factor for the v o r t i c i t y at the surface of the s o l i d object. The authors provide a detailed discussion of the problems involved in sel e c t i n g grid spacing and outer boundary radius. Their results for e l l i p t i c a l cylinders with aspect ratios of 0.995 compared well with recent theoretical results for c i r c u l a r cylinders [39-41]. Chapter 3 THEORY Navi er-Stokes Equation for Floy/ Around a Cylinder The flow of an incompressible Newtonian Fluid around a cylinder is described by the Navier-Stokes and Continuity Equations, subject to appropriate boundary conditions. When the flow is steady the equations may be written in dimension-less form as follows: v. • V v = -^Vp + •—- V2_v (3-1) V • v = 0 (3-2) where 2RrpU0 In this work the dimensionless quantities are based on the cylinder radius R and the free stream v e l o c i t y , U 0. 27 28 The pressure p is defined in terms of the dimensional pressure p1, and a reference pressure p1 : P ' - P ' p ^pU 02 (3-3) For two-dimensional flow around a c y l i n d e r , Equations (3-1) and (3-2) represent a system of three simultaneous p a r t i a l d i f f e r e n t i a l equations. These equations may be com-bined by means of a stream function^ ty3 and a v o r t i c i t y 3 c defined as: vr = 1 ii r 36 11 ar (3-4) C = V 2 ty (3-5) By virtue of i t s d e f i n i t i o n , ty automatically s a t i s f i e s the Continuity Equation. The Navier-Stokes Equation thus becomes: 29 Re 2r U . K . M . i s . 8r " 86 " 38 8r V2 (3-6) c = V 2 ^ (3-7) Figure 3-1 i l l u s t r a t e s the cylinder and surrounding f l u i d . Due to the dimensionless representation, the radius of the cylinder is unity. In devising a numerical technique for solving Equations (3-6) and (3-7) i t is necessary to consider the nature of the f l o w f i e l d as a function of p o s i t i o n . Close to the c y l i n d e r , TJI and ? have large radial gradients, while farther away radial changes are more gradual. Hence i t is desirable to have a fine grid close to the c y l i n d e r , and a coarser mesh at large values of r . Such a grid spacing is achieved by using the exponential transformation: r = e (3-8) Substitution of Equation (3-8) into (3-6) and (3-7) y i e l d s : 31 Re a ifj 3z i ? 96 36 IS. 3z 32<; 32? (3-9) 2z 3Z 36 (3-10) The above equations are second order, and four separate boundary conditions are required to define t h e i r s o l u t i o n s . Boundary Conditions The boundary conditions of the Navier-Stokes Equation may be put in two categories: those pertaining to the cylinder surface and axis of symmetry, and those which apply some distance from the cylinder surface. The f i r s t group presents no d i f f i c u l t i e s and w i l l be discussed f i r s t . A l l boundary conditions are written for the case of f l u i d flow perpendicular to the axis of the cylinder (Figure 3-1). On the surface of the cylinder: ty = 0 C = V2 ty - a t r = l or z = 0 (3-11) 32 Along the axis of symmetry (AD in Figure 3-1) * = ? = 0 Along AB 6 = 0 (3-12) ^ = £ = 0 Along CD 6 = ir Kuwabara [36] and Happel [29] have respectively proposed the zero v o r t i c i t y and free-surface models which define £ on the outer boundary. These models assume iden-t i c a l p a r a l l e l cylinders in a randomly spaced assemblage. Each cylinder is associated with a concentric f l u i d "envelope" of radius R^  , and the s o l i d s c o n c e n t r a t i o n , c, is given by: the boundary conditions i t is possible to obtain flo w f i e l d s around cylinders in assemblages of various concentrations. In this work, two main values of R were studied, R^  = 3.0, 100. These correspond, r e s p e c t i v e l y , to s o l i d s con-centration of 0.111 and 10_ l f. It may be noted that at Rm = 100 the system is a good approximation to a cylinder in an i n -f i n i t e l y d i l u t e array, that i s , a single cylinder in an i n f i n i t e medium. (3-13) Hence, by changing the numerical value of R in 33 The values of ty and z; must be defined on the outer c i r c u l a r boundary. Since Kirsch and Fuchs' [45] experimental data agreed better with the predictions of Kuwabara than with those of Happel, the zero v o r t i c i t y model was adopted in this work. However, for comparison purposes some com-putations were also made with Happel's model. The assumption of streaming p a r a l l e l flow on the outer boundary gives: Zoo ty = R sin 6 = e sin e at r = R o r z = Z (3-14) Since Kuwabara postulates zero values of v o r t i c i t y on the outer boundary: C = 0 at r = R o r z = Z (3-15) Numerical Solution of Navier-Stokes Equation For the purpose of numerical s o l u t i o n , the flow-f i e l d was divided into a grid with angular spacing A6 and radial spacing Az. Figure 3-2 represents such a grid in the e-z plane. The number of angular lines is N , and the number of radial lines is N , giving a total of NQ x Nr Figure 3-2. Grid system in the 6-z plane. 35 grid points. The increments A6 and Az are related to N a and N R as fol1ows: A6 = ( N I 1 } (3-16) a A z - (N r ! l) (3-17) Let the l e t t e r s I and J refer to the subscripts of a general grid point ( I , J ) . The point (I,J) therefore l i e s at the interse c t i o n of the I angular lin e and the Jt h radial l i n e . It follows that 1 < I < N and a 1 < J < N^. Using this nomenclature, the 0-z co-ordinates of the point ( I , J) are: 9(1) = (1-1) A8 (3-18) z(I) = ( J - l ) Az (3-19) The values of the stream function and v o r t i c i t y at (I,J) are s i m i l a r l y designated as i^(I,J) and c;(I,J). 36 The p a r t i a l derivatives at (I,J) can be approxi-mated by second order central difference expressions. For example: 3^ 3z (I.J) M l . J + 1) - ^(I,J-1) 2 Az (3-20) 3 2iJ; 3z : (I.J) i^d.j+D - 2^( I , J ) + i f ) ( i , j - i ) hz1 (3-21) Jenson [35] found that fourth order approximations gave results nearly i d e n t i c a l to those obtained with second order formulations. Hence the use of fourth order approxi-mations is not warranted. F i n i t e Difference Formulation of the  Navier-Stokes Equation The Navier-Stokes Equation may be expanded in f i n i t e difference form by substitution of second order central difference expressions into Equations (3-9) and (3-10). These equations y i e l d , on rearrangement: 37 / T _ A6 2« AZ 2 j ? U ' J ) ~ 2(A6 2+AZ 2) ' Az- c( i , J+i) + d i , j - i ) A6 : C(I+1,J) + c ( I - l , J ) Re 8A6AZ Ml + l ,J) - 1^(1-1 ,J) d i , j + i ) - ? ( I , J - D *(I,J+I) - Mi,J-D C ( I + I , J ) - ? ( I - 1 , J ) (3-22) and A6 2 • Az 2 2(A6 Z+AZ 2) " e 2 Z?(I,J) + Az MI,J + 1) + <!»(!,J-D + p r l l - d + l ,J) + ^ ( 1 - 1 ,J) (3-23) 38 Fi n i t e Difference Formulation of Boundary Conditions The boundary condition (of Equation (3-11)): C = V2 ty at r = 1 or z = 0 must also be written in f i n i t e difference form. U t i l i z i n g the boundary condition for ty on the cylinder surface s namely, . = dty _ d2ty _ Q at r = 1 or z = 0 (3-24) Equation (3-11) becomes z = 0 324> dzJ z = 0 (3-25) The stream function near the surface may be ex-panded in a Taylor s e r i e s : z=Az z=0 A z 2 3 2iJ; z=0 2 3z AZ 33_£ z=0 6 3Z + z=0 (3-26) z=2Az 1» + 2AZ M z=0 9 2 z=0 a Z 8Az3 33 z=0 6 3z + z=0 (3-27) 39 Since v = v. = 0 on the surface: r 9 8 z = 0 z = 0 (3-28) Neglecting terms of order greater than 3, Equations (3-25) to (3-28) y i e l d : z = 0 3 z : z = 0 1 - * Az 2Az-(3-29) or in terms of I and J i 2AZ2 8^(1,2) - 3) (3-30) Relaxation Method Equation (3-21) is nonlinear and p a r t i c u l a r l y unstable at Rec > 0.10. Previous authors [40,43] have successfully solved s i m i l a r equations using the i t e r a t i v e Relaxation Method, which was also adopted for this study. If ij> *(I,J) and ? *(I,J) denote the results obtained from Equations (3-21) and (3-22) after the n i t e r a t i o n , the process of relaxation modifies these results as follows: 40' ^ n ( I , J ) = ^ n _ T ( I , J ) + « H*n* ( I' J ) " ^ n - l ( I ' J ) (3-31) ? n ( i , J ) = c n - 1 Cl .0) + a, C n*(I,J) - ?n_-, (I ,J) (3-32) where a. = relaxation factor for stream function = relaxation factor for v o r t i c i t y As soon as the new values ^ (I,J) and c (I,J) are a v a i l a b l e , they replace the old values ^ n -j (I , J) and ? n _ - j ( I , J ) . The method for obtaining the s t a r t i n g values ipo and Co is described in the next s e c t i o n . The relaxation factors determine the amount by which the grid values are modified between successive i t e r a -t i o n s . If = = 1 , Equations (3-31) and (3-32) repre-sent the Gauss-Seidel Method, which involves no r e l a x a t i o n . was generally selected in the range 1.7-1.8, but for small values of Rro i t was necessary to reduce cc^ to 1 . The solution of the v o r t i c i t y Equation (3-21) for the cylinder surface is even less stable than that for the bulk of the f l u i d . For this reason Masliyah [43] introduced 41 an additional relaxation f a c t o r , a , for obtaining vor-t i c i t i e s on the cylinder surface. The values of and a suggested by Masliyah were used as a basis for this work*(see Figure 3-3). I n i t i a l Values for Relaxation Method The solution procedure for Equations (3-21) and (3-22) requires a set of i n i t i a l values, ^ 0 and r,0. If i n e r t i a l terms are omitted, the Navier-Stokes Equation for a cylinder may be written: V*if» = 0 (3-33) Both Kuwabara [36] and Happel [29] have presented p a r t i c u l a r solutions to Equation (3-33). Kuwabara's solution which s a t i s f i e s the boundary condition r, = 0 at r = R , i s: sin e A i r 2 + A 2r + A3 r l n r + r (3-34) where * The relaxation factors used in this work are l i s t e d in Appendix IV. 42 43 4A 5R 2 (1 - Rep"2) 2 A 5 " 3 n A5 (1 - %R0Q-2) 2 A 5 As - A6 + R " 2 - %R. ~ h CO CO A5 = In R Happel's p a r t i c u l a r solution to Equation ( 3 - 3 3 ) n 9 I Hi r 3 + 1 H2 l l n r - 2 + Hs r + — r 44 The c o e f f i c i e n t s Hx to H4 can be evaluated from the boundary condi t i ons f or ii; stated previously (Equations (3-11), (3-12) and (3-14)), along with Kuwabara's zero v o r t i c i t y condition (Equation (3-15)). Substitution of these equations into Equation (3-35) y i e l d s the following simul-taneous equations: (3-36) (3-37) i H i R3, oo H oo (3-38) CO H„ = - HiR2 CO (3-39) Equations (3-36) to (3-39) were solved for the c o e f f i c i e n t s Hi to h\: 45 H i = 8 Up Rco2 H, = 8 Up Rp 2 UpRco2 u _ UpRcoMl/R^2 - 1)  n* G G = 4 R " InJ- - R * + CO |^ CO 1 - 2 R U t i l i z i n g the fact that c = V 2 ^ , Equation (3-35) gives the following expression for v o r t i c i t y : ? = G 1 8 sin 0 - 1 (3-40) In this work, Equations (3-35) and (3-40) were used to estimate ty0 and ? 0 at Re = 0.5. Once thi s grid was 46 s u f f i c i e n t l y converged, the new values of ty and z, in the grid were used as s t a r t i n g conditions for a higher value of Re c. S i m i l a r l y , when the values in the second grid had been it e r a t e d to convergence, they were used as s t a r t i n g values for larger Reynolds numbers. This process was con-tinued u n t i l a converged grid had been obtained at Rec = 40.0. Convergence C r i t e r i a As mentioned in the previous section the numerical procedure starts from an i n i t i a l guess of ty and cj, and con-verges to a f i n a l solution after a certain number of i t e r a t i o n s . It is very d i f f i c u l t to estimate, a p r i o r i , the number of i t e r a t i o n s which are needed to converge a given grid of values. Consequently, the computations must be continuously monitored and terminated once certain con-vergence c r i t e r i a are s a t i s f i e d . In this work, the most relevant c r i t e r i a are the p a r t i c l e impaction e f f i c i e n c i e s . For a given g r i d , repre-sentative e f f i c i e n c i e s were calculated every 100 i t e r a t i o n s . When these e f f i c i e n c i e s varied by less than a s p e c i f i e d amount over a given "cycle" of 100 i t e r a t i o n s , the c a l c u l a -tions were terminated. E f f i c i e n c i e s calculated for grids having R = 100 were constrained to show no variation in 47 the 5t h s i g n i f i c a n t d i g i t , and at Rm = 3, constancy in the t tl 4 s i g n i f i c a n t d i g i t was required. Numerous other convergence c r i t e r i a may be sug-gested. Before presenting them i t is convenient to define some additional v a r i a b l e s . The changes in stream function and v o r t i c i t y values per i t e r a t i o n may be denoted by: A n * (I,J) = * n * ( I , J ) - V-|(I,J) (3-41) A n c (I,J) = C n * ( I , J ) - C ^ d . j ) (3-42) A n C b ( I , J ) = c n * d , J ) - ?n_-,(i,o) (3-43) A n C s ( I , l ) = C n * ( I , D - ?n_-,(i,D (3-44) The convergence c r i t e r i a which were monitored in this study are the following: 48 Maximum changes in stream function and v o r t i c i t y : max 1 < I < N (3-45) 1 < J < N, Mn max 1 5 I < N. An C b(I,J) (3-46) 2 < J < N. n s max 1 < I < N. (An 5 s ( I . l / (3-47) These c r i t e r i a are s i m i l a r to those of Hamielec and Raal [ 4 0 ] , who considered t h e i r grids to be converged when: M„a ty < 10"* n Mna £ < IO"1* 49 where max 1 < I < N A / ty(l,J) (3-48) 1 < J < N and a * n(i,J) - V i ( I , J ) (3-49) That i s , these authors compared changes in the relaxed values of ty aid £. Hence, for values of a < 1.0, t h e i r c r i t e r i a are less stringent than those employed in this work. Sum of a l l stream function and v o r t i c i t y  values: S_ ty = I I ,(I,J) (3-50) 1=1 J = l N a N r Sn h = I I ? n - l ( I ' J ) ( 3 " 5 1 ) 1=1 J=2 Na 1 = 1 n-1 (1,1) Sum of a l l changes in stream function and  v o r t i c i t y values per i t e r a t i o n : N N, I I A *(I,J) 1=1 J=l " N A N R Dn % 1 I A C(I,J) 1=1 J=2 " Fractional changes in the sum of stream  function and v o r t i c i t y changes per i t e r a t i o n Q * = " i 51 n b Qn Cs - (3-58) n s Sum of f r a c t i o n a l changes over 100 i t e r a t i o n s N+9 9 Q n1 0 0 * = I Qn * ( 3 " 5 9 ) n=N N+9 9 n=N N+9 9 Q 1 0 0 ? = J Q e (3-61) yn ^s, yn ss v ' n=N Several authors [31,39-41] have reported theoretical and experimental values for stagnation pressures and drag 51 c o e f f i c i e n t s for flow around c y l i n d e r s . Their results can * be compared with values calculated in this study as further tests of convergence. In a d d i t i o n , the pressures and drag c o e f f i c i e n t s calculated during relaxation tend to s t a b i l i z e near convergence. Frontal stagnation pressure: Po = 1 + Re 86 dz 6 = 0 ( 3 - 6 2 ) Rear stagnation pressure: rTT TT Po + oZ de ( 3 - 6 3 ) Skin drag c o e f f i c i e n t : The equations for the stagnation pressures and drag c o e f f i c i e n t s are derived in Appendix I I . 52 •TT Jo re Rc sin 6 de 'DS R c(^ pU 02) rTT Re C S sin 6 de (3-54) Form drag c o e f f i c i e n t : 'DF pQ' cos 9 Rc de R c(^ P U O 2 ) Po + Re 3£s 3Z de cos 6 de (3-65) 53 Ratio of drag c o e f f i c i e n t s : It was useful to combine the drag c o e f f i c i e n t s in the form of a r a t i o D^S DRAG RATIO = f— (3-66) DF Integration of Equations (3-62) to (3-65) was performed by a Simpson's three-point method. Calculation of P a r t i c l e Trajectories Equation of P a r t i c l e Motion The motion of a spherical p a r t i c l e subjected to a drag force £ p , is governed by Newton's second law of motion which may be written as: dy_' MP = ID (3"67> or dy.D' mp = CD rr Rp2 \ p | v ' - V p ' |(v'-v p') (3-68) 54 The drag c o e f f i c i e n t , Cp, is defined for a spherical p a r t i c l e as follows: (3-69) 77 V \ P Equation (3-68) takes the following dimension-less form: whe re: dv Re _zP_ - P_ dt 24 (3-70) m. P = U, 6 TT y R R, 2 (Pp'p) RP2 U° 9 y R_ R e . 2 p R „ |v'-vpM P y The dimensionless group P is variously referred to as the p a r t i c l e i n e r t i a l parameter3 i n e r t i a l parameter, 55 or simply, the "Stokes Number." When defined in terms of diameters instead of r a d i i , as is sometimes the case, P d i f f e r s by a factor of \ from the d e f i n i t i o n used in this work. P r e f l e c t s the magnitude of the i n e r t i a l and viscous forces present, and is p a r t i c u l a r l y s i g n i f i c a n t in the Stokes' regime (where CD = 24/Re p). In this case i t is the sole dimensionless group characterizing the p a r t i c l e t r a j e c t o r i e s . P can also be written as: P = PjTP Re„ K2 c (3-71) to i l l u s t r a t e i t s dependence on the density r a t i o , (p p-p)/p , and on the size parameter, K. Drag Coefficients For small values of the p a r t i c l e Reynolds number, Re << 1 , Stokes' Law for spheres may be applied: (3-72) 56 At larger values of Rep the drag c o e f f i c i e n t may be expressed by any of a number of empirical and theoretical formulae. A simple expression which gives good agreement with experimental data in the range 0 < Rep < 300 was suggested by Klyachko [46] 24 Re_ RepV, (3-73) This equation is continuous in i t s range of a p p l i c a b i l i t y and deviates only marginally from Stokes' Law at low Re p. Integration of P a r t i c l e Equations In order to obtain the p a r t i c l e t r a j e c t o r i e s , Equation (3-70) must be integrated twice, subject to the i n i t i a l conditions: t = 0 v - 1 v = 0 py (3-74) 57 i . e . the p a r t i c l e s are s t a r t e d at the outer boundary of the f l u i d f l o w f i e l d with the f r e e stream v e l o c i t y . The i n t e g r a t i o n i s performed i n a stepwise manner by d i v i d i n g the p a r t i c l e motion i n t o i n t e r v a l s of d u r a t i o n At. Equation (3-70) i s i n t e g r a t e d a n a l y t i c a l l y over each i n t e r v a l by a s s i g n i n g a p p r o p r i a t e constant values to the f l u i d v e l o c i t y y_, and to C*D Re p/24 P. If the s u b s c r i p t "0" denotes the c o n d i t i o n s at the beginning of an i n t e r v a l , i n t e g r a t i o n of Equation (3-70) over At y i e l d s : v = v px x v - v I x p x 0 J •BAt (3-75) v = v py y V - V l y pyoj l e - B A t (3-76) x p = x P o + V x A t + B r " V - v ' e - 6 A t - l ' , x px 0. (3-77) y = y + v A t + — J P Po y B v - v y py 0 e " * A t - l (3-78) 58 where 3 = (3-79) The values of the f l u i d v e l o c i t y , y_, at the s t a r t and end of the previous time interval were l i n e a r l y extrapolated by At/2 into the next i n t e r v a l . This gave v, the constant value of f l u i d v elocity used in the an a l y t i c a l integration of the trajectory segment (see Appendix I ) . The summation of these trajectory segments produced the f u l l trajectory of the p a r t i c l e . performed so as to r e f l e c t the varying nature of the flow-f i e l d . When a p a r t i c l e entered a grid c e l l of radial dimension Ar (Figure I-l) with a v e l o c i t y |v_pl> At was determined from: Selection of the variable time step At was At = Ar (3-80) 3 * IVpl Hence, up to three trajectory segments were c a l -culated for each grid c e l l , depending on the d i r e c t i o n of the p a r t i c l e motion. Equation (3-80) has the advantageous 59 property that i t produces small time steps near the cylinder surface, where velocity gradients are steepest and high accuracy is required. Test calculations were performed using five inte-gration steps per grid c e l l , to detect any changes in accuracy due to the use of shorter trajectory intervals. Fibonacci Search for Cr i t i c a l Trajectory As mentioned in the Introduction, the efficiency, E, for a particular set of conditions was determined from y n + » the i n i t i a l c r i t i c a l position of the particle above the centre line. It was possible to estimate an accurate value of E with a minimum number of trajectory calculations by using the Fibonacci sequential search scheme [47]. Optimum search techniques like the Fibonacci method locate maxima, minima or discontinuities in uni-model functions with considerable precision. For the purposes of efficiency calculations i t was necessary to describe the process of "hitting" or "missing" the cylinder in terms of a simple function. Arbitrarily assigning a functional value of 1 to every y p giving a miss, and a value of 0 to every y giving a hit, the 60 c o l l i s i o n process may be described by a step function (Fi gure 3-4) . Figure 1-3 i 11 ustrates three p a r t i c l e t r a j e c t o r i e s , each s t a r t i n g at a d i f f e r e n t value of y . These same points are plotted in Figure 3-4. Because the search pro-cedure requires an independent variable scaled from 0. to 1., values of y p were scaled with d i v i s i o n by 1 + K. The transformed value of y p for the Nt h i t e r a t i o n , y p N » is denoted A^, where X^ = yr)^/^ + ^- Consequently, the c r i t i c a l value A„ . + is i d e n t i c a l to the desired value of the c r i t e f f i c i e n c y , E. The accuracy of the estimate of E was expressed in terms of L^, the "interval of uncertainty" remaining after N separate trajectory c a l c u l a t i o n s . L^ is defined such that the estimated value of E d i f f e r s from the true value of the e f f i c i e n c y by less than L^. If 6 represents the smallest separation of two values of y giving two d i s t i n c t t r a j e c t o r i e s , is given by the following formula: L N = -F- + • 6 (3-81) t- Vi Where F w represents the N Fibonacci number and I CO CO cc O X Ol 0 Xj X2 J L X3 X Figure 3-4. C o l l i s i o n function for Fibonacci search 62 Fo = Fr = 1 FN FN-1 + FN-2 N = 2,3,-«-N (3-82) Assuming that 6 = 0 for the numerical c a l c u l a -tions involved, Equation (3-81) becomes: LN = F N (3-83) Since the e f f i c i e n c i e s were expected to be lowest at low values of P, the highest accuracy was required in that range. Consequently, for P < 1 , N was set equal to 31, while for P > 1, N = 21 . The corresponding inter v a l s of uncertainty, or the errors in the e f f i c i e n c i e s were: 1 1 • 2 1 3 1 2 1 3 1 17711 = 5.64 x 10 - s 1 2178309 = 4.59 x 10' (3-84) These estimates of error do not include contribu-tions re s u l t i n g from inaccuracies on the f l o w f i e l d s , or cumulative machine e r r o r . Chapter 4 RESULTS AND DISCUSSION The main results of this work are presented and discussed in the present chapter. For the sake of conven-ience the chapter i s divided into three major sec t i o n s . The f i r s t section contains a l l findings p e r t a i n -ing to p a r t i c l e behaviour. In p a r t i c u l a r , i t comprises the important results on p a r t i c l e impaction e f f i c i e n c i e s as functions of: Reynolds number, Re , p a r t i c l e i n e r t i a l parameter, P, and p a r t i c l e size parameter, K. The results are compared with those of other workers whenever po s s i b l e . Fluid f l o w f i e l d s , as predicted by the numerical solution of the Navier-Stokes Equation, Davies' Bessel function expression, and by Thorn are presented in the second s e c t i o n . The differences evident between these fl o w f i e l d s help to explain the discrepancies between p a r t i c l e e f f i c i e n c i e s calculated in this work and those found by other authors. 63 64 The th i r d and f i n a l section is devoted to the convergence c r i t e r i a which were monitored during the r e -laxation solution of the Navier-Stokes Equation. The c r i t e r i a are compared with one another, and the c r i t e r i o n best suited for determining the convergence of the numerical solution is ind i c a t e d . Typical P a r t i c l e Trajectories Trajectories obtained by means of Equations (3-75) to (3-78) are depicted in Figures 4-1 and 4-2 , for * Rec = 0.2 and Rec = 40, re s p e c t i v e l y . In both cases the pa r t i c l e s started at the same points in the f l o w f i e l d and with the local f l u i d v e l o c i t y . At the smaller Reynolds number the p a r t i c l e s possessed less momentum than at Rec = 40, and consequently diverged less from the stream l i n e s . When wakes were present behind the c y l i n d e r , p a r t i c l e s were never observed to enter them. This observation is only v a l i d provided p a r t i c l e transport occurs s o l e l y due to an i n e r t i a l mechanism. In the presence of other f o r c e s , such as gravity and e l e c t r o s t a t i c a t t r a c t i o n , p a r t i c l e s can conceivably enter the wake region. * A l l figures and tables are located at the end of the chapter, s t a r t i n g on pages 93 and 153, re s p e c t i v e l y . 65 The curves in Figure 4-3 show how the p a r t i c l e Reynolds number, Re , varies with position for the case of Rec = 40. Clearly Stokes Law is not suitable for ca l c u -l a t i n g the drag c o e f f i c i e n t for the p a r t i c l e s in question. This fact demonstrates the necessity of using Equation (3-73) to calculate CQ for the p a r t i c l e s , since i t gives v a l i d predictions both in Stokes regime, as well as at higher values of Re . It is also i n t e r e s t i n g to note the twin P maxima which occur for trajec t o r y 2. The values of Rep for the t r a j e c t o r i e s shown in Figure 4-1 for Rec = 0.2 have not been p l o t t e d , but in this case they were very small (Rep << 0.1) and well within the range of Stokes Law. Since Cg must be calculated in both Stokes regime and at higher p a r t i c l e Reynolds numbers, Equation (3-73) is quite suitable since i t gives continuous predictions for the desired range. At no time during the trajec t o r y calculations did Rep exceed 30, so that the upper l i m i t of a p p l i c a b i l i t y of Equation (3-73) was never reached. PARTICLE IMPACTION EFFICIENCIES The results are summarized in Tables 4-1 to 4-2. Figures 4-4 to 4-15 show the variation of e f f i c i e n c y with 66 p a r t i c l e i n e r t i a l parameter for most of the numerical flow-f i e l d s . Graphs have not been prepared for a l l values of Rec since most of the curves are f a i r l y s i m i l a r . Before proceeding with the discussion of the results i t i s expedient to comment on the size parameter K. One basic assumption inherent in the determina-tion of p a r t i c l e t r a j e c t o r i e s is that the presence of the p a r t i c l e does not influence the f l u i d f l o w f i e l d i t s e l f . It i s very d i f f i c u l t , however, to estimate the value of K for which this assumption becomes i n v a l i d and when s i g n i f i -can error i s thereby introduced. Because the predicted e f f i c i e n c e s showed uniform increase with K, curves have been presented to a maximum p a r t i c l e size of K = 1.0. Although the p o s s i b i l i t y exists that these p a r t i c l e s are too large to be handled properly by the present solution procedure, they have been included to i l l u s t r a t e the e f f e c t of p a r t i c l e size on impaction e f f i c i e n c y . The problems that would accompany an attempt to solve both the f l u i d and p a r t i c l e motion equations simultaneously are beyond the scope of this work. Apart from the small uncertainty regarding e f f i c i -ency prediction at K = 1.0, those values calculated at K < 0.5 are f e l t to be quite accurate. 67 The curves of e f f i c i e n c y given for K = 0.001 deserve special mention, since they are actually curves of E j , the impaction e f f i c i e n c y due to i n e r t i a l effects alone. They were obtained by s l i g h t modifications of the t r a j e c -tory c a l c u l a t i o n s : K was considered to have the f i n i t e value 0.001 for the purpose of ca l c u l a t i n g Cp, but c o l l i s i o n with the cylinder was taken to occur only when the centre of the small p a r t i c l e coincided with the surface of the cy l i n d e r . Alternately expressed, the interceptive effect was e n t i r e l y neglected for p a r t i c l e s having K = 0.001. Effect of I n e r t i a l Parameter on E R = 100. co The following general observations can be made on the appearance of the curves at ROT = 100: a) A l l of the curves are s i m i l a r and have an S-type shape. b) For large values of P the impaction e f f i c i e n c e s tend to unity as expected from physical considerations. c) For small P, the values of E approach E^ .. d) There i s an i n t e r v a l of the i n e r t i a l parameter, generally one order of magnitude wide, i n which the e f f i c i -encies r i s e r a p i d l y from approximately E„ to unity. The curves of e f f i c i e n c y versus P for Rm = 3 have a very s i m i l a r shape to those obtained at = 100. However, for a given Reynolds number the e f f i c i e n c e s are considerably higher than those calculated from flow f i e l d s having R^  = 10.0. Two further differences are evident: a) The effect of Reynolds number on E i s much less pronounced at = 3 than at R^ = 100. This i s not unexpected when i t i s r e a l i z e d that the flow-f i e l d s at RTO = 3 are not strongly dependent on the Reynolds number, Re . c 69 b) The i n t e r c e p t i v e e f f i c i e n c e s , E^, are s i g n i f i c a n t l y higher at R M = 3 than at R = 100. Once again t h i s i s a consequence of the more c l o s e l y spaced stream l i n e s at R = 3-Comparison v/ith Davies Rec = 0.2. Table 4-3 contains the efficiencies calculated for Re = 0.2 on the basis of the numerical flowfield ob-c tained at R^  = 200. Figure 4-16 compares these efficiences with those presented graphically by Davies and Peetz [7 ]. It should be noted that K = 0.001 corresponds to their condition of an i n f i n i t e l y small particle. The agreement is generally quite good, although Davies and Peetz' predictions are somewhat higher, especially in the range 1 < P < 40. This discrepancy is attributable to the inaccurate expression used by these authors in dis-cribing their flowfield in the vicinity of the cylinder (cf. Equation (2-2 ) ) . A more detailed examination of their flowfield is made in the second section of this chapter. 70 Rec = 10. Figure 4-17 provides a comparison of the e f f i c i -encies calculated in this work at Re„ = 10 and R = 100 c 00 with those obtained by Davies and Peetz using Thorn's data. Their results are considerably higher, except for the case of K = 0.001, and Thorn's approximate repre-sentation of the f l o w f i e l d accounts in part for these di f f e r e n c e s . A further reason which explains t h e i r higher e f f i c i e n c e s i s the fact that these workers started t h e i r p a r t i c l e s close to the c y l i n d e r , i . e . , at x p = -5.0, in contrast with x p * -100 in this study. Pa r t i c l e s whose t r a j e c t o r i e s begin close to the cylinder experience the upward f l u i d forces for a much shorter time than those started far away from the c y l i n d e r . Consequently, even with i d e n t i c a l flowfields the e f f i c i e n c y calculated for a p a r t i c l e s t a r t i n g at x p = -5.0 would be higher than for one s t a r t i n g at x p ~ -100. This e f f e c t decreases as P increases since heavy p a r t i c l e s undergo less d e f l e c t i o n . This argument is supported by the fact that the e f f i c i e n c y curves do approach each other as P increases ( c f . Figure 4-17). Thorn's f l o w f i e l d is described in more det a i l in the second section of this chapter. 71 Experimental Impaction E f f i c i e n c i e s Subramanyam and Kuloor [12] reported experimental values of the impaction c o e f f i c i e n t , e, for 4 < Rec £ 240. These results have been replotted as e f f i c i e n c e s , E, in Figure 4-18 . Since the size parameter of th e i r p a r t i c l e s ranged from K = 0 to K = 0.1, the e f f i c i e n c i e s were c a l -culated on the basis of both these values. As a q u a l i t a t i v e i n d i c a t i o n of how t h e i r data compare with the predictions in this work, the curve of E vs. P calculated at K = 0.1, Re„ = 40 and R = 100 has ' c °° been given as a s o l i d l i n e in Figure 4-18 . The reason that the impaction e f f i c i e n c e s for Rec = 40 were shown is that they correspond to the highest Reynolds number for which a numerical grid was c a l c u l a t e d . It is seen that the curves are of very s i m i l a r shape. The good agreement suggests that the experimental f l o w f i e l d may have been close to that obtained by solution of the Navier-Stokes Equation at Rec = 40. The above observations therefore give q u a l i t a t i v e support to the predictive technique used in this study. Potential E f f i c i e n c i e s The computer programme used to calculate the e f f i c i e n c i e s for the numerical flowfields was modified to 72 duplicate the assumptions used by Davies and Peetz [ 7 ] , Householder and Goldschmidt [13], and Subramanyam and Kuloor [12]. E f f i c i e n c e s were then calculated for potential flow on the basis of Equation (2-6 ) . Since the authors mentioned above used the same equations in t h e i r predic-tions of impaction e f f i c i e n c i e s , comparison of t h e i r results with the "duplicates" calculated in this work gave an i n -dication of the programme accuracy. These comparisons are shown in Table 4-4 . The values ascribed to Davies and Peetz and Householder and Goldschmidt were read from curves presented by these authors. A dash ("-") has been inserted where the e f f i c i e n c e s were not a v a i l a b l e . The l a s t column in the table contains e f f i c i e n c i e s calculated in the normal manner for this study, i . e . , Xp = -100 and Klyachko's formula. The f u l l set of potential e f f i c i e n c i e s is contained in Table 4- 5 , and plotted in Figure 4-19. It is clear that the agreement is f a i r l y good in a l l cases, with Davies and Peetz' results showing the largest d i f f e r e n c e s . These authors did not have the benefit of modern computers to aid th e i r c a l c u l a t i o n s , and may have used excessive step sizes in t h e i r work. The predicted 73 e f f i c i e n c i e s in the other two cases d i f f e r by less than 2%, the numbers being v i r t u a l l y i d e n t i c a l for the predic-tions of Householder and Goldschmidt. Therefore, i t is f e l t that this test s u b s t a n t i a l l y confirms the v a l i d i t y of the solution techniques which were employed in this work for ca l c u l a t i n g the p a r t i c l e t r a j e c -t o r i e s and impaction e f f i c i e n c i e s . The e f f e c t of i n i t i a l s t a r t i n g p o s i t i o n , x p, i s also demonstrated in Table 4-4 . For p a r t i c l e s having the same P and K, the e f f i c i e n c y increases as the p a r t i c l e s are started closer to the c y l i n d e r . This observation has been made in connection with Davies1 e f f i c i e n c i e s at Rec =10. C r i t i c a l P a r t i c l e I n e r t i a l Parameter As mentioned previously, the e f f i c i e n c i e s at K = 0.001 are actually the e f f i c i e n c e s due to interception alone, E^. In order to estimate the c r i t i c a l p a r t i c l e i n e r t i a ! parameters, P , the curves of E vs. P at K = 0.001 were extrapolated to E = 0, in the range 0.2 < Rec < 40. The r e s u l t s , shown in Figure 4-20, can only be regarded as approximate, due to the uncertainties involved in the extrapolation. 74 Davies and Peetz' predictions of P for Re = 0.2 and Rec = 10 were also included for comparison purposes. Their c r i t i c a l values are seen to be s i g n i f i c a n t l y lower than those predicted in this work. Effect of Reynolds Number on E Figures 4-21, to 4-26 show how the impaction e f f i c i e n c i e s change with increasing Reynolds number, Re The e f f e c t of Reynolds number was studied for both R = 100 and R = 3. Figures 4-21 to 4-23 indicate the relatio n s h i p between e f f i c i e n c y and Reynolds number for 0.2 < Rec < 40. The increase of E with Rec i s f a i r l y gradual, being most pronounced at lower values of P. Although the curves are plotted to a maximum Reynolds number of Rec = 40, the e f f i c i e n c i e s ultimately approach the values obtained under potential flow conditions(Figure 4-19 and Tables 4-4 and 4-5). The intermediate values were, however, not found because the f l o w f i e l d becomes unsteady for Rec > 40 and consequently Equation (3-1) f a i I s . 75 The e f f e c t of Rec on e f f i c i e n c y at Rm = 3 i s i l l u s t r a t e d by Figures 4-24 to 4-26. The increase of e f f i c i e n c y with Reynolds number for any given values of P is very small. Furthermore the e f f i c i e n c i e s at R^ = 3 are higher than at R^  = 100 for a given P and K. Comparison with Householder and Goldschmidt Figures 4-27 and 4-28 compare e f f i c i e n c i e s given by Householder and Goldschmidt's expression (Equation (2-5)) with those calculated in this work. In general their e f f i c i e n c y curves are s i g n i f i -cantly lower than those of this work, except in the case of P = l.and K = 0.1, where t h e i r curve is higher. Since the agreement is shown to be poor for both K = 0.1 and K = 1.0 ( c f . Figures 4-27 and 4-28) i t i s clear that Equation (2-5) i s unsuitable for e f f i c i e n c y p r e d i c t i o n . Contour Plots of E f f i c i e n c y The dependence of impaction e f f i c i e n c y on Reynolds number and p a r t i c l e i n e r t i a l parameter may be compactly presented by contour diagrams such as those in Figure 4-29 76 to 4-34. The lines of constant e f f i c i e n c y , E, are plotted with Reynolds number, Rec> as the abscissa, and /p as the ordinate. These curves are redrawn from o r i g i n a l contour plots obtained from the UBC "Calcomp" ink p l o t t e r . The UBC contouring subroutine was used to contour both these e f f i c i e n c y diagrams, as well as the stream lines and equi-v o r t i c i t y lines shown in the next s e c t i o n . Each figure is drawn for a p a r t i c u l a r value of K and outer boundary radius. The contour diagrams for K = 0.10 and R = 100 and R = 3 are not given because they CO CO J J were almost i d e n t i c a l to the ones shown at K = 0.001. This s i m i l a r i t y is to be expected from the close spacing of the e f f i c i e n c y curves at lower values of K in the E versus P p i o t s . The d i s t i n c t r i s e of the E contours at low values of Rec is due to the fact that the impaction e f f i c i e n c i e s increase more rapidly with Reynolds number at lower values of Re c. The dependence of E on K is seen to be greatest for the lower contour l i n e s , which move closer to the horizontal axis as K increases. This too.was evident from the plots of E vs. Re„ at d i f f e r e n t values of K. 77 Effect of Solids Concentration In order to determine the eff e c t of so l i d s f r a c -tion on impaction e f f i c i e n c y , additional f l o w f i e l d s were calculated at Re = 0.2 and Re = 40. The outer boundary r a d i i for these grids were: R = 50, R = 25 and R = 10. CO CO oo Together with the basic grids at R^  = 100 and R^  = 3, flowfi e l d s for the following so l i d s fractions were there-fore a v a i l a b l e : c = 10"\ 4 x IO"3, 1.6 x IO"3, 10"2 and 0.111. When c = 10"1* the conditions correspond very c l o s e l y to a cylinder in a f l u i d of i n f i n i t e extent. It i s seen from Figures 4-35 to 4-40* which show E vs. /c , that E increases with increasing so l i d s concen-t r a t i o n , but that the rate of increase is not p a r t i c u l a r l y r a p i d . The greatest rate of increase is apparent at lower values of P, as was previously found for the curves of E versus Re £. As c + 0 the e f f i c i e n c i e s approach constant values corresponding to an i n f i n i t e l y d i l u t e array. At the other extreme, c increases to a maximum value imposed by the physical arrangement of the c y l i n d e r s . The maximum soli d s concentration for cylinders in a triangular array may be shown to be c = Tr/3/6 * 0.907. Kuwabara's and Happel's models i m p l i c i t l y assume that there exists a f i n i t e distance between the surfaces of See also Tables 4-6 and 4-7. 78 the randomly spaced cylinders in the array. This assumption is not violated for real arrays of c y l i n d e r s , so long as they are not in contact with one another. Si evi ng As the sol i d s concentration is increased the average distance between the cylinders i s reduced. Hence there i s a concentration, denoted by c m , for which the distance exactly equals the p a r t i c l e diameter. Under these conditions the p a r t i c l e s are retained by sieving and the c o l l e c t i o n e f f i c i e n c y is unity. Neither the i n e r t i a l im-paction nor the interception effects are s i g n i f i c a n t when sieving occurs. Raising the sol i d s concentration above c m does not re s u l t in any further increases of the e f f i c i e n c y , The sieving mechanism is i l l u s t r a t e d in Figure 4-41 and i t i s easy to show that The concentration, c m , is indicated in Figures 4-35 to 4-40 by an a s t e r i s k . As a f i r s t approximation the 79 e f f i c i e n c y curves may be extrapolated to the asterisks i f i t i s desired to obtain values for c > 0.111. In e r t i a l and Interceptive Mechanisms In the Introduction, p a r t i c l e impaction was d i s -cussed in reference to the i n e r t i a l and interceptive mechanisms. Although these effects generally occur simul-taneously, i t was pointed out that they could be considered i ndependently. The e f f i c i e n c i e s due to i n t e r c e p t i o n , E^, were calculated by modifying Equations (3-75) to (3-78) so that they calculated the paths of f l u i d streamlines. These streamlines represent the t r a j e c t o r i e s of f i n i t e - s i z e d p a r t i c l e s having P = 0, which impact on the cylinder by virtue of t h e i r size alone. Table 4-8 contains E^ results for both R = 100 and R = 3. CO CO The a d d i t i v i t y of the i n e r t i a l and interceptive effects i s tested in Figures 4-42 and 4-43, where e f f i c i -encies are plotted versus Reynolds number for the following cases: i n e r t i a l impaction e f f i c i e n c y , E j j interceptive e f f i c i e n c y , E^; the arithmetic sum, Ej + E^, and the im-paction e f f i c i e n c y for the combined effects of i n e r t i a l impaction and i n t e r c e p t i o n , E. 80 It is clear that the effects are not l i n e a r l y a d d i t i v e , since the dashed lines representing the sums Ej + E K l i e above the curves for E. Both Ej and E^ increase with increasing Re , although this behaviour is more evident at R = 100 than at R = 3 . For both R = 100 and R = 3 C O C O C O C O the e f f i c i e n c y due to i n e r t i a l e f f e c t s , E j , is seen to exceed the e f f i c i e n c y due to i n t e r c e p t i o n , E^. The cal c u l a t i o n of E^ for the case of K = 0.001 afforded another opportunity to determine the accuracy of the computer programme used in this work. It is clear that for a p a r t i c l e having P = 0 and experiencing no interceptive e f f e c t the predicted e f f i c i e n c y should be zero. The largest value of E K under these conditions i s 0.0014 and occurs at Re„ = 40 and R = 100 ( c f . Table 4-8 C 0 0 This amounts to an error of 0.0014. In view of the good agreement with the known theoretical l i m i t of E^ = 0.0 for K = 0.001, the v a l i d i t y of the programme is considered to have been demonstrated. Comparison with Wong et at. Figure 4-44 compares values of E^ predicted by Equation (2-4 ) , as given by Wong, Ranz and Johnstone [10], with the interceptive e f f i c i e n c i e s calculated in this work. 81 The agreement i s f a i r l y good up to Rec z 1.0, at which point the predictions of Equation (2-4 ) increase r a p i d l y . The f a i l u r e of Equation (2-4) at higher values of Rec > 1 is expected, since i t is based on Equation (2-3) which is v a l i d only for creeping flow. Selection of Numerical Grid Angular and Radial Divisions In order to determine the optimum grid spacing for e f f i c i e n c y c a l c u l a t i o n s , the Navier-Stokes Equation was solved for two additional grids at Re£ = 40, one being f i n e r than the "standard" grid (N = 49, N = 93), and the other, a r coarser (N3 = 33, N^ =79). This comparison was performed a r at Rec = 40, since f i n i t e difference representation of the f l u i d f l o w f i e l d was most d i f f i c u l t at this maximum Reynolds number. The results of this comparison are shown in Table 4-9 , from which i t is seen that the test grids yielded s l i g h t l y higher values of E than the standard g r i d . Since these differences were quite small, the spacing afforded by the standard grid (N, = 33, N^ = 93) was f e l t to be adequate a r for this work. Hence, for R = 100, Az z 0.05 and CO AG ~ 0.0982 rad. or 5 . 6 ° . 82 Steps Per Cell As described in the Theory Chapter, the p a r t i c l e t r a j e c t o r i e s were integrated on the basis of three steps per grid " c e l l . " To ascertain the e f f e c t of decreasing the step s i z e , the e f f i c i e n c i e s at Rec = 40 were r e c a l c u -lated using f i v e steps per grid c e l l . It is clear from Table 4-9 that the results for three steps and fi v e steps per c e l l are v i r t u a l l y i d e n t i c a l . On the basis of this f i n d i n g , a l l the e f f i c i e n c y c a l c u l a -tions were performed with three steps per c e l l . Happe1 ' s Model Table 4-9 also contains the e f f i c i e n c i e s c a l c u -lated from a f l o w f i e l d which was obtained by solving the Navier-Stokes Equation subject to Happel's boundary condition. The Reynolds number and outer boundary radius were 40 and 100, r e s p e c t i v e l y . Happel's model is seen to predict somewhat higher e f f i c i e n c i e s than given by Kuwabara's zero v o r t i c i t y condi-t i o n . Kuwabara's model therefore provides more conservative estimates of impaction e f f i c i e n c i e s . Since the l a t t e r model also predicts drag c o e f f i c i e n t s for cylinders which are in better agreement with experimental data (Kirsch and Fuchs 83 [45]) than Happel's model, Kuwabara's zero v o r t i c i t y boundary condition was used throughout this work. FLOWFIELDS Figures 4-45 to 4-47 show representative stream-lines and equi-vorticity lines for some flowfields used for e f f i c i e n c y c a l c u l a t i o n s . The symmetry of the stream func-tion and v o r t i c i t y values is apparent at low values of Re c. As the Reynolds number increases, the f l o w f i e l d s become progressively more asymmetric, and a wake appears behind the cylinder between Re =5 and Re = 10. With a c c further increase in Rec the wake grows and the angle of separation is seen to advance. R = 3. CO Due to the d r a s t i c a l l y decreased radius of the outer boundary, the streamlines and e q u i - v o r t i c i t y lines l i e much closer to the cylinder ( c f . Figure 4-48 to 4-50). Furthermore, at high Reynolds numbers the streamlines are 84 more symmetric for = 3 than for Rm = 100. It may also be noted that the wakes are considerably smaller, and appear only when Rec > 10. The values of v o r t i c i t y are much higher at RM = 3 than at R^  = 100 numbers. This is expected since the gradient of stream function is much steeper for R^  = 3 than for R = 100. C O Effect of R . i •.—I. . i • • . . . CO Figures 4-51 to 4-54 show how the shapes of the flowfi e l d s at Rec = 50, 25 and 10 change from those shown for R = 100 to those at R = 3. The v o r t i c i t y increases CO CO with decreasing R o t, whereas the streamlines become more symmetric and move closer to the c y l i n d e r . There i s l i t t l e d i s c e r n i b l e change in the flowfields between Rot = 50 and R = 25, but i t is quite marked between R = 25 and R_ = 10. OO * C O oo C h a r a c t e r i s t i c Parameters Table 4-10 summarizes the stagnation pressures and drag c o e f f i c i e n t s calculated for the various Reynolds numbers and outer boundary r a d i i . Figure 4-55 shows the excellent agreement between the values of CQ-J- calculated in this study and those obtained experimentally and numerically by other authors. 85 The other calculated parameters in Table 4-10 also agree quite well with those determined by others [39-41]. However, the values of p at Re = 20 and Re = 30 are rTT c c larger than expected. The explanation for this behaviour is that these two flowfields were not f u l l y converged with respect to the rear stagnation pressures. In certain cases p^ was found to approach a constant value only long after the e f f i c i e n c i e s had s t a b i l i z e d . Since the present work was primarily concerned with e f f i c i e n c y calculations i t was not considered necessary to execute the additional lengthy i t e r a t i o n s to converge the stagnation pressures. Davies' Flowfield at Re = 0.2 c As mentioned e a r l i e r , Davies1 Bessel representation of the f l u i d f l o w f i e l d at Re = 0.2 (Equation (2- 2)) was somewhat inaccurate. The errors were p a r t i c u l a r l y apparent close to the surface of the cy l i n d e r . Figure 4-56 shows the same streamlines f o r : (a) the numerical f l o w f i e l d at Rec = 0.2 and R^ , = 200 and (b) the f l o w f i e l d predicted by Equation (2-2 ). Davies [9 ] recog-nized that Equation (2-2 ) was incorrect in the area behind the c y l i n d e r , though he f e l t i t was adequate on the upstream sid e . It i s clear from Figure 4-56 that Equation (2-2) 86 i s i n v a l i d on the downstream side of the c y l i n d e r , but a closer examination of the f l o w f i e l d shows discrepancies on the upstream side as w e l l . Figure 4-57 depicts some stream lines correspond-ing to low values of i>, which actually terminate on the surface of the c y l i n d e r , a condition which i s p h y s i c a l l y impossible. Furthermore, the stream lines which do not contact the cylinder surface approach i t more c l o s e l y than those calculated numerically. Since the integration of the p a r t i c l e t r a j e c t o r i e s involved the rectangular components of the f l u i d v e l o c i t y , v and v , these quantities have been plotted for constant x y values of the y-co-ordinate ( c f . Figure 4-58 and 4-59) at Re = 0.2. Davies' predictions of v are higher than those derived from the numerical f l o w f i e l d , while those for v y are lower. Such differences explain the higher e f f i c i e n c i e s generally predicted by Davies. Tables 4-11 to 4-14 contain the calculated values of v and v f o r : a) Davies' equa-x y tion and (b) the numerical f l o w f i e l d in the region near the upstream half of the c y l i n d e r . Davies' Flowfield at Rec = 10 Table 4-15 contains the values of the rectangular f l u i d v e l o c i t y components given by Davies [7 ] on the basis 87 of Thorn's [ 6 ] data. These, together with the v e l o c i t i e s interpolated from the numerical f l o w f i e l d at Re£ = 10, are plotted in Figures 4-60 and 4-61 for y = 1.0, and y = 0.6. Tables 4-16 and 4-17 displays v x and v y calculated from the numerical f l o w f i e l d at Re = 10. c Thorn's predictions of v and v are considerably x y higher than the v e l o c i t i e s calculated from the numerical f l o w f i e l d . The errors in his v e l o c i t i e s are not surp r i s i n g when one considers the approximate method he used to i n -tegrate the equations of motion of the f l u i d . CONVERGENCE CR ITER IA As mentioned in the Theory Chapter, several c r i t e r i a were monitored during the convergence of the numer-i c a l solutions of the Navier-Stokes Equation. The r e s u l t for f i v e t y p i c a l cases (Re,, = 0.2, 0.5 and 40, R = 3 and 100) are summarized in Tables 4-18 to 4-22. It must be stressed that the s p e c i f i c behaviour of the convergence c r i t e r i a is strongly dependent on the s t a r t -ing grid values, tyQ and Co. In order to afford a standard basis of compari-son for one s o l u t i o n , the i n i t i a l values of ty and r, for 88 Re£ = 0.5 and = 100 were calculated from Equations (3-35) and (3-40). The following discussion outlines some general properties of the convergence c r i t e r i a i l l u s t r a t e d by the f i v e cases cited above. a) Maximum changes in stream function and v o r t i c i t y , Mn ty, Mn ? b , Mn ? s: In most cases these values decreased appreciably over the f i r s t few hundred i t e r a t i o n s and then decreased slowly while displaying an o s c i l l a t o r y behaviour. Mn ^ and M„ t were larger than M„ ty in most cases, n ^s 3 n r b) Sum of a l l stream function and v o r t i c i t y values, S„ ty, S„ t;. , S„ c : nr n b n s These sums did not change appreciably over the course of several thousand i t e r a t i o n s , although s l i g h t increases or decreases were noted in some cases. 89 c) Fractional changes in the sum of stream function and v o r t i c i t y changes per i t e r a t i o n , Qn % gb' Qn V A l l of these parameters decreased during the relaxation of each g r i d . This i s to be expected as the stream function and v o r t i c i t y approach t h e i r f i n a l constant values throughout the g r i d . d) Sum of f r a c t i o n a l changes over 100 i t e r a t i o n s ,  Hr\  y > Hu sb* vn ^s The sums related to the v o r t i c i t y changes decreased with increasing number of i t e r a t i o n s . The same was true in most cases for the sums of stream function changes. e) Frontal stagnation pressure, p 0: This parameter changed very slowly, generally de-creasing s l i g h t l y during the r e l a x a t i o n . f) Rear stagnation pressure, p^: The magnitude of p^ changed more noticeably than p 0, and i t sometimes converged very slowly to i t s f i n a l value. 90 g) Skin drag c o e f f i c i e n t , C Q S: The skin drag c o e f f i c i e n t showed very l i t t l e change with the number of i t e r a t i o n s performed on a given g r i d . h) Form drag c o e f f i c i e n t , C^: This drag c o e f f i c i e n t was somewhat more se n s i t i v e to the number of i t e r a t i o n s than C D^. This resulted mainly from the fact that i t was based on the v o r t i c i t y values in the v i c i n i t y of the cylinder surface, which converged more slowly than the stream function in the same area. i) Ratio of drag c o e f f i c i e n t s , Cr j s/ CD F: In most cases this r a t i o showed a steady decrease with increasing number of i t e r a t i o n s , but the change was qui te gradual. Since the stream lines moved closer to the c y l i n -der with increasing Reynolds number, one would anticipate a s i m i l a r increase in S„ \b, S„ c. and S„ r . Examination n r n b n s of Tables 4-18 to 4-22 c l e a r l y demonstrates this e f f e c t . 91 In a d d i t i o n , Sn ty decreased with decreasing at a given Reynolds number, whereas Sn ^ and Sn ? increased under the same conditions. Selection of Standard C r i t e r i o n Q 1 0 0 i s the best of the mentioned c r i t e r i a for n monitoring a numerical relaxation solution of the Navier-Stokes Equation. The primary advantage of Q n1 Q 0 i s that i t was a direct measure of the amount of change occurring at a l l points in the g r i d . Because the sums S„ ty, S„ and S„ c did not n T n b n s change appreciably with the number of it e r a t i o n s performed, Q n1 Q 0 e s s e n t i a l l y represented a normalized measure of the changes in ty, c b and c g. Since Q n 1 0 0 was based on the changes over 100 i t e r a t i o n s i t was preferable to Qn> which merely gave the normalized change during one i t e r a t i o n . Q n1 0 0 almost invariably decreased as the values in the numerical f l o w f i e l d approached near th e i r f i n a l values. In some instances Q n1 0 0 decreased by two orders of magnitude, as in the case of ReQ = 40 and = 3. was used by Hamielec and Raal [40] to decide when th e i r numerical solutions of the Navier-Stokes Equation were converged. However, this c r i t e r i o n was much less 92 r e l i a b l e than Q n1 0 0, since i t decreased very slowly during most of the i t e r a t i o n s , and displayed considerable o s c i l l a -tory behaviour. As a consequence of these v a r i a t i o n s , i t is conceivable that drops below some pre-set minimum value before the f i n a l , converged solution i s attained. RECOMMENDATIONS FOR FUTURE WORK The techniques developed in the present work can readily be extended to include g r a v i t a t i o n a l , e l e c t r o s t a t i c , Brownian and radiometric e f f e c t s , in addition to the i n e r t i a l and interceptive mechanisms. It would also be useful to investigate the nature of the p a r t i c l e cylinder i n t e r a c t i o n at the instant of contact, since i t determines the c o l l e c -t i o n e f f i c i e n c y . A s i m i l a r study of p a r t i c l e impaction on spheres at intermediate Reynolds numbers is also of p r a c t i c a l i n t e r e s t . This could be effected by u t i l i z i n g the same approach adopted for c y l i n d e r s . Experimental research should be performed in order to test the various assumptions made in the present theore-t i c a l study. In p a r t i c u l a r this could provide information on the extent to which p a r t i c l e s of f i n i t e size disturb the f l u i d f l o w f i e l d . 93 0 -! -2 -3 -4 X Figure 4-3. P a r t i c l e Reynolds number as a function of x and s t a r t i n g p o s i t i o n , y (K = 0.1, P = 10, Rer = 40, R = 100. Figure 4-4. Impaction e f f i c i e n c y as a function of i n e r t i a l parameter. Figure 4-5. Impaction ef f i c i e n c y as a function of i n e r t i a l parameter. Figure 4 - 6 . Impaction effici e n c y as a function of i n e r t i a l parameter. Figure 4-7, Impaction ef f i c i e n c y as a function of i n e r t i a l parameter. Figure 4-8. Impaction e f f i c i e n c y as a function of i n e r t i a l parameter. 0 1 ! 1 0 1 0 0 1 0 0 0 P Figure 4-9. Impaction e f f i c i e n c y as a function of i n e r t i a l parameter. o o Figure 4-10. Impaction e f f i c i e n c y as a function of i n e r t i a l parameter. Figure 4-12. Impaction e f f i c i e n c y as a function of i n e r t i a l parameter. 10 0-8 I 1 1 — 0-6 0 ^ / / / Rec=20 0 4 °v// — 02 — 0 0 1 1 1 111 III 1 1 IIIIIII 1 I 1 I 11 ul 1 1 1 I II II 01 I 10 100 1000 p Figure 4-14. Impaction ef f i c i e n c y as a function of i n e r t i a l parameter. Figure 4-15. Impaction e f f i c i e n c y as a function of i n e r t i a l parameter. 01 I 10 100 j o o o p Figure 4-16. Comparison of impaction e f f i c i e n c i e s as predicted by: (i) Davies and Peetz [ 7 ] , ( i i ) this work. o Figure 4-17. Comparison i f impaction e f f i c i e n c i e s as predicted by: (i) Davies and Peetz [ 7 ] , ( i i ) this work. o ca 10 0-8 06 0 4 0-2 K=0l K=000l - SUBRAMANYAM a KUL00R - THIS WORK. Rec=40,R00=IOO 0 0 0 Figure 4-18, 4 8 P Comparison of impaction e f f i c i e n c i e s as given by; (i) Subramanyam and Kuloor [12] (experimental), (i i ) this work. 10 o Figure 4-19. Impaction ef f i c i e n c y as a function of i n e r t i a l parameter for potential flow. Figure 4-20. C r i t i c a l i n e r t i a l parameter as a function of Reynolds number. Figure 4-21. Impaction ef f i c i e n c y as a function of Reynolds number. K=OI R^=IOO Figure 4-22. Impaction e f f i c i e n c y as a function of Reynolds number. 10 0-8 0-6 0 4 02 01 K=I0 R^IOO P=40 OOL 1—i—i M i n i i i i i i i 111 i i i i i I 10 100 Re, Figure 4-23. Impaction ef f i c i e n c y as a function of Reynolds number-10 P=40 08 0-6 04 0-2 01 10 I 10 Re, K=000l Roo= 3 0 0' 1 1 1 I I I M l 1 i i i i i 111 i 100 Figure 4-24. Impaction e f f i c i e n c y as a function of Reynolds number. 10 — 1 1 P=40 10 0-8 — • 5 0 6 2 0 4 — 1 — — — _ 0-2 K=0l Roo= 3 — 0 0 J—i—i_J-111 i i i i i i 1 1 1 i I 1 i MM 01 I 10 100 Rec Figure 4-25. Impaction ef f i c i e n c y as a function of Reynolds number. 10 P=40 0-8 0-6 04 0 2 01 K=I0 R oo=3 001 1 1—I I M i l l I I I I I I M 1 i I l l l I I I 10 100 Re, Figure 4-26. Impaction ef f i c i e n c y as a function of Reynolds number. E 1-0-0-8 0-6 04 0-2 K=0l, R^IOO HOUSE, a GOLD. THIS WORK 00 01 J i i i i i 11 J I I M i l l I I l i i i ' I 10 IOO Re, Figure 4-27 Comparison of impaction e f f i c i e n c i e s predicted by: (i) Householder and Goldschmidt [13], ( i i ) this work. CO 0 0 0 1 K = I 0 , Ro l^OO - HOUSE, a GOLD. - THIS WORK J J ' I « I M I I J ! I I I I I I I J » ' t I i l 1 0 1 0 0 Re, Figure 4-28. Comparison of impaction e f f i c i e n c i e s predicted by: (i) Householder and Goldschmidt [13], (i i ) thi s work. 120 Figure 4-29. Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter. 1 21 VP 0 10 20 30 40 Figure 4-30. Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter. 122 Figure 4-31 . Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter. 1 23 -Ol 03-0-5 K=000l RQQ=3 E=0-9 0-8 10 20 Re, 30 40 Figure 4-32. Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter. 124 10 20 Re c 30 40 Figure 4-33. Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter. 1 25 0 10 20 Rec 30 40 Figure 4-34. Impaction e f f i c i e n c y as a function of Reynolds number and i n e r t i a l parameter. Figure 4-35. Impaction effici e n c y as a function of solids concentrati on. Figure 4-36. Impaction ef f i c i e n c y as a function of solids concentration . Figure 4-37. Impaction effici e n c y as a function of solids concentration. 1 32 Figure 4 -41 . Sieving mechanism at high solids concentration. >-o 10 0-8 UJ o LL n 0-6 UJ 0 4 0-2 001 01 Roo=IOO K=I0, P=5 J l l M i l l . 10 Re, E J + EK I 1 1 I I ' » I 1 I I | M i l l 100 Figure 4-42. Impaction e f f i c i e n c i e s (Ej., E R t E + E R , E) as a function of Reynolds number. \ 134 AON3IOIJd3 Figure 4-44. Comparison of impaction e f f i c i e n c i e s due to interception as predicted by: (i) Wong et al. [10], ( i i ) this work. 137 2 139 Figure 4-48. Streamlines and e q u i - v o r t i c i t y lines at R = 3. CO 140 141 142 143 Figure 4-52. Streamlines and e q u i - v o r t i c i t y lines at Rec = 0.2 . 144 145 Figure 4-54. Streamlines and e q u i - v o r t i c i t y lines at Rec = 40. Co 2 0 1 5 -1 0 O L 0 1 I I I I I 111 1 TRITTON X TAKAMI a KELLER A HAMIELEC a RAAL e DENNIS a CHANG 0 THIS WORK J J I I t 1 1 1 1 J 1 M i l l I 1 0 1 0 0 Figure 4-55. Rec Comparison of drag c o e f f i c i e n t s as given by T r i t t o n [31], Takami and K e l l e r [39], Hamielec and Raal [ 40], Dennis and Chang [41], and t h i s work. Figure 4-56. Streamlines at f o r : Peetz [ 7 ] , (b) this (a) Davies work. and 148 Figure 4-57. Streamlines in the vi for : (a) Davies and work. c i n i t y of the cylinder Peetz [ 7 ] , (b) this Figure 4-58. Fluid v e l o c i t y , v , as a function of x f o r : ( i ) Davies and Peetz [ 7 ] , ( i i ) this work. I 0 -I -2 - 3 X Figure 4-59. Fluid v e l o c i t y , v , as a function of x for : ( i ) Davies and Peetz [ 7 ] , ( i i ) this work. 151 Figure 4-60. Fluid for: work. v e l o c i t y , v , as a function of x (i ) Davies and Peetz [ 7 ] , ( i i ) this 1 52 0-6 0 4 Vv 0-2 00 T R e ^ O . R ^ l O O DAVIES THIS WORK -I -2 -3 Figure 4-61 Fluid v e l o c i t y , v y , as a function of x for : ( i ) Davies and Peetz [ 7 ] , ( i i ) this work. Table 4-1 Impaction E f f i c i e n c i e s at R = 100 p K 0.2 RE= 0.5 RE= 1.0 RE= 3.0 RE = 5.0 G E e E £ E e E e E 0.01 0. 001 0. 000 1 0. 0001 0.0002 0.0002 0. 0002 0.0002 0.0003 0.0003 0. 0005 0.0005 0. 10 0. 001 0. 0002 0. 0002 0.0002 0.0002 0.0002 0.0002 0.0004 0.0004 0. 0CO6 0.0006 0. 15 0 . 001 0. 0002 0. 0002 0.0002 0.0002 0. 000 3 0.0003 0.0005 0.0005 0. 0007 0.0007 0.25 0. 001 0. 0002 0. 0002 0.0002 0.0002 0. 0003 0.0003 0.0006 0.0006 0. 0C09 0.0009 0.50 0. 001 0. 0002 0. 0002 0.000 3 0.0003 0. 0005 0.0005 0.001 1 0.0011 0. 00 18 0.00 18 0.75 0. 001 0. 000 3 0. 0003 0.0004 0.0004 0. 0007 0.0007 0.0020 0.0020 0. 0C4 1 0.0041 1.00 0 . 001 0. 0004 0. 0004 0.0006 0.0006 0. 0010 0.0010 0.0047 0.0047 0. 0234 0.0234 2. 00 0. 001 0. 00 10 0. 0010 0.0029 0.0029 0 . 0414 0.04 13 0.2008 0.2006 0. 260 1 0.2598 3.00 0. 001 0.0082 0. 0082 0. 1023 0.1022 0. 1962 0. 1960 0.3 3 73 0.3 369 0. 3 909 0. 3905 5. 00 0. 001 0. 1740 0. 1738 0.2805 0.2802 0. 370 7 0.3703 0.4963 0.49 58 0. 54 21 0.54 16 7.50 0. 001 0. 3069 0. 3066 0.4141 0.4 136 0. 4971 0.4967 0.6069 0.606 I 0. 6458 0.6452 10. 00 0. 001 0. 3990 0. 3986 0.50 18 0.5013 0. 5779 0.577 3 0.6753 0.6747 0. 7C91 0.7084 40.00 0. 001 0. 75 22 0. 7515 0.8096 0.8088 0. 8465 0.8456 0.8887 0.8878 0. 9022 0.90 13 100.00 0. 001 0. 0(127 0. 881 8 0.9125 0.9116 0. 9307 0.9297 0.9506 0.9 49 7 0. 9569 0.9559 1000.00 0. 001 0. 9867 0. 9 857 0.9903 0.9893 0. 9924 0.9914 0.9947 0.99 37 0. 995 3 0.9943 0.01 0. 010 0. 0002 0. 0002 0.0002 0.0002 0 . 0003 0.0003 0.0005 0.0005 0. 0006 0.0006 0. 10 0. 010 0. 0002 0. 0002 0.0003 0.0003 0. 0004 0.0003 0.0006 0.0006 0. 0C08 0.0008 0. 15 0. 010 0. 0002 0. 0002 0.0003 0.0003 0. 0004 0.0004 0.0007 0.0007 0. 0009 0.0009 0.25 0. 010 0. 0002 0. 0002 0.0003 0.0003 0. 0004 0.0004 0.0008 0.0008 0. 00 12 0.0012 0.50 0. 010 0. 0003 0. 0003 0.0004 0.0004 0. 0006 0.0006 0.0014 0.0014 0. 0022 0.0022 0.75 0. 010 0. 0004 0. 0004 0.0005 0.0005 0 . 0009 0.0009 0.0024 0.0024 0. 0C4 9 0.0048 1.00' 0. 010 0. 0004 0. 0004 0.0007 0.0007 0. 0013 0.0012 0.0055 0.0055 0. 0256 0.0253 2. 00 0. 010 0. 0012 0. 0012 0.0034 0.0034 0 . 0436 0.0432 0.2030 0.2010 0. 2623 0.2597 3.00 0. 010 0. 0093 0. 0092 0. 1038 0.1027 0. 1985 0. 196 6 0.3403 0.3 369 0. 3940 0. 390 1 5. 00 0. 010 0. 1762 0. 1744 0.28 35 0.2807 0. 3743 0.3706 0.5006 0.4956 0.5465 0.5411 7.50 0. 010 0. 310 1 0. 3072 0.4182 0.4 14 1 0. 50 19 0.4969 0.6124 0.6063 0. 65 13 0.6448 10.00 0. 010 0. 4031 0. 3991 0.5067 0.5017 0. 58 3 3 0.5775 0.6814 0.6747 0. 7154 0.7083 i»0. 00 0. 010 0. 7597 0. 752 1 0.8176 0.8095 0. 8548 0.8463 0.8972 0.8884 0. 9108 0.90 18 100.00 0. 010 0. 89 15 0. 8826 0.9215 0.9124 0. 9 399 0.9306 0.9601 0.9506 0. 9663 0.9567 1000.00 0 . 010 0. 9965 0. 9866 1.0002 0.9903 1. 0023 0.9924 1.0045 0.99 46 1. 005 3 0.9954 oo o -» o o o -tr -• o o xr oooo»juiwto-.oooooo OOOO-OLP^K>—»oooooo OOOOUIOOOO-JUIIO—*-*o OOOOUIOOOO-JUINJ—*-*o ooooooooo u o ui m o J oooooooooLnoouio—• "0 OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO NjKXcrotsjwwtorowNJPowKJN; —' — —»—•——._._.—»—._._» OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO :* -.—.OOOOOOOOOOOOO -OOOOOOOOOOOOOOO —.oo-crwroooooooooo O ^ C O C ^ S J O O O O O O O O O UJOIOUIO—.—.uj-^ -rro—•—*—*—» uio^^^o^-iU'ruuuuu vO—.O.te—.^jO~Js£>0>O^JjrKIO U^ UJ^ rON^ OlO<<JjMo-' a) ra I 0 . to OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO \Ott^JJ=w—.OOOOOOOOO v0£0-Jtuj-*OOOOOOOOO a I ui o - jjssro—• -. o o o o ca a ui o ^  e s -.0000000 c*toro-fcuitocccc*?N>o*o%o-0>o C1 fo w -» o M -•O^WWJ^MMM CT»C*l-C=Ul\O^JW —i O W C2 CO Ul ^  W CTiJl^McnOwlNUlaUOOCOCO PI —»—.OOOOOOOOOOOOO -.-.OOOOOOOOOOOOO —. 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COMOuiUUl^JfCOOCOO —• (Jl %£> —»—.CCsOCC-ujroO^OjrCC ra —•—»OOOOOOoOOOOO —»—-OOOOOOOOOOOOO —• —» 0 ct >j a> ^  ui -» 0 0 0 0 0 0 ooo~j-j<-n— roooooooo iC^ XUlyJLICCt^ lfl^ CWWW U^10^ JO^ >l-JOCCCNJ—.—.—.O fsioo^^co^^in^^cc^w j^O^ co^ w^ jcin-iWaoJ: WU^ O^CC-.\C£CCO|oCOCiUl CCCNJujO'O^ IOvlNirC^ Jl n RE= 5.0 OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO l^O^ sj^ uicw-oOoooo vooo^ jcinwroooooooo Jl.lCOCCOOMCCUlW^ UW O U I O O C C J V J N W - ' - ' - ' O O uic*,°cocigo>-*c-;ra'WOsi u~. 0 w —•^ ocoo^ fooocc W^ UlCgUQUIUIOO^ IUUl^  ^N'SJ>J^U1U1C^^O<J-'UJC1^J rlj RE= 5.0 t?g i Table 4-1 (Conti nued) p K 0.2 BE= 0.5 RE= 1.0 RE= 3.0 R E= 5.0 c E e E e E e E £ E 0.01 0. 500 0.0578 0.0386 0.0708 0.0472 0.0877 0.0584 0. 1348 0.0899 0. 1682 0.1122 0. 10 0.500 0. 05(37 0.0391 0.0726 0.0484 0.0908 0.0606 0. 1418 0.0946 0. 1779 0.1186 0. 15 0.500 0.0593 0.0395 0.0737 0.0491 0.0928 0.06 19 0. 14 64 0.0976 0. 184 3 0.1228 0.25 0. 500 0.0606 0.0404 0.0764 0.0509 0.0974 0.0649 0. 1 568 0.1045 0. 1908 0.1 325 0.50 0. 500 0.06 5> 0.0434 0.0848 0.0565 0. 11 18 0.0745 0. 1 8 95 0.1263 0. 2432 0.1621 0. 75 0.500 0.07 10 0.0473 0.09 56 0.0638 0. 1 302 0.0868 0. 2299 0.1533 0. 2952 0.1968 1. 00 0.500 0.078 1 0.0521 0.1089 0.0726 0.1527 0.1018 0. 2 757 0. 1838 0. 34 97 0.2331 2. 00 0.500 0. 1206 0.0804 0.1870 0. 1247 0.2757 0. 1838 0. 4592 0.3061 0. 54 29 0.3 620 3.00 0.500 0. 1056 0. 1237 0.29 15 0.1943 0.4075 0.2717 0. 60 31 0.4021 0. 6 828 0.4552 5. 00 0.500 0.3 393 0.2262 0.48 38 0.3225 0.6109 0.407 2 0. 7945 0.5 29 7 0. 86 3 2 0.5754 7.50 0. 500 0.5025 0.3350 0.6552 0.4368 0.7759 0.5 173 0. 9 3 85 0.6 256 0. 9 96 3 0.6642 10. 00 0. 500 0.6255 0.4170 0.7745 0.5163 0.0862 0.5908 1. 0307 0.6872 1. 0C09 0.7206 40. 00 0.500 1.13 10 0.7540 1.2163 0.8109 1.2712 0.8475 1. 3340 0.8894 1. 3538 0.9025 100.00 0. 500 1.3245 0.883 0 1.3689 0.9126 1 . 39 61 0.9307 1. 4259 0.9506 1. 4 34 9 0.9566 1000.00 0.500 1.4799 0.9866 1.4854 0.9902 1.4886 0.9924 1. 4919 0.9946 1. 4929 0.9 9 53 0.01 1.000 0. 1903 0.0952 0.2321 0.1160 0.2858 0. 1429 0. 4278 0.2139 0. 5205 0.2 60 2 0. 10 1.000 0. 1924 0.0962 0.2361 0.1181 0.2924 0. 1462 0. 4407 0.2204 0. 5366 0.2683 0. 15 1.000 0. 19 37 0.0968 0.2387 0.1193 0.29 66 0. 1483 0. 4486 0.2243 0. 5465 0.2733 0.25 1.000 0. 1966 0.0 9 83 0.2442 0. 1221 0.3056 0. 1528 0. 4658 0.2329 0. 5677 0.28 3 8 0.50 1.000 0.2059 0. 1029 0.2608 0.13 04 0.3319 0. 1660 0. 514 1 0.2571 0. 6250 0.3125 0.75 1.000 0. 2173 0.1087 0.2808 0. 1404 0.3626 0. 18 13 0. 5660 0. 28 30 0. 6834 0.34 17 1.00 1.000 0.2306 0. 1153 0.3033 0. 1516 0.39 64 0.1982 0. 61 84 0.309 2 0. 7396 0.3 69 8 2. 00 1.000 0.2978 0.1489 0.4106 0.2053 0.5439 0. 2720 0. 8074 0.40 37 0. 9288 0.4644 3.00 1.000 0.3792 0.1896 0.5277 0.2639 0.6843 0.3422 0. 9542 0.4771 1. 0678 0.5339 5. 00 1.000 0.5(499 0.2 74 9 0.7374 0.3687 0.9056 0.4528 1. 1 577 0.5789 1. 2548 0.6274 7.50 1.000 0.7344 0.3672 0.9335 0.4667 1 .0946 0.5473 1. 31 78 0.6 589 1. 3994 0.6997 10. CO 1.000 0.8797 0.4398 1.07 56 0.5378 1.2254 0.6127 1. 4239 0.7120 1. 4 942 0.7471 40.00 1.000 1.5 156 0.7578 1.6296 0. 8148 1.7036 0.85 18 1. 7891 0.8945 1. 0 166 0.9083 100.00 1.000 1.76 8 4 0.8842 1.8278 0.9139 1.8642 0.9321 1. 9050 0.9525 1. 9176 0.9588 1000.00 1.000 1.97 35 0.9868 1.9807 0.9904 1.9851 0.9925 1. 9895 0.9948 1. 99 10 0.9955 1 56 o • o It w a CO i n r i u > o o ^ c i o ? f , ' » o c o * 2 3 »-(7'v03c ,c ,*-or % ' ixi-jirr , 'or v o o c o ^ ' - j ' O ' - a c - a r N ^ t r c c o o r N T C N ' ~ ~ f \ r N ^ i v j ^ o o o o o ^  •— ^ioohh{r ,P{r ooooc^*»~ r n ^ , * 0 r^r*CT*c s 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 u l ^ r c ^ O O « - O c > - J W 3 ) C , i r i C O 1 -0.^in(M f"OOt>lN^fMOf>.fs oocors i rgco*— — r « ro o r oooor-d*—cro*— =r !•-•> r-< r-» o C o o o c T v ' ^ ^ C M ^ c r a . cr o o o o o «— •— =?tn-cr-r-crcr.o o o o c o o o o o c o o o o o o o o o o o o o o o c o o o - — c • o Ii to c o o o « — c x c o - ^ ^ r - o j o o ocoo«—oxccc .^*Nr*<M-wr* o o o o o •— ^-^.or^r^crc^c^ o o o o o * - ' • r i p ^ ^ r x c , c r o o o o o o o o c o o o o o o o o o o o o o o o o o o o o o u ooco*—occcr^a-r^r^rg,wC> oooo*-occcr . •—zrm^rrr-o 00000^ "^ "^ "—^  r-r— er cr o c o o o -^ «— r-tinor-r^cr-cro o o o o o o o o o o o o o o o 00000000000000--o • o rsj II cr: ooooor*-vc r-cT ' O J ' -o r , | \cc o o o c — r**vr:r*corM«— ^pg^cr o o o o o o *— na-or^r-^cncr o o o c o o * - r^ .^ -v^ Dr—r-crcrcr 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 to oi/>3»ovD(M^ r)ir. tr*j p» vo -^ if* ^c>nn<?Nf-rrnccrM(V)r*in^ ooooor-*vcr-cir^*— o r\i o ooco^*P ,*or»cT'rv]r^r^mr^o o o o o o o «— mzror^r^trcrc^ o o o o o o * - r n c r ^ r ' - p ^ o c r - 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 — O •> m n O L to O f-NvO '-c-3-^NrcO j L ', i »-'-'~(Nr,-(NO>fNriOOJrs-olno o o o o o L n ^ r i n ^ - ' — O I J - . rvocr> ooooOu'i^Lnr-'*—o^c^ocr*-oooooo*—mrror^r-crcrcr o o o o o o *— m-a-^ or^ r^ c-crcr 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 <J c o ( N 3 , » - n r t v o a c ^ c ^ ^ ^ ^ r^^e^^^o^^^xt f i r s i r^o™ o^* -rvj^D*—o^-^^^cT''* - <— *—«—c\jr-<—1 «-~ co o c i/i r iri occoc^^inNfo^Nor o o c o o LTI 3" LT* r«- •— o »c r^  o 000000*— ^ -^siDr^r^c^jTC o o o c o o *— r, ^ ^ ^ ^ o e o o o o o o o o o o o o o o o o o o o o c o o o o o o o o c — o t o If cc to r- cr Lnco«— inr^OinO'^r*'-'- o^rsisyt^tnco-rr^r^rr'— cr — ceo oO'*TCr*cr,0\o*"arMiA o»-f_»-rT'-r-inrs-(r,_'N(\'irf\vc occoorsiorsj=!'Crcorr*-vccr- C C O O O N O ^ ? 3 3 * ? ' " ^ ^ oocooo*-fi^-i/ior—eerier* oocooo«—'**rj'ir**cf^cr-criCT,» 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 u p * C ' * i n r T - v « c ?>* t>J o o crrNj^-cr»i/^oir*ofsar,"orr^i-**'-o c o o o c^Jo^ N ^ r c r c c zr*~"ocr ooooofNOfM'j*. crcr-jncNr^-o OOCOCO-—r^^-L-i^r:r^cfcC'cr o o C c O C *— J I ^ M T C C o o o o o o o o o o o o o o o o o o o c o o o o o o o c 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 o o o c o o o o o o o o c o c o o o o c o o o o o o o c c o cu ^ - O L n i n o u n o o o o o o o o c - - o m m o u - j o o o o o o c o o o »— fMirir^ ocJooinococ o «— «— oj ir ir-^ooooj-ioooo o o o o o o «— (Nminp-oooo o c o o o o ^ - r N c n i n r - o o o o *- .=x o o *-3"00 «- o *- o o -» o o -* 0000»JU»UJ(SJ-*0O0000 O O O O ^ O O O O - * J L T I N J - ^ - * O ooooooooou^o^oo—» o -» o o c -* oooo- j cnwro—'OOOOOO ooooooooot-noui<-no—* OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO OOOOOOOOOOOO — _*oa>co-^Jwij?(o-*_.oooo -* _ » - * o o o o o o o o o o o o O O O C D « J O ^ C U _ » o o o o o o vO v o -» u*» -c oouncowro—*--—» 50 u OOOOOOOOOOOOOOO O^DO-OO^lJlCW—«-^00000 v D C 0 - ' O M 3 ^ - ^ - - ' v J N J W i \ ) O OOOOOOOOOOOOOOO * « • « « • • • • • • • • • • •XJi-DvD-^jOL/T-X^^—'OOOOOO _»_»_»oooooooooooo i^vOO^i^C\;-» —» O O O O » c y i o o x : u c i J y i j i J r o C D ( M 7 \ u i —>-»-»oooooooooooo OOOCD*JCT»^;»J— . — » o o o o o CCsJMXr-JCOCXrCO-iWOOOi I cr OOOOOOOOOOOOOOO U)v£vi3slsJl>CUlW-*_»0000 OOOOOOOOOOOOOOO VDVOIONJC^O^CW-I-IOOOOO fD 4* I • —*oo o o o o o o o o o o -i-i-.iflCCsJcjvCWW-'OOOO u i c c w o a o o v c c j o M J c c i v u i _*-»_»oooooooooooo o o o o ^ j ^ u i r w - » o o o o o U l K > U ' > O v j O c > > N J —j [7 s J - » -C M O OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO ^O^-J-JO^W-» —iOOOOO • ^ O f o O ^ o— i C D ^ c o r o ^ r s j r o r o - ^ u C T » i j j v O | \ j f r C C | s j £ : 0 ^ i \ j y i U i - i V l to o o o Z3 —»_»—»oooooooooooo —»-*—'Oogircuw-*—kOoo _*_i-»OOOOO0O0O000 OOOco^JC^tJl^'f'^-^^OOOO rt I OOOOOOOOOOOOOOO OOO-J**J0 NC* ,jrK;NJ-*OOoO soc^rsjo'*—» u i —» u~' o t o *o 7> OOOOOOOOOOOOOOO ^^vOsJ^CM^ijW-»00000 J\ w -J LP C: -»_»Oxr^rji*Jc_t - o o o o o o o o o o o o — » _ » — * v O C D ' s j 7 , t _ n U J N J _ » _ . — » o o -*-^-*oooooooooooo OOOCsJCT'JlCW-» o o c o o 50 I 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 O s j s J i y L n u i M - » O o O O O s 3 - i N ) ^ _ » i J / O C - * a i O ? r W N N ) o o - c : o ^ j — » c ^ c c - ^ c r . c c — » — • c c - ^ Table 4-1 (Continued) p K RE=10.0 8E=15.0 RE=20.0 RE=30.0 RE=40.0 e E e E e E E E E E 0.01 0.500 0.2272 0.1515 0.2676 0.1704 0.2988 0. 1992 0.3415 0.2277 0.36 95 0.2463 0. 10 0.500 0.2408 0.1606 0.2033 0.1889 0.3159 0.2 106 0.3599 0.2 399 0.3802 0.2508 0. 15 0. 500 0.2499 0.1666 0.29 38 0.1950 0.3271 0.2 100 0.3717 0.2470 0.4000 0.2667 0.25 0.500 0.2703 0.1002 0.3169 0.21 13 0.3516 0.2 34 4 0.3970 0.26 47 0.4250 0.2013 0.50 0.500 0.3292 0.2195 0.30 00 0.2538 0.4171 0.270 1 0.4621 0.3000 0. 4 00 1 0. 32 54 0.75 0.500 0.39 12 0.2608 0.44 45 0.2964 0. 4006 0.3204 0.5232 0.3 408 0.54 65 0.3643 1.00 0. 500 0.4507 0.30 05 0.5036 0.3358 0.53 85 0.3590 0.5783 0.3055 0.5991 0.3994 2. 00 0. 500 0.64 14 0.4276 0.6875 0.4583 0.7161 0.4774 0.74 60 0.49 7.3 0. 75 9 1 0. 50 61 3.00 0.500 0.77 16 0.5144 0.0113 0.5408 0.0351 0.5567 0.8584 0.5723 0.86 7 3 0.5782 5. 00 0.500 0.9 36 0 0.624 0 0.96 67 0.64 4 5 0.9 04 0 0.6565 1.0010 0.6673 1.0056 0.6704 7.50 0. 500 1.0559 0.7039 1.08 0 1 0. 720 1 1.09 41 0.7294 1.10 5 7 0.7 372 1. 1000 0.73 0 7 10.00 0.500 1. 13 15 0.7544 1. 1517 0.76 7 0 1.16 31 0.7754 1.1723 0.7015 1.17 36 0.7024 U0.00 0.50 0 1. 37 27 0.9151 1. 37 9 6 0.9197 1.3836 0.9224' 1.3 065 0.924 1 1.3 866 0.9244 100.00 0. 500 1.4 4 36 0.9624 1.4467. 0.9644 1.4405 0.9657 1.4497 0.9665 1.4490 0.9666 1000.00 0.500 1. 4 9 39 0.9959 1. 49 42 0. 996 1 1 . 4944 0.996 3 1.4 94 5 0.99 6 3 1. 4946 0.9964 0.01 1.000 0.66 51 0.3325 0.7 49 5 0.3747 0.00 73 0.40.36 0.8744 0 .4 372 0.9103 0.4551 0. 10 1.000 0.6846 0.3423 0.7699 0. 3 849 0.0277 0.4 139 0.8944 0.4472 0.9296 0.4 64 8 0. 15 1 .000 0.6 96 3 0.34 81 0.70 19 0.3909 0.0 39 6 0.4 190 0.9057 0.4 529 0. 94 05 0.4702 0.2 5 1.000 0.7207 0.3604 0.80 65 0.4 033 0.0 63 7 0.4.319 0.92 85 0.46 4 3 0. 96 2 2 0.4011 0.50 1.000 0. 78 32 0.3916 0.86 7 7 0.43.39 0.9227 0.4614 0.9035 0.4917 1.0139 0. 50 69 0. 75 1.000 0.84 29 0.4214 0.9248 0.4624 0.9771 0.4805 1.0337 0.5168 1. 06 10 0.5305 1.00 1.000 0.8978 0.4 4 89 0.97 6 6 0.4803 1.0261 0.5 1.31 1.0780 0 . 5 39 4 1. 1036 0.5518 2.00 1.000 1. 07 25 0.53 6 3 1. 1391 0.5696 1 .1797 0.5090 1.2201 0.6 10 1 1.237 1 0.6106 3.00 1.000 1. 1959 0.5980 1.2529 0.6265 1.2071 0.6 4 36 1.3199 0.6 599 1.1323 0.6 6 62 5. 00 1.000 1.359 2 0.6796 1.40 35 0.7017 1.4296 0.7 140 1.4532 0.7 266 1. 46C4 0.7302 7.50 1.000 1.48 4 3 0.7422 1.5192 0.7596 1.5394 0.7697 1.5566 0.7703 1.5609 0.7005 10. CO 1.000 1.566 1 0. 783 0 1.59 50 0.7975 1.6115 0.0058 1.62 52 0.0 126 1.6279 0.8139 UO. 00 1.000 1.84 3 3 0.92 17 1.8529 0.9265 1.0584 0.9 29 2 1. 862.3 0.9311 1.06 27 0.9313 100.00 1.000 1.9295 0.9047 1.9 3 39 0.9669 1.936.3 0. 960 1 1.93 78 0.9609 1.93 7 9 0.9 690 1000.00 1.000 . 1.99 22 0. 996 1 1.9920 0.9964 1.9930 0.9965 1.9931 0.9966 1. 99.3 1 0.9966 Table 4-2 Impaction E f f i c i e n c i e s at R =3 p K RE= 0.2 RE= 0. 5 RE= 1.0 RE= 3.0 RE* 5.0 e E e E e E e E e E 0.01 0.001 0. 0001 0. 0001 0.0001 0.0001 0. 0001 0.000 1 0. 0001 0.0001 0. 000 1 0.0001 0. 10 0.001 0. 0002 0. 0002 0.0002 0.0002 0. 0002 0.0002 0. 0002 0.0002 0. CC02 0.0002 0. 15 0.001 0. 0002 0. 0002 0.0002 0.0002 0. 0002 0.0002 0. 0002 0.0002 0. 0003 0.0003 0.25 0.001 0. 0003 0. 0003 0.0003 0.0003 0. 0003 0.0003 0. 0004 0.0004 0. 0C05 0.0005 0.50 0.001 0. 00 10 0. 00 10 0.00 1 1 0.0011 0. 0013 0.0013 0. 0022 0.0022 0. 004 2 0.0042 0.75 0.001 0. 1054 0. 1053 0.1142 0.114 1 0. 1295 0. 129 4 0. 1 650 0. 1640 0. 1604 0. 10 02 1.00 0.001 0. 26 0 3 0. 2600 0.26 39 0.2636 0. 2709 0.2706 0. 2904 0.2901 0. 3057 0. 30 54 2. 00 0.001 0. 5 182 0. 5176 0.5199 0.5194 0. 5235 0.5229 0. 5342 0.5337 0. 54 35 0.5429 3.00 0.00 1 0. 6352 0. 6346 0.6366 0.6359 0. 6394 0.6308 0. 6479 0.6 472 0. 6553 0.654 6 5. 00 0.001 0. 75 15 0. 7507 0.7525 0.7517 0. 7 54 6 - 0.7539 0. 7610 0.7602 0. 7664 0.7657 7.50 0.001 0. 0213 0. 8205 0.8221 0.82 13 0. 0238 0.8229 0. 8287 0.0 279 0. 83 28 0.8320 10. 00 0.001 0. 0602 0. 05 94 0.8609 0. 860 1 0. 8623 0.86 14 0. 8663 0.0654 0. 8697 0.0608 40. 00 0.001 0. 96 12 0. 9602 0.96 14 0. 9605 0. 9610 0.9609 0. 963 1 0.9621 0. 964 1 0.9631 100.CO 0.001 0. 9041 0. 903 1 0.9042 0.9832 0. 9844 0.9034 0. 9049 0.9040 0. 9E54 0.9044 1000.00 0.001 0. 998 4 0. 9974 0.9984 0.9974 0. 9984 0.9974 0. 9 9 84 0.9974 0. 9 906 0.9976 0.01 0.010 0. 0004 0. 0004 0.0004 0.0004 0. 0004 0.0004 b. 0004 0 .0004 0. 0005 0.0005 0. 10 0.010 0. 0004 0. 0004 0.0005 0.0005 0. 0005 0.0005 0. 0005 0.0005 0. CC06 0.00 0 6 0. 15 0.010 0. 0005 0. 0005 0.0005 0.0005 0. 0005 0.0005 • 0. 0006 0.0006 0. 0007 0.0007 0.25 0.010 0. 0007 0. 0007 0.0007 0.0007 0. 0007 0.0007 0. 0009 0.0009 0. 00 1 1 0.001 1 0.50 0.010 0. 0021 0. 002 1 0.0023 0.0023 0. 0027 0.0026 0. 0045 0.0044 0. 0004 0.0083 0.75 0.010 0. 1142 0. 113 1 0.1221 0.1209 0. 1362 0. 1349 0. 1693 0. 1676 0. 19 18 0. 1099 1.00 0.010 0. 266 1 0. 2634 0. 269 3 0.2667 0. 2760 0.2733 0. 2 94 7 0.2917 0. 3 095 0. 30 64 2.00 0.010 0. 5253 0. 52 01 0.52 68 0.5216 0. 530 1 0.5 248 0. 5399 0.5346 0. 54 06 0.5432 3.00 0.0 10 0. 64 31 0. 6367 0.6442 0.6378 0. 6468 0.6404 0. 654 5 0.6400 0. 66 14 0.6548 5. 00 0.010 0. 7600 0. 7525 0.7608 0.7533 0. 7620 0.7552 0. 7685 0.7609 0. 7736 0.7 6 59 7.50 0.010 0. 8302 0. 8220 0.8309 0.8226 0. 8 324 0.8 24 2 0. 0368 0.0 20 6 0. 04 07 0.0324 10. 00 0.010 0. 06 94 0. 8608 0.0699 0. 86 13 0. 8712 0.8625 0. 0748 0.0661 0. fc779 O.H69 3 40.00 0.0 10 0. 9709 0. 9613 0.97 11 0. 96 15 0. 9715 0.96 19 0. 9726 0.96 30 0. 9736 0.9639 100.00 0.010 0. 9940 0. 9842 0.9941 0.9843 0. 9943 0.9844 0. 9 94 7 0.9040 0.9952 0.9053 1000.00 0.010 1. 000 4 0. 9985 1.0084 0.9905 1. 0084 0.9985 1. 0084 0.9905 1. 0004 0.9905 Table 4-2 (Continued) p K RE= 0.2 RE= 0.5 RE= 1.0 RE= 3 .0 RE= 5.0 E E E E e E e E £ E 0. 01 0. 100 0.0204 0. 0186 0.0204 0.0186 0.0208 0.0190 0. 0219 0.0199 0. 0239 0.02 17 0. 10 0. 100 0.0210 0. 0191 0.0213 0.0193 0.0220 0.0200 0. 0243 0.0220 0. 0270 0.0245 0. 15 0. 100 0.0229 0. 0208 0.0233 0.0212 0.0243 0.0220 0. 0272 0.0248 0. 03 06 0.0279 0. 25 0. 100 0.0293 0. 0266 0.0300 0.0272 0.0315 0.0207 0. 0366 0.0333 0. 04 2 2 0.0304 0. 50 0. 100 0.0747 0. 0679 0.0777 0.0706 0.0045 0.0760 0. 1061 0.0964 0. 1262 0.1147 0. 75 0. 100 0.2139 0. 1945 0.2173 0. 1976 0.2252 0.2047 0. 2468 0.2244 0. 2647 0.2406 1. 00 0. 100 0.3385 0. 3077 0.3403 0.3094 0.3455 0.3 140 0. 3604 0.3276 0. 3738 0.3398 2. 00 0. 100 0.5983 0. 54 3 9 0.5988 0.5443 0.60 10 0.546 3 0. 6083 0.5530 0. 6 154 0.5595 3. 00 0. 100 0.7 196 0. 6542 0.7 190 0. 654 3 0.7214 0.6558 0. 7266 0.6605 0. 7320 0.6654 5. 00 0. 100 0.8406 0. 7642 0.8407 0.7642 0.0410 0.7653 0. 0454 0.7 60 5 0. 04 92 0.7720 7. 50 0. 100 0.9133 0. 8303 0.9133 0. 83 03 0.9 142 0.8311 0. 9169 0.0336 0. 9 197 0.0361 10. 00 0. 100 0.95 38 0. 0671 0.9539 0.8672 0.9546 0.8670 0. 9567 0.0697 0. 9590 0.0718 40. 00 0. 100 1.059 3 0. 9630 1.0593 0.9630 1.0 594 0.9631 1. 0601 0.9637 1. 0607 0.9643 100. CO 0. 100 1.0833 0. 9 84 8 1.03 33 0. 9848 1.0834 0.9849 1. 0 63 7 0.9052 1. Ofc'40 0.9055 1000. 00 0. 100 1.0983 0. 9985 1.0983 0.9985 1.0983 0.9905 1. 0 983 0.9905 1. 0984 0.99 85 0. 01 0. 200 0.07 48 0. 0623 0.0748 0.0623 0.0761 0.0634 0. 0795 0.0662 0. 0857 0.0715 0. 10 0. 200 0.0757 0. 063 1 0.0764 0.0637 0.0787 0.0656 0. 0853 0.0711 0. 0935 0.0779 0. 15 0. 200 0.0810 0. 0675 0.0820 0.06 8 3 0.0848 0.0707 0. 0 93 1 0.0776 0. 1020 0.0056 0. 25 0. 200 0.0982 0. 0819 0.0999 0.0032 0. 10 39 0.0866 0. 1 1 63 0.0969 0. 12 95 0. 1079 0. 50 0. 200 0. 1876 0. 1563 0. 1909 0. 159 1 0.1988 0.1657 0. 2214 0.1045 0. 24 23 0.2019 0. 75 0. 200 0.3231 0. 2693 0.3254 0.2712 0.3321 0.2767 0. 3511 0. 2926 0. 3600 0.30 74 1. 00 0. 200 0.4 369 0. 364 1 0.4301 0.3650 0.4427 0.3609 0. 4566 0. 3805 0. 4 7 06 0.39 2 2 2. 00 0. 200 0.69 16 0. 5763 0.69 15 0.5762 0.6932 0.5777 0.6993 0.5827 0. 7C66 0.50 09 3. 00 0. 200 0.8136 0. 6700 0.0 131 0.6776 0.8143 0.6706 0. 8181 0.60 10 0. 02 3 3 0.60 61 5. 00 0. 200 0.9359 0. 7799 0.9355 0.7796 0.9361 0.780 1 0. 93 65 0.7021 0. 94 10 0.7848 7. 50 0. 200 1.0096 0. 0413 1.0092 0.84 10 1 .009 6 0.84 14 1. 0 112 0.0427 1. 0136 0.8446 10. 00 0. 200 1.05 00 0. 8757 1.0504 0.0753 1.0 50 8 0.8757 1. 0520 0.0767 1. 05 30 0.0702 40. 00 0. 200 1. 1502 0. 9652 1. 1502 0. 9652 1.1502 0.9652 1. 1586 0.9655 1. 159 1 0.9 659 100. 00 0. 200 1. 1829 0. 9057 1.1829 0.9857 1.1829 0.9857 1. 1 £3 1 0.90 59 1. 1832 0.9 0 60 1000. 00 0. 200 1. 190 3 0. 9986 1. 1983 0.9906 1.1983 0.9986 1. 1903 0.9986 1. 1903 0.9986 CTl O Table 4-2 (Continued). p K RE= 0.2 BE= 0.5 RE= 1.0 RE = 3.0 RE= 5.0 e E e E e E e E e ' E 0.01 0. 500 0.U057 0.2705 0.4050 0.2700 0.4089 0.2726 0.4102 0.2708 0.4392 0.29 2 8 0. 10 0.500 0.U002 0.2668 0.40 18 0.2679 0.4086 0.2724 0.4275 0.20 50 0. 4532 0.3021 0. 15 0.500 0.1* 1 13 0.2742 0.4135 0.2757 0.4213 0.2009 0.4434 0.2956 0. 4707 0.3130 0.25 0.500 . 0.1*489 0.2993 0.4519 0.3012 0.4 60 9 0.3073 0.4866 0.3244 0.5156 0. 3438 0.50 0.500 0.580 1 0.3867 0.5824 0.3882 0.5908 0.3930 0. 6149 0.4 100 0. 64 13 0. 42 75 0.75 0. 500 0.7039 0.4693 0.7047 0.4690 0.7107 0.4738 0.7286 0.4058 0. 74 97 0.4990 1.00 0.500 0.80 14 0.5343 0.00 12 0. 534 1 0.8053 0.5369 0. 8183 0.5455 0.0350 0.55o7 2. 00 0.500 1.0254 0.6836 1.0237 0.6025 1.0248 0.6832 1.0291 0.6061 1.0377 0.69 1e 3.00 0.500 1. 136 3 0.7575 1.1345 0.7563 1.1347 0.7565 1.1363 0.7575 1. 14 10 0.7612 5. 00 0.500 1.2492 0.8328 1.2475 0. 03 16 1.2472 0.8314 1.2471 0.0314 1. 25C 1 0.0334 7.50 0.500 1.318 1 0.8787 1.3166 0. 8770 1.3162 0.8775 1.3156 0.8771 1.3 175 0.8783 10. 00 0.500 1.3569 0.904 6 1.3558 0. 9038 1.3553 0.9036 1. 3 54 6 0.9031 1.3560 0.9040 40.00 0. 500 1.4596 0.9730 1.4591 0. 9728 1.4589 0.9726 1.4585 0.9724 1.4507 0.9 72 5 100.00 0.500 1.4834 0.9890 1.48 32 0.9808 1.4831 0.9000 1. 4 82 9 0.9K06 1.4030 0.9087 1000.00 0.500 1.4 98 3 0.9989 1. 4983 0.9989 1.4983 0.9909 1.4903 0.9989 1.4903 0.99 89 0. 01 1.000 1.3002 0.6501 1.2973 0.6486 1.2984 0.6492 1.3011 0.6506 1.3208 0.6604 0. 10 1.000 ' 1. 2798 0.6399 1. 2793 0.6397 1.2042 0.6421 1.2987 0.6493 1.3243 0.6621 0.15 1.000 1.2846 0.6423 1. 20 48 0.6424 1.2907 0.6453 1.3081 0.6540 1.3 34 0 0.6674 0.25 1.000 1.3131 0.6565 1.3133 0.6567 1.3193 0.6597 1. 33 7 9 0.6609 1.3639 0.6020 0.50 1.000 1.4 08 1 0. 704 1 1. 4069 0.7034 1.4104 0.7052 1.4236 0.7 110 1.4440 0.7220 0.75 1.000 1.4060 0.7430 1.48 30 0.74 19 1.4854 0.7427 1.4938 0.7469 1.5C97 0.7548 1.00 1.000 1.5451 0.7726 1.5422 0.7711 1.5428 0.7714 1.5480 0.7740 1. 5607 0.7003 2. CO 1.000 1.6826 0.8413 1.6791 0. 8396 1.6782 0.039 1 1.6778 0.0309 1. 6047 0.0424 3.00 1.000 1.7533 0.8767 1.7500 0.8750 1.7483 0.8741 1.7465 0.07 32 1. 75 1 1 0.0755 5. 00 1.000 1.8274 0.9137 1.8248 0.9124 1.8232 0.9 1 16. 1. 82 01 0.9 100 1. 8224 0.9112 7.50 1.000 1.87 36 0.9368 1.8712 0.9356 1.8 69 6 0.9340 1.8673 0.9 337 1. 06 89 0.9 3 44 10. 00 1.000 1.9002 0.9501 1.09 02 0. 94 9 1 1.89 69 0.9485 1.8946 0.9 47 3 1.0954 0.9477 1.0.00 1.000 1.9715 0.9 857 1.9708 0. 9854 1.9703 0.9852 1.9693 0.9846 1.9694 0.90 4 7 100.00 1.000. 1.9083 0.9942 1.9880 0.9940 1.9877 0.9939 1.9873 0.9937 1.9873 0.9937 1000.00 1.000 1.9980 0.9994 1.9987 0. 9993 1.9987 0.999 3 1.9987 0.9993 1.9987 0.9993 Table 4.-2 (Continued) p K RE= 10.0 RE= 15. 0 RE=20. 0 RE= 30.0 RE=40.0 e E e E e E e E e E 0.01 0. 001 0. 0002 0.0002 0.0002 0.0002 0.0002 0.0002 0. 0003 0.0003 0. 0003 0. 0003 0. 10 0. 001 0. 0003 0.0003 0.0003 0.0003 0.0004 0.0004 0. 0005 0.0005 0. CCC6 0. 0006 0. 15 0. 001 0. 0004 0.0004 0.0005 0.0005 0.0005 0.0005 0. 0007 0.0007 0. 0000 0. 0008 0.25 0. 001 0. 0007 0.0007 0.0009 0.0009 0.0011 0.0011 0. 0016 0.0016 0. CC20 0. 0020 0.50 0. 001 0. 03U0 0.0380 0.0007 0.0806 0. 1006 0. 1005 0. 1272 0. 1271 0. 14 15 0. 1414 0.75 0. 001 0. 2236 0.2234 0.2430 0.2435 0.2545 0.2542 0. 2717 0.2714 0. 2820 0. 281 7 1.00 0. 001 0. 3321 0.3317 0.3402 0.3479 0.3570 0.3566 0. 3715 0.3712 0. 3004 0. 3000 2. 00 0. 001 0. 5606 0.5600 0.5715 0.5710 0.5776 0.5770 0. 58eo 0.5874 0. 594 1 0. 59 3 5 3.00 0. 001 0. 6689 0.6682 0.6777 0.6770 0.6025 0.6018 0. 6908 0.6901 0. 6 958 0. 69 51 5. 00 0. 001 0. 776 5 0.7757 0.70 30 0.7822 0.7064 0.7856 0. 7923 0.79 16 0. 7959 0. 7951 7.50 0. 001 0. 8404 0.8396 0.0453 0.8445 0.0479 0.0470 0. 8523 0.8515 0. 8549 0. 0541 10. 00 0. 001 0. 8757 0.8749 0.0796 0.8708 0.8817 0.8008 0. 0853 0.0844 0. 6 674 0. 0065 40.00 0. 001 0. 9660 0.9650 0.9671 0. 966 1 0.9677 0.9667 0. 9687 0.9670 0. 9693 0. 9 604 100.00 0. 001 0. 986 1 0.9852 0.9866 0.9856 0.9 0 69 0.9059 0. 9073 0.90 6 3 0. 9675 0. 9 0 65 1000.00 0. 001 0. 9986 0.9976 0.9987 0.9977 0.9907 0.9977 0. 9987 0.9977 0. 9907 0. 9977 0.01 0. 010 0. 0006 0.0006 0.0007 0.0007 0.0008 0.0008 0. 0009 0.0009 0. 0010 0. 0010 0. 10 0. 010 0. 0000 0.0008 0.0009 0. 0009 0.00 10 0.0010 0. 0012 0.0012 0. CO 14 0. 00 14 0. 15 0. 010 0. 0009 0.0009 0.00 11 0.0011 0.0013 0.0013 0. 0016 0.0016 0. 00 19 0. 00 19 0.25 0. 010 0. 00 15 0.0015 0.0020 0.0020 0.0024 0.0024 0. 0033 0.0033 0. 0C4 1 0. 0041 0.50 0. 010 0.0467 0.0462 0.0860 0.0852 0.1040 0.1030 0. 12 84 0. 1271 0. 14 15 0. 1 40 1 0.75 0. 010 0. 2262 0.223 9 0.2456 0.243 1 0.2555 0.2530 0. 2715 0.2600 0. 2806 0. 2779 1.00 0. 010 0. 3 3 49 0.3316 0.3502 0.3468 0.3502 0.3546 0. 3714 0.3677 0. 3 7 90 0. 3752 2. 00 0. 010 0. 5647 0.5591 0.57 50 0.5693 0.5802 0.5745 0. 5892 0.5034 0. 5943 0. 50 85 3.00 0. 010 0. 6742 0.66 75 0.6823 0.6755 0.6863 0.6796 0. 6934 0.6065 0. 6972 0. 69 0 3 5. 00 0. 010 0. 7830 0.7752 0.7808 0.7810 0.79 1 7 0.7030 0. 7967 0.7800 0.7994 0. 79 15 7.50 0. 010 0. 8478 0.8394 0.8521 0.8437 0.8 54 3 0.8458 0. 8500 0.0 49 5 0. 06 00 0. 8515 10. 00 0. 010 0. 80 36 0. 8749 0.8871 0.8783 0.8888 0.8000 0. 8917 0.8829 0. 893 3 0. 0044 UO.00 0. 010 0. 9753 0.9656 0.9763 0.9666 0.9768 0.967 1 0. 9777 0.9680 0. 9701 0. 9 604 100.00 0. 010 0. 9959 0.9060 0.9963 . 0.9064 0.9965 0.9866 0. 9968 0.9069 0. 9970 0. 9871 1000.00 0. 010 1. 0086 0.9986 1.0086 0.9986 1.0086 0.9906 1. 0087 0.9987 1. 0087 0. 9987 Table 4-2 (Conti nued) p K RE«10.0 FE=15.0 RE=20.0 RE=30.0 RE*40.0 e E e E E E E E £ E 0.01 0. 100 0.0207 0.0261 0.0 3 30 0.0300 0.0362 0.0329 0.0427 0.0 380 0. 04 0 1 0.0437 0. 10 0.100 0.0332 0.03 02 0.0305 0.0350 0.0424 0.0 306 0.0499 0.0453 0.0550 0.0507 0. 15 0. 100 0.03(14 0.0349 0.0450 0. 0409 0.0 49 9 0.045 3 0.0 590 0.0 5 36 0. 066 1 0 .0 60 1 0.25 0.100 0.0551 0.0501 0.06 61 0.0600 0.0739 0.0672 0.0802 0.0002 0.0905 0.009 6 0.50 0. 100 0. 16 22 0.1475 0. 1845 0.1677 0.1964 0. 1706 0.2156 0. 19 60 0.2266 0.20 60 0.75 0. 100 0.2952 0. 2 6 04 0.3134 0. 2049 0.3223 0. 29 30 0.3371 0.30 6 5 0.3450 0.3136 1.00 0. 100 0.3970 0.3617 0.4125 0.3 75 0 0.4191 0.30 10 0.4307 0 . 39 1 5 0.4 363 0. 39 66 2. 00 0. 100 0.6 29 5 0.5723 0.6381 0.5001 0.6411 0.5 0 20 0.64 6 9 0.500 1 0.64 00 0. 5098 1.00 0.100 0.7424 0.6749 0.7407 0.6 0 06 0. 7 50 5 0.6023 0.7542 0.6057 0.755 0 0.60 64 5. 00 0. 100 0.056 3 0.77 05 0.0604 0.7022 0.8614 0.7031 0.0635 0.7 0 50 0.06 35 0.70 50 7.50 0.100 0.9 250 0.04 09 0.9 2 70 0.04 35 0.9204 0.0440 0.9296 0.0451 0.92 9 3 0.0449 10. 00 0.100 0.9631 0.0755 0.9654 0.8776 0.9 657 0.0779 0.9666 0.0707 0.9663 0.0 7 04 40. 00 0. 100 1. 06 20 0.9655 1.0626 0.9660 1.0626 0.9660 1.0 62 0 0.96 62 1.06 26 0.96 60 100.00 0. 100 1.00 4 5 0.9 05 9 1.00 4 6 0.9060 1.0047 0.906 1 1.0 04 8 0.90 62 1.0646 0.90 60 1000.00 0. 100 1.0905 0.9906 1.0905 0.9906 1.0905 0.9906 1.0 9 05 0.9906 1.0905 0.9906 0.01 0.200 0. 1015 0.0 04 6 0.1155 0.0962 0.1253 0. 1044 0. 14 54 0.1211 0. 16 16 0.1347 0.10 0.200 0. 1122 0.0935 0. 1200 0.1066 0. 1391 0. 1 159 0.1602 0. 13 35 0. 1765 0.1471 0. 15 0.200 0. 1243 0.1035 0. 14 19 0.1 102 0.1541 0.1204 0.1760 0. 1474 0. 19 30 0. 1615 0.25 0. 200 0. 1579 0. 1316 0.1790 0.1499 0.1941 0. 16 17 0.2195 0. 10 29 0.2372 0.1977 0.50 0. 200 0.2006 0.2330 0.3057 0.2547 0.3194 0.266 2 0.3432 0.2060 0.3579 0.2903 0. 75 0.200 0.4015 0.334 6 0.4224 0.3520 0.4 32 6 0.3605 0.4512 0. 37 60 0.46 19 0. 3049 1.00 0. 200 0.4973 0.4144 0.5144 0.4 2 06 0.52 20 0.4 350 0.53 65 0.4471 0.5442 0.4 53 5 2. 00 0.200 0.7217 0.6014 0.7310 0.6092 0.7339 0.6 1 16 0.74 04 0.6 170 0.7420 0.6190 3.00 0.200 0.H319 0.694 9 0.0403 0.7003 0.8416 0.7013 0.H4 52 0.7044 0.04 50 0.7049 5. 00 0.200 0.9405 0.7905 0.9524 0.7937 0.9526 0.79 38 0.9541 0.7951 0.9515 0.79 4 6 7.5 0 0. 200 1.0103 0.04 06 1.0200 0.85 07 1 .0206 0.8505 1.02 12 0.0510 1.02 04 0 . 0 50 3 10. CO 0.200 1.057 5 0.0012 1.0594 0.0020 1.0 591 0.0026 1.0 593 0.00 27 1.C5 04 0.0020 40.00 0. 200 1. 16 00 0.966 7 1. 1604 0.9670 1. 1603 0.9669 1. 1601 0.9660 1. 15 97 0.9 6 64 100.CO 0. 200 1. 10 36 0.9 863 1.1838 0.9065 1. 1038 0.906 5 1.1036 0.9063 1. 1634 0.9 0 62 1000.00 0.200 1. 1903 0.99 06 1. 1903 0.9986 1.1983 0.99O6 1.1983 0.9906 1. 1903 0 .9 9 0 6 Table 4-2 (Continued) p K RE= 10.0 HE=15.0 RE=20.0 RE=30.0 RE=40.0 e E e E e E e E G E 0.01 0.500 0.4940 0.3293 0.5401 0.3601 0.5662 0.3775 0.6230 0.4153 0.6623 0.4415 0. 10 0.500 0.5 128 0.3419 0.5603 0.3735 0.5878 0.39 18 0. 643 1 0.4287 0.6801 0.4534 0. 15 0. 500 0.5320 0.3547 0.5796 0.3864 0.6068 0.4045 0.6606 0.4404 0.696 1 0.4 641 0.25 0.500 0.5777 0.3 851 0.6239 0.4 160 0.6497 0.4331 0. 6997 0.4665 0. 73 18 0.4879 0.50 0. 500 0.6959 0.4639 0.7348 0.4899 0.7548 0.5032 0.7947 0.5 298 0. 8190 0.5460 0.75 0.500 0.7944 0.5296 0.8262 0.5508 0.8413 0.5609 0.8729 0.58 19 0. 89 16 0.5944 1.00 0. 500 0.87 18 0.5812 0.8981 0.5987 0.9096 0.6064 0.9354 0.6 2 36 0.9500 0.6333 2. 00 0. 500 1.058 5 0.7056 1.07 33 0.7155 1.0782 0.7 188 1.092 1 0.7281 1.0990 0.7327 3.00 0.500 1. 1559 0.7706 1. 1661 0.7774 1.1684 0.7790 1.1775 0.7850 1.10 13 0.7875 5. 00 0.500 1.2586 0.8391 1.2647 0. 8432 1.2652 0.8435 1. 2700 0.8467 1. 2 7 16 0.8477 7.50 0.500 1. 3231 0.8821 1. 3271 0.8847 1.3268 0.8845 1. 32 96 0.8864 1.3301 0.8867 10. 00 0.500 1.3600 0.9067 1. 36 29 0.9086 1.3624 0.908 3 1. 3642 0.9095 1.3644 0.9096 40.00 0.500 1.4597 0. 973 1 1.4603 0.9735 1.4 599 0.9732 1.4 601 0.9734 1.4 5 99 0.9732 100.CO 0.500 1.4834 0.9889 1.48 36 0.9891 1.4834 0.9889 1.4834 0.9890 1.4833 0 .9889 1000.00 0.500 1.4 98 3 0.9989 1. 49 8 3 0.9989 1.4983 0.9989 1.4 9 83 0.9989 .1.4 983 0.9909 0. 01 1.000 1.3819 0.6910 1.4308 0.7154 1.4514 0.7257 1.5027 0.7514 1.5336 0.7668 0. 10 1.000 1.3903 0.6952 1.4395 0.7198 1.4606 0.7303 1.5093 0.7546 1.5300 0.7690 0. 15 1.000 1.4007 0.7003 1.4488 0.7244 1.4694 0.7347 1.5162 0.7581 1.54 36 0.7718 0.25 1.000 1.4 266 0.7133 1.47 13 0.7356 1.4902 0.7451 1. 5329 0.7665 1.5575 0.7707 0.50 1.000 1.4943 0.7472 1.5302 0. 765 1 1.5444 0.7722 1.5777 0.7888 1.5964 0.7982 0.75 1.000 1.5507 0.7753 1.5801 0.7900 1.5906 0.7953 1. 6175 0.8087 1.6320 0.8160 1.00 1.000 1.5953 0. 7976 1.6201 0.8101 1.6283 0.8 14 1 1. 6506 0.8253 1.6623 0.8311 2. CO 1.000 1.7062 0.8531 1.7217 0.8608 1.7252 0.8626 1. 7385 0.8693 1. 74 4.3 0.8722 3.00 1.000 1.76 66 0.8 83 3 1.7781 0. 889 1 1.7797 0.8899 1. 7891 0.8945 1.7926 0.8963 5. CO 1.000 1.8329 0.9164 1.8402 0.9201 1.8406 0.9203 1.84 6 3 0.9231 1. 8479 0.9240 7.50 1.000 1.87 59 0.9380 1.88 12 0. 94 06 1.8805 0.9402 1.8844 0.9422 1. 885 1 0.9426 10. 00 1.000 1.9008 0.9504 1.9047 0.9523 1.9044 0.9522 1.9071 0.9536 1.9C75 0.9537 UO.OO 1.000 1.9706 0.9853 1.97 18 0.9859 1.9715 0.9857 1. 9 72 1 0.9860 1. 97 10 0.9 8 59 100.00 1.000 1.9877 0.9939 1.9883 0.9942 1.9880 0.9940 1.9883 0.9942 1.9883 0.9942 1000.00 1.000 1.9987 0.9993 1.9987 0.9993 1.9987 0.9993 1.9987 0.999 3 1.9987 0.9993 Table 4-3 Impaction E f f i c i e n c i e s at Re = 0.2, R = 200 p K = 0. 001 K-0. 010 ' K=0. 100 K = 0. 200 K=0. 500 K = 1. 00 3 c E E !•: E E £ E E E E R 0 . 0 1 0. 000 2 0. 00 0 2 0. 0002 0.0002 0.03 29 0. 0026 0. 0103 0.0086 0.054 0 0. 3 3 60 0. 1777 3. 3 00 9 0. 10 0.000 2 0. 0002 0. 000 2 0.0002 0.0329 0. 0027 0. 3 105 0.0 08 7 0.054 0 0. 0 3 65 0. 17 96 0. 0 ii 9 8 0 . 15 0. 000 2 0. 0002 0. 000 3 0.0002 0.00 30 0. 0027 0. 0106 0.0 080 0.0553 3. 3 3 69 0. 1000 3. 0904 0. 25 0. 0002 0. 0002 0. 0003 0.00O3 0.00 31 0. 00 20 0. 0 109 0.0091 0.0565 0. 0 3 76 0. 10 34 .3. 001 7 0 . 50 0. 000 3 0. 0001 0. 000 3 0.0003 0.03 34 0. 00 31 0. 3 120 0.0 100 0.06 03 3. 3402 0. 19 15 3. 0957 0. 7 5 0. 0001 0. 0003 0. 0004 0.0004 0.0039 0. 00 35 0. 3134 0.0112 0.0654 0. 0 4 3 6 0. 20 14 0. 1007 1. 00 0. 00 OU 0. 0 0 04 0. 000 5 0.0 005 0.0345 0. 004 1 0. 3 153 0.0 127 0.07 15 0. 3 4 77 0. 2 130 3. 1065 2. 00 0. 00 10 0. 00 10 0. 00 12 0.00 12 0.0395 0. 0086 0. 3 2 00 0.0 240 0.107 3 0. 0715 0. 27 14 3. 13 5 7 3. 00 0. 004 7 0. 0047 0. 00 53 0.0052 0.0321 0. 0 29 2 0. 3 661 0.0551 0.1627 0. 13 05 0. 34 29 3. 1715 5. 00 0. in 6 2 0 . 14 00 0. 140 2 0. 1467 0. 1703 0. 1548 0. 1 994 0. 1662 0.3005 0. 2003 0. 4973 0. 24 06 7. 50 0. 2718 0. 2735 0. 2760 0.2 74 0 0. 3D 54 0. 2777 0. 3390 0.28 32 0. 4540 0 . 30 27 0. 6702 3. 3 3 5 1 10. 00 0. 36 14 0. 363 0 0. 3o71 0.3 63 5 0.4022 0 . 3656 0. 4420 0.369 0 0.5727 0. 3018 0. 0097 0 . 404 0 no. 00 0. 7 10 3 0. 71 76 0. 7254 0.7103 0.7900 0. 7102 0. 0622 0.7 185 1.0000 3. 7200 1. 44 7 3 3. 72 3 7 100. 00 0. 05 9 4 0. 0'jtt5 0. 0679 0.0593 0.9451 0. 6592 1. 0310 0.8592 1. 2 09 2 0 . 0 59 5 1. 7214 0 . 0607 1 0 0 0 . 00 0. 9830 0 . 9 82 0 0. 9920 0.9830 1.0813 0 . 9830 1. 1794 0.9829 1. 4744 3. 9029 1. 9660 3. 9830 CTl Table 4-4 Comparison of Impaction E f f i c i e n c i e s for Potential Flow p K XP= -5 XP =" B x p = -40 xp = -100 Davies & Peetz [7] this work Subr. & Kuloor [12] this work House. & Gold. [13] this work this work 1 .5 10 40 0.001 0.001 0.001 0.001 0.38 0.77 0.84 0.91 0.4000 0.8020 0.8923 0.9712 0.38 0.77 0.88 0.3875 0.7856 0.8812 0.9677 — 0.3839 0.7757 0.8724 0.9639 0.3801 0.7729 0.8700 0.9622 1 5 10 40 0.10 0.10 0.10 0.10 0.47 0.79 0.85 0.93 0.4884 0.8269 0.9055 0.9746 0.46 0.80 0.89 0.4453 0.7900 0.8814 0.9672 — 0.4689 0.8003 0.8855 0.9673 0.4384 0.7786 0.8714 0.9627 1 5 10 40 1.0 1 .0 1.0 1 .0 — 0.8299 0.9337 0.9628 0.9898 — 0.8017 0.9030 0.9412 0.9825 0.80 0.91 0.94 0.98 0.8003 0.9077 0.9439 0.9830 0.7899 0.8913 0.9313 0.9782 CT) Table 4-5 Impaction E f f i c i e n c i e s for Potential Flow p K=0. 001 K=0 0 10 K = 0 100 K = 0 200 K = 0. 5 00 K = 1. 000 e K c K e E e F e E e E 0.0 1 0. 000 1 0.0001 0. 0100 0.0099 0. 1702 0. 1620 0 .3586 0.2900 0. 0 30 7 0.55 30 1. 4 9 93 0.7 497 0. 10 0. 0004 0.0004 0. 0122 0.0 121 0. 1642 0. 14 93 0 .3419 0. 2 04 9 0. 8230 0 .5406 1. 4 976 0.74U0 0. 15 0. 0007 0.0007 0. 0202 0.0200 0. 17 8 3 0. 162 1 0 .3516 0.2930 0. 0263 0.5 5 09 1. 4991 0 .7496 0. 25 0. 0 3 76 0.0 37 5 0. 058 9 0.0 50 3 0. 2171 0. 1 9 74 0 . 30 2 3 0. 3106 0. 0 40 5 0.56 0 3 1. 50 61 0.7 526 0.50 0. 1 040 0.18 39 0. 1951 0.19 33 0.32 20 0. 2934 0 .460 5 0.3905 0. 0094 0.5929 1. 52 06 0.7643 0.75 0. 2 '16 0 0.29 6 5 0. 3 04 2 0.30 12 0.4 1 15 0. 3 74 1 0 .5434 0.4520 0. 9 376 0.6 25 1 1. 554 7 0.7774 1.00 0. 3 HO5 0. 3801 0. 3 064 0.3026 0.40 22 0. 4 3 04 0 .6047 0.5039 0. 9 79 7 0.6532 1. 57S9 0.7099 2.00 0. 5741* 0.57 38 0. 5780 0.5731 0.65 07 0. 5988 0 .7643 0.6 36 9 1. 1001 0.7 3 34 1. 66 90 0.8299 3.00 0. 6724 0.6718 0. 677 3 0.670 6 0.7541 0. 6 855 0 .0542 0.71 10 1. 1 740 0.7027 1. 7143 0.0571 5.00 0. 7737 0.7729 0. 77 96 0.7718 0.0565 0. 7786 0 .9534 0.7945 1. 2 60 6 0.84 0 4 1. 7026 0.09 13 7.60 0. 33 50 0.8360 0. 84 25 0.8342 0.9214 0 . 8376 1 .0176 0.8400 1. 3193 0.0795 1. 03 2 0 0.9160 10.00 0. 0709 0.8700 0. 878 1 0.8694 0.9506 0. 8714 1 .0549 0. 879 1 1. 3545 0.9030 1. 0627 0.9313 1*0.00 0. 9632 0.9622 0. 97 2 3 0.9627 1.0509 0. 9627 1 .1573 0.964 4 1. 4558 0.9705 1. 9563 0.9782 100.00 0. 9648 0.9838 0. 994 4 0.9846 1.0830 0 . 9 84 6 1 .1822 0 . 9852 1. 4814 - 0.9876 1. 9814 0.9907 1000.00 0 . 9984 0.9974 1. 0084 0.9985 1.098 3 0 . 9985 1 . 1981 0.9985 1. 4981 0.9987 1. 9981 0.9990 4^ Table 4-6 Impaction E f f i c i e n c i e s at Re = 0.2 \ p K niNF= 100.0 RI NF= 50. 0 R INF= 25.0 RINF= 10.0 RI NP = 3.0 e E e E e E E E E E 0.01 0.001 0.0001 0.0001 0.00 01 0.0001 0.0001 0.0001 0.0001 0.0001 0.CC01 0.0001 0. 10 0.001 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.000 1 0.0001 0.0C02 0.0002 0. 15 0.001 0.0002 0.0002 0.0001 0.0001 0.0001 0.000 1 0.0001 0.0001 0.0002 0.0002 0.25 0.001 0.0002 0.0002 0.0002 0.0002 0 .000 1 0.000 1 0.0002 0.0002 0.0C03 0.0003 0.50 0.001 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.00 10 0.00 10 0.75 0.001 0.0003 0.0003 0.0003 0.0003 0.000 3 0.0003 0.0004 0.0004 0.1C5U 0.1053 1.00 0.001 0.ooou O.OOOU 0.0003 0.C003 0.0004 0.OOOU 0.0006 0.0006 0.26 03 0.2 600 2.00 0.001 0.00 10 0.0010 0.00 11 0.0011 0.0022 0.0022 0.1394 0. 139 3 0.5 102 0.5176 3.00 0.001 0.008 2 0.0002 0.0407 0. 04 07 0.12 60 0.1259 0.3181 0.3170 0.6552 0.6346 5. 00 0.001 0. 17U0 0.1738 0.2270 0.2268 0.3225 0.3222 0.50fc5 0.5000 0. 75 15 0.7507 7.50 0.001 0.3069 0.30 6 6 0.367 3 0.3669 0.4670 0.U66 5 0.6322 0.6316 0. 82 13 0.0205 10.00 0.001 0.3990 0.3 9 06 0.46 19 0.46 14 0.5590 0.5 50 4 0.7054 0.7046 0. 0602 0.0594 UO.OO 0.00 1 0. 7522 0.7515 0.7994 0.7986 0.0 52 7 0.85 18 0.9121 0.9 112 0. 96 12 0.9 602 100.00 0.001 0.00 27 0.0818 0.9090 0.9089 0.9365 0.9 356 0.9633 0.9623 0.9E4 1 0.98 31 1000.00 0.001 0.9067 0.9857 0.9902 0.9892 0.9933 0.9923 0.9962 0.9952 0. 9904 0.9 9 74 0.01 0.010 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.00 02 0.0002 0. 0004 O.OOOU 0. 10 0.010 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0. 0CC4 O.OOOU 0. 15 0.010 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0. 0005 0.0005 0.25 0.0 10 0.0002 0.0002 0.00 0 2 0.0002 0.0002 0.0002 0.0003 0.000 3 0.CCC7 0.0007 0.50 0.010 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 O.OOOU O.OOOU 0.0021 0.0021 0.75 0.010 O.OOOU O.OOOU 0.0004 0.0003 0.0004 0.0004 0.0005 0.0005 0. 1 142 0.1131 1.00 0.0 10 0.OOOU O.OOOU 0.0004 0.0004 0.000 5 0.0005 0.0009 0.0009 0. 266 1 0.263U 2. 00 0.010 0.0012 0.0012 0.00 14 0.0014 0.0027 0.0027 0. 14 2 1 0. 140 7 0.5253 0. 5201 3.00 0.010 0.0093 0.0092 0.0427 0.0422 0.12 79 0. 1266 0.322 1 0.3109 0.641 1 0.6367 5. 00 0.0 10 0. 1762 0 . 1 74 4 0.2298 0.2275 0.3261 0.3228 0.51U0 0.5089 0.7600 0.7525 7.50 0.010 0. 3 103 0.3072 0.37 12 0.3675 0.4710 0.467 1 0.63 60 0.6 32 5 0.0302 0.8220 10.CO 0.010 0.U011 0.3991 0.4666 0.46 19 0.564 7 0.559 1 0.7126 0.70,55 0.06 94 O.OtOfc UO.OO 0.010 0.7597 . 0. 752 1 0.80 7 3 0.7993 0.0612 0.85 27 0.921 1 0.9120 0.9 7 09 0.9613 100.00 0.010 0.89 15 0.8826 0.9189 0.9 090 0.9 4 50 0.9 36 4 0.9729 0.96 3 3 0.9 94 0 0.9042 1000.00 0.010 0.9965 0.9066 1.0001 0.9902 1.0032 0.9933 1.0062 0.99 6 3 1.OOOU 0.9985 Table 4-6 (Conti nued) p K 100.0 RINPa 50. 0 . RINF= 25.0 R IN F = 10.0 RI N F = 3.0 e E E E E E E E E E o.oi 0 . 100 0.0031 0.0028 0.0035 0.0032 0.0042 0.0039 0.0065 0.0059 0 . 0204 0.0186 0 . 10 0.100 0.0031 0.0028 0.0036 0.0032 0.0043 0.0 0 39 0.0066 0.0060 0.02 10 0.0191 0 . 15 0 . 100 0.0032 0.0029 0.0036 0.0033 0.0044 0.0040 0.0060 0.00 62 0.0229 0.0200 0.25 0. 100 0.00 3 3 0.0030 0.00 30 0.0034 0.0046 0.0042 0.0073 0.0067 0.0293 0.0266 0.50 0. 100 0.0037 0.0033 0.0043 0.0039 0.0054 0.0049 0.0095 0.0087 0.0747 0.0 679 0.75 0. 100 0.00'I2 0.0038 0.0050 0.0046 0.0067 0.006 1 0.0 133 0.0121 0.2 139 0.1945 1.00 0. 100 0.0050 0.0045 0.0061 0.0055 0.0004 0.0077 0.0200 0.0102 0.3305 0. 30 77 2.00 0. 100 0.01.12 0.0102 0.0162 0.0147 0.0349 0.0317 0.184 4 0.1676 0.5 50 3 0.5439 3.00 0.100 0.0457 0.0415 0.0004 0. 073 1 0.1586 0. 144 1 0.3630 0. .3 300 .7196 0.6542 5. CO 0.100 0.2002 0.1020 0.2576 0.2342 0.3615 0.3207 0.5658 0.5144 0. 64 C6 0.7642 7.5 0 0. 100 0. 314 17 0. 3106 0.4078- 0.3707 0.5171 0.4701 0.6996 0.6 360 0.9 133 0. 0.30 3 10. CO 0. 100 0. U14 1 3 0.UO 12 0.5102 0.4638 0.6172 0.5611 0.7709 0.7001 0.95 30 0.0671 40.00 0. 100 0.(3 27 3 0. 752 1 0.0794 0.7994 0.9382 0.0529 1. 0040 0.9120 1. 05 9.3 0.9 630 100.00 0. 100 0.9707 0.0025 1.0006 0.9096 1.0300 0. 9364 1.0599 0.96 36 1. 0033 0.904 0 1000.00 0. 100 . 1.005.3 0.9 066 1.0892 0.9902 1.0926 0.9933 1.0959 0.99 6 3 1. 090.3 0.9905 0.01 0.200 0.01 10 0.0092 0.0125 0.0104 0.0153 0.0120 0.0235 0.0196 0.0740 O.OC23 0 . 10 0.200 0.01 12 0.00 914 0.0127 0.0106 0.0156 0.0130 0.0240 0.0200 0.0757 0.0631 0 . 15 0. 200 0 . 0 1 in 0.0095 0.0129 0.0107 0.0159 0.0 132 0.0245 0.0204 0. 06 10 0.0675 0.25 0.200 ' 0.01 17 0.0090 0.0133 0.011 1 0.0165 0.0 130 0.0261 0.0217 0. 0902 0.0019 0.50 0.200 0.0129 0.0100 0.0149 0.0125 0.0190 0.0 159 0.0327 0.0272 0.1676 0. 1563 0.75 0.200 0.0 116 0.0122 0.0172 0.014 4 0.0227 0.0109 0.0 4 34 0.0 362 0.323 1 0.2 69 3 1 . 0 0 0.200 0.01f>9 0. 01 '1 0 .0202 0.0169 0.0279 0.0232 0.0607 0.0 50 6 0.4369 0.3641 2.00 0.200 0.0337 0.02 01 0.0457 0. 030 1 0.000 1 0.0667 0.2380 0. 190 3 0.6 9 16 0.5763 3. 00 0.200 0.00 19 0.0601 0.1180 0.090 3 0.1904 0. 165 3 0.4150 0.3459 0. 8 136 0.6700 5.00 0.200 0. 231.3 0. 1927 0.2928 0.244C 0.4049 0. 3 374 0.6274 0.5220 0.9359 0.7799 7.5 0 0.200 0.3791 0.3159 0.4509 0.3757 0.5699 0.4749 0.7690 0.6415 1.0CS6 0.0413 10.00 0.200 0.14 0 6 4 0. (4 04 5 0.5604 0.4 670 0.6769 0.564 1 0.0546 0.7122 1. 05 00 0.0757 UO. CO 0.200 0.9029 0.7524 0.9 59 7 0.7990 1.0241 0.0534 1.0964 0.9137 1. 1502 0.9652 100.00 0. 200 1.059 1 0.0026 1.09 17 0.9098 1. 1240 0. 9366 1.1560 0.9640 1.1029 0.9057 1000.CO 0.200 1. 10 39 0.9066 1. 1802 0.9902 1.1920 0.9933 1. 1956 0.9963 1.1983 0.9986 CTl CO Table 4-6 (Continued) p K HINF* 100.0 RINF= 50. 0 RINF = 25.0 RI NF = 10.0 RINF* 3.0 e E E E e E e E G E 0.01 0.500 0.0570 0.0306 0.0655 0.04 36 O.OBOU 0.0536 0.123U 0.00 2 3 0.4C57 0.2705 0 . 10 0.500 0.0507 0.0391 0.0665 0.0443 0.0817 0.0545 0.1251 0.08 34 0.4002 0.2668 0 . 15 0.500 0.050 3 0.0395 0.0672 0. 0440 0.0027 0.055 1 0.1269 0.00 4 6 0.4 1 13 0.2742 0.25 0.500 0.06 06 0.040U 0.0609 0. 0459 0.0851 0.0567 0. 1321 0.000 1 0.4409 0.299 3 0.50 0. 50 0 0.0651 0.0414 0.0746 0. 0497 0.0937 0.06 25 0. 152 7 0. 1010 0.5fc01 0.306 7 0.75 0.500 0 . 07 10 0.0473 0.00 2 3 0.05U8 0.10 56 0.0704 0. 182 1 0 . 1214 0.7039 0.4 69 3 1. CO 0. 500 0.07 0 1 0.052 1 0.09 17 0.06 1 1 0.120 5 0.000 3 0.2197 0. 146 5 0.0 0 14 0.5343 2.00 0.500 0.1206 0.0 004 0. 1407 0.0991 0.2115 0. 14 10 0.4184 0.2790 1.025U 0.60 3 6 3. 00 0. 500 0.1056 0.1237 0.2334 0. 1556 0.3333 0.2222 0.5976 0. 3904 1. 136 3 0.7575 5. 00 0. 50 0 0.3393 0.2262 0.4143 0.2762 0.5516 0.36/7 0.82 95 0.5530 1.2U92 0.0328 7. 50 0. 500 0.5 0 25 0.3350 0.59 1 1 0.3 9U0 0.7390 .0.4927 0.9922 0.66 1 5 1.3181 0.0787 10.00 0. 500 0.6 255 0.4 170 0.7109 0.4793 0.06U5 0.5 76 3 1.0910 0.7 27 3 1.3509 0.90 4 6 UO. 00 0.500 1.13 10 0.7540 1. 2025 0.80 17 1.2039 0.8560 1.3764 0.9176 1.U5S6 0.9 7 30 100.00 0. 500 1.32U5 0.0030 1. 3657 0.910U 1. 40 6-3 0.9 376 1.4482 0.9655 1.U83U 0.9090 1000.00 0.500 1.47 9') 0.9 066 1.4054 0.9903 1.4902 0.993U 1.4947 0.9964 1.U983 0.9909 0.01 1.000 0 . 1903 0.0952 0.2153 0. 1076 0.2 64 5 0. 1323 0.4052 0.2026 1..3C02 0 . 6 50 1 0 . 10 1.000 0. 192U 0.0962 0.2177 0.1008 0.2673 0.13 37 0.4082 0.20 41 1.2790 0.6399 0. 15 1.000 0 . 1937 0.0960 0.2192 0.1096 0.2693 0. 13U7 0.41 13 0.2O57 1.2£U6 0.64 2 3 0.25 1.000 0. 1966 0.0903 0.2228 0. 11 1U 0.2 742 0. 137 1 0.4203 0.2101 1.3 13 1 0.6565 0.50 1.000 0.2059 0. 1029 0.2344 0. 1172 0.2907 0.1U53 0. 454 7 0.2273 1.4 00 1 0.7041 0.75 1.000 0.2173 0 . 1007 0 . 24119 0 . 1245 0.3120 0 . 1560 0.4999 0.2499 1. 4 06 0 0.7430 1. CO 1.000 0.2 306 0.1153 0.26 59 0. 1329 0.3370 0.1605 0.5515 0.27 58 1. 54 5 1 0.7726 2.00 1.000 0.2970 0.1409 0.3510 0.1755 0.4 59 8 0.2299 0.7720 0. 30 (.0 1.6 0 26 0 . 8 •; 1 j 3. 00 1.000 0.3792 0.1096 0.4517 0.2259 0.59 52 0.2976 0.9590 0.479 5 1.75 3 3 0.8767 5. 00 1.000 0.5499 0 . 2749 0.6504 0.3252 0.8330 0.4 169 1.2117 0.6059 1. 8274 0.9137 7.50 1.000 0.73UU 0 . 3 6 72 0.8520 0.U260 1 .0494 0.52U7 1. 3965 0.6982 1.fc736 0.9368 10.00 1.000 0.0797 0.4390 1.00 39 0.5019 1. 1992 0.5 996 1.5108 0.7554 1. 9002 0 .9 50 1 UO. CO 1 .000 1.5 156 0.75 78 1.6118 0.0059 1.7222 0.06 1 1 1.0504 0.9252 1. 97 15 0.9057 100.00 1.000 1.7604 0.0842 1.8 2 40 0.9120 1.8793 0.9 396 1. 93 72 0.960 6 1. 900.3 0.9942 1000.CO 1.000 1.97 35 0.9860 1.9808 0.990U 1.9872 0.9936 1.9935 0.99 68 1.9908 0.99 94 Table 4-7 Impaction E f f i c i e n c i e s at Re = 40 p K RINF= 100.0 RINF= 50. 0 RIHF= 25.0 R INF= 10.0 RINF= 3.0 e E e E e E E E £ E 0.01 0.001 0.0015 0.0015 0.0011 0.0011 0.0008 0.0008 0.0005 0.0005 0.0C03 0.0003 0. 10 0.001 0.0023 0.0023 0.0018 0.0018 0.0013 0.0013 0.0009 0.0009 0.0CO6 0.0006 0. 15 0.001 0.0029 0.0029 0.0022 0.0022 0.0017 0.0017 0.0012 0.0012 0.0008 0.0008 0.25 0.001 0.OOU6 O.O0U6 0.0036 0.0036 0.0028 0.0028 0.0022 0.0021 0.0C20 0.0020 0.50 0.001 0.0270 0.0270 0.0260 0.0260 0.0260 0.0260 0.0U06 0.0U05 0. 14 15 0. 1 U1 U 0.75 0.001 0. 1200 0.1199 0.12U9 0. 12U7 0.1321 0. 1320 0.1615 0.1613 0.2820 0.2817 1.00 0.001 0.2001 0. 1999 0.2066 0.206U 0.2159 0.2156 0.2501 0.2U98 0.3804 0.3800 2.00 0.001 O.UOUO 0.U036 0.U130 0.U126 0.U256 0.U252 0.U670 0.4666 0.594 1 0.5935 3.00 0.001 0.5 189 0.518U 0.5287 0.5282 0.5U2U 0.5U 19 0.58U3 0.5837 0.6958 0. 69 51 5. CO 0.001 0.6U7U 0.6U67 0.657U 0.6568 0.6716 0.6709 0.7102 0.7095 0.7959 0.7951 7.50 0.001 0.7323 0.7316 0.7U20 0.7U13 0.7556 0.75U9 0.789U 0.7886 0.8549 0.85U1 10. CO 0.001 0.7828 0.7820 0.7921 0.7913 0.8050 0.80U 1 0.83UU 0.8336 0.6674 0.8865 UO.OO 0.001 0.9299 0.9290 0.9358 0. 93U8 0.9U23 0.9U 1U 0.9535 0.9525 0.9693 0.9684 100.00 0.001 C.9695 0.9686 0.9727 0.9717 0.9760 0.9750 0.9809 0.9799 0.9875 0.9865 1000.00 0.001 0.9968 0.9958 0.9972 0.9962 0.9975 0.9966 0.9981 0.9971 0.9987 0.9977 0.01 0.010 0.0021 0.0021 0.0017 0.0017 0.001U 0.0013 0.0011 0.0011 0.0010 0.00 10 0. 10 0.010 0.0030 0.0029 0.002U 0.002U 0.0019 0.0019 0.0015 0.0015 0.OC 14 0.0014 0. 15 0.010 0.0036 0.0036 0.00 30 0.0029 0.002U 0.002U 0.0019 0.0019 0.0019 0.0019 0.25 0.010 0.0055 0.005U 0.00U5 0.00U5 0.0037 0.0037 0.003 1 0.0031 0.0C4 1 0.0041 0.50 0.010 0.0282 0.0279 0.027U 0.0271 0.0277 0.0275 0.0U25 0.0U20 0. 14 15 0. 140 1 0.75 0.010 0. 1191 0.1179 0.12U1 0. 1229 0.1315 0. 1302 0.1608 0.1592 0.2806 0.2779 1.00 0.010 0. 1990 0.1971 0.2055 0.2035 0.21U7 0.2126 0.2U88 0.2U63 0.3790 0.3752 2. CO 0.010 0.4030 0.3990 0.U120 O.UOOO 0.U2U7 O.U 205 0.4662 0.4616 0.59U3 0.5885 3.00 0.010 0.5189 0.5137 0.5288 0.5236 0.5U25 0.5371 0.58U8 0.5790 0.6972 0.6903 5. 00 0.010 0.6U92 0.6U28 0.6592 0.6527 0.6735 0.6668 0.7126 0.7056 0.799U 0.7915 7.50 0.010 0.7357 0.728U 0.7U5U 0.7381 0.7592 0.7517 0.793U 0.7856 0.86C0 0.8515 10. 00 0.010 0.7872 0.7795 0.7966 0.7887 0.8096 0.8016 0.8396 0.8312 0.8933 0.8844 UO.OO 0.010 0.9380 0.9287 0.9U38 0. 9344 0.9506 0.9U12 0.9620 0.9524 0. 978 1 0.9684 100.00 0.010 0.9787 0.9690 0.98 19 0.9722 0.9853 0.9755 0.9902 0.9804 0.9970 0.9871 1000.00 0.010 1.0067 0.9967 1.0071 0.9971 1.007U 0.997U 1.0080 0.99 80 1.0087 0.9987 Table 4-7 (Continued) p K RINF«100.0 RIHF« 50.0 RINF= 25.0 RINP= 10.0 RINF=» 3.0 e E e E E I e E E B 0.01 0.100 0.0267 0.0243 0.0272 0.0247 0.0276 0.0251 0.0309 0.0281 0. 0401 0.0437 0. 10 0. 100 0.030B 0.0280 0.0315 0.0286 0.0320 0. 0291 0.0359 0.0326 0.0550 0.0 50 7 0. 15 0.100 0.0346 0.0314 . 0.0354 0.0322 0.0361 0.032U 0.0409 0.0372 0. 066 1 0.0 601 0.25 0. 100 0.01)53 0.0412 0.0466 0.0423 0.0 4 80 0.04 36 0.0558 0.0507 0.0905 0.0896 0.50 0. 100 0.0978 0.0809 0.1014 0.0922 0. 1062 0.0966 0. 1274 0.1150 0.2266 0.2060 0.75 0. 100 0. 17 17 0. 1561 0. 1773 0. 1612 0.1851 0. 1603 0.2158 0. 1962 0.3450 0.3136 1.00 0.100 0.21 10 0.2191 0.2401 0.2255 0.2578 0.2343 0.2941 0.2674 0.4 36 3 0.3966 2.00 0. 100 0.4 379 0.3901 0.4475 0.4068 0.4609 0.4 190 0.5060 0.4600 0.6400 0.5098 3. CO 0.100 0.5568 0.5062 0.5673 0.5158 0.5820 0.529 1 0.6263 0.5712 0.765 0 0.60 64 5.00 0. 100 0.6950 0.63 19 0.7059 0.64 18 0.7213 0.6557 0. 7648 0.6953 0.0635 0.70 50 7.50 0.100 0.7093 0.7176 0.7999 0.7272 0.0150 0.7409 0.8534 0.7750 0.9293 0.0449 10.00 0. 100 0.8464 0.7695 0.0567 0.770B • 0.0709 0.7917 0.904 0 0.H225 0.9663 0.0784 40. 00 0.100 1.0169 0.9245 1.0234 0.9304 1.0310 0.9 37 3 1.044 1 0.9492 1.06 26 0.9 6 60 100.00 0. 100 1. 06 30 0.9671 1.0674 0.9704 1.0712 0.97 30 1.0770 0.9791 1.0046 0. 9 0 60 1000.00 0. 100 1.0961 0.9965 1.0966 0.9969 1.0970 0.9973 1.0976 0.9979 1.0985 0.9986 0.01 0.200 0.0079 0.0733 0.0896 0.0747 0.09 18 0.0765 0. 1035 0.0062 0. 16 16 0.1347 0. 10 0.200 0.0975 0.0812 0.0995 0.0029 0. 10 21" 0.0051 0. 1 149 0.09 50 0. 1765 0.1471 0. 15 0.200 0. 1052 0.0877 0. 1075 0.0096 0.1 106 0.0921 0. 1248 0. 1040 0. 1918 0.1615 0.25 0.200 0. 1244 0. 1037 0. 1275 0.1062 0.1316 0. 1096 0.1497 0.1248 0.2372 0.1977 0.50 0.200 0. 1071 0.1559 0.1922 0.1602 0.1992 0. 1660 0.22EO 0. 1900 0.3579 0.2903 . 0.75 0.200 0.25 49 0.2124 0.2618 0.2182 0.2711 0.2260 0.3078 0.2565 0.46 19 0. 30 49 1. 00 0.200 0.3177 0.2648 0.3258 0.2715 0.3 3 69 0.2008 0.37C8 0.3157 0.5442 0. 4 53 5 2.00 0.200 0.50 37 0.4198 0.5145 0.4287 0.5291 0.4409 0.5796 0.40 30 0.74 28 0.6190 3. 00 0.200 0.6214 0.5178 0.6330 0.5275 0.6490 0.5400 0.7009 0.50 40 0.8450 0.7049 5.00 0.200 0.76 26 0.6355 0.7746 0.6455 0.7914 0.6595 0.0402 0.7002 0.95 35 0.79 4 6 7.50 0.200 0.86 13 0.7178 0.8730 0.7275 0.0095 0.7412 0.9328 0.777 3 1.C204 0.8503 10.00 0.200 0.9221 0.76 84 0.9334 0.7778 0.9491 0.7909 0.9074 0.0229 1.05 04 0.0020 HO. CO 0.200 1.1076 0.9230 1. 1140 0.9290 1.1232 0.9 360 1. 1382 0.9 40 5 1.1597 0.9t64 100.00 0.200 1.1596 0.9664 1. 1636 0.9697 1.1678 0.9732 1.1745 0.97U8 1. 1834 0.9862 1000.00 0.200 1.1957 0.9964 1. 1962 0.9968 1.1967 0.997 3 1. 1974 0.9979 1.1S83 0.9986 Table 4-7 (Continued) p K RINF=100.0 RINF= 50. 0 RINF" 25.0 RINF = 10.0 R l NF = 3.0 e B G E e E c E C E 0. 01 0.500 0.3695 0.2463 0.3771 0.2514 0.3066 0.2577 0.4326 0.2884 0.6623 0.4415 0. 10 0.500 0.3882 0.2588 0. 3964 0.2642 0.4066 0.2711 0.4538 0.3025 0.680 1 0.4534 0. 15 0.500 0.4000 0.2667 0.4085 0.2723 0.4192 0.2795 0.4675 0.3117 0. 696 1 0.4641 0.25 0.500 0.4250 0.2833 0.4341 0.2894 0.4458 0. 2972 0.4966 0.3311 0.73 18 0.4879 0.50 0.500 0.4 801 0.3254 0.4986 0.3324 0.5124 0.34 16 0.5694 0.3796 0.8190 0.5460 0.75 0.500 0.5465 0.3643 0.5583 0.3722 0.5737 0.3025 0.6354 0.4236 0.89 16 0.5944 1. 00 0.500 0.5991 0.3994 0.61 17 0.4078 0.6284 O.U 189 0.6934 0.4622 0.95C0 0.6333 2.00 0.500 0.7591 0.5061 0.7737 0.5158 0.7933 0.5289 0. 864 1 0.5761 1.0990 0.7327 3. 00 0.500 0.8673 0.5782 0.8827 0.5085 0.9034 0.6023 0.9743 0.6495 1. 1813 0.7875 5.00 0.500 1.0056 0.6704 1.0210 0.6806 1 .0424 0.6 949 1.1088 0.7 392 1. 27 16 0.8477 7.50 0.500 1. 1080 0.7387 1.1229 0. 7486 1.1442 0.7620 1.2032 0.8021 1.3301 0.8067 10.00 0. 500 1. 1736 0.7824 1. 1880 0.7920 1.2083 0.0055 1.2608 0.0405 1.3644 0.9096 40.CO 0.500 1.3866 0.9244 1.39 58 0.9305 1.4069 0.9380 1. 4278 0.9519 1.4699 0.9732 100.00 0.500 1.4498 0.9666 1.4550 0.9700 1.4 60 5 0.9737 1.4699 0.9799 1.4833 0.9809 1000.00 0.500 1.4946 0.9964 1.4953 0.9968 1.49 59 0.997 3 1.4970 0.9980 1.4983 0.9989 0.01 1.000 0.9103 0.4551 0.9274 0.4637 0.9491 0.4746 1.0503 0.5252 1.5336 0.7668 0. 10 1.000 0.9296 0.4648 0.9470 0.4735 0.9 69 4 0.4847 1.0702 0.5351 1.5380 0.7690 0. 15 1.000 0.9405 0.4702 0.9580 0.4790 0.9807 0.4904 1.0816 0.5408 1.5436 0.7718 0.25 1.000 0.9622 0.4811 0.9800 0.4900 1.0034 0.5017 1. 1048 0.5524 1.5575 0.7707 0.50 1.000 1.0139 0.5069 1.0326 0.5163 1.0573 0.5287 1.1604 0.5802 1. 5964 0.7982 0.75 1.000 1.06 10 0.5305 1.0803 0. 54 02 1.1062 0.5531 1.2106 0.6053 1.6320 0.8160 1. 00 1.000 1. 1036 0.5518 1.1233 0.5616 1.1500 0.5750 1.2554 0.6277 1.6623 0.8311 2.00 1.000 1. 237 1 0.6186 1.2579 0.6289 1.2068 0.6434 1.3925 0.6963 1.7443 0.8722 3.00 1.000 1.3323 0.6662 1.3531 0.6765 1.3829 0.69 14 1.4860 0.7430 1. 7926 0.8963 5.00 1.000 1.4604 0.7302 1.4008 0.74 04 1.5109 0.7554 1.6062 0.8031 1.8479 0.9240 7.50 1.000 1.5609 0.7805 1.5806 0.7903 1.6099 0.8049 1.6947 0.8474 1.6851 0.9426 10.00 1.000 1.6279 0.8139 1.6468 0.8234 1.6749 0.8375 1.7506 0.8753 1.9075 0.9537 40.CO 1.000 1.86 27 0.9313 1.8748 0.9374 1.8905 0.9452 1.9216 0.9608 1. 97 18 0.9859 100.00 1.000 1. 9379 0. 9690 1.9449 0.9725 1.9529 0.9764 1.9669 0.9035 1.9003 0.9942 1000.00 1.000 1.9931 0.9966 1.99 42 0.9971 1.9951 0.9976 1.9966 0.9983 1.9987 0.9993 CO Table 4-8 Interception E f f i c i e n c i e s RB RINP K=0 .001 K=0. 010 K=0. 100 K=0.200 K = 0 .500 K=1. 000 e E e E e E e E e E C E 0.2 100.0 0.0001 0.0001 0.0002 0.0002 0.0031 0.0028 0. 0110 0.0092 0.0578 0.0385 0. 1901 0.0950 0.5 100.0 0.0002 0.0002 0.0002 0.0002 0.0038 0.003U 0. 0135 0.0 112 0.0706 0.0U70 C.2317 0.1158 1.0 100.0 0.0002 0.0002 0.0003 0.0003 0.00U7 0.00U3 0. 0168 0.01140 0.087U 0.0582 0. 285 1- 0. 1U25 3.0 100.0 0.0003 0.0003 0.0005 0.0005 0.007U 0.0067 0. 0262 0.0218 0. 13U 1 0.089U CU 26<! 0.2132 5.0 100.0 0.0004 o.ooou 0.0006 0.0006 0.009U 0.0086 0. 0332 0.0277 0. 1673 0.1115 0.5 188 0.259U 10.0 100.0 0.0006 0.0006 0.0009 0.0009 0.013U 0.0121 0. 0U6U 0.0387 0.2259 0.150 6 C.6631 0.3315 15.0 100.0 o.oooa o.oooa 0.00 12 0.00 11 0.016U 0.01U9 0. 056U 0.0U70 0. 266 1 0.177U 0.7U7 3 0.3737 20.0 100.0 0.C01O 0.0010 0.00114 0.0013 0.0190 0.0 17 3 0. 06U7 0.0539 0.2972 0.1981 C.8051 0.U025 30.0 100.0 0.0012 0.0012 0.0017 0.0017 0.0232 0.0211 0. 0776 0.06U6 0.3398 0.2266 0.8723 0.U361 140.0 100.0 0.00H4 0.001« 0.0020 0.0020 0.0265 0.02U1 0. 0873 0.0728 0.3677 0.2U51 C.9082 0.U5U1 0.2 3.0 0.0001 0.0001 O.OOOU 0.OOOU 0.0207 0.0 188 0. 0757 0.0631 0.U093 0.2729 1.305U 0.6527 0.5 3.0 0.0001 0.0001 0.00014 O.OOOU 0.0207 0.0188 0. 0756 0.0630 0.U083 0.2722 1.3022 0.651 1 1.0 3.0 0.0001 0.0001 O.OOOU 0.OOOU 0.0210 0.0191 0. 0768 0.06U0 0.U1 17 0.27UU 1.3028 0. 65114 3.0 3.0 0.0001 0.0001 O.OOOU 0.OOOU 0.0220 0.0200 0. 0797 0.066U O.U 196 0.2797 1.3038 0.6519 5.0 3.0 0.0001 0.0001 0.0005 0.0005 0.0239 0.0217 0. 0859 0.0716 0.U399 0.2933 1.322U 0.6612 10.0 3.0 0.0002 0.0002 0.0006 0.0006 0.0287 0.026 1 0. 1015 0.08U6 0.U9U 1 0.329U 1.3825 0.6913 15.0 3.0 0.0002 0.0002 0.0007 0.0007 0.0331 0.0301 0. 1156 0.096U 0.5U0 1 0 . 3 60 1 1.U 310 0.7155 20.0 3.0 0.0002 0.0002 0.0008 0. 0007 0.03614 0.0331 0. 1255 0.10U5 0.5658 0.3772 1.U5 15 0.7257 140.0 3.0 0.COO3 0.0003 0.0010 0.0010 0.0U86 0.0UU2 0. 1625 0. 1354 0.6623 0.UU15 1.53U2 0.7671 30.0 3.0 0.0002 0.0002 0.0009 0.0009 0.01430 0.0391 0. 1U59 0.1215 0.6226 0.U152 1.5030 0.7515 •p. Table 4-9 Impaction E f f i c i e n c i e s Calculated at Re = 40, R = 100 under Various Conditions p KU W ABA Ii A •S MODEL HAT-PEL'S MODEL K STEl'5/CELL= 3 STEPS/CELL= 3 STEPS/CELL= 3 STEPS/CELL= 5 STEPS/CELL= 3 NA = 3 3 «R = 93 N A= 49 NR=9 3 NA=33 NR = 79 NA = 33 NR=93 NA = 33 NR=93 e n £ E • £ E E E E E 0.0 1 0.001 0.0015 0.00 15 0.00 14 0.0014 0.0021 0.0021 0. 00 16 0.0016 0.00 16 0.00 16 0.10 0.001 0 .0023 0.0023 0.0021 0.0021 0.003 1 0.0031 0.0 024 0.0024 0.0025 0.0025 0. 15 0.001 0.00 29 0.0029 0.0026 0.0026 0.0039 0.0039 0.0C30 0.0030 0.0031 0.003 1 0.25 0.001 0 .0046 0.0 04 6 0.00 42 0.0042 0.0061 0.0061 0. 0040 0.0048 0.0051 0.0051 0.50 0.001 0.0270 0.02 7 0 0.0265 0.C265 0.0324 0.0 324 0.0274 0.0271 0.0348 0.0347 0.75 0.001 0. 1200 0.1199 0.1224 0. 1223 0.1243 0. 1242 0. 1190 0.1197 0. 136 1 0. 1359 1 . 00 0.001 0.2001 0. 1999 0.20 36 0.2034 0.2044 0.2042 0.2CC0 0.1998 0. 2 195 0.2193 2.00 0.001 O.UOUO 0.4036 0.40 90 0.4 08 6 0.4085 0.4081 0. 4039 0.4035 0.4 301 0.4297 3 . 00 0.001 0.5189 0.5184 0.52 43 0.5238 0.52 3 5 0.5229 0.5180 0.5103 0.5479 0. 54 74 5 .00 0.001 0,fiU7U 0.6467 0.6525 0.65 19 0.6516 0.6510 0. 6473 0.6467 0.6708 0.6781 7.50 0.001 0.7323 0.73 16 0.7369 0.7 36 2 0.7361 0.7 354 0.7323 0.7315 0.7646 0.7639 10.00 0.001 0.7828 0. '820 0.7869 0.786 1 0.7862 0.7855 0.7027 0.7819 0.8 152 0.8144 UO.OO 0.001 0.9299 0.9290 0.93 17 0.9 308 0.93 16 0.9 306 0.9299 0.9290 0.9554 0.9544• 100.00 0.001 0.969 5 0. 9686 0.9 70 4 0.9695 0.9704 . 0.9694 0.96 95 0.9 686 0.9848 0.9038 1000.00 0.001 0.9968 0.9 95 8 0.99 69 0.9959 0.9968 0.9958 0.9968 0.9958 0.9909 0.9979 0.0 1 0.010 0.0021 0. 002 1 0.00 19 0.00 19 0.0027 0.0027 0.0022 0.0021 0.0022 0.0022 0.10 0.010 0.00 10 0.0029 0.0027 O.C027 0.0039 0.0038 0. 003 1 0.0031 0.0032 0.0032 0.15 0.010 0.0036 0.0036 0.00 33 0.003 3 0.0047 0.0046 0.0038 0.0037 0.0039 0.0039 0.25 0.010 0.00 55 0.0054 0.00 51 0.0050 0.0071 0.0070 0. 0 05 7 0.0057 0.006 1 0.0060 0.50 0.0 10 0.0282 0.0279 0.0 2 78 0.0275 0.0332 0.0329 0.02eo 0.0285 0.0357 0.0353 0.7 5 0.010 0.119 1 0. 117 9 0.12 15 0.120 3 0.1232 0.1220 0. 1 192 0.1100 0. 1351 0.1338 1 .00 0.010 0. 1990 0.197 1 0.2026 0.2006 0.2032 0.2012 0. 1990 0.19 71 0.2 102 0.2160 2.00 0.0 10 0.40 30 0.3990 0. 40 8 1 0.4 04 1 0.4 0 76 0.40 36 0. 4030 0.3990 0.4 291 0.424 0 3.00 0.010 0.5189 0.5 137 0.5243 0.5191 3.5234 0.5 183 0.5 109 0.5137 0.5480 0.5425 5 .00 0.010 0.6 49 2 0.6428 0.6544 0.6480 0.6535 0.6 470 0. 64 9 1 0.6427 0.6807 0.6 73 9 7.50 0.010 0.7 3 57 0.7284 0.7404 0.73 50 0.73 95 0.7322 0.7357 0.7284 0.7681 0.7605 10.00 0.010 0.7872 0.7795 0.79 13 0.70 35 0.7907 0.7829 0.7872 0.7 79 5 0. 0 199 0.8117 UO.OO 0.010 0.9380 0.9287 0.9399 0.9 306 0.9 3 97 0.9 30 4 0.9380 0.9287 0.96 38 0.9542 100.00 0.010 0.9787 0.96 90 0.9796 0.9699 0.9796 0.9699 0.9787 0.9690 0.9941 0. 9 04 3 1000.00 0.010 1.0067 0.9967 1.00 68 0.9968 1.0068 0.9968 1.0C67 0.9967 1.0087 0.9987 Table 4-9 (Continued) p K KUWABARA •S M3DEL HAPPEL'S MODEL STEPS/CELl= 3 STETS/CELL= 3 5TEPS/CELL= 3 STEPS/CELL= 5 STEPS/CELL= 3 NA=33 NR = 93 N A= 49 NR=9 3 N A=33 NR = 79 NA = 33 N R=93 NA = 3 3 NR = 93 e E e E e E e E c E 0.01 0.100 0.0267 0.0243 0.0265 0.024 1 0.0276 0.0251 0.0268 0.0244 0.0201 0.0256 0.10 0. 100 0.030H 0.0280 0.0306 0.0278 0.0320 0.0 291 0.0309 0.0281 0.0327 0.0297 0. 15 0.100 0.0346 0. 03 14 0.0344 0.031 3 0.0350 0.0326 0.0347 0.0316 0.0369 0.0335 0.25 0.100 0.0453 0. 04 12 0.0453 0.04 12 0.0468 0.0425 0.0455 0.0414 0.0490 0.0446 0.50 0.100 0.0970 0.0089 0.0993 0.0902 0. 1004 0.09 13 0.0982 0.0892 0. 1002 0.0 9E4 0.7 5 0. 100 0. 17 17 0. 156 1 0.1747 0. 1588 0.1752 0.159 3 0.1720 0.1563 0. 1076 0.1706 1.00 0. 100 0.2410 0.2191 0.2449 0.2227 0.2450 0.2228 0.24 12 0.2193 0.2606 0.2369 2.00 0. 100 0.4379 0. 3 98 1 0.4435 0.4031 0.4426 0.4024 0.4379 0.3981 0.4651 0.4228 3.00 0. 100 0.5568 0.5062 0.562 7 0.5 115 0.5617 0.5 107 0.5568 0. 50 62 0.587 3 0. 533 9 5.00 0. 100 0.6950 0. 63 19 0.7009 0.6372 0.69 97 0.6361 0.6950 0.6319 0.7205 0.6623 7.50 0. 100 0.7093 0. 7176 0.79 4 6 0.7223 0.793 6 0.7214 0.7893. 0.7175 0.024 1 0. 74 92 10.00 0.100 0.8464 0. 76 95 0.0511 0.7738 0.8502 0.7729 0.8464 0.7694 0.0015 0.80 14 40.00 0.100 1.0169 0.9245 1.0191 0.9265 1.0188 0.9262 1.0169 0.9245 1.0453 0.9503 100.00 0. 100 1.06 38 0. 967 1 1.0649 0.960 1 1.0648 0.9600 1.0638 0.9671 1.0809 0.9826 1000.00 0.100 1.0961 0.9965 1 .09 62 0.9966 1.0962 0.9966 1. 096 1 0.99 65 1.0905 0.9 9 66 0.0 1 0. 200 0.0879 0.0733 0.00 8 8 0.0740 0.0892 0.0744 0.0880 0.0733 0.0924 0.0770 0.10 0.200 0.09 7 5 0.0812 0.0984 0.0820 0.0 9 90 0.0025 0.0976 0.0813 0. 10 29 0.0658 0.15 0. 200 0.10 52 0.0077 0. 10 63 0. 08:3 6 0.1069 0.009 1 0.1C53 0.0878 0.1114 0.0929 0. 25 0.200 0.1244 0. 1037 0. 12 60 0. 1050 0.12 65 0.10 54 0. 1246 0.1038 0.1326 0.1105 0.50 0.2 00 0.1071 0. 1559 0.1899 0. 1583 0.1901 0 . 150 4 0.107 3 0. 1561 0.2008 0.1673 0.75 0. 200 0.25 49 0. 2 124 0.2509 0.2 158 0.2 5 87 0.2156 0.2552 0.2126 0. 27 32 0.2277 1. 00 0.200 0.3177 0.2 640 0.32 2 6 0.2688 0.3220 0.2683 0.3 179 0.2 64 9 0. .3 39 4 0.2620 2.00 0.200 0.50 37 0. 4198 0.5101 0.4251 0.5009 0.4241 0.5038 0.4199 0.5 3 30 0.4442 3 .00 0.200 0.6214 0.5178 0.6201 0.5234 3.6267 0.5223 0.62 15 0 . 5179 0.65 4 2 0. 5452 5.00 0.200 0.76 2 6 0.6355 0.7 691 0.6 4 09 0.7676 0.6 39 7 0. 76 26 0.6 3 55 0.7907 0.6656 7.50 0.200 0.06 1 3 0.7170 0.0673 0.7220 0. 0661. 0.7217 0. 86 13 0.7178 0.8990 0. 74 91 10.00 0.200 0.9221 0.7604 0.9275 0.7729 0.9264 0.7720 0. 922 1 0.7684 0.9603 0.0002 40.00 0.200 1. 1076 0.9230 1.1101 0.925 1 1.1098 0.9248 1. 1C76 0.9230 1.1386 0. 94 89 100.00 0.200 1. 159 6 0.9664 1. 160 8 0.9673 1.1607 0.9672 1. 1596 0.9664 1. 1784 0.9820 1000.00 0.200 1. 1957 0.9964 1. 19 59 0.9966 1.1950 0.9965 1. 1957 0.9964 1. 1982 0.9965 J Table 4-9 (Continued) p K KUWABA RA •S MODEL HAFPEL'S MODEL STEPS/CELl= .1 STEPS/CELL= 3 STEPS/CELL= 3 STEPS/CELL= 5 STEPS/CELL= 3 N A = 3 3 NR = 93 NA=49 NR=93 H A= 3 3 NR=79 NA=33 N R=9 3 NA = 3.3 NR = 93 C E e E e E e E e E 0.01 0. 50 0 0 . 3695 0.246 3 0.3750 0.2500 0.3734 0.2489 0.3694 0.2463 0.3864 0.2576 0. 10 0. 500 0. 3882 0.2588 0.3940 0.26 27 0.3923 0.2615 0.3883 0.2589 0.4066 0.2711 0. 15 0. 500 0.4000 0.2667 0.40 60 0.2707 0.4043 0.2695 0.4C02 0.2668 0.4 194 0.2796 0.25 0. 500 0.4250 0.2833 0.43 13 0.2875 0.4295 0.2863 0.4252 0.2835 0.4462 0.2974 0.50 0. 500 0.48 8 1 0.3254 0.49 52 0.3 301 3.4932 0.3288 0.4883 0.3255 0.5 133 0.3422 0.75 0. 500 0.5465 0. 364 3 0. 5543 0.3696 0.5522 0.3682 0.5468 0.3 64 5 0.5751 0.3834 1.00 0. 500 0.5991 0.3994 0.60 74 0.4049 0.6050 0.4034 0.5993 0.3995 0.6303 0.4202 2.00 0. 500 0.7 591 0. 506 1 0.7683 0.5 122 0.7658 0.5105 0.7593 0. 50 62 0.7969 0.5313 .1.00 0. 500 0.8673 0.5782 0.8765 0.5844 0. 874 1 0.5827 0. 8674 0.5783 C.9000 0.6059 5.00 0. 500 1.0056 0.6704 1.0143 0.6762 1.012 1 0.6747 1.0056 0.6704 1.0506 0.7004 7.50 0. 500 1. 1080 0.7387 1.1159 0.7439 1.114 1 0.7 427 •1. 1080 0.7387 1. 1549 0.7699 10.00 0. 500 1. 1736 0.7824 1. 1808 0.7872 1.1793 0.7862 1.1736 0.7824 1.2212 0.014 1 UO.OO 0. 500 1 . 3866 0.9244 1.3899 0.9266 1.3 896 0.9264 1.3 866 0.9244 1.4259 0.9506 100.00 0. 500 1.4498 0.9666 1.4513 0.96 75 1.4513 0.9675 1.4498 0.9 666 1.4740 0.9 82 7 1000.00 0. 500 1.4946 0.9964 1 . 49 4 7 0.9965 1.4948 0.9966 1. 4 946 0.99 64 1.4 979 0.9 9 86 0.0 1 1.000 0.9103 0.455 1 0.92 31 0.46 16 0.9188 0.4 59 4 0.9 102 0.4551 0.9475 0.4737 0. 10 1.000 0.9 29 6 0.4 64 8 0.9424 0.47 12 0.9381 0.4 690 0. 9296 0.4648 0.96 0 0 0. 4 04 0 0. 15 1.000 0.0110 5 0.4702 0.9534 0.4767 0.9490 0.4745 0.94C5 0.4 70 3 0.97 97 0.4 LS9 0. 25 1.000 0.96 22 0.4811 0.9 7 50 0.4375 0.9 70 8 0.4854 0. 9622 0.4811 1.0026 0.50 13 0.50 1.000 1.0139 0.5 06 9 1.0 2 70 0.5135 1.0229 0.5115 1.014 1 0. 50 70 1.057 5 0.52 £7 0.7 5 1.000 1.06 10 0.53 05 1.0743 0.5 37 1 1.0 703 0.5351 1. 06 1.3 0.530 6 1. 107 1 0.553 5 1.00 1.000 1.1036 0.55 18 1.1169 0.5584 1.1129 0.5 56 5 1. 1 C.l 8 0.5519 1. 15 17 0.5759 2.00 1.000 1.237 1 0. 6186 1.2504 0.6252 1.2469 0.6234 1.2374 0.6187 . 1.2911 0.6456 1.00 1.000 1 . 3323 0.666 2 1.34 50 0.6725 1.34 18 0.6709 1. 332.3 0.6662 1.3894 0.6947 5.00 1.000 1 . 46 0 4 0.7302 1.4720 0.7360 1. 4 696 0.7348 1.4 6 04 0.7302 1.5211 0.7606 7.50 1.000 1.5609 0.7805 1.5713 0.7856 1.5693 0.7846 1.56C9 0.7805 1.6 237 0.ei 19 10.00 1.000 1.6 279 0.8139 1. 6372 0.8 186 1.6358 0.8 179 1.6279 0.8139 1.69 16 0.0458 UO.OO 1.000 1.8627 0. 93 13 1.8666 0.9 3.3 3 1.8666 0.9333 1-8627 0.9313 1.9159 0.9579 100.00 1.000 1.9379 0. 9690 1.9400 0.9700 1.9401 0.9700 1.9379 0.9690 1.97 12 0.9856 1000.CO 1.000 1.99.31 0.9966 1.9934 0.9967 1,9934 0.9967 1.9932 0.9966 1.9978 0.9969 Table 4-10 Stagnation Pressures and Drag Coefficients BE H I NF P-ZEUO P-BEAH CDSKIM CDFOKM CDS/CDF CDTOTAL 0.2 100. 0 12.4746 -11.2923 18.6 3 20 18.6693 0.9900 3 7.3013 0.5 100. 0 6.3393 -5. 1553 8.9444 9.0223 0.9914 17.9667 1.0 1 00. 0 4.0705 -2.9527 5.3813 5. 5002 0.9784 10.00 15 3.0 100. 0 2.2916 - 1.3310 2.5783 2.7917 0.9236 5.36 99 5.0 100.0 1. 8575 -0.9727 1.8703 2. 1354 0.8 7 59 4.0057 10.0 100.0 1.4842 -0 .6910 1.2234 1. 5592 0.7046 2.7027 15.0 100.0 1. 3427 -0.6133 0.9553 1.3427 0.7115 2.2 90 1 20.0 100.0 1.2672 -0.5513 0.8022 1. 2022 0.667 3 2.0045 30.0 100.0 1. 1851 -0.5113 0.6221 1.0574 0.588 3 1.6795 40.0 100.0 1.1390 -0.5735 0.5255 1.0,362 0.5071 1.56 16 0.2 3.0 39.8528 -155.9388 122.3 114 153. 72 86 0.7956 276.0398 0.5 3. 0 16.4796 -6 1 .89 39 48.9497 61. 5427 0.7954 110.4924 1.0 3. 0 8.7176 -31.0064 24.9073 31.194 1 0.790 5 56. 10 13 3.0 3.0 3. 5405 - 10.20 30 8.4207 10.7602 0.7020 19.1090 5.0 3.0 2.5177 -6 .9 304 5.3 155 7.3744 0.7200 12.6099 10.0 3.0 1.7673 -4 .5321 3. 0185 4.7074 0 .6 30 5 7.0059 15.0 3.0 1.5193 -3.0959 2.2406 3.9771 0.5654 6.2258 20.0 3.0 1.3930 -3.6087 1.83 24 3.6603 0.500 6 5.4 927 30.0 3.0 1.2 66 8 -2.9315 1.3576 2.9204 0.46 49 4.2700 1*0.0 3.0 1.2027 - 1 . 7 336 1.1125 2.0441 0.5443 3. 1666 0.2 200. 0 11.8113 - 10 .4274 17.4483 17.4655 0.9990 34.9 130 0.2 100. 0 12.4746 - 11 .2923 18.6320 10.6693 0.9900 3 7.3013 0.2 50. 0 13.7751 - 13.1261 21.04 6 7 21.13 0 7 0.99 60 42.1774 0.2 25. 0 16.1542 - 16 .9694 25.8024 26.00 00 0.9952 51.0904 0.2 10. 0 21.7583 -30.2086 39.9402 40.90 0 5 0.9765 HO. 04 07 0.2 3.0 39.0528 -155.9388 122.3 114 153. 7206 0.7956 276.0390 40.0 100. 0 1.1390 -0.5735 0.5255 1.0362 0.5071 1.56 16 40.0 50. 0 1.1447 -0.5238 0.5253 0.99 93 0.5257 1.5246 1*0.0 25. 0 1.1483 -0.5818 0.5474 1.0 3 86 0.5271 1.5860 40.0 10.0 1.1586 -0.8610 0.6270 1.2358 0.5080 1.0636 40.0 3.0 1.2027 -1.7336 1. 1125 . 2.0441 0.5443 3. 1566 Table 4-11 Fluid Velocity, v , According to Davies and Peetz [7] at Re = 0.2 X -3.0 -2.9 -2. 8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2. 1 X 0.0 0.24128 0.23247 0. 22344 0. 21419 0.20470 0.194 96 0. 1 8498 0. 17475 0. 16426 0. 15350 0.1 0.24164 0.23286 0. 22385 0. 21462 0.20516' 0.19546 0. 18551 0. 17531 0. 16486 0. 15415 0.2 0.24272 0.23400 0. 22507 0. 21592 0.20654 0.19693 0. 18709 0. 17700 0. 16667 0. 156C9 0.3 0.24451 0.23590 0. 22709 0. 21806 0.20883 0. 19937 0. 18969 0. 17979 0. 16965 0. 15928 0.4 0.24698 0.23853 0. 22988 0. 22104 0.21199 0.20275 0. 19329 0. 18363 0. 17376 0. 16368 0.5 0.25013 0.24186 0. 23342 0. 22480 0.21600 0.20701 0. 19784 0. 18849 0. 17895 0. 16922 0.6 0.25 39 1 0.24587 0. 23767 0. 22932 0.22080 0.21212 0. 20328 0. 19429 0. 185 13 0. 17582 0.7 0.258 30 0.25052 0. 24259 0. 23454 0.22634 0.21801 0. 20955 0. 20096 0. 19224 0. 18339 0.8 0.26325 0.25576 0. 24814 0. 24041 0.23257 0.22462 0. 21657 0. 20841 0. 200 17 0. 19183 0.9 0.26874 0.26155 0. 25426 0. 24608 0.23942 0.23188 0. 22426 0. 21658 0.20083 0. 20103 1.0 0.27471 0.26784 0. 26090 0. 25390 0.2468 3 0.23972 0. 23255 0. 22535 0.21813 0. 2ioea 1.1 0.28 1 1 1 0.27459 0. 2 6 801 0. 26139 0.25474 0.24806 0. 24136 0. 23466 0.22796 0. 22127 1.2 0.28791 0.28173 0. 2 7553 0. 269 30 0. 26 307 0.25603 0. 25061 0. 24440 0.23823 0. 232 10 1.3 0.29506 0.28924 0. 2834 1 0. 277 58 0.27176 0.26597 0. 26021 0. 25450 0.24885 0. 24327 1.4 0.30251 0.29704 0. 29159 0. 20616 0.28076 0.27540 0. 2701 1 0. 26488 0.25973 0. 25468 1.5 0.31023 0. 305 1 1 0. 30002 0. 29498 0.20999 0.28507 0. 28022 0. 27546 0.27080 0. 26626 1.6 0.31815 0.31339 0. 30867 0. 30401 0.29941 0.29490 0. 29049 0. 28618 0.28199 0. 27793 1.7 0.32626 0.32183 0. 31747 0. 31318 0.30897 0.30486 0. 30086 0. 29698 0. 29323 0. 28962 1.8 0.33450 0.33 04 1 0. 32639 0. 32246 0.31862 0.3 1409 0. 31128 0. 30780 0.30446 0. 30128 1.9 0.34285 0.33908 0. 33539 0. 33180 0.32031 0.32494 0. 32170 0. 31860 0.31565 0. 31285 2.0 0.35128 0.34701 0. 34444 0. 34117 0.33801 0.33498 0. 33209 0. 32934 0. 32674 0. 32430 2. 1 0.35974 0.35657 0. 35349 0. 35053 0.34769 0.34498 0. 3424 1 0. 33998 0.33771 0. 33560 2.2 0. 36822 0.36533 0. 362 54 0. 35986 0.357 32 0. 35490 0. 35262 0. 350 50 0.34853 0. 34672 2.3 0.37670 0.37407 0. 37154 0. 36914 0.36686 0.36472 0. 36272 0. 36087 0.35917 0. 35762 2.4 0.38514 0.38276 0. 38049 0. 37833 0.376 31 0.37442 0. 37267 0. 37107 0.36962 0. 3 6 83 1 2.5 0. 39 354 0.39139 0. 38935 0. 38743 0.38564 0.30399 0. 38247 0. 38109 0.37986 0. 37877 2.6 0.40180 0.39994 0. 39812 0. 39 642 0.39484 0.39340 0. 39209 0. 39092 0.38989 0. 389C0 2.7 0.410 13 0.40040 0. 40678 0. 40 528 0.4 0390 0.40265 0. 40154 0. 40055 0.39970 0. 39897 2.8 0.41830 0.41675 0. 41532 0. 41400 0.4 1280 0.41 173 0. 41079 0. 40997 0.40928 0. 4087 1 2.9 0.42636 0.42499 0. 42373 0. 42258 0.42155 0.42064 0. 41985 0. 41918 0.4 1863 0. 41819 3.0 0.43432 0.433 11 0. 43200 0. 43101 0.43013 0.42936 0. 42871 0.42818 0.42775 0. 42744 ^4 VO Table 4-11 (Continued) X -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 - 1.2 -1. 1 X 0.0 0. 14249 0.13120 0.11964 0. 10780 0.09565 0.083 13 0.07012 0.05633 0.04 119 0.02339 0.1 0. 14318 0.13195 0. 12044 0. 10866 0.09656 0.08410 0.07114 0.05740 0.04230 0.02459 0.2 0. 1U526 0.134 17 0.12282 0.11120 0. 09927 0.08697 0.07417 0.06059 0.04566 0.02819 0.3 0 . 14867 0. 13783 0.12674 0.11538 0. 10373 0.09171 0.07919 0.06590 0.05127 0.03426 0.4 0. 15338 0.14287 0.13213 0. 12114 0. 10988 0.09825 0.08614 0.07327 0. 059 13 0.04279 0.5 0. 15931 0. 14920 0.13090 0. 12838 0 . 11760 0. 10648 0.09491 0.08260 0.06914 0.05372 0.6 0 . 16636 0. 15673 0.14694 0. 13697 0. 12676 0.1 1626 0. 10533 0.09375 0. 08 115 0.06687 0.7 0. 17443 0.16534 0.15613 0.14676 0. 13721 0 . 12740 0.11724 0. 10650 0 . 09490 0.08193 0.8 0. 18 3111 0.17490 0. 1663 1 0. 15760 0. 14876 0. 13972 0. 13039 0. 12059 0. 1 1010 0.09854 0.9 0. 19318 0.18528 0.17734 0. 16933 0. 16123 0.15300 0.14454 0.13574 0. 126 4 1 0. 11628 1.0 0.20361 0.19635 0. 18907 0 . 18177 0. 17444 0.16702 0. 15947 0.15167 0 . 14350 0.13475 1. 1 0.21160 0.20796 0.20135 0 . 19477 0. 18819 0.18159 0. 17492 0.16811 0 . 16 105 0. 1536 1 1.2 0.22602 0.22000 0.21405 0.20816 0.20232 0. 1965 1 0 . 19069 0. 18481 0. 17880 0.17255 1.3 0.23777 0.23236 0.22704 0.22182 0.21668 0.21162 0.20660 0.20158 0. 19652 0. 19133 1.4 0. 24971* 0.24490 0.24019 0.23560 0.23113 0.22676 0.22248 0.21824 0.21403 0.20977 1.5 0.26 185 0.25756 0.25342 0.24942 0.24556 0. 24183 0.23821 0'.23468 0.23 120 0.22775 1.6 0.27U0 1 0.27024 0.26663 0.26317 0.25987 0.25671 0.25368 0.25077 0.24795 0.24517 1.7 0 . 286 17 0.28287 0.27974 0.27678 0. 27398 0.27.134 0.26084 0.26647 0.26420 0.26199 1.8 0.29825 0.29540 0.29271 0.29019 0.28705 0.28566 0.28362 .0.28171 0.27991 0.27818 1.9 0.31022 0.30776 0.30548 0.30336 0.3014 1 0.29963 0.29798 0.29647 0.29506 0.29374 2.0 0.32203 0.31993 0.31800 0.31625 0.31465 0.3 1321 0.31191 0.31074 0.30966 0. 30866 2. 1 0.33366 0.33 100 0.33027 0.32883 0.32754 0.32640 0.32539 0.32450 0.32370 0.32296 2.2 0.3U506 0.34358 0.34225 0.34108 0. 34006 0.33918 0.33842 0.33777 0.337 19 0.33667 2. 3 0..35624 0.35501 0.353 93 0.35300 0.35222 0.35155 0.35101 0.35055 • 0.35016 0.34982 2.<4 0.36716 0. 366 16 0.36530 0. 36459 0.36400 0.36352 0.36315 0.36286 0.36 26 2 0. 36242 2.5 0.37783 0.37703 0.37637 0.37503 0.37541 0.37509 0.37487 0.37471 0. 37460 0.37450 2.6 0. 38821* 0.30761 0.38711 0.38673 0. 38646 0. 38628 0.38617 0.38612 0. 386 11 0.38611 2.7 0.39838 0.39791 0.39755 0. 39731 0.39715 0.39708 0. 39708 0.39712 0. 397 18 0.39725 2.8 0.40825 0.40792 0.40 769 0 . 40755 0.40751 0.40753 0.40760 0.40772 0.40784 0.40796 2.9 0.4 1787 0.41765 0.41752 0.41749 0.4 1753 0.41762 0.41777 0.41793 0.418 11 0.4ie27 3.0 0.42722 0.427 10 0.42707 0.42711 0.42722 0.42738 0.42758 0.42779 0.42000 0.42819 Table 4-1-1 (Continued) X -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0. 1 Y 0.0 0.00006 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.0 0.1 0.00144 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.00564 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0. 0 0.0 0.3 0.01265 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.0 0.4 0.02246 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.03496 0.0104 1 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.6 0.04987 0.02848 0.00006 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.7 0.06683 0.04847 0.02529 0.0 0. 0 0. 0 0.0 0.0 0.0 0.0 0.8 0.08536 0.06984 0.05107 0.02806 0.00005 0.0 0.0 0.0 0.0 0.0 0.9 0. 10496 0.09203 0.07693 0.05921 0. 03865 0.01558 0.0 0.0 0.0 0.0 1.0 0. 12519 0.11453 0.10247 0.08881 0.07353 0.05699 0.04008 0.02422 0.01124 0.00284 1.1 0. 14563 0.13693 0.12736 0.11682 0.10537 0.09330 0.08119 0.06988 0.06039 0.05366 1.2 0. 16597 0.15894 0.15138 0. 14324 0. 13461 0.12567 0.11680 0. 10851 0. 10136 0.09590 1.3 0. 18596 0.18032 0. 1 7438 0. 16810 0. 16155 0.15485 0.14823 0. 1 4200 0. 13647 0.13196 1.4 0.2054 3 0.20095 0.19630 0. 19145 0. 18645 0.18137 0.17635 0.17155 0. 167 17 0.16338 1.5 0.22427 0.22074 0.21710 0.21337 0.20953 0.20564 0.20176 0. 19799 0. 19443 0. 19118 1.6 0.24242 0.23964 0.23683 0.23394 0.23098 0.22796 0.22492 0.22190 0.2 1894 0.2161 1 1.7 0.25983 0.25768 0.25550 0.25327 0.25 098 0.2486 1 0.24618 0.24371 0. 24120 0.23870 1.8 0.27651 0.27405 0.27317 • 0.27145 0.26966 0.26779 0.26582 0.26375 0.26 159 0.25934 1.9 0.29246 0.291 19 0.28991 0.28858 0.20718 0.28567 0.28404 0.28228 0. 28038 0.27834 . 2.0 0.30770 0.30675 0.30578 0.30476 0.30364 0. 3024 1 0.30105 0.29952 0.29782 0.29594 2.1 0.32226 0.32 156 0.32084 0.32004 0. 31915 0.31814 0.31697 0.31562 0.3 1408 0.31234 2.2 0.336 18 0.33568 0.33514 0.33453 0.33381 0.33295 0.33193 0.33073 0.32932 0.32769 2.3 0.34949 0.349 14 0.34 875 0.34827 0.34768 0.34695 0.34605 0.34496 0.34365 0.34212 2.4 0.36222 0.36199 0.36171 0.36134 0.36085 0.36022 0.35940 0.35840 0.35717 0.35572 2. 5 0.37441 0. 37428 0.37408 0.37379 0.37338 0.37281 0.37207 0.37113 0.36990 0.36860 2.6 0.30609 0.38603 0.38590 0.38 567 0.38531 0.38480 0.38412 0.38323 0.38214 0.36082 2.7 0.39729 0.39729 0.39 720 0.39702 0.39671 0.39624 0.39560 0.39476 0.39372 0.39245 2.8 0.40805 0.40809 0.40804 0.40789 0.40760 0. 40717 0.40656 0.40576 0.40476 0.40354 2.9 0.41039 0.4 1845 0.41843 0.41830 0.41804 0.4 1763 0.41705 0.41629 0.4 1532 0.414 14 3.0 0.42834 0.42842 0.42842 0.42830 0.42806 0.42767 0.42711 0.42637 0.42543 0.42430 182 LOOmm—J ^ rna)r^^ox(NCT'-CDor^o^corninm r^^  I OOOOOOOOOOOO*— t— r-f-r-t— r-f- f—<-«— t-«—f-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 I OOOOOOOOOOOO*-*— r-*-«-*-»— •—«-«-«-*-«— «— «- «— 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 Ooo<^c4»^v^no^^alno^•coN^ao(N^v0^coc^Of•lN ©ooooooooooo«-»-«~'-*-^*-^«~«-<-^^^»-«-»-«-«-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 c ^ c ^ ^ c n a o r ^ u ^ ^ o j ^ ^ r N C O f n c o i ^ P , " T - c r p*ofN=J ,vDcocro»— CN<N oooooooooooo'-'-'-«-«-'-«-«-«-'-'-'-«-«-'-«-«-'-»-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 I ©©00©000©000«-t-«~«-«-«-^«-*-*-"-^«-'-«-*— «~«--*-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 I tO C7> ^  O 3" d" mi— IALT)T— (N o fl .fN CO *"Ov0CT'OC0U1^'"«-O ©*— c^ rn^ u^ f^^ r^ ccc^ cno'—^ CNrNrsir^ m © O O © © O O O O G 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 I © «— CMrn^ 3-mvop*r»coc)>cr. o o «— ^ N ^ f n n m s a - a ^ ^ i / . i r , i/iui o©oo©ooooooo©f-'-«-«-«-«-«-'-*-r-.j-«-t-«~.J— — «- «-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 I 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 t o^fsinju^LONor^Lnujo^oo^r-cNnjiNn OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO i O r ^ c ^ o ^ ^ h c c r ^ ^ c ^ c r * o c o n r vo fN i> «- ^ s f ^ c o o c ^ o c h v 0 2 ) c o s ^ o n o ^ t / ^ • - 3 l n t f « - v O D c o ^ l n J ( N c x ^ ^ o v o ^ f f l ^ o ^ c ^ ' , ? ^ ' - J ^ c ^ l n ^ o ' " l N n ©oo©©oo©oooo©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 o OOOOOOOOOO^^^^^^^^^^r\|!NfNiN(N(N<N<NO|lNn Table 4-12 (Continued) X -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 - 1.2 -1. 1 X 0.0 -0.0 -0.0 -0.0 -0. 0 -0. 0 -0.0 -0.0 -0.0 -0.0 -0. 0 0. 1 0.01070 0.01081 0.01086 0. 01082 0.01065 0.01028 0.00963 0.00857 0.00689 0. 00425 0.2 0.02131 0.02153 0.02164 0. 02156 0.02123 0.02054 0.01931 0.01730 0.01414 0. 00925 0.3 0.03175 0.03208 0.03224 0. 03214 0.03169 0.03073 0.02904 0.02630 0.02203 0. 01552 0.4 0.04194 0.04236 0.04258 0. 04248 0.04194 0.04080 0.03878 0.03556 0.03060 0. 02320 0.5 0.05179 0. 0523 1 0.05257 0. 05248 0.05190 0. 05065 0.04845 0.04498 0.03974 0. 03209 0.6 0.06 124 0.06184 0.06215 0. 06208 0.06148 0. 06018 0.05792 0.05439 0. 049 18 0. 04 174 0.7 0.07024 0.07089 0.07123 0. 07118 0.07058 0.06927 0.06703 0.06359 0.05858 0. 05162 0.8 0.07873 0.0794 1 0.07977 0. 07972 0.07913 0.07784 0.07566 0.07235 0.06765 0. 06126 0.9 . 0.08668 0.08736 0.08772 0. 08766 0.08706 0.08578 0.08367 0.08052 0. 076 1 3 0. 07029 1.0 0.09405 0.09471 0.09504 0. 09495 0.09432 0.09305 0.09098 0.08796 0.08384 0. 07846 1.1 0. 10085 0.10146 0.10173 0. 10157 0. 10090 0.09959 0.09754 0.09461 0.09069 0. 08565 1.2 0. 10706 0.10759 0.10777 0. 10753 0. 10678 0. 10542 0. 10335 0. 10046 0.09666 0. 09185 1.3 0. 11269 0. 113 12 0. 11320 0. 11284 0. 11199 0. 1 1054 0. 10841 0.10551 0. 10176 0. 09709 1.4 0. 11776 0. 11807 0.11801 0. 11753 0. 11654 0.1 1498 0. 11277 0. 10982 0. 10607 0. 10145 1.5 0.12230 0.12247 0. 12226 0. 12163 0. 12050 0.11880 0. 11647 0.11344 0. 10964 0. 10504 1.6 0. 12633 0.12634 0.12597 0. 12517 0. 12388 0.12204 0. 11958 0.1 1644 0. 1 1257 0. 10793 1.7 0. 12988 0. 12972 0.12919 0. 12822 0. 12676 0. 12476 0. 12216 0.1 1890 0. 1 1494 0. 11025 1.8 0.13299 0.13266 0.13195 0. 1 30 80 0. 12918 0.12701 0. 12426 0.12008 0. 1 1683 0. 11208 1.9 0.13570 0. 13519 0. 13430 0. 1 3298 0. 13118 0.12886 0.12597 0. 12246 0. 11831 0. 1 1349 2.0 0.11003 0. 13735 0.13628 0. 13479 0. 13282 0.13035 0.12731 0.12370 0. 11946 0. 1 1457 2. 1 0. 14002 0.139 17 0.13 793 0. 13627 0. 13415 0. 13 15 2 0.12836 0.12463 0. 12031 0. 11537 2.2 0.14172 0.14070 0.13929 0. 1 3747 0. 13520 0. 13244 0. 12916 0.12533 0. 12093 0. 11594 2. 3 0.14314 0. 14196 0. 14040 0. 13842 0. 13601 0.13312 0.12973 0. 12582 0.12136 0. 11634 2.4 0.14431 0. 14298 0.14127 0. 13916 0. 13662 0. 1336 1 0.13013 0.12614 0. 12162 0. 11658 2.5 0.14527 0. 14380 0. 14195 0. 13971 0. 13705 0.13394 0. 13037 0.12632 0. 12177 0. 11671 2.6 0.14604 0.14443 0.14246 0. 14010 0. 13733 0.134 13 0. 13049 0.126 38 0. 12180 0. 11675 2.7 0.14664 0.14491 0.14282 0. 1 40 35 0. 13749 0.13421 0. 13050 0.12636 0. 12176 0. 1167 1 2.8 0. 14710 0.14525 0.14305 0. 14048 0. 13754 0.134 19 0. 13044 0. 12626 0. 12166 0. 11662 2.9 0.14742 0.14547 0. 14317 0. 140 52 0. 13750 0.134 10 0. 13030 0.12611 0. 12150 0. 1 164 9 3.0 0. 14764 0.14558 0.14320 0. 14047 0. 13739 0.13394 0. 13012 0.12591 0. 12131 0. 11632 CO CO Table 4-12 (Continued) X -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0. 1 Y 0.0 -0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.00015 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.00173 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.00572 0.0 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 O.U 0.01233 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.02117 0.00590 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.6 0.03144 0.01755 -0.00055 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.04226 0.03007 0.01482 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.8 0.05290 0.04234 0.02956 0.01492 -0. 00057 0.0 0.0 0.0 0.0 0.0 0.9 0.06282 0.05361 0.04277 0.030 63 0.01797 0.00603 0.0 0.0 0.0 0.0 1.0 0.07170 0.06355 0.05410 0.04365 0. 03274 0.02221 0.01299 0.00603 0.00182 0.00014 1.1 0.07943 0.07203 0.0 63 52 0.05415 0.04430' 0.03450 0.02530 0.01725 0.01066 0.00537 1.2 0.08600 0.07909 0.07121 0.06252 0.05328 0.04384 0.03454 0.02572 0.01757 0.01009 1.3 0.09147 0.08488 0.0 773 9 0.06912 0.06023 0.05095 0.04154 0.03220 0.02308 0.01424 1.4 0.09 59 6 0.08956 0.08231 0.07428 0.06559 0.05642 0.04691 0.03724 0.02755 0.01790 1.5 0.09959 0.09330 0.00619 0.07831 0.06975 0.06064 0.05110 0.04126 0.03 124 0.021 15 1.6 0. 10250 0.09626 0.08923 0.08145 0. 07299 0.06394 0.05441 0.04452 0.03435 0.02405 1.7 0. 10480 0.09858 0.09160 0.08390 0.07552 0.06656 0.05709 0.04721 0.03703 0.02666 1.8 0.10659 0.1C038 0.09344 0.08580 0.07752 0.06865 0.05927 0.04948 0.03937 0.02905 1.9 0. 10798 0.10176 0.09486 0.08729 0. 07910 0.07035 0.0 6109 0.05143 0.04 144 0.03 124 2.0 0. 10902 0.10282 0.09595 0.08846 0.08037 0.07174 0.06264 0.05313 0.04331 0.03328 2. 1 0. 10980 0. 1036 1 0.09679 0.08938 0.08140 0.07291 0.06397 0.05464 0. 04502 0.03519 2.2 0.11036 0. 104 19 0.09742 0.09010 0.08225 0.07391 0.06514 0.05601 0.04660 0.03698 2. 3 0. 11075 0.10461 0.09791 0.09068 0.08295 0.07477 0.0 661 8 0.05726 0.04806 0.03e67 2.4 0. 11100 0. 10489 0.09827 0.09114 0.08355 0.07553 0.06713 0.05841 0.04944 0.04028 2.5 0. 11115 0. 105 08 0.09053 0.09152 0.08406 0.07620 0.0 6799 0.05948 0. 0507 3 0.04181 2.6 0.11121 0. 10520 0.09873 0.09183 0.08451 0.07682 0.06880 0.060 50 0.05 197 0.04327 2.7 0.11121 0. 10526 0.09688 0.09209 0.08491 0.07739 0.06955 0.06146 0.05315 0.04468 2.8 0. 11116 0.10527 0.09899 0.09231 0.08528 0.07792 0.07027 0.06237 0.054 28 0.04603 2.9 0. 11107 0. 105 26 0.09907 0.09251 0.08562 0.07842 0.07095 0.06325 0.055 37 0.04734 3.0 0. 11096 0. 10522 0.09913 0.09269 0. 08594 0.07890 0.07161 0.06410 0. 05642 0.04860 Table 4-13 Fluid Velocity, v , as Interpolated from the Numerical Flowfield at Re„ = 0.2, R = 200 c 00 X -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2. 1 1 0.0 0.20665 0. 19716 0.18753 0. 17759 0. 16740 0.15701 0. 14630 0.13532 0. 124 11 0. 11267 0. 1 0.20715 0. 19770 0.18812 0. 17821 0. 16807 0. 15774 0.14707 0.13615 0. 12501 0. 11363 0.2 0.20794 0.19856 0.18905 0. 17920 0. 16914 0.15889 0. 14829 0.13746 0. 12642 0. 115 14 0.3 0.20912 0.2C001 0.19083 0. 18126 0. 17154 0. 16162 0.15137 0. 14092 0. 13027 0. 11938 0.4 0.21250 0.20356 0.19440 0.18507 0. 17552 0. 16574 0.15568 0.14544 0.13500 0. 12426 0.5 0.21615 0.20738 0. 19043 0. 18921 0. 17987 0.17027 0. 1 6077 0.15112 0. 14125 0. 13 1 15 0.6 0.22011 0.2 1 183 0.2033 1 0.19463 0. 18584 0. 17675 0.16754 0.15819 0. 14057 0. 13E81 0.7 0.22568 0.2 1766 0.20932 0.20092 0.19233 0.18354 0.17467 0. 16558 0. 15677 0. 14787 0.8 0.23149 0.22365 0.21550 0.20749 0. 19924 0.19122 0.18307 0. 17474 0. 166 35 0. 15786 0.9 0.23751 0. 23010 0.22272 0.21525 0.20761 0.19997 0.19209 0. 18410 0. 17623 0. 16 843 1.0 0.244 01 0.23772 ' 0.23069 0.22345 0.21620 0.20003 0.20142 0.19444 0. 10728 0. 18014 1.1 0.25227 0.24552 0.23877 0.23184 0.22522 0.21857 0.21201 0.20535 0. 19868 0. 1 9200 1.2 0.25908 0.25348 0.24 73 3 0.24123 0.23505 0.22086 0.22268 0.21642 0.21047 0. 20466 1.3 0. 260 15 0.26247 0.25663 0.25090 0. 24501 0.23929 0.23352 0.22820 0.22283 0. 21756 1.4 0.27693 0.27 146 0.26603 0.260 53 0.25517 0. 2501 1 0.24511 0.24017 0.23526 0. 23052 1.5 0.20579 0.28054 0.2 754 9 0.27066 0.26601 0.26127 0.25673 0.25211 0. 2479 1 0. 24379 1.6 0.29455 0.29003 0.28554 0.28115 0.27673 0.27248 0.26819 0.26441 0. 26064 0. 257 1 1 1.7 0. 30 399 0.29907 0.29563 0.29157 0.287 50 0.28370 0.20012 0.27669 0.27339 0. 27024 1.0 0.31349 0.30956 0.30 575 0.30189 0.29856 0. 295 13 0.29201 0.28808 0. 20602 0. 2 B 3 3 6 1.9 0.32302 0.3 1927 0.31586 0.31261 0.30951 0.30657 0.30366 0.30107 0.29057 0. 29644 2.0 0.33233 0.32923 0.32615 0.32333 0.32045 0.31781 0.31532 0.31307 0.31109 0. 309 15 2.1 0.34200 0,33921 0.33643 0.33378 0.33136 0.32897 0.32695 0.32500 0.32327 0. 32 182 2.2 0.35167 0.34904 0.34665 0.34421 0.34 209 0.34018 0.33832 0.33679 0.33532 0. 334 C4 2.3 0.36 125 0.35085 0.35660 0.35472 0. 35278 0.35112 0.34964 0.34025 0. 347 14 0. 346 10 2.4 0.37062 0.36864 0.36667 0.36494 0.36 34 2 0.36190 0. 36069 0.35959 0. 35860 • 0. 35796 2.5 0.38001 0.37024 6.37669 0.37 508 0. 37 37 5 0.37262 0.37147 0.37062 0.36999 0. 36946 2.6 0.30945 0.38700 0.3063 9 0.30517 0. 38 39 5 0.38295 0.38219 0.30144 0. 38 102 0. 36078 2.7 0.39861 0.39731 0.39 60 1 0. 39495 0.39404 0.39315 0.39253 0.39219 0.39 102 0. 39152 2.8 0.407/0 0.40657 0.40 557 0.40456 0.4 0 380 0.40327 0.40275 0.40251 0.40230 0. 402 13 2.9 0.41672 0.4 1568 0.41482 0.41411 0.4 1343 0.4 1302 0.41287 0.41251 0.4 1237 0. 41252 3.0 0.42560 0.42469 0.42391 0.42337 0.42300 0.42266 0.42242 0.42236 0.42237 0. 42257 Table 4 -13 (Continued) X -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 - 1.2 -1. 1 X 0.0 0. 10097 0.08905 0.07699 0.06485 0.05273 0.04080 0.02936 0.01879 0.00959 0. 00294 0.1 0. 10199 0. 090 14 0.07015 0.06607 0.05400 0.04211 0.03065 0.02003 0.01068 0. 00364 0.2 0. 10366 0.092 16 0.00052 0.06878 0.05705 0.04543 0.0341 1 0.02348 0.01388 0. 00587 0.3 0. 10B23 0. 09691 0.08543 0.07384 0.06222 0.05069 0.03972 0.02911 0. 01925 0. 01033 0.4 0.11330 0.10266 0.09180 0.08082 0.06977 0.05871 0.04773 0.0 370 3 0.02696 0. 01727 0.5 0.12090 0.11049 0.09993 0.08923 0.07867 0.06833 0.05795 0.04756 0. 037 17 0. 027C4 0.6 0. 12090 0.11924 0.10951 0.09966 0.08972 0.07968 0.06981 0.06007 0.05007 0. 03969 0.7 0.13885 0. 12964 0.12033 0.11092 0. 10192 0.09287 0.08366 0.07429 0. 0650 1 0. 055 1 1 0.8 0.149 19 0. 14061 0. 13235 0. 12402 0.11556 0.10699 0.09880 0.09040 0.08165 0. 07267 0.9 0.16077 0.153 10 0.14532 0.13747 0. 13002 0. 12260 0. 1 1508 0.10757 0.09994 0. 09183 1.0 0.17302 0.16576 0.15887 0. 15216 0. 14546 0. 13863 0. 13215 0.12560 0. 1189 1 0. 1 1204 1.1 0.10560 0.17940 0.17330 0.16711 0. 16 115 0.15549 0.14986 0.14415 0. 13050 0. 13274 1.2 0.19095 0.193 35 0.18769 0.18253 0. 17752 0. 17246 0.16764 0.16302 0. 15835 0. 1535 1 1.3 0.21240 0.20737 0.20275 0. 19819 0. 19373 0.18965 0.18563 0.18176 0. 17800 0. 1 7425 1.4 0.22606 0.22188 0.21771 0.21376 0. 21019 0.20667 0.20349 0.20031 0. 197 34 0. 19454 1.5 0.23997 0.23617 0.23267 0.22944 0. 226 39 0.22368 0.22092 0.21861 0. 216 28 0. 214C4 1.6 0.25365 0.2505 1 0.24759 0.24490 0.24248 0.24015 0.23821 0.23636 0.23467 0. 23308 1.7 0.26737 0.26474 0.26232 0.26016 0.25814 0.25643 0.25499 0.25353 0. 25236 0. 25134 1.8 0.20099 0.27877 0.27683 0. 27503 0.27352 0.27233 0.27111 0.27022 0.26959 0. 26880 1.9 0.29435 0.29262 0.29097 0.28963 0.28058 0.28758 0.28687 0.28635 0. 28592 0. 26546 2.0 0.30762 0.30605 0.30487 0. 30 391 0.30309 0.30247 0.30203 0.30189 0.30 15 1 0. 30163 2. 1 0.32038 0.31934 0.31041 0.31777 0.3 1717 0.31680 0.31693 0.31656 0. 3 1666 0. 31710 2.2 0.33307 0.332 19 0.33171 0.33110 0.33079 0.33096 0.33075 0.33084 0.33141 0. 33140 2. 3 0.34535 0.34496 0.34434 0.34409 0.34417 0.34420 0.34429 0.34472 0.345 13 0. 34535 2.4 0.35746 0.35699 0.35680 0.35675 0.35701 0.35711 0.35736 0.35801 0.35833 0. 35880 2.5 0. 3691 3 0.36887 0.36079 0.36909 0.36926 0.36944 0.37004 0.37074 0. 37096 0. 37164 2.6 0.30040 0. 38036 0.38069 0.38085 0.38103 0.38160 0.38236 0.38263 0.38322 0. 3 84 24 2.7 0.39154 0.39163 0.39201 0.39220 0.39260 0.39322 0.39386 0.39438 0.39508 0. 3959 1 2.8 0.40223 0.40265 0.40289 0.40314 0.4C374 0.40469 0.40499 0.40555 0.40648 0. 40712 2.9 0.41283 0.413 10 0.41337 0.41395 0.4 1465 0.41522 0.41572 0.41657 0.4 1760 0. 41780 3.0 0.42305 0.42333 0.42364 0.42426 0.42518 0.42564 0.42619 0.42705 0.42779 0. 42825 Table 4-13 (Continued) X -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0. 1 It 0.0 0.00000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.00012 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.00079 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0. 0 0.0 . 0.3 0.00291 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.00706 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.5 0.01637 0.00459 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.0 0.6 0.02068 0.01590 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.7 0.04432 0.03 190 0.01670 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.8 0.06274 0.05142 0.03790 0.02116 0.0 0. 0 0.0 0.0 0. 0 0.0 0.9 0.00 31 1 0.07331 0.06184 0.04823 0.03203 0.01297 0.0 0.0 0.0 0.0 1.0 0. 10471 0.09643 0.08715 0.07645 0.06424 0.05088 0.03638 0.02241 0.01041 0.00266 1.1 0. 12665 0. 12015 0.11266 0. 10452 0.09574 0.00579 0.07579 0.06635 0.05862 0.05290 1.2 0. 14073 0. 14354 0.13 796 0.13170 0. 12524 0.11830 0.11136 0.10456 0.09090 0.09512 1.3 0. 17032 0.16655 0.16210 0. 15784 0. 15207 0. 14795 0.14316 0.13857 0. 134 30 0. 13125 1.4 0. 19147 0.1186 1 0.18543 0. 10218 0. 17886 0.17526 0.17175 0.16051 0. 16555 0. 163C9 1.5 0.21211 0.20970 0.20780 0. 20519 0. 20293 0.20049 0. 19775 0.19563 0. 19310 0. 1912 1 1.6 0.21161 0.23003 0.22 857 0.22696 0.22519 0.22303 0.22157 0.21996 0.21805 0.21635 1.7 0.25024 0.24940 . 0.24834 0.24749 , 0.24607 0.24522 0.24352 0.24219 0.24080 0. 2392 1 1.0 0.26022 0.26764 0.26704 0.26637 0.26572 0.26408 0.2 63 85 0.26266 0.26 168 0.26014 1.9 0.20541 0.20497 0.28490 0.28429 0.20422 0.28327 0.28278 0.28164 0.28099 0.27946 2.0 0.3015B 0.30143 0.30156 0.30134 0.30140 0.3 0 056 0.30050 0.29934 0.29896 0. 2974 1 2.1 0.31600 0.31729 0.31734 0.31733 0.31730 0.31688 0.31604 0.31595 0.3 1558 0. 31404 2.2 0.33170 0.33243 0.33221 0.31257 0.3 324 3 0.33238 0.33223 0.33141 0.33096 0.32956 2.3 0.34605 0.34629 0. 34 64 6 0.34718 0.34690 0.34696 0.34654 0.34596 0.34549 0.34422 2.4 0.35960 0.35969 0.36021 0.360 70 0.36044 0.36073 0.3C020 0.35981 0.35930 0.35813 2.5 0. 37235 0.37268 0.37324 0.37341 0.37 34 4 0.37395 0.37329 0.37307 0.37245 0.37133 2.6 0.304 50 0.30491 0.18575 0.30574 0. 30601 0. 3863 1 0.38564 0.38544 0.38472 0.36368 2.7 0. 396 10 0.3 96 06 0.39765 0.39751 0.39779 0.39790 0.39741 0.39734 0.39655 0.39559 2.8 0.40757 0.40826 0.40867 0.40075 0.40922 0.40918 • 0.40084 0.40005 0.4 079 2 0.40699 2.9 0.41036 0.4 1928 0.41948 0.41974 0.42021 0.41988 0.41956 0.41960 0.4 1860 0.41774 3.0 0.42902 0.42981 0.42976 0.43005 0.43067 0.43019 0.42998 0.4 301 1 0.42905 0.42824 Table 4-14 Fluid Velocity, v , as Interpolated from the Numerical Flowfield at Re„ = 0.2, R = 200 C ' C O X -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2. 1 Y 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.00925 0.00948 0.00973 0.00997 0.01023 0.01049 0.01075 0.01100 0.01126 0.01150 0.2 0.01049 0.01895 0.01944 0.01993 0.02044 0.02096 0.02148 0.02200 0.02251 0.02300 0.3 0.02769 0.02835 0.02904 0.02974 0.03046 0.03120 0.03192 0.03266 0.03338 0.03408 0 . 1 4 0.03653 0.0374 1 0.0 3 033 0.03926 0.04022 0.04 120 0.04218 0.04317 0.044 16 0.04508 0.5 0.04532 0.04642 0.04756 0.04872 0.0499 4 0.05113 0.05221 0.05331 0.05437 0.05536 0.6 0.05402 0.05522 0.05643 0.0 57 68 0.05098 0.06027 0.06159 ' 0.06293 0.06421 0.06546 0.7 0.06205 0.06344 . 0.06485 0.06632 0.06782 0.06933 0.07090 0.07242 0. 0736 1 0.07478 0.B 0.07000 0.07158 0.07318 0.07407 0.07645 0.07789 0.07935 0.00077 0.00220 . 0.0836 1 0.9 0.07707 0.07944 0.00096 0.00253 0.08411 0.08576 0.00738 0.00904 0.09072 0.09189 1.0 0.08406 0.00648 0.00819 0.08991 0.09169 0.09351 0.09526 0.09662 0.09709 0.09914 1.1 0.09166 0.09344 0.09531 0.09719 0.09890 0.10035 0.10107 0.10336 0. 10484 0. 10635 1.2 0.09036 0.10032 0.10186 0. 10345 0. 10505 0.10669 0. 10836 0. 1 1002 0. 1 1138 0. 11220 1.3 0. 10437 0.10602 0.10 767 0. 10940 0. 11113 0.11295 0. 11449 0.11560 0. 1 1665 0. 11768 1.4 0. 10986 0.11 160 0.11340 0. 11523 0. 11699 0.11820 0.1 1943 0.12066 0.12106 0. 123 10 1.5 ' 0. 11527 0. 117 10 0.11899 0. 12026 0. 12160 0. 1229 1 0. 12429 0.12561 0. 12659 0.12689 1.6 0.120 55 0. 12 193 0.12329 0. 12470 0. 126 10 0.12757 0.12901 0.12963 0. 130 12 0. 13063 1.7 0. 12466 0.126 11 0.12 754 0. 12904 0.13054 0.13 171 0. 13237 0. 1 3302 0. 13365 0. 1 3426 1.8 0. 12868 0. 13018 0.13173 0. 13328 0. 1341 3 0. 13486 0.13564 0.13636 0. 137 12 0.13676 1.9 0. 11265 0. 134 19 0.13547 0. 1 3629 0. 137 1 3 0.13790 0.13800 0.13951 0. 1392 1 0. 13893 2.0 0. 13649 0.1374 3 0.13830 0. 1 3922 0.14009 0.14 102 0.14155 0.14140 0. 14125 0. 14100 2. 1 0. 13922 0. 14017 0.14110 0. 14204 0. 14304 0. 14340 0.14338 0.14330 0. 14 320 0. 143C9 2.2 0. 14187 0. 14204 0.14386 0. 14404 0. 14510 0. 145 16 0. 14516 0.14519 0. 145 14 0. 143B9 2. 3 0. 14447 0. 14548 0.14650 0. 14666 0. 14675 0.14606 0.14695 0. 14700 0. 1459 1 0. 14472 2.4 0. 14700 0. 14700 0.14805 0. 14023 0. 14842 0.14854 0.14 869 0. 14771 0. 14665 0. 14557 2.5 0. 149 10 0. 14935 0.14960 0. 14980 0. 1500 1 0.15023 0. 14932 0.14040 0. 1474 3 0. 14638 2.6 0.15056 0. 15002 0.15109 0. 15137 0. 15160 0.15001 0.15000 0. 14910 0. 140 18 0. 14684 2.7 0.15197 0.15229 0.15258 0. 15200 0. 15219 0.15142 0. 1 5064 0. 14904 0. 14876 0. 14669 2.8 0.153 37 0.15372 0.15407 0. 15341 0. 15275 0.15206 0. 15130 0. 15052 0. 1486 3 0. 14 66 1 2.9 0.15477 0.155 13 0.15457 0.15398 0. 153 33 0.15266 0.15199 0.15037 0. 14849 0. 14656 3.0 0.15614 0.1556 3 0.1550 8 0. 15452 0. 15394 0.15329 0.15198 0. 1 5024 0. 1484 1 0. 14652 Table 4-14 (Continued) X -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 - 1.2 -1. 1 1 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0. 0 0. 0 0. 1 0.01172 0.01190 0.01203 0.01206 0.01197 0.01168 0.01107 0.00998 0.00818 0. 00519 0.2 0.02343 0.02375 0.02396 0.02401 0.02380 0.02323 0.02210 0.02005 0.0 1667 0. 01 127 0.3 0.03469 0.03521 0.03557 0.03571 0. 03551 0.03477 0.03312 0.03038 0.02576 0. 01 858 0.4 0.04591 0.04648 0.04686 0.04697 0.04666 0.04575 0.04396 0.04082 0.03558 0. 02739 0.5 0.05629 • 0.05709 0.05770 0.05802 0.05770 0.05654 0.05446 0.05106 0.04574 0. 03735 0.6 0.06665 0.06743 0.06788 0.06801 0.06771 0.06680 0.0 6471 0.06109 0.05585 0. 04 8C8 0.7 0.07586 0.07675 0.07746 0.07789 0.07733 0.07613 0.074 14 0.07097 0.06556 0. 05829 0.8 0.00488 0.08587 0.08626 0.08641 0.08611 0.08529 0.08288 0.07955 0.07507 0. 06805 0.9 0.09284 0.09371 0.09436 0.09474 0.09405 0.09278 0.09095 0.08784 0.08315 0. 07735 1.0 0. 10037 0. 10142 0.10172 0.10158 0. 10118 0.10028 0.09778 0.09467 0. 09064 0. 08470 1.1 0. 10724 0. 10771 0.10810 0. 10820 0. 10754 0. 10597 0.10407 0. 10097 0. 09652 0. 09155 1.2 0. 1 1300 0.11374 0.11428 0.11364 0. 11278 0. 11153 0. 10914 0. 10593 0. 10222 0. 09645 1.3 0. 1 1070 0.11909 0.11885 0.11841 0. 11778 0.1 1506 0. 11336 0.1 1058 0. 10599 0. 10097 1.4 0. 12324 0.12333 0. 12324 0. 12299 0. 12137 0.1 1945 0.11743 0.11353 0. 10938 0. 10430 1.5 0. 12722 0. 1274 1 0.12722 0.12597 0. 12455 0.12306 0.11973 0. 11632 0. 11209 0. 10658 1.6 0.13106 0.13085 0.12991 0.12087 0. 12773 0.12492 0. 12201 0.11863 0. 11379 0. 10863 1.7 0. 1 3400 0.13332 0.13255 0.13171 0. 12931 0.12678 0. 124 10 0. 1 1978 0. 1 1520 0. 1099 1 1.8 0. 13630 0.13574 0.13514 0.13306 0. 13007 0.12856 0.12492 0.12081 0. 1 16 47 0. 11054 1.9 0. 13053 0.13814 0. 13 631 0. 13440 0. 13238 0.12936 0.12565 0. 12172 0. 11669 0. 11095 . 2.0 0. 14078 0. 13915 0.13748 0.13570 0. 13 32 4 0.12984 0.12626 0.12209 0. 11676 0. 11134 2. 1 0. 14165 0. 14020 0.13861 0.13665 0. 13353 0.13025 0. 12688 0.12190 0. 11603 0. 1 1 138 2.2 0. 14260 0. 14 1 18 0.13968 0. 1 3678 0. 13 378 0. 13073 0. 12649 0.12175 0. 11692 0. 11086 2.3 0. 14349 0. 14223 0.13969 0. 1 3693 0. 13409 0.13060 0. 12617 0. 12161 0. 11652 0. 1 1039 2.4 0. 14441 0. 14230 0.13977 0.13710 0.13431 0.13016 0.12585 0.12145 0. 11588 0. 1 0994 2.5 0. 14468 0. 14231 0.13982 0. 13729 0. 13376 0. 12969 0.12556 0.12089 0. 1 1522 0. 10 948 2.6 0. 14459 0. 14230 0.13996 0.13701 0.13320 0.12932 0.12531 0. 12008 0. 1 1463 0. 10910 2.7 0.14458 0. 14234 0.14000 0. 13643 0. 13274 0.12894 0.12455 0. 1 1938 0. 1 1408 0. 10 848 2.8 0. 14453 0. 1424 1 0.13937 0.13586 0. 13228 0.12864 0.12373 0.1 1067 0. 11355 0. 10770 2.9 0.14456 0.14205 0.13875 0. 13538 0. 13189 0. 12775 0.12295 0. 11807 0. 11307 0. 10693 3.0 0. 14454 0. 14 143 0.13819 0.13409 0. 13149 0.12693 0.12224 0.11747 0. 11219 0. 10624 CO vo Table 4 -14 (Continued) X -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0. 1 X 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.00029 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.2 0.00227 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ,0.3 0.00719 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.01505 0.0 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.5 0.02532 0.00734 0.0 0.0 0.0 0.0 0.0 .0.0 0. 0 0.0 0.6 0.03652 0.02097 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.04018 0.03479 0.01779 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.8 0.05930 0.04775 0.03393 0.01758 0. 0 0.0 0.0 0.0 0.0 0.0 0.9 0.06904 0.05943 0.04739 0.03415 0.02029 0.00699 0.0 0.0 0. 0 0.0 1.0 0.07793 0.06873 0.05862 0.04718 0.03533 0.02412 0.01410 0.00662 0.00207 0.00027 1.1 0.08450 0.07684 0.06721 0.05712 0.04666 0.03587 0.02602 0.01748 0.01055 0.00499 1.2 0.09038 0.00264 0.07425 0.06448 0.05457 0.04438 0.03436 0.02488 0.01631 0.00853 1.3 0.09454 0.08749 0.07901 0.07020 0.06025 0.05023 0.04016 0.03016 0.02050 0.01131 1.4 0.09793 0.09097 0.08278 0.07395 0.06450 0.05442 0.04420 0.03388 0. 02365 0.01361 1.5 0. 10085 0.09334 0.08578 0.07669 0.06744 0.05745 0.04708 0.03669 0.02603 0.01556 1.6 0. 10234 0.09526 0.0874 1 0.07075 0. 06937 0.05968 0.04920 0.03872 0.02796 0.01726 1.7 0. 10344 0.09672 0.08059 0.08026 0.07076 0. 06 1 18 0.05078 0.04029 0.02956 0.01679 1.8 0. 10436 0.09724 0.0 894 1 0.08093 0.07178 0.06214 0.05198 0.04154 0.03093 0.02019 1.9 0. 10493 0.09755 0.09003 0.08137 0.07253 0.06282 0.05293 0.04257 0.03213 0.0215C • 2.0 0. 10484 0.09769 0.09015 0.08165 0. 07 297 0.063 3 2 0.05371 0.04344 0.03321 0.02273 2.1 0. 10461 0.09779 0.09002 0.08175 0.07305 0.06371 0.05424 0.04421 0. 034 18 0.02389 2.2 0.10439 0.09771 0.08978 0.08178 0.07308 0. 06402 0.05467 0.04488 0.03506 0.02500 2.3 0. 10418 0.09716 0.08951 0.08180 0.07307 0.06425 0.05501 0.04548 0.03589 0.02608 2.4 0. 10390 0.09665 0.08927 0.08152 0.07299 0.06443 0.05533 0.04606 0.03669 0.02712 2.5 0.10325 0.096 18 0.08898 0.08117 0.07292 0.06462 0.05564 0.04663 0.03747 0.02814 2.6 0.10256 0.09567 0.00872 0.08088 0. 07287 0.06469 0.05592 0.04715 0.03820 0.02912 2.7 0. 10189 0.09524 0.08033 0.08058 0.07278 0.0647 1 0.05620 0.04767 0.03893 0.03008 2.8 0.10130 0. 09481 0.08781 0.080 30 0.07274 0.06477 0.05649 0.04820 0.03965 0.03103 2.9 0. 10071 0.09443 0.08737 0.08007 0.07271 0.06482 0.05676 0.04070 0.04034 0.03195 3.0 0. 10021 0. 094 00 0.08694 0.07903 0.07268 0.06489 0.05706 0.04921 0. 04104 0.03287 o Table 4 - 1 5 Fluid Velocities According to Davies and Peetz [ 7 ] at Rec = 10 V X X 0 -0 . 2 -0 . 4 -0 . 6 -0 . 8 - 1 . 0 -1 .2 -1 . 4 -1 . 6 -1 . 8 - 2 . 0 - 2 . 2 - 2 . 4 -2 . 6 -2 . 8 -3 . 0 y 0 • • m # . 0 . 12 0 . 24 0 . 35 0 . 46 0 . 5 4 0 . 6 1 0 . 6 6 0 . 6 9 5 0 . 7 2 5 0 . 75 0 . 2 • • 0 . 0 . 0 1 5 0 . 13 0 . 26 0 . 38 0 . 48 0 . 5 6 0 . 6 3 0 . 6 7 5 0 . 71 0 . 74 0 . 76 0 . 4 • « • 0 0 . 0 4 0 . 20 0 . 33 0 . 44 0 . 52 0 . 5 9 0 . 6 5 0 . 6 9 0 . 73 0 . 7 5 5 0 . 78 0 . 6 • 0 0 . 2 3 0 . 35 0 . 43 0 . 51 0 . 57 0 . 6 3 0 . 6 8 0 . 7 2 0 . 75 0 . 7 7 5 0 . 7 9 5 0 . 8 • • 0 0 . 30 0 . 4 3 0 . 50 0 . 55 0 . 60 0 . 64 0 . 6 8 0 . 7 1 0 . 7 4 0 . 77 0 . 79 0 . 81 1 . 0 0 0 . 01 0 . 05 0 . 45 0 . 55 0 . 6 0 0 . 64 0 . 67 0 . 69 0 . 71 0 . 7 3 0 . 7 5 0 . 7 7 5 0 . 80 0 . 82 0 . 8 3 5 1 . 2 0 . 40 0 . 47 0 . 60 0 . 72 0 . 76 0 . 7 7 0 . 78 0 . 78 0 . 7 8 0 . 7 8 5 0 . 7 9 0 . 8 0 0 . 8 2 0 . 8 4 5 0 . 8 6 5 0 . 88 1 . 4 0 . 71 0 . 80 0 . 86 0 . 91 0 . 9 2 0 . 9 1 5 0 . 90 0 . 88 0 . 86 0 . 85 0 . 8 4 5 0 . 8 5 0 . 8 7 0 . 89 0 . 91 0 . 92 1 . 6 0 . 95 1. 0 4 1. 05 1. 0 4 1. 0 2 1 . 0 0 0 . 98 0 . 9 5 5 fi. 93 0 . 9 0 0 . 8 9 0 . 9 0 0 . 9 2 0 . 9 3 5 0 . 95 0 . 96 1 . 8 1. 10 1. 18 1 . 17 1. 12 1. 06 1 . 0 3 1. 00 0 . 97 0 . 95 0 . 9 2 5 0 . 9 1 5 0 . 9 2 5 0 . 9 1 5 0 . 96 0 . 9 7 5 0 . 99 2 . 0 1 . 17 1. 25 1. 21 1. 13 1. 07 1 . 0 0 • • • • • • . a • * v / 0 • 0 0 0 0 0 0 0 0 0 0 0 0 . 2 • • 0 . 1 1 0 . 14 0 . 14 0 . 125 0 . 0 9 5 0 . 0 7 5 0 . 0 6 0 . 0 5 0 . 04 0 . 0 3 5 0 . 0 3 0 0 . 4 • • * 0 . 2 2 0 . 26 0 . 26 0 . 23 0 . 18 0 . 1 4 0 . 1 2 0 . 1 0 0 . 08 0 . 0 6 5 . 0 . 0 5 5 0 . 6 • o ' 0 . 3 3 0 . 37 0 . 36 0 . 32 0 . 25 0 . 2 0 0 . 1 6 5 0 . 1 4 0 . 1 1 5 0 . 0 9 5 0 . 08 0 . 8 • • o " 0 . 28 0 . 4 1 0 . 46 0 . 43 0 . 38 0 . 30 0 . 2 5 0 . 2 0 5 0 . 1 7 0 . 14 0 . 115 0 . 0 9 5 1 . 0 0 o ! 01 0 . 13 0 . 27 0 . 40 0 . 4 7 5 0 . 51 0 . 47 0 . 41 0 . 33 0 . 2 7 5 0 . 2 2 0 . 1 9 0 . 16 0 . 13 0 . 105 1 .2 0 . 06 0 . 12 0 . 28 0 . 39 0 . 4 6 5 0 . 5 1 0 . 52 0 . 4 8 5 0 . 42 0 . 3 4 5 0 . 2 8 0 . 2 3 0 . 1 9 o • 0 1 . 4 0 . 12 0 . 21 0 . 35 0 . 45 0 . 49 0 . 5 1 0 . 51 0 . 47 0 . 41 0 . 34 0 . 2 8 0 . 2 3 0 0 » 1 . 6 0 . 18 0 . 28 0 . 38 0 . 46 0 . 4 9 5 0 . 5 0 0 . 49 0 . 44 0 . 39 0 . 33 0 . 2 8 0 . 2 3 • « • 1 . 8 2 . 0 0 . • 23 0 . • 32 0 . • 39 0 . « 44 0 . • 47 0 . 4 8 • 0 . • 46 0 . 41 0 . « 37 0 . • 3 1 5 0 . 2 6 5 • 0 . 2 3 « e • 0 • m co Table 4-16 Fluid Velocity, v , as Interpolated from the Numerical A Flowfield at Rec = 10, = 100 X -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2. 1 y 0. 0 0.67960 0.6614514 0. 64832 0.63092 0.61184 0. 59068 0. 56762 0.54238 0.51461 0. 48397 0. 1 0.67931 0.66U30 0. 64819 0.63090 0.61192 0. 59092 0. 56805 0.54304 0.5 1554 0. 48522 0.2 0.67952 0.661458 0. 64865 0.63158 0.61272 0. 59200 0. 5 6947 0.54484 0.51781 0. 468C2 0. 3 0.68025 0.66589 0. 65063 0.63434 0.61616 0. 59644 0. 57507 0.55179 0.5 26 30 0. 49823 0. 4 0.601463 0.67077 0. 65608 0.64027 0.62270 0. 60381 0. 58338 0.56117 0.53666 0. 5097 1 0. 5 0.68938 0.67607 0. 66202 0.64660 0.62982 0. 61 191 0. 593 12 0.57252 0.54998 0. 52557 0. 6 0.691452 0.68220 0. 66924 0.65479 0.63950 0. 62322 0. 60553 0.58617 0.56535 0. 54289 0. 7 0.70163 0.69010 0. 67764 0.66417 0.64997 0. 63479 0. 61808 0.60034 0.58225 0. 56294 0. 8 0.70098 0.69826 0. 68629 0.67302 0.66 100 0. 64723 0. 63276 0.6 1756 0.60151 0. 56396 0. 9 0.71652 0. 706514 0. 69 60 3 0.60523 0.67353 0. 66115 0. 64828 0.63470 0.6 1998 0. 60540 1. 0 0.725514 0.71622 0. 70 674 0.69679 0.68608 0. 675 10 0. 66370 0.65220 0.64046 0. 62 647 1. 1 0.731434 0.72593 0. 71747 0.70811 0.69901 0. 69001 0. 60034 0.67063 0.66088 0. 65044 1. 2 0.714315 0.73565 0. 72 824 0.7 20 62 0.71306 0. 70484 0. 69671 0.68860 0.68039 0. 67296 1. 3 0.75263 0.7146140 0. 73973 0.73324 0.72630 0. 71940 0. 71286 0.70653 0.70060 0. 6950 1 1. 4 0.76252 0.75673 0. 75104 0.74521 0.73940 0. 73443 0. 72916 0.72436 0.7 1979 0. 71537 1. 5 0.77197 0.76690 0. 76203 0.75743 0.75 327 0. 74888 0. 74493 0.74121 0.73004 0. 73608 1. 6 0.70111 0.77735 0. 77342 0.76982 0.76621 0. 76280 0. 75985 0.75766 0.75632 0. 755C9 1. 7 0.79093 0.70769 0. 70450 0.78155 0.77866 0. 77648 0. 77475 0.77300 0. 77297 0. 77305 1. 8 0.80052 0.79763 0. 79522 0.79267 0.79121 0. 78976 0. 78905 0.70853 0.70080 0. 79CC9 1. 9 0.809146 0.80729 0. 80539 0.00414 0.80305 0. 80249 0. 80220 0.80263 0.80399 0. 806 13 2. 0 0.81815 0.81685 0. 81570 0.81492 0.81442 0. 81432 0. 81485 0.8 1622 0.81821 0. 62088 2. 1 0.82715 0. 826 13 0. 02560 0.82510 0.02517 0. 82572 0. 82708 0.82888 0.83 136 0. 83448 2. 2 0.83560 0.83507 0. 83475 0.03487 0.83540 0. 83682 0. 83840 0.84072 0.84349 0. 64 74 0 2. 3 0.814367 0.84353 0. 84355 0.84431 0. 84549 0.84698 0. 84914 0.85158 0.855 15 0. 85925 2. 4 0.85116 0.85 155 0. 85236 0.85333 0.05478 0. 85664 0. 05890 0.86217 0. 86580 0. £702 1 2. 5 0.05890 0.85959 0. 86052 0.06192 0.86350 0. 86560 0. 86847 0.87178 0.87578 0. 88020 2. 6 0.86619 0. 867 15 0. 86 83 2 0.06982 0.07178 0. 87427 0. 87731 0.88084 0.88482 0. 68954 2. 7 0.87320 0. 8714 24 0. 87567 0.87734 0.07965 0. 88245 0. 88553 0.88915 0.89334 0. 89812 2. 8 0.87975 0.88103 0. 88252 0.08468 0.88713 0. 88994 0. 89325 0.89689 0.90123 0. 906 14 2. 9 0.885914 0.88738 0. 08931 0.89150 0.09411 0. 89703 0. 90031 0.90423 0.90859 0. 91340 3. 0 0.89188 0.09359 0. 89565 0.89799 0.90060 0. 9036 1 0. 90710 0.91100 0.9 1537 0. 92018 to ro Table 4-16 (Continued) X -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 - 1.2 -1. 1 Y 0.0 0.45008 0.41262 0.37131 0.32603 0.27686 0.22422 0. 1 6881 0.11293 0. 06 071 0. 01868 0.1 0.45172 0.4 1470 0.37392 0.32923 0.28073 0.22880 0.17412 0. 11876 0.06640 0. 0228 1 0.2 0.45536 0.41984 0.38078 0.33805 0.29172 0.24210 0.18980 0. 13612 0. 08373 0. 03746 0.3 0.46724 0.43330 0.39616 0.35537 0.31102 0.26352 0.21473 0.16396 0. 11272 • 0. 06332 0.« 0.48036 0.449 18 0.41523 0.37837 0.33851 0.29520 0.24913 0.20108 0. 15248 0. 10190 0.5 0.49908 0.47027 0.43897 0.40502 0.36915 0.33160 0.29133 0.24797 0.20186 0. 15326 0.6 0.51861 0.49330 0.46630 0.43721 0.40592 0.37232 0.33726 0.300 64 0.26036 0. 21491 0.7 0.54229 0.51984 0.49588 0.47030 0.44458 0.4 174 1 0.38820 0.35645 0.32206 0. 28435 0.8 0.56540 0. 546 16 0.52682 0.50645 0.48494 0.46215 0.43944 0.41489 0.38764 0. 35690 0.9 0.59057 0.575 16 0.55913 0.54219 0.52576 0.50892 0.49123 0.47273 0.45269 0. 42949 1.0 0.6 1599 0.60207 0.59042 0.57822 0.56588 0.55332 0.54141 0.52877 0.5 1480 0. 4983 1 1.1 0.64058 0.63 132 0.62228 0.61315 0.60449 0.59663 0.58888 0.58108 0.57269 0. 56302 1.2 0.66588 0.65072 0.65179 0.64640 0.64170 0.6375 1 0.63355 0.62999 0.62642 0. 62 12 1 1.3 0.609 32 0.68468 0.68133 0.67825 0.67586 0.67489 0.67490 0.67503 0. 67462 0. 67425 1.4 0.71248 0.71013 0.70830 0.70754 0.70837 0.70962 0.71201 0.71526 0.71839 0. 72084 1.5 0.73438 0.73 350 0.73391 0.73516 6.73766 0. 74 15 1 0.74570 0.75108 0.75719 0. 76183 1.6 0.75482 0. 7556 1 0.7574 8 0.76067 0.76450 0.76975 0.77631 0.78319 0.79057 0. 79820 1.7 0.77413 0.77627 0.77949 0.78344 0.78915 0.79549 0.80322 0.81139 0. 82015 0. 82 94 1 1.8 0.79230 0.795 35 0.79928 0.80477 . 0.81102 0. 81884 0.82690 0.83643 0.84629 0. 85593 1.9 0.80900 0.81274 0.81782 0.82383 0.83102 0.03904 0.84824 0.85817 0.86850 0. £76 89 2.0 0.82433 0.82901 0.83462 0.84118 0.84880 0.85736 0.86702 0.87720 0.80796 0. 89900 2.1 0.83876 0.84388 0.84985 0.85686 0.86475 0.07302 0.88354 0.89394 0.90473 0. 916 18 2.2 0.85201 0.85740 0.86373 0.87093 0.87927 0.88820 0.89003 0.90837 0.91955 0. 9307 1 2. 3 0.86413 0.86978 0.87629 0.88384 0.89197 0.90109 0.91075 0.92117 0.93229 0. 94292 2.4 0.87522 0.88108 0.88785 0.89522 0.90355 0.91242 0.92203 0.93257 0. 94 208 0. 95370 2.5 0.88546 0.89149 0.89815 0.90567 0.91374 0.92255 0.93219 0.94209 0.95229 0. 963 19 2.6 0.89489 0.90090 • 0.90767 0.91493 0.92295 0.93170 0.94095 0.95051 0.96067 0. 971 13 2.7 0.90355 0.90956 0.91612 0.92340 0.93126 0.93970 0.94068 0.95801 0.96774 0. 97789 2.8 0.91 146 0.91739 0.92397 0.93101 0.93868 0.94698 0.9 554 9 0.96450 0.97406 0. 98367 2.9 0.91875 0.92460 0.93098 0.93794 0.94542 0.95328 0.96156 0.97036 0.97955 0. 96650 3.0 0.92539 0.93119 0.93749 0.94419 0.95142 0.95901 0.96703 0.97545 0. 904 11 0. 99284 t o c o Table 4-16 (Continued) X -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0. 1 Y 0.0 0.00000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.00071 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.00497 0.0 0.0 0.0 0. 0 0.0 0.0 0.0 0.0 0.0 0.3 0.01836 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.04872 0.0 0.0 0.0 0.0 0. 0 0.0 0.0 0. 0 0.0 0.5 0.09738 0.02823 0.0 0.0 0. 0 0. 0 0.0 0.0 0.0 0.0 0.6 0. 16235 0.09380 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0.0 0.7 0.23024 0.17747 0.09545 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.8 0.31927 0.2694 1 0.20381 0.11502 0.0 0.0 0.0 0.0 0. 0 0.0 0.9 0.4001 1 0.36279 0.31204 0.24631 0. 16356 0.06503 0.0 0.0 0.0 0.0 1.0 0.47822 0.45053 0.41502 0.36799 0.30852 0.23926 0.16586 0.09811 0.04243 0.00997 1.1 0.54940 0.53 2 27 0.50608 0.47285 0.43145 0.38202 0.32897 0.27587 0.23019 0.19554 1.2 0.61407 0.60323 0.58773 0.56400 0.53564 0.50097 0.45984 0.41833 0.37939 0.34412 1.3 0.67 194 0.66702 0.65669 0.64306 0.62276 0.59707 0.56859 0.53413 0.49985 0.46788 • 1.4 0.72256 0.72196 0.71727 0.70978 0.69654 0.67830 0.65568 0.63005 0.60108 0.57168 1.5 0.76720 0.76903 0.76982 0.76550 0.75795 0.74561 0.72858 0.70929 0.68439 0.65927 1.6 0.804 39 0.81015 0.81248 0.81316 0.80866 0.80204 0.78983 0.77447 0.75447 0.73259 1.7 0.0 370 1 0.04544 0.84921 0.0 53 52 0.85153 0.84038 0.83996 0.82892 0.81362 0.794 94 1.8 0.06544 0. 074 18 0.80065 0.88581 0.88760 0.80624 0.88203 0.87420 0. 86325 0. 6476 1 1.9 0.88966 0. 09 06 3 0.90743 0.91325 0.91779 0.91788 0.91717 0.9 1148 0.9046 1 0.89178 2.0 0.90978 0.91965 0.92909 0.93650 0.94261 0.94424 0.94637 0.94250 0.93883 0.92657 2.1 0.92687 0.93769 0.94711 0.95557 0.96213 0.96612 0.96945 0.96822 0.96649 0.95899 2.2 0.94151 0.95284 0.96247 0.97156 0.978 4 2 0.98425 0.98838 0.90910 0.90808 0.96399 2.3 0.95423 0. 965 16 0.97492 0.98496 0.99202 0.99922 1.00368 1.00596 1.00709 1.00443 2. 4 0.96525 0.97532 0.98553 0.99536 1.00335 1.01 124 1.01582 1.01966 1.02187 1.02108 2. 5 0.97 30 7 0.98400 0.99455 1.00406 1.01235 1.02070 1.02571 1.03075 1.03382 1.03460 2.6 0.98124 0.99151 1.00218 1.O1089 1.01956 1.02789 1.03375 1.03970 1. 04345 1.04552 2.7 0.98773 0.99783 1.00 778 1.01658 1.02556 1.03371 1.04026 1.04677 1.05082 1.05377 2.8 0.99315 1.00299 1.01245 1.02139 1.03050 1. 0382 1 1.04495 1.05177 1.05623 1.05S99 2.9 0.99782 • 1.00749 1.01647 1.02510 1.03406 1. 04154 1.04862 1.05575 1.06050 1.06489 3.0 1.00193 1.01108 1.01955 1.02819 1.03700 1.04427 1.05154 1.05885 1. 06381 1.06860 Table 4-17 Fluid Velocity, v , as Interpolated from the Numerical Flowfield at Re„ = 10, R = 100 C ' C O X -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2. 1 Y 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.01426 0.01546 0.01679 0.01827 0.01995 0.02185 0.02397 0.02636 0.02904 0.03202 0.2 0.02842 0.03080 0.03345 0.03639 0.03973 0.04349 0.04770 0.05243 0.05774 0.06367 0.3 0.01239 0.04589 0.04976 0.05407 0.05897 0.06446 0.070 60 0.07748 0. 085 19 0.09379 0.4 0.05580 • 0.06037 0.06542 0.07106 0.07747 0. 0846 1 0.09261 0.10156 0. 1 116 2 0. 1228 1 0.5 0.06887 0.07444 0.08059 0.08753 0.09533 0.10399 0. 11349 0.12415 0. 13605 0. 14920 0.6 0.08 148 0.08786 0.09490 0. 10288 0. 11173 0. 12156 0.13258 0. 14493 0. 15863 0. 17380 0.7 0.09308 0.10025 0.10826 0.11722 0. 12713 0. 13817 0.15061 0. 16437 0. 17912 0. 19537 0.8 0. 10(412 0.1 1199 0.12092 0. 130 73 0. 14143 0. 15330 0.16639 0. 18084 0. 19601 0.21449 0.9 0. 11M 57 0.123 04 0.13236 0. 14255 0. 15397 0. 1666 1 0. 18053 0.19 594 0.21307 0.231 14 1.0 0.12371 0. 13275 0.14257 0. 15341 0. 16551 0.17876 0.19333 0.20891 ' 0.22590 0.2444 1 1.1 0.13233 0. 14176 0.15196 0. 16347 0. 17572 0.18891 0.20358 0.21955 0.23696 0.25615 1.2 0.140 32 0. 15002 0.16038 0. 17171 0. 18398 0.19772 0.21261 0.22884 0. 24633 0.26452 1.3 0. 14716 0. 15675 0.16741 0.17885 0. 19159 0.20542 0.22018 0.23591 0.25278 0.27098 1.U 0. 1530 1 0. 16296 0.17366 0.18539 0. 19810 0.21120 0.22572 0.24126 0. 258 1 1 0.27638 1.5 0. 158(42 0.16847 0.17919 0. 19054 0.20258 0.21594 0.23020 0.24568 0.26 195 0.27848 1.6 0. 16333 0. 17283 0.18330 0. 19444 0.20663 0.21974 0.23389 0.24831 0.26344 0.27992 1.7 0. 1668 3 0.17619 0.18674 0. 19783 0.20992 0.22242 0.23561 0.24942 0.26455 0.26039 1.8 0. 16978 0.17945 0.18954 0.20073 0.21172 0.22385 0.23647 0.25028 0.26473 0.27905 1.9 0. 17255 0.18193 0.19186 0.20199 0.21298 0.22456 0.23710 0.25015 0. 26 317 0.27678 2.0 0. 1740 1 0.18354 0.19297 0.20293 0.21361 0.22497 0.23673 0.2 48 59 0.26 107 0.27422 2. 1 0. 17506 0.18460 0.19360 0.20 3 53 0.21380 0.22447 0.23522 0.24669 0.25868 0.2713 1 2.2 0. 17678 0. 185 19 0.19420 0. 20 359 0.21323 0.22296 0.23356 0.24444 0.25607 0.26692 2. 3 0.17739 0.18554 0.19432 0.20291 0.21190 0.22154 0.23139 0.24214 0. 25205 0.2625 1 2.a 0.17761 0. 1856 1 0. 19336 0.20179 0.21051 0.21967 0.22941 0.23841 0.248 15 0.25797 2.5 0.17750 0.18476 0.19247 0.20034 0.20893 0.21774 0. 22615 0.23501 0. 24 395 0.25353 2.6 0. 17684 0. 18381 0.19123 0. 19903 0.20699 0.21492 0.22295 0.23128 0.24006 0.24856 2.7 0. 17509 0.18281 0.18986 0.19739 0.20450 0.21185 0.21971 0.22769 0. 2357 1 0.24339 2.8 0. 17494 0.18 152 0.18851 0. 19501 0.20190 0.20907 0.21630 0.22403 0.23107 0.23806 2.9 0. 17390 0.18022 0.18631 0. 19272 0. 19921 0.20603 0.21314 0.219 69 0. 22635 0.2332 1 3.0 0. 17261 0.17835 0. 18416 0. 19021 0. 19660 0.20306 0.20929 0.21556 0.22182 0.22816 Table 4-17 (Continued) X -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1. 1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 1 0.03533 0.03093 0.04275 0.04667 0.05043 0.05362 0.05533 0.05427 0.04815 0.0333 1 0. 2 0.07020 0.07723 0.0B469 0.09231 0.09961 0. 10580 0. 10949 0.107 69 . 0.09735 0.07027 0. 3 0. 10328 0.11360 0. 12463 0. 13594 0. 14686 0.15627 0.16174 0. 16023 0. 14708 0. 11358 0. U 0. 13511 0. 1(40 13 0.16196 0. 17621 0. 19016 0.20210 0.21024 0.21120 0. 19735 0. 1 6330 0. 5 0. 1f>'365 0.1793K 0. 19607 0.21342 0.22995 0.24413 0.25431 0.25659 0.24704 0.2V475 0. 6 0. 190148 0.20798 0.22637 0.24532 0.26401 0.201 18 0.29355 0.29795 0.29 100 0.26529 0. 7 0.21317 0.232'114 0.25295 0.27430 0.29 39 5 0.31195 0.32659 0.33455 0.329 16 0.31007 0. a 0.23379 0.25'4 27 0.27506 0.29653 0.31001 0.33044 0.35289 0.36192 0.36312 0.34792 0.9 0.250U8 0.2712B 0.29342 0.31645 C.33759 0.35710 0.37437 0.30528 0.30702 0. 37978 1. 0 0. 261457 0.2 06 20 0.30799 0.329 04 0.35182 0.37320 0.30902 0.40095 0.40646 0.39973 1. i 0.27606 0.29668 0. 31 04 5 0.34106 0.36259 0.30175 0.39914 0.41175 0.4 1603 0.4 1597 1. 2 0. 2U>40U 0.30496 0.32 704 0.34767 0.368 34 0. 3006 1 0.40480 0.4 1671 0.42499 0.42268 1. 3 0.29070 0.31000 0.3 30 03 0.35151 0.37 239 0.39020 0.40620 0.42003 0.4255 1 0.42691 1. 14 0.29439 0.3 1314 1 0.33332 0. 15374 0.37 190 0.30943 0.40602 0.41689 0.42457 0.426 1 1 1. 5 0.29 6 29 0. 3 1'199 0.3 3 395 0.35109 0.36 90 3 0.3 0766 0.40088 0.41249 0.4207 3 0.42196 1. 6 0.29720 0.3 TI6 1 0.33 176 0.34925 0. 16700 0. 3 014 9 0.39521 0.40605 0.4 1317 0.4 1670 1. 7 0.29623 0.31227 0.32006 0.34598 0.36070 0. 37506 0.30076 0.39799 0.405 12 0.40885 1. 8 0.29 387 0.30938 0.32 54 8 0. 33981 0.35416 0.36027 0.37944 0.30090 0.39664 0.3 9 897 1.9 0.29113 0. 30606 0.31970 0.3 3353 0.34739 0.35956 0.37004 0.37956 0.3056 1 0.38878 2.0 0.20797 0.3C075 0.31379 .0.32709 0.33950 0.35035 0. 36064 0.369 15 0. 37465 0.3766 1 2. 1 0.211 3 1<4 0.29530 0. 30 783 0.32003 0.33003 0.34 124 0.35120 0.3 500 7 0.36 300 0.36774 2. 2 0.27820 0.20905 0.30161 0.31205 0.32221 0. 33240 0.34051 0.34720 0. 35324 0.35611 2. 3 0. 27 3214 0. 2014 27 0.29434 0 . 30 40 5 0.31400 0.32209 0.33015 0.33687 0.34215 0.34471 2. 14 0. 268 2 3 0.27706 0.28704 0.29654 0.30576 0.31317 0.32023 0.32674 0.3 3007 0. 33377 2. 5 0.26 261 0.27 129 0.20021 0.20898 0.29600 0.3 03 89 0.3 1056 0.31628 0.320 10 0.32320 2. 6 0.25676 0.265 05 0.27324 0.20130 0.20021 0.29484 0.30143 0.30596 0.30972 0.313CC 2. 7 0.2510 3 0.25001 0.26677 0.27336 0.2799 1 0.28639 0.29186 0.29605 0.29990 0. 30285 2. 8 0.2'»550 0.25291 0.25943 0.26586 0. 27 209 0.27007 0.20261 0.20675 0.29044 0.29266 2. 9 0.24002 0. 2146614 0.25271 0.25060 0.264 41 0.26963 0.27307 0.27774 0.20 137 0. 2 83 10 3.0 0.23U79 0.214 0414 0.24595 0.25169 0.257 2 4 0.26143 0.26542 0.26929 0.27222 0.27392 vO cn Table 4-17 (Continued) X - 1 . 0 - 0 . 9 - 0 . 8 - 0 . 7 - 0 . 6 - 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 - 0 . 1 Y 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 .0 0 . 0 0. 0 0 . 0 0 . 1 0 .00196 0 . 0 0 .0 0 . 0 0. 0 0 . 0 0 .0 0 .0 0. 0 0 . 0 0 . 2 0 .01499 0 . 0 0 .0 0 . 0 0 .0 0 . 0 0 .0 0 . 0 0 .0 0 . 0 0 . 3 0 .04710 0 .0 0 .0 0 .0 0 . 0 0 . 0 0 .0 0 .0 0. 0 0 . 0 0 . it 0.09441 0 . 0 0 .0 0 . 0 0. 0 0 . 0 0 .0 0 .0 0 .0 0 . 0 0 . 5 0. 15295 0 .04642 0 .0 0 . 0 0 .0 0. 0 0 .0 0 . 0 0. 0 0 . 0 0 . 6 0 .21368 0. 12666 0 .0 0 .0 0. 0 0. 0 0 .0 0 .0 0. 0 0 . 0 0 . 7 0 .27010 0 .20072 0 .10373 0 . 0 0 .0 0 . 0 0 .0 0 .0 0. 0 0 . 0 0 . 8 0 .31764 0 .26366 0 . 1 9 0 4 1 . 0 .09819 0. 0 0 . 0 0 .0 0 . 0 0 . 0 0 . 0 0 . 9 0 .35422 0 .31502 0 . 2 5 6 1 8 0 .18550 0. 10867 0 . 0 3 6 4 7 0.0 0 . 0 0. 0 0 . 0 1. 0 0 .38 384 0 .35081 0 .30663 0 .24983 0. 18620 0 . 12343 0 . 0 6 9 5 9 0 . 0 3 1 0 7 0 .00936 0 . 0 0 1 2 5 1. 1 0 .40 1 38 0 .37849 0 .34031 0 .29383 0 .24050 0 . 1 8 4 7 7 0. 13247 0 . 0 8 7 5 6 0 .05277 0 . 0 2 7 4 6 1. 2 0 .41456 0 .39380 0 .36496 0 .32466 0.27941 0 . 2 2 9 7 9 0. 1 7951 0 .13319 0. 09272 0 . 0 5 8 8 2 1. 3 0 .4 1964 0 .40434 0 .37881 0 .34713 0 .30649 0 . 2 6 2 0 8 0 . 2 1 6 2 2 0 .17051 0. 12033 0 .09C7U 1. tt 0 .42098 0 .40803 0 . 3 8 7 6 7 0.36041 0 .32566 0 . 2 8 6 2 9 0 . 2 4 3 9 5 0 . 2 0 1 1 5 0. 15978 0 . 12119 1. 5 0 .41996 0.4 085 3 0 .39255 0.36002 0 .33835 0 . 3 0 3 5 2 0 . 2 6 5 4 4 0 . 2 2 5 9 0 0. 18642 0 .14 888 1. 6 0 .41376 . 0 .406 12 0 .39 155 0.37181 0. 34564 0 .3 1568 0 . 2 8 1 7 8 0 . 2 4 5 7 0 0 .20802 0 » 1 7 3 0 7 1. 7 0 . 4 0 6 5 7 0 .40160 0 . 3 8 8 3 9 0 .37262 0 .349 39 0 .3234 1 0 . 2 9 3 1 8 0 .26111 0 . 2 2 7 3 3 0 . 1 9 3 8 5 1. 8 0 .39866 0 .39366 0 . 3 8 3 5 9 0 . 369 54 0 .35035 0 . 3 2 7 3 3 0 . 3 0 1 1 0 0 . 2 7 2 5 6 0. 24 215 0.21 120 1. 9 0 .38956 0 . 3 0 4 8 0 0 .37756 0 .36486 0 .34907 0. 3286 1 0 . 3 0 6 0 9 0 .28048 0 .25358 0 . 2 2 5 2 2 2. 0 0 .37874 0 .37554 0 . 3 6 9 4 9 0 .35900 0. .34579 0 .32776 0 . 3 0 8 6 0 0 . 2 8 5 6 5 0 .26 194 0 . 2 3 6 1 2 2. 1 0 .36776 0 .36604 0 .36042 0.35192 0. 34032 0 .32524 0 . 3 0 8 5 3 0 .28849 0 .26746 0 . 2 4 4 2 2 2. 2 0 . 3 5 6 8 7 0 .356 06 0 . 3 5 1 0 7 0 .34424 0. 3 3.394 0 .32 140 0 .30672 0 . 289 19 0 .27059 0. 24 985 2. 3 0 .34623 0 .345 18 0 . 3 4 1 3 0 0 .33616 0 .32692 0 . .31654 0 .30344 0 . 2 8 8 1 7 0 .27 177 0 . 2 5 3 3 7 2. <t 0 .33582 0. 3.3434 0 . 3 3 1 6 0 0 .32705 0. 3 1946 0 .31076 0 . 2 9 8 9 6 0 . 2 8 5 8 6 0 .27139 0 . 2 5 5 1 2 2. 5 0 .32472 0 .32382 0 .32202 0 . 3170.3 0 .31150 0 .30425 0 . 2 9 3 7 5 0 .28254 0 .26976 0 . 2 5 5 4 0 2. 6 0 .31396 0 .31364 0 . 3 1 2 5 6 0 . 308 50 0 .30333 0. 29701 0 . 2 8 8 0 2 0 . 2 7 8 4 3 0. 267 12 0 . 2 5 4 4 9 2. 7 0 .30363 0 .30375 0 . 3 0 2 7 7 0.299.33 0 .29518 0 .28955 0 . 2 8 1 9 0 0 . 2 7 3 6 7 0. 26353 0 . 2 5 2 3 5 2. 8 0 .29374 0 .29420 0 . 2 9 3 1 0 0 . 290 39 0 .20706 0 .28191 0 . 2 7 5 3 0 0 . 2 6 8 1 9 0. 259 16 0 . 2 4 9 3 3 2. 9 0 .28430 0 .20498 0 . 2 0 3 7 6 0.28161 0 .27890 0 . 2 7 4 2 1 0 . 2 6 8 5 6 0 .26248 0. 2544 1 0 . 2 4 5 7 7 3.0 0 .2751B 0 .27595 0 . 2 7 4 7 8 0 .27310 0. 27092 0 . 2 6 6 6 2 0 .26181 0 . 2 5 6 5 9 0 .24936 0 . 2 4 1 7 3 Table 4-18 Convergence C r i t e r i a for Re = 0.2, R = 100 n Q x 10 3 x 1 0 " x 10 3 x 1 0 c x 1 0 5 x 1 0 3 100 0 3662109 0 61 3331 8 0 1939535 32424 .47 223 3393 11 88349 7. 150173 . 4. 935801 0 1898850 200 0 2593994 0 6181002 0 3061295 32391 .41 222 8590 11 88748 3. 972983 3 503687 0 2537882 300 0 1831055 0 7987022 0 2726912 32377 .59 222 8292 11 95545 1. 595299 3 997266 0 2987888 400 0 1678467 0 7551908 0 3043496 32378 .21 223 1770 12 02909 1 . 291838 4 018430 0 2217368 500 0 2136230 0 6848574 0 2304316 32386 .64 223 6526 12 08207 2. 006573 3 606286 0 2223956 600 0 1525879 0 4304134 0 2557039 32397 .70 224 0821 12 10978 2. 104382 2 627323 0 2312208 700 0 1678467 0 7200241 0 2989173 32408 .15 224 3369 12 11779 1. 786049 1 861549 0 2352977 800 0 1373291 0 4565716 0 2302527 3241 5 .90 224 5573 12 11320 1. 231336 1 225793 0 2362006 900 0 1678467 0 5465746 0 2422333 32420 .19 224 6109 12 10024 0. 7276921 1. 025761 0 2059342 1000 0 1525879 0 4553795 0 2218485 32421 .61 224 5710 12 08541 0. 4844510 0. 9798929 0 2050424 1100 0 1831055 0 4017353 0 2691746 32421 .28 224 4960 12 07522 0. 4265002 0 8423536 0 2810527 12C0 0 1373291 0 6759167 0 2540946 32419 .88 224 4247 12 0691 3 0. 5417496 0 7821969 0 2007642 1300 0 2288818 0 5650520 0 2390742 3241 7 .80 224 3533 12 06310 0. 5757742 0 6369250 0 2144844 1400 0 1220703 0 9381771 0 2876520 324 16 .00 224 3000 12 0621 7 0. 4796247 0 6948230 0 2852161 1500 0 1525379 0 4756451 0 1652241 3241 4 .73 224 2751 12 06344 0. 4365410 0 3656165 0 1687109 1600 0 1678467 0 3957748 0 2100468 32414 .04 224 2698 12.06556 0. 3567274 0 3559429 0 1580623 1700 0 1220703 0 3451109 0 1873374 3241 3 .76 224 2748 12 06791 0. 3003355 0 3094173 0 1306260 1800 0 1 220703 0 3689528 0 2254248 3241 3 .82 224 2839 12 06886 0. 3126982 0 2457092 0 1201635 1900 0 1220703 0 4374981 0 2340674 3241 4 .03 224 2961 12 07075 0. 2898445 0 3256616 0 1373836 2000 0 1373291 0 3033876 0 2139807 3241 4 .30 224 3054 12 07086 0. 3109455 0 2737409 0 1552640 2100 0 1525379 0 3212690 0 1 379251 32414 .56 224 3123 12 07135 0. 31 20705 0 2546480 0 1 1 73576 2200 0 1678467 0 5418062 0 3687143 3241 4 .80 224 3187 12 07148 0 . 3070587 0 2364612 0 1578440 2300 0 1520879 0 3 74 31 72 0 2601147 3241 4 .87 224 3233 12 07162 0. 3418558 0 2197179 0 1453026 2400 0 1525879 0 4512072 0 3088713 3241 4 .90 224 3243 12 07140 0. 3265565 , 0 2537884 0 1826439 2500 0 1678467 0 1829863 0 1275539 324 1 4 .91 224 3243 12 07180 0. 3420255 0 1811919 0 1034351 2600 0 2441406 0 4172325 0 1881719 3241 4 .94 224 3252 12 071 82 0. 3508134 0 2339178 0 1409571 2700 0 1678467 0 3790855 0 1763701 324 14 .96 224 3252 12 07171 0 3017497 0 2079946 0 1376554 2600 0 1220703 0 3379583 0 1655817 3241 4 .98 224 3256 12 07195 0 2806894 0 2379599 0 1353326 2900 0 1525879 0 3951788 0 1761913 3241 4 .98 224 3258 12 07171 0 3272932 0 2240903 0 1456900 30C0 0 1220703 0 481 0095 0 1909733 3241 4 .99 224 3264 12 07202 0 3035752 0 2377925 0 1246146 3100 0 1670467 0 29 74272 0 1407266 3241 4 .98 224 3267 12 07178 0. 2820640 0 2298317 0 1 340814 3200 0 1373291 0 3820658 0 2023578 3241 5 .00 224 3258 12 07147 0. 3344360 0 2571139 0 1714804 3300 0 1373291 0 4 30 3455 0 2081 394 3241 4 .99 224 3265 12 07161 0. 3218735 0 2336772 0 1622145 3400 0 1220703 0 2563000 0 0621042 3241 5 .00 224 3264 12 07185 0. 3179489 0 1556114 0 1018527 3500 0 1678467 0 31 82888 0 2275109 3241 4 .99 224 3267 12 07186 0. 3451950 0 2527984 0 1557258 3600 0 1373291 0 2658367 0 2171397 32415.01 224 3264 12 07168 0. 3050212 0 2068497 0 1332243 3700 0 1220703 0 3492832 0 1794696 3241 5 .04 224 3278 12 07179 0. 2826210 0 2592765 0 1598873 3800 0 1373291 0 5745888 0 2851486 32415.02 224 3272 12 07166 0. 2844763 0 2651960 0 1727926 3900 0 1220703 0 3701448 0 1338720 32415 .04 224 3277 12 07162 0. 3015327 0 1805770 0 1025956 4000 0 1525879 0 3558397 0 1871162 32415.01 224.3274 12.07203 0.2884764 0 2658353 0 1645718 CO Table 4-18 (Conti nued) n Q / 0 0 * Q n <.b Qr ] o a -ts Po PTI CDS CDF CDS/CDF CD TOTAL x 10" x 10 2 x 10' 100 7 .732573 8 .026108 0. 2888260 12 42114 -10 88319 18 34061 18 .29985 1 002227 36 64046 200 5 615565 3 .867237 0. 2382388 12 39877 -10 97000 18 34688 18 .34366 1 000175 36 69054 300 2 .604036 3 .789320 0. 2537769 12 39844 -11 40244 18 45369 18 .69798 0 986935 37 15167 400 1 .236813 4 .133679 0. 2434398 12 41647 -1 1 .62570 18 56787 18 .89192 0 982847 37 45979 500 1 .660638 3 .906574 0. 2416733 12 44124 -11 65694 18 64868 18 .92371 0 985467 37 57239 600 2 . 105945 3 .174808 0. 2257891 12 46280 -11 58049 18 69147 18 .87633 0 990207 37 56779 700 1 .971503 2 .281506 0. 2252967 12 47843 -11 50123 18 70422 18 .84224 0 992675 37 54646 800 1 .517206 1 .473901 0. 2051357 12 48667 -11 35229 18 69496 18 .72481 0 998406 37 41977 900 0 954553 1 068371 0. 2130742 12 48917 -11 328)3 18 67744 18 .69952 0 998819 37 37697 1000 0 585362 1 .039642 0. 2374525 12 48693 -11 26943 18 65321 18 .65115 1 000110 37 30437 1100 0 439911 0 952705 0. 2241366 12 48299 -11 24377 18.63721 18 .63858 0 999926 37 27579 1200 0 478394 0 81 3143 0. 2196897 12 47928 -11 26189 18 62828 18 .65472 0 998582 37 28300 1300 0 .560640 0 .740319 0. 2192370 12 47559 -11 23158 18 61845 18 .61946 0 999946 37 23792 1400 0 .518971 0 .666703 0. 2393850 12 47287 -11 26669 18 61732 18 .65097 0 998196 37 26830 1500 0 .453138 0 .467445 ' 0. 1920039 12 47189 -11 25685 18 61839 18 .63422 0 999151 37 25261 1600 0 369607 0 334040 0. 1485944 12 47161 -11 29150 18 62247 18 .66212 0 997875 37 28459 1700 0 346567 0 313880 0. 1505162 12 47176 -11 29163 18 62576 18 .66394 0 997954 37 28970 leoo 0 314561 0 287733 0. 1417457 12 47232 -11 31389 18 62825 18 .68796 0 996805 37 31621 1900 0 306497 0 303364 0. 1504800 12 47293 -11 30577 18 63020 18 .67574 0 997562 37 30594 2000 0 312265 0 275238 0. 1406657 12 47334 -11 31006 18 63098 18 .68289 0 997221 37 31387 2100 0 304181 0 251536 0. 1355204 12 47371 -11 29468 18 63121 18 .66769 0 998046 37 29890 2200 0 307096 0 249181 0. 1375719 12 47401 -11 32909 18 63261 18 .70255 0 996261 37 3351 6 2300 0 320471 0 246206 0. 1380551 12 47426 -11 3071 1 18 63203 18 .67764 0 997558 37 30968 2400 0 320482 0 228877 0. 1365416 12 47448 -11 30654 18 63193 18 .68332 0 997249 37 31525 2500 0 327679 0 240385 0. 1480852 12 47449 -11 30057 18 63191 18 .67282 0 997809 37 30473 2600 0 338027 0 216020 0. 1299474 12 47453 -11 29 31 3 18 63196 18 .67061 0 9979 30 37 30257 2700 0 300572 0 230483 0. 1411159 12 47433 -11 30625 18 63205 18 .67685 0 997601 37 30890 2800 0 296888 0 241187 0. 1482385 12 47445 -11 29723 18 63196 18 .66417 0 998274 37 2961 3 2900 0 319610 0 244308 0. 1489319 12 47443 -11 29459 18 63181 18 .66756 0 998085 37 299 36 3000 0 307807 0 228157 0. 1396955 12 47435 -11 30437 18.63246 18 .67850 0 997535 37 31096 3100 0 309469 0 234688 0. 1438734 12.47436 -11.28905 18 63184 18 .66170 0 998400 37 29353 3200 0 317932 0 236364 0. 1470463 12 47443 -11 32988 18 63242 18 .69502 0 996651 37 32744 3300 0 311189 0 225638 0. 1368260 12 47446 -11 29414 18 63173 18 .66534 0.998199 37 29707 3400 0 .325372 0 .233173 0. 1424393 12 47449 -11 30502 18 63216 18 .67316 0 997804 37 30531 3500 0 . 319297 0 .218356 0. 1339688 12 47436 -11 29507 18 63202 18 .66586 0 998187 37 29788 3G00 0 .315672 0 .225532 0. 1373813 12 47450 -11 28871 18 63168 18 .66212 0 998369 37 29381 3700 0 289290 0 .230780 0. 1404415 12 47423 -11 30902 18 63237 18 .67828 0 997542 37 31065 3300 0 296593 0 .250369 0. 1520862 12 47448 -11 31734 18 63228 18 .68742 0.997049 37 31970 3900 0 .310010 0 .229651 0. 1417075 12.47447 -11 .30644 18 63219 18 .67889 0 997499 37 31108 4000 0 .314439 0 .232862 0. 1463900 12 47462 -1 1 .29218 18.63202 18.66928 0 .998004 37 30130 to to Table 4-19 Convergence C r i t e r i a for Re = 0.2, R = 3 n Mn * M n <b H n r-s "b Q n * «n <b c n h x 10 6 x 10'* x IO1* x IO1* x IO 3 x IO 3 100 7. 688999 1 696348 2 .508163 1771 311 3282 .504 80 .56285 3 665644 6.043103 4. 237801 200 12 .39 777 2 307892 3 .385544 1770 941 3283 .510 80 .43411 5 622692 5 .585801 6. 273467 300 15 .25879 2 794266 4.053116 1770 505 3284 .180 80 .29771 7 066622 5 .300507 8. 070458 400 16 .74891 3 175735 4 .653931 1770 045 3284 .537 80 .16118 8 031356 5 .103365 9. 384096 500 17 .58337 3 .356934 4 .796982 1769 548 3284 .607 80 .03004 8 614443 4 .934125 10 .94751 600 18 .11981 3 356934 4 .796982 1769 048 3284 .444 79 .90666 8 877232 4 .754219 IC .15857 700 18 . 11981 3 309250 4 .682541 1766 548 3284 .094 79 .79274 8 889337 4 .544154 9. 955030 800 17 .28535 3 118515 4 .348755 1768 052 3283 .589 79 .68860 8 678266 4 .298594 9. 293891 900 17 .28535 2 861023 3 .881454 1767 577 3282 964 79 .59460 8 315190 4 022188 8. 588515 1000 16 .21246 2 574921 3 .576279 1767 125 3282 263 79 .51059 7 895411 3 722415 7. 799447 1100 15 .25879 2 279282 3 .261566 1766 689 3281 483 79 .43639 7 375022 3 413866 6. 850116 1200 14 .54353 1 9931 79 2 .737045 1766 302 3280 684 79 .37161 6 776687 3 101599 5.963568 1300 13 .35144 1 707077 2 241135 1765 932 3279 865 79 .31569 6 145390 2 792990 .5. 033828 1400 12 .09974 1 468658 2 .031 326 1765 605 3279 049 79 .26810 5 549570 2 .495288 4. 29 3259 1500 10 .72834 1 220703 1 .468658 1765 304 3278 .2 38 79 .22929 4 927446 2 .210183 3. 496339 1600 9. 596348 1 039505 1 .392365 1765.045 3277 .465 79 .19786 4 344992 1 944807 2. 957584 1700 8. 583069 0 877380 1 .125336 1764 809 3276 .728 79 .17294 3 785610 1 .699361 2. 220203 1800 7. 450581 0 743866 0 .84 8770 1764 611 3276 .035 79 .15366 3 262402 1 .473492 1 . 813659 1900 6. 675720 0 619888 0 619888 1764 438 3275 395 79 .13992 2 795525 1 268199 1 . 332181 2000 5. 722046 0 524521 0 514984 1764 292 3274 .800 79 .13092 2 361546 1 083920 1 . 028249 2100 4. 768372 0 438690 0 .391007 1764 177 3274 .260 79 .12613 1 978497 0 918955 0. 777392 2200 3. 814697 0 371933 0 .381470 1764 080 3273 .783 79 .1 2431 1 600771 0 .774262 0. 500194 2300 3. 039837 0 314713 0 .257492 1764 on 3273 358 79 .12560 1 302906 0 645247 0. 356382 2400 2. 980232 0 267029 0 .457764 1763 952 3272 .982 79 .12926 0 979475 0 537941 0. 353353 2500 2. 205372 0 219 345 0 .3051 76 1763 926 3272 .671 79 .13528 0 780561 0 .449259 0. 294953 2600 1 . 907349 0 190735 0 .352860 1763 902 3272 .401 79 .14275 0 557911 0 .376650 0. 441258 2700 1 . 907349 0 162125 0 .333786 1763 892 3272 .180 79 .15096 0 403196 0 .322402 0 427356 2800 1 . 907349 0 1 43051 0 .295639 1763 891 3271 .991 79 .15974 0 206111 0 .282809 0. 433784 2900 0. 953674 0 190735 . 0 314713 1763 909 3271 843 79 .16890 0 085670 0 250734 0. 387055 3000 0. 952674 0 162125 0 .333786 1763 922 3271 .729 79 .17813 0 057209 0 226068 0. 412003 3100 0. 953674 0 238419 0 333786 1763 932 3271 .632 79 .18567 0 071355 0 207723 0. 412716 3200 0. 834465 0 181 198 0 305176 1763 947 3271 .551 79 .19330 0 093325 0 193708 0. 405075 3300 0. 9536 74 0 171661 0 .314713 1763 961 3271 .507 79 .20090 0 110211 0 .178787 0. 371105 3400 0. 953674 0 152588 0 .276566 1763 980 3271 .478 79 .20828 0 135777 0 .169710 0. 342841 3500 1. 072884 0 162125 0 .238419 1764 000 3271 .460 ' 79 .21455 0 157709 0 .155822 0. 309255 3600 1. 132488 0 114441 0 .305176 1764 018 3271 .458 79 .22069 0 140339 0 .141678 0 354224 3700 1. 907349 0 123978 0 .286102 1764 037 3271 .467 79 .22652 0 150980 0 .1 32230 0 344342 3800 1. 907349 0 143051 0 .381470 1764 058 3271 .474 79 .23099 0 153984 0 .119577 0 289707 3900 1. 907349 0 104904 0 257492 1764 081 3271 .499 79 .23566 0 176497 0.112916 0. 303380 4000 1. 072884 0 114441 0 .333786 1764 106 3271 521 79 .23984 0 148587 0 .102654 0. 269666 4100 1. 132488 0 095367 0 276566 1764 126 3271 .557 79 .24324 0 140848 0.092805 ' 0. 276273 4200 0. 953674 0 114409 0 219345 1764 147 3271 .586 79 .24649 0 116996 0 .086500 0. 230156 4300 0. 953674 0 104904 0 .276566 1764 162 3271 .621 79.24994 0 102239 0 .082807 0. 227814 4400 0. 953674 0 667572 0 .190735 1764 175 3271 .666 79 .25195 0.073288 0 .070994 0 185390 4500 0. 953674 0 667572 0 .247955 1764 187 3271 .706 79 .25290 0.068793 0 .060974 0. 115294 Table 4-19 (Continued) n Qn100 # Po PIT C C DF C 3S/CDF CD rOTAL x 1 0 " x 1 0 3 x i o 3 100 2 583549 6. 444227 3 193853 39 85486 -158 .7242 124 .3777 156 .0578 0 .7969976 280 .4353 200 4 698345 5 792599 5 342774 40 06970 -157 1782 124 .1856 155 .0768 0 .8008001 279 .2622 300 6 396144 5 432308 7 184137 40 23824 -155 7266 123 .9795 154 0817 0 .8046347 278 .0610 400 7 589322 5 196106 8 652471 40 3641 4 -154 4993 123 .7698 153 .1913 0 .8079424 276 .9609 500 8 352662 5 016956 9 659093 40 45186 -153 5629 123 . 5652 152 .4798 0 8103712 276 .0449 600 8 764211 4 845861 10.13161 40 5061 5 -152 8267 123 .3702 151 .8947 0 .8122089 275 2646 700 8 88931 3 4 651256 10.10136 40 53261 -152 2967 123 .1886 151 4564 0 8133605 274 .6448 800 8 791771 4 422347 9 688854 40 53636 -151 9308 123 .0218 151 .1395 0 .8139619 274 .1611 900 2 515895 4 160725 9 009130 40 52269 -151 7114 122 .8709 150 9311 0 .8140863 273 8020 1000 8 114893 3 872286 8 165877 40 49586 -151 5990 122 . 7357 150 .8038 0 .8138767 273 5393 1100 7 616449 3 567281 7 267587 40 45953 -151 5893 ' 122 .6162 150 7545 0 8133500 273 3706 1200 7 055230 3 255383 6 352384 40 41701 -151 6490 122 .5121 150 7645 0 8126054 273 2764 1300 6 460263 2 944244 5 469970 40 37074 -151 7663 122 .4220 150 8139 • 0 8117424 273 . 2358 1400 5 844426 2 640916 4 644755 40 32275 -151 9428 122 .3456 150 91 36 0 81 06995 273 . 2590 1500 5 235441 2 350033 3 878897 40 27455 -152 1974 122 .2834 151 0839 0 8093740 273 3672 1600 4 633926 2 074864 3 198716 40 22731 -152 4465 122 . 2330 151 2432 0 8081884 273 4761 1700 4 061419 1. 819421 2 591989 40 18199 -152 7222 122 .1929 151 4261 0 8069476 273 6189 1800 3 524331 1 582745 2 053485 40 1 3930 -152 9903 122 .1621 151 6074 0 8057794 273 7693 1900 3 020708 1 367905 1 598944 40 09952 -153 2954 122 .1400 151 8227 0 8044911 273 9626 2000 2 566229 1 173433 1 207811 40 06300 -153 5595 122 .1255 152 0043 0 8034342 274 1296 2100 2 150231 0. 998216 0. 875003 40 02977 -153 8629 122 .1178 152 2230 0 8022295 274 3406 2200 1 768756 0. 844695 0 613017 40 00000 -154 1187 122 .1146 152 4035 0 8012586 274 5181 2300 1 438465 0. 708735 0 415187 39 97359 -154 3553 122 .1163 152 5716 0 8003870 274 6877 2400 1 139820 0. 590515 0 297954 39 95039 -154 5942 122 .1220 152 7442 0 7995196 274 8660 2500 0. 830219 0. 490525 0. 282180 39 93025 -154 8362 122 1313 152 9243 0 7986385 275 0554 2600 0. 657482 0. 411033 0 320917 39 91295 -155 0470 122 1431 153 0802 0 7979029 275 2231 2700 0. 464176 0. 347852 0. 353551 39 89830 -155 2213 122 1558 153 2054 0 7973334 275 3611 2800 0. 287206 0. 300964 0. 392682 39 88612 -155 3615 122 1694 153 3089 0 7968841 275 4783 2900 0. 140709 0. 264753 0 420449 39 8761 7 -155 5067 122 1835 153 4131 0 7964340 275 5964 3000 0. 069986 0. 238843 0. 448121 39 86809 -155 6129 122 .1979 153 4876 0 7961414 275 6853 3100 0. 068452 0. 218061 0. 456232 39 86177 -155 6595 122 .2090 153 5168 0 7960630 275 7256 3200 0. 092637 0. 201865 0. 475389 39 85718 -155 7427 122 2208 153 5824 0 7957994 275 8030 3300 0. 103976 0. 186498 0.43631 6 39 85368 -155 8118 122 .2324 153 6346 0 7956041 275 8669 3400 0. 119918 0. 174016 0. 419280 39 85155 -155 8751 122 2439 153 6873 0 7954066 275 9312 3500 0. 132400 0. 160776 0. 378339 39 84978 -155 8913 122 2534 153 6932 0 7954380 275 9465 3600 0. 149762 0. 147814 0. 356970 39 84892 -155 9336 122 .2630 153 7300 0 7953101 275 9929 3700 0. 1 3931 2 0. 138158 0. 337809 39 84851 -155 9559 122 2720 153 7457 0 7952873 276 0176 3800 0. 152425 0. 126041 0. 290798 39 84851 -155 9583 122 .2787 153 7460 0 7953297 276 0247 3900 0. 161051 0. 117114 0. 294745 39 84872 -155 9595 122 .2860 153 7448 0 7953831 276 0308 4000 0. 155760 0. 107053 0. 269724 39 84946 -156 0029 122 2926 153 7840 0 7952229 276 0764 4100 0. 140247 0. 097784 0. 243714 39 85031 -155 9814 122 2976 153 7692 0 7953321 276 0667 4200 0. 133075 0. 091302 0. 249651 39 85126 -155 9710 122 .3025 153 7591 0 7954160 276 0615 4300 0. 113824 0. 083672 0. 227649 39 85213 -156 0180 122 .3079 153 8000 0 7952398 276 1077 4400 0.094787 0. 076321 0. 183333 39 85310 -155.9713 122 .3105 153 7562 0 7954832 276.0667 4500 0.075530 0. 065425 0. 120520 39 85422 -155 9299 122.3115 153 7231 0 .7956607 276 .0344 ro o Table 4-20 Convergence C r i t e r i a for Re = 0.5, R = 100 n V n cb Mn <s Sn Qn * Q n Cb Q n h x IO3 X 10" x 1 0 3 . x 1 0 6 X IO5 X IO3 100 9. 521484 308.9738 69 .45044 32152 98 269 6206 12 48976 192.2936 598.1970 78 .60362 ZOO 5. 813599 72 .93880 • 11 .64007 32897.04 217 3437 11 56137 135.9953 378.1964 20 .20574 300 3 005981 15 .23376 1 . 390696 33521 07 248 5018 16 95943 73 .88487 158.3455 1 . 337505 400 1 . 464844 5. 1 4 3881 0. 960767 33836 05 263 7090 16 82074 34 .05107 61 .77011 0. 956267 500 0. 74 7681 5. 266070 0. 942051 33974 82 267 5125 15 95480 14 .89648 41 .41983 1 . 089754 500 0. 366211 3. 955364 0. 911474 34024 83 266 6531 15 23694 6. 696414 32 .34337 0. 917320 700 0. 196457 3. 103614 0. 573337 34023 62 264 2664 14 74446 4. 349235 26 .1 8395 0. 495385 800 0. 320435 2. 298355 0. 394404 • 33988 12 261 5579 14 41931 7. 746299 21 .33438 0. 358446 900 0. 389099 1 . 533628 0. 418961 33935 96 259 0725 14 22863 8. 805154 16 .11329 0. 299834 1000 0. 342369 a. 569820 0. 345707 33885 04 257 1851 14 16297 7. 489736 10 .49571 0. 199975 , 1100 0. 274658 0. 652075 0. 409603 33846 32 256 0806 14 19449 4. 979044 6. 285653 0. 218003 1200 0. 167847 l . 133084 0. 356 31 6 33823 99 255 7011 14 27831 2. 504750 5. 082808 0. 290933 1300 0. 167847 l . 282692 0. 422418 '33816 34 255 8445 14 37438 1 .081979 4. 848797 0. 337939 1400 0. 106812 0. 735521 0. 413001 3381 9 36 256 2695 14 45832 1 . 197126 4. 547894 0. 187760 1500 0. 122070 0. 736713 0. 300169 33827 48 256 7605 14 51092 1 . 808348 3. 623804 0. 259597 1600 0. 103106 0. 713468 0. 413179 33837 37 257 1919 • 14 53942 1 . 768719 . 2. 749849 0. 197124 1700 0. 183106 0. 494719 0 365496 33846 16 257 4912 14 54451 1 . 426763 1 . 841213 0. 214781 1800 0. 183106 0. 478625 0. 422001 33852 08 257 6426 14 53440 0. 995388 1 . 156043 0. 212080 1900 0. 122070 0. 505447 0. 275791 33855 00 257 6755 14 51743 0. 581042 1 . 062523 0. 198199 2000 0. 228882 0. 723004 0. 222564 33855 89 257 6230 14 49941 0. 417295 1 . 044971 0. 182862 2100 0. 1 2;'0 70 0. 582337 0. 360787 33855 43 257 5464 14 48388 0. 428392 0. 928156 0. 267273 2200 0. 213C23 0. 404716 0. 309050 33853 83 257 4719 14 48084 0. 516841 0. 786435 0. 180125 2300 0. 122070 0 . 604987 0. 259578 33851 95 257 3989 14 47639 0. 459557 0. 774381 0. 201942 2400 0. 137329 0. 600219 0 . 341415 33850 60 257 3545 .14 47627 0. 416522 0. 543386 0. 221883 2500 0. 167847 0. 324249 0. 259042 33849 67 257 3345 14 47839 0. 395265 0. 357152 0. 113529 2600 0. 152588 0. 309 348 0. 209153 33849 21 257 3303 14 48089 0. 31 39 78 0. 305742 0. 1 76529 2 700 0. 1 831 OC 0. 318885 0. 192285 33849 12 257 3359 14 48322 0. 337376 0. 230977 0. 153277 2800 0. 137329 0. 535318 0. 340045 33849 25 257 3494 14 48550 0. 311518 0. 254344 0. 1 4 39 1 7 2900 0. 152588 0. 320077 0. 249744 33849 43 257 3604 14 48582 0. 319694 0. 254052 0. 145055 3000 0. 137229 0. 510812 0. 220895 33849 63 257 3696 14 48641 0. 31 3066 0. 229429 0. 154955 3100 c. 137329 0. 313520 0. 180053 33849 81 257 3748 14 48655 0. 271504 0. 199335 0. 132320 3200 0. 198364 0. 417233 0. 190496 33849 93 257 3779 14.48654 0. 299809 0. 158379 0. 093568 3300 0. 16 784 7 0. 287890 0. 198245 33850 00 257 3818 14 48684 0 264300 0. 185080 0. 105418 3400 0. 152588 0. 439382 0. 142992 33850 05 257 3826 14 48667 0 271085 0. 214491 0. 1 2581 5 3500 0. 137329 0. 409484 0. 155151 33850 06 257 3848 14 48735 0. 315248 0. 199007 0. 108058 3600 0. 183106 0. 367761 0. 225425 33850.07 257 3845 14 48654 0. 300786 0. 202527 0. 170830 3700 0. 167847 0. 281334 0. 121892 33850 07 257 3845 14 48670 0. 280235 0. 147138 0. 102530 3800 0. 152588 0. 486970 0 206471 33850 09 257 3843 14 48687 0 307100 0. 185090 0. 124333 3900 0. 167847 0. 277758 0. 137448 33850.08 257 3848 14.48711 0. 262374 0. 174157 0 108831 4000 0. 183106 0. 392199 0. 303388 33850 09 257 3848 14.48661 0.286841 0. 191796 0. 132210 Table 4-20 (Conti nued) n % 10 0 r *•!} \ 10 0 - Po P* CDS/C0F TOTAL X 10" X 103 x 10 1 100 777.9241 2594.948 604.9518 6 399675 36 .34631 7. 469881 -23 .85773 -0.313101 -16 38783 200 155.8785 562.1548 45 .24591 5 401373 -17 .31380 7. 211985 • 17 .85170 0.403994 25 06367 300 103.8590 263.4616 7. 496189 6 116107 -14 . 2301 1 •• IC .47842 15 .97412 ' 0.655963 26 45255 400 51 .71347 98 .06842 6. 254554 6 454000 -8. 026783 IC .38021 11 .36821 0.913091 21 74841 500 23 .07153 49 .45197 1 .060185 6 538251 -5. 374091 9 8451 18 9. 352885 1 .052629 19 . 19800. 600 10 .25873 36 .62514 0. 922711 6 521586 -4. 425424 9. 402945 8. 589066 1 .094757 17 .99200 700 5. 033973 29 .15219 0. 658928 6 473050 -4. 177400 9. 101320 8. 360942 1 .088552 17 46225 800 6. 010483 23.67815 0. 467425 6 418550 -4. 151567 8. 901600 8. 301298 1 .07231 3 17 .20290 900 8. 527758 18 .77093 0. 323148 6 369474 -4. 292788 8. 784275 8. 373407 1 .049067 17 .15767 1000 8. 320517 13 .30759 0. 233420 6 332848 -4. 573229 8 744663 8. 562129 1 .021318 17 .30678 1100 6. 293631 8. 077320 0. 218993 6 011744 -4. 869937 8 764620 8. 782026 0.998018 17 .54665 1200 3. 713081 5. 580425 0. 243026 6 305036 -5. 121823 8 816704 8. 969275 0.982990 17 .78596 1300 1 . 688944 5. 011052 0. 248738 6 308683 -5. 282665 8 876005 9. 096009 0.975813 17 .97200 1400 1 . 029053 4. 753985 0. 229748 6 317655 -5. 339795 8 927345 9 149370 0.975733 18 •07671 1500 1. 566322 4. 138451 0. 213067 6 327747 -5. 329376 8 959652 9. 151669 0.97901 8 18 .11131 1600 1. 827121 3. 254870 0. 206023 6 336169 -5. 285035 8 976837 9. 121782 0.984110 1 8 .09862 1700 1. 625662 2. 260068 0. 198879 6 341838 -5. 232705 8 979706 9. 081871 0.988751 18 .06157 1800 1 . 180066 1 . 429558 0. 186788 6 344737 -5. 174661 8 973278 9. 039791 0.992642 18 .01306 1900 0. 741426 1 . 074603 0. 184246 6 345247 -5. 143297 8 963215 9 017463 0.993984 17 98067 2000 0. 497338 1 . 064901 0. 21 8978 6 344159 -5. 108975 8 951845 8. 987653 0.996016 17 .93948 2100 0. 408959 0. 936774 0. 205980 6 342561 -5. 122100 8. 945486 8. 9981 37 0.994149 1 7 .94362 2200 0. 482327 0. 797232 0. 201985 6 341143 -5. 141502 8 941284 9. 015608 0.991756 17 .95688 2300 0. 505006 0. 727014 0. 206333 6 339540 -5. 140205 8. 938021 9 009363 0.992081 17 .94737 2400 0. 454986 0. 569552 0. 210985 6 338715 -5. 141205 8 937981 9 012220 0.991762 17 .95020 2500 0. 409966 0. 396451 0. 170176 6 338330 -5. 141486 8 939159 9 011545 0.991967 17 .95070 2600 0. 358159 0. 293308 0. 137626 6 338263 -5. 145746 8 940617 9 014179 0.991839 17 .95479 2700 0. 332902 0. 27399 3 0. 134507 6 338476 -5. 169041 8 942518 9 032464 0.990042 17 .9 7498 2800 0. 300416 0. 276702 0. 136252 6 338636 -5. 152096 8 943373 9 017553 0.991774 17 .96092 2900 0. 289496 0. 254124 0. 129680 6 338923 -5. 160241 8 944004 • 9 027002 0.990806 1 7 .97099 3000 0. 308181 0. 226922 0. 124626 6 339016 -5. 151320 8 944212 9 023125 0.991254 17 .96733 3100 0. 293890 0. 203207 0. 123117 6 339200 -5. 151319 8 944172 9 019409 0.991658 17 .96358 3200 0. 295796 0. 199876 0. 127157 6 339210 -5. 157525 8 944424 9.025557 0.991011 17.96997 3300 0. 286259 0. 195110 0. 123262 6 339327 -5. 148466 8 944311 9 016989 0.991940 17 .96129 3400 0. 284780 0. 194784 0. 128825 6 33931 6 -5. 1 5481 3 8 944427 9 023410 0.991247 17 .96783 3500 0. 31 3081 0. 188242 0. 123294 6 339327 -5. 149818 8 944548 9 018152 0.991838 17 .96269 3600 0. 316996 0. 196212 0. 132730 6 339297 -5. 155107 8 944427 9 022446 0.991353 17.96687 3700 0. 287831 0. 186650 0. 129220 6 339365 -5. 146738 8.944244 9 016602 0.991975 17.96085 3800 0. 295443 0. 185980 0. 128660 6 339423 -5. 151116 8 944328 9 018015 0.991829 17 .96234 3900 0. 297497 0. 185464 0.125916 6 339297 -5. 153989 8.944567 9 022766 0.991333 17.96733 4000 0. 279591 0. 183802 0. 129112 6 339336 -5. 155262 8.944403 9.022274 0.991369 17 .96667 Table 4-21 Convergence C r i t e r i a for Re = 40, R = 100 n Mn <s S n <b Qn +. «n C n <s x 10 1 x 1 02 x 1 02 x 1 06 x 1 03 X i o2 100 0 1983643 0 6668091 1 .230431 1 291594 2 102468 0. 295458 200 0 1831055 0 6532669 1 170254 1 273185 2 059035 0. 280922 300 0 1525879 0 6475449 1 201344 1 298084 2 016782 0. 267310 400 0 2441406 0 6332397 1 192570 1 375002 1 974082 0. 249072 5 0 0 0 1678467 0 6280899 1 128387 1 414542 1 9371 51 0. 230943 600 0 1678467 0 6093025 1 089191 1 519139 1 900999 0. 222784 700 0 21 36230 0 5880356 1.047039 1 580679 1 868643 0. 221859 800 0 2136230 0 5641937 0 977135 1 736079 1 845623 0. 215696 900 0 1831055 0 5435944 0 952911 l 690156 1 838715 0. 215468 1000 0 1983643 0 5089760 0 908279 1 798760 1 828332 0. 214261 1100 0 2746582 0 4788399 0 877857 1 859271 1 819822 0. 214214 1200 0 2238818 0 4512787 0 806522 1 807392 1 809245 0. 207264 1300 0 2441406 0 4092216 0 722885 1 815648 1 788809 0. 196578 1400 0 1831055 0 3764153 0 682396 1 888188 1 763372 0. 191568 1500 0 2441406 0 3447533 0 714564 l 850942 1 732493 0 185764 1600 0 2593994 0 3204346 0 689363 1 850673 1 693715 0. 1 70193 1700 0 2136230 0 3405273 0 717467 1 826818 1 646181 0. 170102 1800 0 2238818 0 3511012 0 735372 1 776798 1 592637 0. 166536 1900 0 1831055 0 3569067 0 724924 1 758803 1 532072 0 156058 2000 0 1678467 0 3628850 0 721192 l 680284 1 467823 0 152744 2100 0 2593994 0 3615916 0 697426 3741 6 02 503.9138 68 02225 1 682060 1 397989 0.1 36730 2200 0 1831055 0 3630996 0 665063 37407 52 503 4678 68 1 5862 l 525117 1 327950 0 1 34 371 2300 0 1983643 0 3550708 0 659019 37399 82 503 0869 68 30487 1 442062 1 257850 0 132097 2400 0 1983643 0 3481805 0 637931 37392 81 502 7581 68 46954 1 363077 1 191895 0 142101 2500 0 2441406 0 3369629 0 584668 37386 56 502 5051 68 65569 l 276649 1 132730 0 138086 2600 0 1331055 0 3193557 0 552690 37381 16 502 3169 68 83795 1 153235 1 080699 0 133840 2700 0 3204346 0 3057718 0 540924 37376 59 502 1904 69 03424 1 .060163 1 046091 0 143475' 2800 0 1525879 0 2368772 0 490093 37372 97 502 1233 69 24002 0 966125 1 021 395 0 139120 2900 0 1933643 0 2861023 0 535011 37370 29 502 1255 69 44005 0 907823 0 9981 51 0 149511 3000 0 1831055 0 2952576 0 603390 37368 54 502 1819 69 64682 0 868960 0 9831 50 0 1 5921 6 3100 0 1983643 0 2941132 0 573158 37367 81 502 2966 69 85341 0 843627 0 9671 26 0 142822 3200 0 2288818 0 3055573 0 584316 37367 91 502 4685 70 06000 0 848838 0 955567 0 149561 3300 0 2136230 0 2968788 0 538063 37368 79 502 6877 70 24905 0 855010 0 933772 0 127209 3400 0 1678467 0 3043175 0 534439 37370 46 502 9517 70 43877 0 813602 0.920256 0 124725 3500 0 2441406 0 2981186 0 555897 37373 07 503 2502 70 61066 0 814779 0 902206 0 119377 3600 0 1678467 0 2924919 0 569439 37376 25 503 5850 70 77069 0 868425 0 887805 0 105677 3700 0 1831055 0 2859116 0 552463 37379 97 503 9451 70 91615 0 959947 0 .875755 0 104559 3800 0 1831055 0 2788544 0 511455 37384 14 504 3279 71 05078 1 005581 0 .867068 0 100364 3900 0 1831055 0 2731323 0 493813 37388.98 504 7217 71 17053 1 065001 0 .863128 0 100258 4000 0 1678467 0 2575874 0 515842 37393 91 505 1252 71 27045 1 .084593 0.863945 0.101495 Table 4-21 (Continued) n " ° lb n y i o o , tl '•b 10 0 , >. s Po C DS/CDF C 3 TOTAL x 1011 x I O 1 100 1 .31 4693 2 .124828 0 2982533 1 . 142896 -0 3307629 0 5247070 0 853658 0 6146572 1 .378365 200 1 .290960 2 .030667 0 2847220 1.142811 -0 3233471 0 5215412 0 846943 0 6157926 1 .368484 300 1 .305476 2 .037089 0 2692951 1 .142702 -0 3174810 0 5185379 0 841432 0. 6162565 1 .359969 400 1 .342586 1 .996216 0 2527016 1 . 1 42573 -0 3141441 0 5157066 0 837885 0. 6154861 1 .353591 500 1 .403603 1 .955230 0 2372945 1 . 142424 -0 31 1 5425 0 51 30401 0 834948 0 6144574 1 .347983 600 1 .474733 1 .917639 0 2242634 1 . 142255 -0.3117952 0 5106079 0 834286 0. 61 20300 1 .344393 700 1 .546225 1 .883624 0 2210818 1 .142069 -0 31 38885 0 5084000 0 835112 0. 6087808 1 .34351 2 800 1 .617093 1 .855219 0 21 88038 1 .141873 -0 3178988 0 5064278 0 837432 0. 6047392 1 .343859 900 1 .690562 1 .842204 0 2216495 1 .141663 -0 3215256 0 5046062 0 839540 0 6010505 1 .344147 1000 1 .739688 1 .831412 0 2124700 1.141444 -0 3292017 0 5031425 0 844843 0. 5955452 1 . 347985 1100 1 .786033 1 .824294 0 2100453 1.141221 -0 3369503 0 5018768 0 850240 0 5902766 1 .352117 1200 1 .828202 1 .815712 0 2081533 1 . 1 40992 -0 3445349 0 5007747 0 855563 0 5853161 1 . 356338 1300 1 .851610 1 .798664 0 1987789 1 . 1 40762 -0 3556004 0 5000126 0 863672 0 5789383 1 .363684 MOO 1 .867506 1 .777012 0 1936976 1 . 140533 -0 3658161 0 4994217 0 871187 0 5732656 1 .370608 1500 1 .878604 1 .748096 0 1868154 1 . 140306 -0 3758278 0 4990127 0 878553 0 5679936 1 .377565 1600 1 .871609 1 .713039 0 1800582 1 .140082 -0 3864622 0 4988132 0 88651 5 0 5626674 1 .385328 1700 1 .854698 1 .670104 0 1709459 1.139865 -0 3995905 0 4983720 0 896459 0 5564917 1 .395330 1800 1 .820566 1.619601 0 1640437 1 .1 39657 -0 41 1 3731 0 4990906 0 905404 0 5512353 1 .404494 1900 1 .780522 1 .561838 0 1557587 1 . 1 39458 -0 4234428 0.4994922 0 914654 0 5460994 1 .414146 2000 1 .734900 1 .500093 0 1489416 1 . 1 39272 -0 43651 58 0 5000812 0 924733 0 5407842 1 .424814 2100 1 .669077 1 .433321 0 1 4239 76 1.139099 -0 4487238 0 5008118 0 934195 0 5360893 1 .435006 2200 1 .538970 1 .362929 0 1396648 1 .1 38941 -0 4620934 0 5017353 0 944640 0 5311383 1 .446375 2300 1 .504886 1 .292013 0 1336139 1 .1 38798 -0 4739456 0 5027458 0.953946 0 5270172 1 .456691 2400 1 .405464 1 .223427 0 1338050 1 .1 38672 -0.4866219 0 5039123 0 963922 0 5227726 1 .467834 2500 1 . 316716 1 .161875 0 1 394890 1 . 1 38570 -0 4996014 0 5052547 0 974183 0 51 86447 1 .479437 2600 1 .217745 1 .105760 0 1348662 1 .1 38484 -0 5094404 0 5065799 0.982006 0 5158623 1 .488585 2700 1 . 108542 1 .061581 0 1433614 1.138416 -0 5208387 0 5080276 0 991100 0 5125898 1 .499126 2800 1 .033478 1 .033845 0 1494408 1 . 1 38371 -0 5312948 0 5095615 0 999505 0 5098140 1 .509066 2900 0 .961960 1 .007423 0 1449962 1 .1 38343 -0 5399694 0 5110664 1 .006564 0 •5077336 1 .51 7631 3000 0 .897825 0 .98871 3 0 1493978 1 .1 38332 -0 5490694 0 5126358 1 013971 0 5055722 1 .526607 3100 0 .859266 0 .975847 0 1485261 1 .1 38344 -0 5563145 0 5142117 1 .019884 0 5041863 1 .534096 3200 0 .836836 0 .960633 0 1483834 1 . 1 38371 -0 5640001 0 5158043 1 026215 0 5026281 1 .542019 3300 0 .816519 0 .944523 0 1350858 1 .13841 7 -0 5675640 0 5172687 1 029318 0 5025354 1 .546586 3400 0 .814615 0 .926681 0 1354721 1 . 138474 -0 5728397 0 51 8751 1 1 033759 0 5018104 1 .552510 3500 0 .830727 0 .910580 0 1224833 1 .1 38546 -0 5750561 0 5201069 1 035835 0 5021135 1 .555942 3600 0 .873907 0 .894979 0 1136836 1 . 1 38629 -0 5764227 0 5213783 1 037218 0 5026698 1 .558596 3700 0 .923325 0 .881451 0 1043551 1 . 1 38721 -0 5769472 0 5225483 1 037971 0 5034322 1 .560519 3800 0 .978368 0 .871982 0 1027506 1 .1 38822 -0 5768309 0 5236400 1 038189 0 5043783 1 .561829 3900 1 .029360 0 .863619 0 1005135 1 .138928 . -0 5762291 0 5246254 1 038030 0 5054049 1 .562654 4000 1 .079816 0 .863020 0 0980801 1 .1 39042 -0 5734720 0 5254627 1 036172 0 5071192 1 .561634 Table 4-22 Convergence C r i t e r i a for Re = 40, R = 3 n Mn * Mn Cb Mn C s S n 4- S n Cb S n Cs Q n * Q n Cb Q n CS x 10" x 10 s x 1.0 1 x IO5 x IO3 x IO3 100 0 1400709 0 9293556 1 .392788 2040 857 3426 031 154 8513 0 3 4 6 7 9 3 3 0 7 8 G 5 S 3 8 1 T65'50T 200 0 1394749 0 9123392 1 .365238 2040 145 3426 019 154 4943 0 3493675 0 7778001 1 . 132417 300 0 1400709 0 8975208 1 .341963 2039 431 3426 029 154 1448 0 3515810 0 7692611 1 118940 400 0 1400709 0 8826971 1 .321894 2038 713 3426 031 153 7998 0 3529487 0 761 04 34 1 102858 500 0 1406670 0 8669615 1 .300365 2037 987 3426 052 153 4611 0 3549284 0 7530260 1 083823 600 0 1406670 0 8669615 1 280403 2037 257 3426 070 153 1312 0 3595524 0 7449502 1 078818 700 0 1418591 0 8384902 1 258981 2036 529 3426 085 152 8044 0 3616924 0 7373017 1 . 044338 800 0 1424551 0 8255858 1- 235676 2035 797 3426 UO 152 4860 0 3626496 0 7296449 1 . 02981 1 900 0 1424551 0 8124582 1 216811 2035 059 3426 144 152 1719 0 3625278 0 7223630 1 . 024658 1000 0 1 43051 1 0 7998999 1 197195 2034 316 3426 178 151 8632 0 3664021 0 7151753 1 . 013332 1100 0 1430511 0 7885221 1 179689 2033 561 3426 216 151 5589 0 3666522 0 7081628 0. 981842 1200 0 1436472 0 7760465 1 159352 2032 815 3426 267 151 2624 0 3686047 0 7011962 0 973452 1300 0 1 448393 0 7640958 1 138419 2032 063 3426 321 150 9705 0 3726473 0 6944281 0 942498 1400 0 1 448393 0 7509828 1 120031 2031 302 3426 383 1 50 6871 0 3750059 0 6075819 0 936943 1500 0 1448393 0 7395923 1 100761 2030 536 3426 448 150 4056 0 3745891 0 6812334 0 914581 1600 0 1454353 0 7262230 1 071843 2029 783 3426 521 150 1301 0 3776234 0 6748337 0.890650 1700 0 1454354 0 7143248 1 058370 2029 Oil 3426 599 149 8630 0 3746976 0 6683898 0 897934 1800 0 1460314 0 7034846 1 041663 2028 240 3426 688 149 5962 0 3769961 0 6623900 0 831124 1900 0 1460314 0 6921146 1 018399 2027 465 3426 .780 149 3348 0 3826011 0 6564409 0 854759 2000 0 1484156 0 6808143 0 9990752 2026 694 3426 872 149 0795 0 3836500 0 6506258 0 846232 2100 0 1478195 0 6697774 0 9827614 2025.922 3426 .979 148 8289 0 3838099 0 6448960 0 835069 2200 0 1484156 0 6586611 0 9587288 2025 138 3427 090 148 5855 0 3855191 0 6389081 0 793672 2300 0 1484156 0 6472766 0 9402931 • 2024 356 3427 205 148 3485 0 3847217 0 6330991 0 791462 2400 0 1484156 0 6363938 0 9288490 2023 568 • 3427 333 148 1121 0 3868251 0 6277746 0 79639 3 2500 0 1496077 0 6263573 0 9104490 2022 790 3427 469 147 8802- 0 3892183 0 6224406 0 771538 2600 0 1496077 0 61631 76 0 89231 1 3 2022 005 3427 607 147 6547 0 3879888 0 6171118 0 753766 2 700 0 1496077 0 6061472 0 8769631 2021 220 3427 751 147 4351 0 3903583 0 6117045 0 728441 2800 0 1496077 0 5960863 0 8579016 2020 429 3427 906 147 2200 0 3890465 0 6064554 0 720968 2900 0 1507998 0 5857356 0 8382022 2019 644 3428 062 147 0084 0 3881885 0 6013145 0 704585 3000 0 1502037 0 5752977 0 8244157 2018 850 3428 228 146 7994. 0 3884565 0 5964944 0 703523 3100 0 1513958 0 5658090 0 8070946 2018 065 3428 400 146 5969 0 3924572 0 5914154 0 669455 3200 0 1513958 0 5566299 0 7899165 2017 279 3428 583 146 4000 0 3907015 0 5864485 0 664696 3300 0 1 51 3958 0 5173375 0 7716715 2016 486 3428 762 146 2032 0 3911441 0 5814182 0 646135 3400 0 1513958 0 5380452 0 7716715 2016 486 3428 762 146 0177 0 3933860 0 5768305 0 6 384 86 3500 0 1537800 0 5288124 0 7384479 2014 912 3429 154 145 8339 0 3907124 ' 0 5719939 0 616114 3600 0 1 531 839 0 5195856 0 7202625 2014 121 34 29 365 145 6534 0 3905066 0 5673298 0 610083 3700 0 1519918 0 5109549 0 7053137 2013 325 3429 .576 145 4767 0 3907352 0 5627447 0 600209 3800 0 1537800 0 5026698 0 6855965 2012 548 3429 786 145 3031 0 3906973 0 5582620 0 585128 3900 0 1519918 0 4943788 0 6685972 2011 760 3430 009 145 1341 0 3870730 0 5537209 0 572760 4000 0 1537800 0 4860520 0 6492674 2010 975 3430 240 144 9709 0 3889631 0 5491019 0 553076 4100 0 1519918 0 4776955 0 6299496 2010 192 3430 471 144 8118 0 3883103 0 5445829 0 544365 4200 0 1561642 0 4692614 0 6130517 2009 419 3430 706 144 6548 0 3864181 0 5402178 0 522291 4300 0 1537800 0 4616737 0 5951881 2008 634 3430 953 144 5063 0 3855380 0 5354874 0 521162 4400 0 1 51 3958 0 4541039 0 5829394 2007 868 3431 203 144 3585 0 3827075 0 5311212 0 488553 4500 0 1537800 0 4464865 0 5692184 2007 094 3431 455 144 2177 0 3843792 0 5265840 0 468675 4600 0 1525879 0 4388988 0.5516112 2006 316 3431 .720 144 0838 0 3822995 0 5217327 0 460922 4700 0 1537800 0 4313171 0 53951 74 2005 554 3431 .997 143 9499 0 3833231 0 5174421 0 459459 4800 0 1507998 0 4240811 0 5245328 2004 795 3432 .265 143 8179 0 3796685 0 5132814 0 457138 4900 0 1496077 0 41 73398 0 5093336 2004.030 3432 .538 143 6884 0 3790741 0.5090935 0 441493 5000 0 1507998 0 4105270 0 4954338 2003 276 3432 .814 143 5669 0 3749216 0 5044720 0 417942 Table 4-22 (Continued) n Qr Cb Q 11 0 0 Cs Po CDS CDF CD TOTAL x I O1 x I O1 x I O1 100 0 3459174 0 790971 5 1 171311 1 203033 -1 470630 1 .187882 1 . 839385 0 6458041 3 027267 200 0 3474574 0. 7821536 1 150740 1 203024 -1 472588 1 . 1 85440 1 . 841661 0 6436797 3 027102 3C0 0 3503470 0 7734537 1 129601 1 203015 -1 474226 1 . 183050 1 843480 0 6417482 3 026530 400 0 3525801 0 7651204 1 117106 1 203006 -1 474968 1 . 180696 1 844854 0.6399944 3 025551 500 0 3546397 0 7569861 1 099405 1 202996 -1 476142 1.178387 1 846170 0 6382870 3 024557 600 0 3567713 0 7488716 1 072800 1 202987 -1 479044 1 .176142 1 848893 0 6361328 3 025035 700 0 3592726 0 7411158 1 065190 1 202979 -1 480267 1 . 1 73922 1 850559 0 6343604 3 024481 800 0 3611837 0 7333755 1 039666 1 .202971 -1 483285 1.171763 1 853428 0 6322142 3 025191 900 0 3638344 0 7259107 1 028283 1 .202963 -1 484738 1 . 1 69633 1 855029 0 6305199 3 024662 1000 0 3652479 0 7187068 1 012322 1 202954 -1 486856 1 . 167543 1 857241 0 6286441 3 024784 1100 0 3672899 0 7116920 0 999432 1 .202947 -1 488900 1 .1 65491 1 859522 0.6267692 3 025013 1200 0 3C96666 0 7045990 0 975957 1 202938 -1 492130 1.163490 1 862404 0 6247250 3 025394 1300 0 3715025 0 6978190 0 .962797 1 202929 -1 495221 1.161 521 1 865046 0 6227842 3 026567 1100 0 .3727691 0 6909060 0.936202 1 202922 -1 499798 1 . 159620 1 869287 0 6203541 3 028908 1500 0 3749619 0 6844074 0 931938 1 202914 -1 502001 1 .1 57726 1 871197 0 6187090 3 028923 16C0 0 3760953 0 67801 1 2 0.913237 1 .202905 -1 505687 1 .1 55876 1 874476 0.6166394 3 030353 1700 0 3779838 0 671 4076 0 .887514 1 .202896 -1 510370 1 .1 54095 1 878699 0 6143051 3 032794 1800 0 3792436 0 6653720• 0 .887965 1 .202888 -1 512737 1 .1 52308 1 880807 0 6126665 3 033114 1900 0 .3801719 0 6594253 0 .871216 1 .202830 -1 516445 1 . 1 50562 1 884013 0 6106976 3 034575 2000 0 .3821435 0 6534636 0 .852447 1 .202370 -1 52041 8 1 .148858 1 887334 0 6087201 3 036192 2100 0 3833349 0 6477034 0 838156 1 .202862 -1 524734 1 .147188 1 890985 0 6066618 3 038173 2200 0 3847270 0 6419486 0 814781 1 202855 -1 531334 1 . 145571 1 896338 0 6040964 3 041908 2300 0 3852774 0 6359029 0 795271 1 202847 -1 536999 ' 1 . 144000 1 900887 0 6018245 3 044887 2400 0 3873182 0 6303358 0 794700 1 202839 -1 539875 1 . 142426 1 902994 0 6003310 3 045421 2500 0 3875643 0 6251007 0 780285 1 202830 -1 544664 1.140887 1 9071 25 0 5982237 3 048012 2600 0 3884404 0 6197350 0 759862 1 202822 -1 550507 1 . 1 39396 1 911698 0 5960122 3 051094 2700 0 3887198 0 6143835 0 740936 1 202814 -1 557132 1 . 1 37940 1 916630 0 5937194 3 054570 2S00 0 3905422 0 6090367 0 726883 1 .202806 -1 563076 1 . 1 3651 8 1 921207 0 5915647 3 057726 2900 0 3899820 0 6038438 0 71 659 7 1 .202799 -1 568626 1 .135119 1 925514 0 5895149 3 060634 3000 0 3906379 0 5988347 0 70831 9 1 202792 -1 573074 1 .1 33738 1 928747 0 5878103 3 062485 3100 0 3912360 0 59 39 701 0 687023 1 202785 -1 580461 1 .1 32405 1 934664 0 5853240 3 067069 3200 0 3913143 0 5888813 0 669330 1 202778 -1 587403 1.131107 1 939739 0 5831234 3 070847 3300 0 3912130 0 5839220 0 653036 1 202771 -1 595273 1 .1 29843 1 .945395 0 5807779 3 075238 3400 0 39 16307 0 5790719 0 649034 1 .202765 -1 600250 1.128590 1 949194 • 0 5790032 3 077784 2500 0 391176 7 0 5743853 0 626237 1 .202757 -1 608343 1.127335 1 955301 0 5765787 3 082686 3500 0 391 8051 0 5696237 0 616994 1 .202750 -1 614834 1.126195 1 959771 0 5746563 3 085966 3 700 0 3903647 0 5650340 0 603678 1 .202744 -1 .622046 1.125038 1 965497 0 5723937 3 090535 3800 0 3905019 0 5604819 0 594000 1 .202737 -1 628523 1.123899 1 969889 0 5705391 3 093787 2900 0 3902896 0 5559565 0 578748 1 .202730 -1 635825 1.122789 1 975150 0 5684577 3 097939 4000 0 3396339 0 5514128 0 5C0217 1 .202724 -1 .644518 1.121723 1 .981 738 0 5660300 3 103461 4100 0 3890502 0 5468003 0 547165 1 .202720 -1 652637 1.120681 1 937164 0 5639597 3 107845 4200 0 3876998 0 5423978 0 .539008 1 .202713 ' -1 .65971 8 1 .1 19649 1 992172 0 5620241 3 111821 4 300 0 3866141 0 5377869 0 .513983 1 .202709 -1 .670073 1 .1 1 8675 1 999290 0 5595361 3 117966 44C0 0 3849147 0 5333875 0 508523 1 .202703 -1 .678151 1.117708 2 005282 0 5573819 3 .122991 4500 0 3855095 0 5288066 0 489750 1 202701 -1 687804 1.116781 2.011660 0 5551541 3 128441 4600 0 3836032 0 5241019 0 463138 1 .202696 -1 699182 1 .1 1 5908 2 019847 0 5524714 3 135755 4700 0 3816742 0 5195352 0 461945 1 .202688 -1 706842 1.115040 2.025583 0 5504783 3 140623 4800 0 3804944 0 5153077 0 456155 1 .202682 -1 714274 1 .114181 2 030828 0.5486335 3.145009 4900 0 3788571 0 5111936 0 .448261 1 .202676 -1 .722136 1 . 1 1 3326 2 035708 0 5468985 3 .1 49035 5000 0 3779216 0 5067693 0 .419957 1 .202672 -1 .733464 1 . 1 1 2537 2 043962 0 5443042 3 .156500 NOMENCLATURE Roman Letters a f r a c t i o n of angular increment A i to A 6 c o e f f i c i e n t s f o r Equation (3-34) s o l i d s c o n c e n t r a t i o n d e f i n e d by Equation (3-13) c minimum s o l i d s c o n c e n t r a t i o n d e f i n e d m by Equation (4-1) C Q t o t a l drag c o e f f i c i e n t C n P form drag c o e f f i c i e n t d e f i n e d by u t" Equation (3-65) Cn<- s k i n drag c o e f f i c i e n t d e f i n e d by U 5 Equation (3-64) C i to C i o c o e f f i c i e n t s f o r Equation (2-5) D ip,D c b.D c sum of changes i n stream f u n c t i o n and v o r t i c i t y values f o r the nth i t e r a t i o n as d e f i n e d by Equations (3-53) to (3-55) impaction e f f i c i e n c y 208 209 impaction e f f i c i e n c y due to p a r t i c l e i nerti a impaction e f f i c i e n c y due to interception drag force on p a r t i c l e Fibonacci number defined by Equation (3-82) c o e f f i c i e n t in Equation (3-40) c o e f f i c i e n t s for Equation (3-35) unit vectors in r , e and £ directions angular subscript of a grid point radia l subscript of a grid point siz e parameter defined by Equation (1-1) modified Bessel functions f r a c t i o n of radial increment inte r v a l of uncertainty v i r t u a l mass of spherical p a r t i c l e defined by (p p - P)4/3TT Rp3 210 M I{J , M Sk>M £ s maximum changes in stream function and v o r t i c i t y values for the nth i t e r a t i o n as defined by Equations(3-45) to (3-47) Mna ty , Mna r, maximum changes in relaxed values of TT stream function and v o r t i c i t y N number of angular grid d i v i s i o n s a Nr number of radial grid d i v i s i o n s dimension!ess pressure defined by Equation (3-3) p0 frontal stagnation pressure defined by Equation (3-62) dimensional reference pressure pfi dimension!ess pressure on the cylinder surface at angle 6 p rear stagnation pressure defined by Equation (3-63) p a r t i c l e i n e r t i a l parameter defined on page 54 P C c r i t i c a l i n e r t i a l parameter point within f l u i d f l o w f i e l d Q ty* Q Sk'Q £ f r a c t i o n a l changes in the sum of stream n n n s function and v o r t i c i t y values for the nth i t e r a t i o n as defined by Equations (3-56) to (3-58) 211 Q 1 0 0 ij> , Q 1 0 0 Q 1 0 0 r sum of f r a c t i o n a l changes in n n D n n stream function and v o r t i c i t y values over 100 i t e r a t i o n s as defined by Equations (3-59) to (3-61) r dimensionless radius cylinder radius radius of spherical p a r t i c l e dimensionless radius of f1uid.envelope cylinder Reynolds number defined on page 27 p a r t i c l e Reynolds number defined on page 54 sums of stream function and v o r t i c i t y values for the nth i t e r a t i o n as defined by Equations (3-50) to (3-52) dimensionless time At time increment defined by Equation (3-80) Uo free stream v e l o c i t y y_ dimensionless f l u i d v e l o c i t y vector v_ dimensionless p a r t i c l e v e l o c i t y R P R CO n ^ ' n bb' n bs dimensionless radial component of flu i d veloci ty dimensionless angular component of flu i d velocity dimensionless x-component of flu i d veloci ty dimensionless y-component of f l u i d veloci ty dimensionless x co-ordinate dimensionless y co-ordinate dimensional y co-ordinate of particle dimensional starting position of c r i t i c a l trajectory transformed radius of fluid envelope transformed radial spacing of grid lines defined by Equation (3-17) transformed radial co-ordinate defined in Equation (3-8) relaxation factor for stream function relaxation factor for vorticity 213 relaxation factor for v o r t i c i t y on the surface of the cylinder parameter defined by Equation (3-79) factor defined by Equation (2-3) minimum separation of s t a r t i n g co-ordi nates "del" operator impaction c o e f f i c i e n t i n e r t i a l impaction c o e f f i c i e n t interception c o e f f i c i e n t dimensionless v o r t i c i t y defined by Equation (3-5) value of v o r t i c i t y without relaxation after n i t e r a t i o n s i n i t i a l values of v o r t i c i t y components of v o r t i c i t y in r , 6. and-£ di r e c t i ons v o r t i c i t y at the surface of the cylinder angle angular spacing of grid lines defined by Equation (3-16) 214 transformed s t a r t i n g position of p a r t i c l e defined on page 60 u f l u i d v i s c o s i t y £ co-ordinate in ^ - d i r e c t i o n p f l u i d density pp p a r t i c l e density x „ dimensionless shear stress re <J> parameter defined by Equation (2-7) ty dimensionless stream function defined by Equation (3-4) ty * value of stream function without relaxa-n tion after n i t e r a t i o n s ty0 i n i t i a l values of stream function Superscri.pt prime denotes dimensional quantity REFERENCES [1] Fuchs, N.A., "The Mechanics of Aerosols," C.N. Davies, Ed., Pergammon Press, Oxford (1964). [2] Dorman, R.G., in "Aerosol Science," C.N. Davies, Ed., Academic Press, London (1966). [3] Pich, J . , in "Aerosol Science," C.N. Davies, Ed., Academic Press, London (1966). [4] L o f f l e r , F. , in "Air Pollution Control," W. Strauss, Ed., Wiley & Sons, New York, N.Y. (1971). [5] Landahl , H.D. and R.G. Herrmann, J . C o l l o i d Sc., 4, 103 (1949). [6] Thorn, A., Proc. Roy. S o c , A 141 , 651 (1933). [7] Davies, C.N. and C V . Peetz, Proc. Roy. S o c , A234, 269 (1956). [8] Davies, C.N. and M. Aylward, P r o c Phys. S o c , B64, 889 (1951). [9] Davies, C.N., Proc. Phys. S o c , B63_, 288 (1950). [10] Wong, J.B., W.E. Ranz and H.F. Johnstone, J . Appl. Phys . , 26_, 244 (1955) . [11] Lamb, H., "Hydrodynamics," 6th ed., Dover (1945). [12] Subramanyam, M.V. and N.R. Kuloor, Ann. Occup. Hyg., 12, 9 (1969). 21 5 216 [13] Householder, M.K. and V.W. Goldschmidt, J . Co l l o i d Sc. , 31_, 464 (1969). [14] May, K.R. and R. C l i f f o r d , Ann. Occup. Hyg., 1_0, 83 (1967). [15] S t a r r , J.R., Ann. Occup. Hyg., 1_0 , 349 (1967). [16] Gregory, P.H., Ann. Appl. B i o l . , .38, 357 (1951 ) . [17] Ranz, W.E. and J.B. Wong, Ind. Eng. Chem., 44, 1371 (1952). [18] Langmuir, I. and K.B. Blodgett, Report No. RL-225 Gen. Elec. Res. Lab., Schenectady, N.Y. (1944-45). [19] Brun, R.J., J.S. S e r a f i n i , and H.M. Gallagher, NACA TN2903 (1953). [20] Brun, R.J. and H.W. Mergler, NACA TN2904 (1953). [21] Davies, C.N., Inst. Mech. Eng., 1_B, 185 (1952). [22] Golovin, M.N. and A.A. Putnam, Ind. Eng. Chem. F u n c , 1 , 264 (1962). [23] Johnstone, H.F. and M.H. Roberts, Ind. Eng. Chem., 41 , 2417 (1949). [24] Sherman, P., J.S. K l e i n , and M. Tribus , U. of Michigan Eng. Res. Inst., P r o j . M992-D, USAF Contract AF 18 (600) - 51 (1952). [25] Tomotoika, S. and T. A o i , Quart. J . Mech. Appl. Math., 3, 140 (1950). [26] Yamada, H., Rep. Res. Inst. Appl. Mech., Kyushu Univ., 3, 11 (1954). [27] Underwood, R.L., J . Fluid Mech., 3_7_, 95 (1969). 21 7 [28] Proudman, I. and J.R.A. Pearson, J . Fluid Mech., 2, 237 (1957). [29] Happel , J . , A.I. Ch. E. J . , 5_, 1 75 (1959). [30] Kawaguti , M. , J . Phys. Soc. Jap., 8_, 747 (1953). [31] T r i t t o n , D.J., J . Fluid Mech., tS_ , 547 (1959). [32] A l l e n , D.N. de G. and R.V. Southwell, Quart. J . Mech. Appl. Math. , 8, 1 29 (1955). [33] Dennis, S.C.R. and M. Shimshoni , Aero. Res. Counc. Lond. Current Paper No. 797 (1965). [34] Son, J.S. and T.J. Hanratty, J . Fluid Mech., 3_5, 369 (1969). [35] Jenson, V.G., Proc. Roy. S o c , A249 , 346 (1959). [36] Kuwabara, S., J . Phys. Soc. Jap., 1_4, 522 (1959). [37] Hamielec, A.E., T.W. Hoffman and L.L. Ross, A. I .C.H.E.J. , 1_3 , 212 (1967). [38] Kawaguti, M. and P. J a i n , J . Phys. Soc. Jap., 21, 2055 (1966). [39] Takami, H. and H.B. K e l l e r , Phys. F l . Suppl. I I , 1_2 , 51 (1969). [40] Hamielec, A.E. and J.D. Raal , Phys. Fl . , 1_2 , 1 1 (1969). [41] Dennis, S.C.R. and G. Chang, J . Fluid Mech., 42_, 471 (1970). [42] Pruppacher, H.R., B.P. Le C l a i r and A.E. Hamielec, J . Fluid Mech. , 4_4 , 781 (1970). [43] Masliyah, J.H., Ph.D. Thesis, University of B r i t i s h Columbia, Vancouver, Canada (1970). 218 [44] Masliyah, J.H. and N. Epstein, Ind. Eng. Chem. Fund., 1_P_, 293 (1971 ) . [45] K i r s c h , A.A. and N.A. Fuchs , Ann. Occup. Hyg., 1_0, 23 (1967). [46] Klyachko, L., O t o p i l . i V e n t i l . , No. 4. Quoted in "Mechanics of Aerosols," C.N. Davies, Ed., Pergammon, Oxford (1964). [47] Wilde, D.J., "Optimum Seeking Methods," Prentice-Hal 1 , Englewood, N.J. (1964). APPENDIX I RECTANGULAR CO-ORDINATES In polar co-ordinates, v r refers to the radial component of f l u i d v e l o c i t y , and v Q to the angular com-ponent. If a rectangular co-ordinate system is located at the centre of the cylinder (Figure 3-1), the f l u i d v e l o c i t y at a point may also be expressed in terms of v and v , i . e . the x and y components of the f l u i d v e l o c i t y ; x y as follows: vx = ve S 1' n 9 " v r c o s 6 v„ = v Q cos 0 + v M sin 9 y 9 r (I-D Here r and 9 are the radial and angular co-ordi nates of the point. Substitution of Equation (3-4) into (I-1) gi ves : 219 220 v.. = | t . s i n e • I f f cose v . = i r t • cos 6 - -r si n 6 11 'y 3r r 36 (1-2) Equation (1-2), when written in f i n i t e difference form for a general point (I,J) becomes: (I,J) = s i n e (I) + 1 t( I + 1,J)-t(I--1 yJ) c o s e ( I )  + z l J l 2A6 C 0 S 0 V i ; v y ( I , J ) • 4>(I,J + 1 ^( I'J"1 ) Cos 6(1) _1 + 1 tJ)-t|»(I-1 ,J) . m i W 2A6 S 1 9 U ) (1-3) At the boundaries the rectangular v e l o c i t y com-ponents are determined by physical considerations: 221 v x ( I , l ) = 0 v y ( I , l ) = 0 r=l or z=0 Cylinder Surface (Zero S l i p Conditions) v y ( l , J ) = 0 v yO,Nr) = 0 6=0 6=TT Axis of Symmetry (No flow across axis of symmetry) v x(I,N r) = 1.0 vyU,Nr) = 0 r=R or z=Z P a r a l l e l Streaming Flow on outer envelope Solution of Equations (3-6) and (3-7) w i l l y i e l d values of ty and r, at a l l i n t e r i o r grid points. The rectan-gular f l u i d v e l o c i t y components at these points can be estimated with the aid of Equation (1-3). However, determi-nation of v and v at a position other than a grid x y point w i l l require i n t e r p o l a t i o n . Figure (I-l) i l l u s t r a t e s a ty p i c a l grid c e l l delimited by angular and radial l i n e s . Consider a point Q inside the c e l l , whose position is defined by the f r a c t i o n a l quantities a and I , as shown in Figure ( I - l ) . 223 Interpolating with respect to angular and radial position y i e l d s the following formulae for the f l u i d v e l o c i t y com-ponents at point Q: = 0 - a ) [ v x ( I , J ) . ( l - l ) + v x ( I , J + l ) . | ] + a£ y x(I + l , J ) - a + v x( 1 + 1 ,J + 1) • a-fj = 0 - a ) [ v y ( I s J ) - ( l - l ) + v y ( I , J + l ) - l ] + a|~vy(I + 1 ,J)-a + v y ( I + l ,J + 1)-a-IJ (1-4) APPENDIX II STAGNATION PRESSURES AND DRAG COEFFICIENTS In order to derive the stagnation pressures and drag c o e f f i c i e n t s i t is useful to write Equation (3-1) in an alternate form. To accomplish t h i s , the following i d e n t i t i e s were used: £ = V x v (II-D V x £ = V x V x v (11-2) V2_V = V(V/v) - V x (V x y_) (11-3) (v'V)v = %V(y_'v) - (v x v x _y_) (11-4) Equation (11 — 1) may be expanded in determinant form, gi vi ng: 224 225 3 1 3 3 3r r 36 3? vr ve (II-5) Since = 0 and 3/3^ = 0 , Equation (11-5) becomes: 5 ^ 3 v„ 1 9 v r 3r r 3 9 ( H - 6 ) The v o r t i c i t y vector £ has in general three components, c r , c Q and r,^ . In the present case of axisymmetric flow around a c y l i n d e r , c r and c 0 are zero. For the sake of brevity the term was abbreviated as simply s . Substitution of Equations (11-1) to (11-4) into Equation (3-1) y i e l d s : 226 W r i t i n g Equation (11-7) in s l i g h t l y expanded form: 7 r \ h v r v6 v ? i J L + i 1 - L + i J L r 3r 8 r 39 £ 3£ 'r 3r 9 r 36 1 £ 35 v: 2 + v 2 + v 2 Re 'r '8  lK JL 1 J L J L 3r r 36 3£ 0 0 <r (II-8) F r o n t a l S t a g n a t i o n Pressure The f r o n t a l s t a g n a t i o n p r e s s u r e , p 0 , i s the pressure on the c y l i n d e r s u r f a c e at 6 = 0 , r = 1 as shown i n Figure (3-1). 227 The r-component of Equation (11-8) is 9 ^ 3r Re r 39 3r[ r Along the centreline ( c f . Figure ( 3 - l ) ) s 6 = 0 and v Q = v r = 0 , hence ifi. = _ i _ 1 i*L 3r " Rec r 36 _3_ 3r 8=0 (11-10) Equation (11-10) may be integrated with the following boundary conditions: P P = v„ = = 0 Po 1 r = R. r = 1 R ( H - l l ) v„ = 0 r = 1 228 The following expression for p0 is obtained Po = 1 + Re l £ dr 3=0 (H-12) Introducing the transformation r = e gives Po = 1 + Re. l £ ae dz e=o (H-13) Rear Stagnation Pressure The rear stagnation pressure, p^ , is the surface pressure at e = TT (Figure (3-1)). The e-component of Equation (II-8) i s : - 2 vr ? r ae Re. l £ + 1 ar r _a_ ae + v, + V, ( 11-14 ) On the surface of the c y l i n d e r , v r = v Q = 0 , a n d r = l , g i v i n g : 229 ^6 4 96 Rec 8r (H-15) where p Q is the pressure on the cylinder surface at 6 = 8. Integrating Equation (11-15) from p 0 to p. and from 8 = 0 to 8 = 8 or f9 P e - Po Re !5l 3r d8 (11-16) Pfi = Po + f9 35. Re 3z de (11-17) Hence the rear stagnation pressure is P = Po Re ' o dz de (11-18) 2 3 0 Skin Drag C o e f f i c i e n t The skin drag c o e f f i c i e n t , C Q S ' 1 S defined: - x re R sin 6 de c r=l 'DS h P u0 ( H - 1 9 ) In terms of dimensionless shear stress, x r Q , Equation ( 1 1 - 1 9 ) becomes: 'DS re s i n e d 9 r=l ( 1 1 - 2 0 ) By d e f i n i t i o n : Tr e = - v r' 3r' r' 36 ( H - 2 1 ) and 231 re Re. 3r A r r 38 (11-22) 8 vr On the surface of the c y l i n d e r , v Q = -r-r-y do = 0 and r = 1 , giving: T r e 4 8 v e Rec 3r (11-23) S i m i l a r l y , on the surface of the c y l i n d e r , Equation (11 -6) becomes: 's 3r (11-24) r=l Substituting Equations (11-23) and (11-24) into Equation (11-20) y i e l d s : TT C S sin 6 de (II-25) o CDS Re„ c 232 Form Drag C o e f f i c i e n t The form drag c o e f f i c i e n t , C^p, is defined: CDF = p.' cos e R de e c \ p U o 2 (11-26) or, in dimensionless form: CDF = p Q cos 6 de (H-27) This is e a s i l y evaluated by s u b s t i t u t i n g Equations (11-13) and (11-17) for p : 233 1 + Re 2\k 38 dz 6=0 (11-28) cos 6 d 6 APPENDIX III HAPPEL'S MODEL Equation (11-22) may be expanded in terms of the stream function, ty: 4 dzty 1 1 Z2ty R ec 3r2 r 3r r2 392 ( I I I - l ) Happel's model s p e c i f i e s that x r Q = 0 on the outer boundary, where r = Rm , Equation ( I I I - l ) then becomes: 3 2 if; 3r< CO + _L_ i l l R 2 392 r = R„ C111-2) In a d d i t i o n , at r = Rm , ty = R^  s in 6 . Hence 234 235 38 32iJ> R cos 8 CO - R sin 9 00 (IH-3) Equation (3-7) may be expanded in terms of r and 8: C V ip 2 r 3 r + 2 3 0 2 (III-4) Substituting Equations (111-2) and (111-3) into (111-4) gives: = 2 r=R 1 3^ R„ 3r 7^— si n 8 r=R. (II1-5) 236 By introducing the exponential transformation, r = ez , Equation (111 - 5) becomes: 2Z, z=Z 3^ 3z - e sin 6 z=Z (III-6) Equation (111 -6) may be written in f i n i t e difference form as: C Na,J 2ZC * I.N. + 4i/j I,N a-1 - 3ip 2 Az (HI-7) - eZ°° sin 8 which i s the desired equation for the v o r t i c i t y on the outer boundary. APPENDIX IV Table IV-1 Relaxation Factors Used in this Work R a c CO ? ?s 0.2 200 1.8 0.75 0.50 0.2 100 1.8 0.8 0.55 0.2 50 1.8 0.8 0.55 0.2 25 1.8 0.8 0.55 0.2 10 1 .8 0.8 0.55 0.2 3 1 .0 0.6 0.45 0.5 100 1.8 0.7 0.5 0.5 3 1 .0 0.6 0.45 1 100 1.8 0.5 0.4 1 3 1 .0 0.5 0.4 3 100 1.8 0 .35 0.25 3 3 1.0 0.35 0.25 5 100 1.8 0.3 0.25 5 3 1 .0 0.25 0.2 10 100 1.8 0.2 0.10 10 3 1.0 0.20 0.10 15 100 1.8 0.08 0.05 15 3 1 .0 0.08 0.05 20 100 1.8 0.07 0.04 20 3 1 .0 0.07 0.08 30 100 1 .8 0.07 0.04 30 3 1.0 0.07 0.04 40 100 1 .8 0.03 0.02 40 50 1 .8 0.03 0.02 40 25 1 .8 0.03 0.02 40 10 1 .8 0.03 0.02 40 3 1 .0 0.03 0.02 237 APPENDIX V Computer Programmes and Sample Output 238 239 Computer Programme used to Solve the Navier-Stokes Equation for Flow Normal to a Circ u l a r Cylinder 1 REAL S(33,93) ,V (33,91) ,FSI!1 (33) . FCOS (33) , RS (93) .THETAG (33) 2 REAL VGX (33. 93) »VGY (33, 93) . P (20) ,X (20) 3 COMMON S ,V,FSIN,FCOS.RG,THETAG,VGX,VGY.P ,K » COMMON RINP,BE.A,B,GNUM,XPSr 5 COMMON NR, MA, NUHBA, JSOB 6 READ(t ,2900) R l N F , H E , R L X S , R L X V , R L X V S ,A,8 7 READ(1,3000) NR,NA,ITER,NCYCL,NUMBA 8 C 9 c »*«***»•»*»»»«»*«•***««»***«»»**»*»*«**«***»*»****»»»«*«***•» 10 C 11 C THIS PROGRAMME CALCULATES THE PLOW FIELD ABOUT A CIRCULAR 12 C CYLINDER. THE EQUATIONS ARE ALL DIMENSIONLESS, AND ARISE FROM THE 13 C HAVIER STOKES EQUATION. THEY ARE EXPRESSED I !l FINITE DIFF EBENCB 1H C FOBS AND SOLVED BY A GAUSS-SEIDEL ITER ATIOH METHOD. 15 C 16 C THESE EQUATIONS, TWO IN NU33EH, INVOLVE VORTIZITT (V) 17 C AND STREAM FUNCTION (S) . 18 C 19 C IT SHOULD BE NOTED THAT RELAXATIOH FACTORS ARE INTRODUCED 20 C TO EXPEDITE THE CONVERGENCE, AND TO GUARD AGAINST DIVERGENCE. IN 21 C A L L , THREE SUCH FACTORS WERE USED: " R L X S " . RELAXATION FACTOR 22 C FOR THE STREAM FUNCTION, " R L X V " , RELAXATION FACTOR FOR TS E BULK 23 C VOBTICITY, AND n R L X V S n , RELAXATION FACTOR FOR THE VORTICITY AT THE 21 C CYLINDER SURFACE. 25 C 26 C 27 C »»«»«*«•«•»*«•**»*«»»«•***•*••**** t******************* *********** 28 C 29 C 30 C A GRID OF POINTS IS ESTABLISHED ABOUT THE UPPER HALF OP 31 C THE CYLINDER, WITH NA POINTS IN THE ANGULAR DIRECTION, AND 8R 32 C IN THE BADIAL DIRECTION. TO OBTAIN FIN S SPACING SEAR THE SURFACE, 33 C THE TRANSFORMATION R=EXP(Z) WAS USED, WHERE Z WAS INCREASED 81 3« C EVEN INCREMENTS. THE ANSULAR STEP SIZE IS CALLED 3, AND THE BADIAL 35 C Z-STEP SIZE IS " A " . 36 C 37 C 38 c **»**»•«*»****«»•*«*»**»»*««•***»**»*«*****•»»•**»*******»*»••*»» 39 C HO WRITE(6,3500) 01 READ(8,3600) NCYCL ,3E ,RLXV,SLXVS <»2 ITEB=100 U3 WRITE(6,1100) RIN F ,RB,NR,NA q<l WRITE(6,1900) RLXS,RLX 7 US WRITE(6,2U00) RLXVS U6 WRITE(6,1200) ITER, NCYCL «7 C H3 c »»*»»***«*««»***««»»**•«*»»»»»»*»»«»»*««»**•***»**»*•*»»*«*«»*»* Q9. C 50 C 51 C A SUMMARY OF NOTATION 52 C 53 C 5« C A= RADIAL INCREMENT IN Z 55 C B= ANGULAR INCREMENT (RADIANS) 56 C ER= CURRENT CHANGE IN POINT VALUES 57 C NA= NUMBER OF ANGULAR GRID DIVISIONS 58 C K R= NUMBER OP RADIAL GRID DIVISIONS HR'l=f ozz oa 911 anminoo OIZ Sit 0"0= (HH'I) A mi (I) HIS4*3«IH= (HR'I) S t i l 0*0= (I 'I) S Zl l *rs'i=i oiz oa III OC Oil ( v s ' <r'i) s) (oi6Z'ti)a?3H 601 8H'i=r ce oa eoi anriiiKo: : OZ £01 <YB*I=I* (r'l) A) (oi6Z'ti)a¥3a 901 HK'i=r cz oa 501 3 601 ***************************************************************** 3 toi D ZOl 3 101 *«I XIHO D 001 EC83 0Y3B 38Y (HK'YH)S ONY (BH'YIOA 30 S3DI8IYR 1Y1IIRI SHI 0 66 3 86 3 £ 6 ***************************************************************** 3 96 3 56 t=a:vasr tie ts OtiZ lb • (O'l-r) • (0" l-HH ) / iKIZ) dT3= (D SH 16 8K'i=r otiZ oa 06 a n R i i K o o otz 68 fllORV= (l)0VJ.3Hi 88 (aiSRY) S03= (I) S03i £8 (310F?) RIS = (I) SIS1 98 tOM-1)* (0*l-YK) /Id=31DHY 58 YH'I=I OEZ oa ti8 B't (008l '9)3IIBH £8 (0" l -YK)/Id=a ze £ 6 / . 6 8 5 t 5 9 Z 6 5 l t l t - E = Id 18 (l-HH)/3KIZ=Y 08 <4BI8) 001V = JKIZ 6 £ 3 6£ ***************************************************************** 3 LL 3 9£ 3 S£ •«R OR Y *8K 'JRI8 IfidHI 3 «£ BO oasva *aaiYiD3TY0 aav u e . ORY =Y M JO sarnvn a H i aaaH 3 C£ 3 ZL 3 KL «•********«********•*•***•****•************** ******************** 3 OL 3 69 3 89 ***************************** ************************************ 3 £ 9 3 99 3 S9 I1I3II80A aDY38nS 80i HOIDYJ K0IIYXYT38 =SAX1H 3 » 9 I113I2.E0A X1C8 80J BOICYJ K0IJYXY13H = AXT8 3 £9 BOIiOHfJi HY3MIS HOi HOlDVi K0IIYXY13H = SX18 3 Z9 soiavs AHvannoa aaiao =JNIB 3 19 assiiriK s.aiosAaa = a a 3 09 OISS S2AC SHCUYH3JI AO 833WQM JK388fl3 =Y0UQN 3 65 241 117 S (1 , J) =0.0 118 S (NA, J) =0.0 119 V(1,J)=0.0 120 y (nA r j ) =0.0 121 220 CONTINUE 122 BRITE(6.2200) 123 DO 90 J=1,NB,JSPACB 12a WRITE ( 6 , 1400) (V ( I , J) , 1= 1, HA, ISPACE) 125 90 CONTINUE 126 WBITE(6,2300) 127 DO 100 J=1,NR,JSPACE 128 WHITE ( 6 , 14 00) (S {I, J) ,1=1, 8A,ISPAZE) 129 100 COHTIMOE 130 C l = (A*A*B*B)/2./(A*A*B*B) 131 C2=1./A/A 132 C3=1./8/B 133 HA1=HA-1 130 NR1 = S8-1 135 DO 60 KK=1 ,NCTCt 136 SERV=0 .0 137 SERVS=0.0 138 SE3S = 0 . 0 139 SUMAX=0.0 140 DO 50 KL=1,ITE8 141 EBS=0 .0 112 EBV=3.3 1U3 E8VS=0.0 149 » T L = 0 . 0 145 VSTL=0.0 146 STL=3 .0 147 ERTS=0.0 148 E B T t=0 .0 149 ERTVS=0.0 150 IERS=0 151 IERV=0 152 ISRVS=0 153 JEBS=0 1 5 » JERV'O 155 JEBVS=0 156 C 157 c « • * * « « » * • * » » • « » « * * • * * « » » * * » » » » * » * * * * * * * * * • * » » * • * « » * » * * * * » * » * * » * » * 158 c 159 c 160 c WITHIH THE NEXT TSO LOOPS, THE ACTUAL GAUSS-S2IDEL 3 EL A X I T I 3 1 161 c TECHS I3UE IS EW PLOYED. A TEST FOR CONVERGENCE IS ALSO INCLUDED 162 c HERE. IH E F F E C T , IT COMPARES THE ^H&HSES IH VALOES AT 38ID POINTS, 163 c FROS ONE CYCLE TO THE NEXT. WHEN X HIS CHANGE IS LESS THAN THE CON-164 c VERGENCE CRITERION, THE OPERATION IS CONSIDERED TO HAVE CONVERSED. 165 c 166 c 167 c •a**************************************************************** 168 c 169 DO 110 1=2,NAt 170 2=0.3 171 DO 00 J=2 .NB1 172 Z = Z*A 173 VTE."1P = C1* ( C 2 » ( V ( I . J - l ) *V ( I . J * 1)) *C3» (V ( I - I . J ) *V ( I * 1 , J ) ) 17H 1 *HE/8./A/B» ( (S ( 1 * 1 . J) -S ( I - I . J ) ) * ( V ( I , J * 1)-V ( I . J - 1 ) ) 2 4 2 175 2 -|S(I,J»1)-S | I , J - 1 ) ) » | - | I H , J ) - ? ( M , J ) ) ) ) 176 VPREV=V( I , J ) 177 V ( I , J) =8LXY* (VTEtlP-V (1,3) ) *1 (1,3) 178 DIFF=ABS(VPS EV-VTEBP) 179 ERTV^ERTV+DIFP 180 VTL=VTL*VPREV 181 I F ( D I P F . L T . E B ? ) GO TO 000 182 ERV=DIPF 183 IEBY=I 18» J E R » = J 185 100 COSTISOE 186 STESP=C1* (-EXP (2 . *Z ) *V ( I , J ) » C 2 * ( S ( I , J » 1 ) *S ( I . J-1) ) • 187 1 C 3 * ( S ( ~ > 1 , J ) » S ( I - 1 , J ) ) ) 188 SPREV = S ( I . J ) 189 S ( I , J ) =RLXS« (STENP-S ( I , J ) ) •» ( I , J ) 190 D IFF = ABS (SPR EV-STEflP) 191 ERTS=ERTS*DIFF 192 S T L = S T L » S P 8 E V 193 I F ( D I F F . L T . E R S ) GO TO 110 191 ERS=DIFF 195 IERS=I 196 J ERS=J 197 M10 C03TIBOB 198 l»0 COSTIHOB 199 V T E I 1 P = ( C 2 / 2 . 0 ) * ( 8 . 0 » S ( I . 2 ) - 3 (1 .3 ) ) 200 VPREV=V( I ,1 ) 201 D IFF = ABS (VPRSV-VTErlP) 202 E R T V S = E R T » S » C I F P 203 I F ( D I F F . L T . E 3 V S ) GO T3 120 204 ERVS=DIFF 205 IEHVS=I 206 JER¥S=1 207 t20 CO3TIB08 208 VSTL=VSTL*VPREV 209 V (1 .1) = RLXVS* (YTESP-V ( I , 1) ) »V ( I , 1) 210 110 CONTIHOE 211 ERTS=ERTS/STL 212 ERTV=ERTV/VTL 213 ERTVS=ERTVS/VSTL 21B S E R S = S E R S » E R T S 215 S E R V = S E R V » B 8 T ¥ 216 S E R V S = S E R ¥ S » E R r » S 217 ERTS=ERTS*100 .0 218 ERTV=ERTV*100 .0 219 ERTVS=ERTVS*100 .0 220 HEVES=SUHBA 221 NUMBA=BUHBA*1 222 N W R I T E = N E V E V 2 5 » 2 S 223 I F ( N S R I T E . EQ.MEVEN) GO TO 300 221 GO TD 310 225 300 WR ITE (6 .1600 ) 226 310 COHTISUE . 2 2 7 WRITE (6 ,US00) N0.1BA , ERS, ER V, ERVS , E RTS, ER TV, E8TYS 228 VR ITE (6,U700) I ERS , J ERS , I E R V , J E R Y , I E B V S , J E RV S , S T L , V T L , V S T L 229 E R S = E H S » R L X S 230 ERV = E R V * R L X » 231 ERVS=ERYS*RLX?S 232 E R T S = E 8 T S » R L X S 243 233 ERTV=ERTV*RLXV 234 ERTVS= EHTV5 *HLIVS 235 C 236 C •**»«*•*•»«*****««»•»*»**»****«*»**«*»***»**•»«*«*******•»•****** 237 C 238 C 239 C HERE THE CONVERGENCE IS TESTED, AND I F "ER" IS LESS T HA B 240 c THE CRITERION (HERE 0 . 0 0 0 1 ) , THE PROGRAMME EXITS TO DISPLAY AND 241 c RECORD THE RESULTS. AS A PRECAUTION AGAINST NON-CONVERGENCE 212 c WITHIN A GIVEN TIME LIMIT, VALUES OF THE VORTICITY AND STREAM 243 c FUNCTION MATRICES ARE STORED EVERY 50 ITERATIONS. 244 c 245 c 246 c ***************************************************************** 247 c 248 249 50 CONTINUE 250 CERS=0.0 251 CERV=0.0 252 CERVS=0.0 253 CSTL=0.0 254 CVTL=0.0 2SS CVSTL=0.0 256 CERTS=0.0 257 CERTV=0.0 258 CERTVS^O.O 259 DO 190 1=2,HA 1 260 Z=0.0 261 DO 200 J=2,NRI 262 Z=Z*A 263 VTEMP = C 1 * ( C 2 * ( V ( I , J - 1 ) » V ( I , J * 1 ) ) * C 3 * ( V ( I - 1 , J ) * V ( I * 1 , J ) ) 264 1 »RE/8./A/B* ( (S ( 1 * 1 , J) -S ( I - 1, J) ) * (V ( I , J» 1) -V ( I . J - 1 ) ) 265 2 - ( S ( I . J * 1 ) - S ( I . J - 1 ) ) » (V ( I » 1 , J ) - V ( I - 1 , J ) ) ) ) 266 VPREV=V(I,J) 267 DIFFV = ABS (VPREV-VTEMP) 268 IF(DIFFV.GT.CEBV) CERV=DIFP? 269 CERTV=CERTV*CIFF? 270 CVTL=CVTL*VPREV 271 STEHP = C1* (-EXP(2.*Z) • V ( I , J) »C2* (S ( I , J » 1 ) *S ( I , J - 1 ) ) • 272 1 C3* (S ( 1 * 1 , J ) *S ( 1 - 1 , J) ) ) 273 SPREV = S ( I , J) 274 DIFFS=ABS (SPREV-STEMP) 275 I F ( D I F F S . G T . CERS) CBHS=DIFPS 276 CERTS=CERTS*DIPFS 277 CSTL=CSTL»SPHEV 278 200 CONTINUE 279 VTEMP= (C2/2. 0) * ( 8 . 0 * S ( I , 2 ) - S ( 1,3)) 280 VPREV = V ( I , 1 ) 281 DIFFV = ABS (VPREV-VTEMP) 282 CSRTVS=CE3TVS*DIFFV 28 3 I F ( D I F F V . GT. CERVS) CERVS=DIFF7 284 CVSTL=CVSTL«VPHEV 285 190 CONTINUE 286 CERTS=CERTS/CSTL*100.0 287 CERTV=CERTV/CVTL«100.0 288 CERTVS=CERTVS/CVSTL*100.0 289 WRITE(6,U200) NUMBA 290 WRITE(6,3900) CERS,CERV,CERVS,CERTS, CE3TV, CERTVS 244 291 CERS=CSRS*BL IS 292 CE3V=CSRV*RLX7 293 CERVS = C E R V S « R L X V S 294 CERTS=CERTS*RLXS 295 CERTV=CERT7*BLX7 296 CERTVS=CERTVS*RL I7S 297 WR ITE (6 ,4000 ) 298 WR ITE (6 ,3900 ) C E R S , C E R V , C E R V S , C E R T S , C E R T V , C E R T V S 299 W3 ITE (6 ,4300 ) 300 WR ITE [6 ,4400 ) S ERS , S ER7 , SER7S 301 SERS=SERS*RLXS 302 SERV=SER7*RLXV 303 S E R V S = S E 3 V S » 8 L X V S 304 WR ITE (6 ,4000 ) 305 WR ITE (6 ,4400 ) S EBS , S EH7 , SER7S 306 REWIND 1 307 WR ITE (1 ,2900 ) R I N F , R E , R L X S . 3 L X V , R L X 7 S , A , B 308 WR ITE (1 ,3000 ) SH , N A , ITER , NCYCL, H 0 . 1 B 1 309 DO 170 J=1,NR 310 WR ITE (1 ,2910 ) (7 ( I , J) , 1= 1 , Hi) 311 170 CONTINUE 312 DO 180 J = 1 , M B 313 H R I T E ( 1 , 2 9 1 0 ) (S ( I , J) , 1=1, Hi) 314 180 CONTINUE 315 REWIND 1 316 CALL DRAG 317 CALL EFFS08 318 60 CONTINUE . 319 C 321 C 322 C 323 C THIS SECTION OF THE PROGRAMME IS OUTS IDE THE GAUSS-S EI DEL LOOPS, 324 C AND F INAL RESULTS ABE BOTH DIS PLAT ED, AND RECORDED 111 A F I L E 7 IA 325 C ONIT 1. 326 C 327 C 329 C 330 120 WR ITE (6 ,2500 ) NOMBA 331 WR ITE (6 ,2600 ) 332 DO 70 J M . S R . J S P A C B 333 WRITE(6 , 1400) (V ( I , J) , 1= 1, NA , ISPAC E) 334 70 CONTINUE 335 WRITE ( 6 ,2700 ) 336 DO 80 J=1 ,NR , J S PACB 337 WRITE (6, 1400) (S ( I , J) , 1= 1, NA , ISPACB) 338 80 CONTINUE 339 1000 FORMAT(5F12 .5 ) 340 1 100 FORMAT(•1 ' ,•BOUNDARY RADIOS = • , F l 0 . 5 , 5 X , ' R E Y N O L D S N 0 . = " , F 1 0 . 5 , 341 1 5X , "RADIAL D I V I S I O N S ' ' , 1 3 . 5 X , "ANGULAR D I V I S I O N S ' 13//) 342 1200 FORMAT(10X , " BO. OF ITERATIONS PER C Y C L E * " , IX , 1 3 , 1 3 1 , 343 1 "NO. OF C Y C L E S = " , 1 X . I 3 / ) 344 1300 FORMAT(6I5) 345 1400 FORMAT(8314 .7 ) 346 1500 F O R M A T ( • 1 • , 2 0 X , • * * » VALUES OF THE STREAM FUNCTION FOR C Y C L E • 347 1 , 1 2 , " • « * ' / / ) 348 1600 F O R M A T C I ' ^ O X , " * * * VALUES OF THE VORTIC ITY FOR CYCLE • 245 349 1 , 1 2 . ' *«*•//) 3 5 0 1 7 0 0 P O R M A T O O X , 'ANGULAR VALUES ARE P R I H T E D BY , , I 2 , " , S V ) 3 5 1 1 8 0 0 FORMAT ( 1 0 X . • 2 - I H C R E 9 EMT= FB. 4, 311 ,' ANGULAR INCREMENT= • , P 8 . 4 / ) 352 1 9 0 0 FORMAT ( 1 0 X , ' R E L A X A T I O N FACTOR FDR STREAM F U N C T I O H = ' , P 1 0 . S , 3 1 , 3 5 3 1 ' R E L A X A T I O N FACTOR FOR BULK V O R T I C I T T = • , P 1 0 . 5/) 3 5 1 2 0 0 0 P O R M A T ( 1 0 X , ' R A D I A L V A L U E S ARE P R I N T E D BY ' ,12,' "S'/) 3 5 5 2 100 F O R B A T ( ' l ' ) 3 5 6 2 2 0 0 F O R N A T ( ' 1 ' , 2 0 X , ' * * * I S I T I A L V A L O E S OF V O R T I C I T Y »**•//) 357 2 3 0 0 FORKAT('1',20X,'*** I S I T I A L V A L 3 E S DF ST3 EAM F U N C T I 3 N »**•//) 358 2 1 0 0 F O R M A T ( 6 1 X , ' R E L A X A T I O N FACTOR FOR S U R F A C E V O R T I C I T i f = • , F 1 0 . 5 / ) 359 2 5 0 0 F O R M A T ( ' 1 ' , 2 0 X , ' A F T E R ' , 1 1 , 1 X , ' I T E R A T I O N S : •//) 3 6 0 2 6 0 0 P O R M A T ( 2 0 X , • *** F I N A L VALUES OF V O R T I C I T Y •*»•//) 3 6 1 2 7 0 0 F O f l M A T l '1 ' , 2 0 X , • « • * F I N A L Y A L U E S DF STREAM ruaCTIDJ **»•//) 3 6 2 2 8 0 0 FORMAT('1'/////20X'CUHaENT ER 80R= • F I 0. 6, 3 X • A FT ER '11,11. 3 6 3 1 ' I T E R A T I O N S ' / ) 3 6 1 2 9 0 0 F O R B A T ( 8 F 1 0 . 6 ) 3 6 5 2 9 1 0 F O R M A T ( 1 7 G 1 4 . 7 ) 3 6 6 3 0 0 0 F O R M A T ( 5 I 5 ) 3 6 7 3 100 FORM AT (//I OX ,' AFTER ' , 1 1 , ' I T E R \ T I O N S , (NO R E L A X & T I D S ) :'//) 3 6 8 3 5 0 0 FORM AT (//,'ENTER THE VALUES OF N C Y C L , RE, RLXV , AMD R L X V S V 369 1 ,'THE FORMAT I S I 2 / G 1 1 . 7 / G 1 1 . 7 / G 1 1 . 7 ; SO THAT MEANS FOUR'/ 3 7 0 2 ,'CDUHT THEM, FOUR L I N E S OF I N P U T ! ! ' / / ) 371 3 6 0 0 F O R M A T ( I 2 / G 1 1 . 7 / G 1 1 . 7 / G 1 1 . 7 ) 3 7 2 3 9 0 0 FORM A T ( 3 0 X , • MAXIMUM CHANGES OVER G R I D : ' , / / , 3 7 3 1 ' STREAM F U N C T I O N ^ ',G11. 7,5X, 'B'JLK V O R T I C I T Y = • , G 1 1 . 7, 5 X , 3 7 1 2 'SURFACE V O R T I C I T Y = • , G 1 4 . 7 / / 3 0 X , • A V E R AGE % CHANGE OVER G R I D : • 3 7 5 3 ,//, 'STREAM F U N C T I O N ^ ', G 11. 7,5X, 'BULK VOR TICITr=', 3 1 1 . 7, S I , 3 7 6 1 'SURFACE V O R T I C I T Y = • , G 1 1 . 7 ) 3 7 7 1 0 0 0 FORMAT!///,'WITH R E L A X A T I O N : ' / / ) 378 4 2 0 0 FORMAT!'1'., 1 0 X , ' C O N V E R G E N C E TE S T OH G R I D A F T E R ' , 1 1 , ' I T E R A T I O N S : • 3 7 9 1 //) 3 8 0 1 3 0 0 FORM A T ( / / / / / 1 0 X , ' A V E R A G I N G THE STEP TO S T E P CHANGES OYER THE L A S T 381 1 100 I T E R A T I O N S : '//) 3 8 2 1 1 0 0 FORMAT ( 1 0 X , • CV ERALL * CHANGES, ( 5 3 R E L A X A T I O N ) : ' / / , 3 8 3 1 'STREAM FUNCTION=',G1 4. 7 , 5 X , ' B U L K V O R T I C I T 1= • , G 1 1 . 7, S X , 3 8 4 2 'SURFACE VORT I C I T Y =•,G11.7//) 3 8 5 1 5 0 0 P O R H A T ( 3 X . I 5 , 2 X . 6 ( 1 X , G 1 1 . 7 ) ) 3 8 6 1 6 0 0 F O R M A T C 1 • , 3 X , ' I T E R ' , 8 X , • M A X STREAM ',8X,'MAX YORT/B 387 1 8X,'MAX VORT / S',8X,•%AVG S T R E A M • , 7 X , • X A V G VORT/B', 388 2 7 X , ' * A V G VOBT/S'//) 389 1 7 0 0 F O R H A T ( 7 X , 3 ( 1 1 X , ' ( ' , I 2 , ' , ' , I 2 , ' ) ') , 2 X , 3 ( ' 4 X , G 1 4 . 7 ) ) 3 9 0 STOP 3 9 1 END BO OH DIBIT RADIUS' 100.00000 REYNOLDS NO." 10.00000 RADIAL DIVISIONS* 93 ANGOLAR DIVISIONS- 33 RELAXATION FACTOR FOR STREAM FOBCTI08« 1.79999 RELAXATION FACTOR FOR BULK VORTKIIX" 0.20003 RELAX All ON FACTOR FOR SURFACE VORTICITY- 0.10000 NO. OP ITERATIONS PER CYCLE" 100 NO. OF CYCLES* S Z-INCREMENT- 0.0501 ANGULAR INCREMENT" 0. 0982 CO a> 3 TO — j fD O C CT TO C < c+ n> -+> s -5 i O GO 3 rr o •o 7T -s ro o 10 IQ -J m CU XI 3 c 3 n> c+ c o to =3 ro CL o CO o — 1 < fD cn fN fN CN fN CN fN f N O o o o O O O O o O O DO CO CO CO CO CO to CO CO CO Cd in tf CO IT* r- tf o CO vO <-> tf o tf CN f— in o m in r-CO r- tf r-*' cn sO m vO cn CO fN in ro in o r— sO in CO o r* o vO rr*« r- CO r- cn CN o o fN p- r- CO o ro vO o CO vO vO vO m tf O cr vO <N (N Cn CO in vO tf tf ro r- - »- CO fN CN cn «— r o CO cr- tf fN i n tf O vO tf r- o tf «"~ vO in tf p<* vO %o «— tf tf o o fN CN •*~ cn in *— vO CN p* fN vO ro "~" fN 33 in CO vO tf *~ CN tN ro <n r-i *n CN CN fN *~ *~ r- tf *~ *~ in in *~ CN in m o o o O O o o © o O O O o O o O o o o o O o o o  1 r. fN tf tf tf ro ro ro tf ro fN f N O o O o o o o o O o O O O M u CO DJ to CO CO CO CO to u DO to CO Cn m tf p- <*n fN rrv r> fN fN rrv f in in fN ro CO ro CN vO tf ^~ ON co u i r— CN O in ro i n tf w— r— CO cn ro O fN tf fN vO CO r» cn vO r- in cn tf o vO \0 o CO in o r- tf o vO en CO in ro tf p- in tf »~ vO O vO CN cn in o in in CO vO CN <n ro r - uO o CN -— tf m r—  CN in o cn fN tf ro p- tf cn in r - r-» ro (N m r» •oO o o Cn o in O vO CO CN CO r-» o CN f N rsi tf in tn in m tf 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 r- fN n*» tf vO in vO in in tf tf tf in ro ro o O O O o o o o o o o o o o O o u CO CO u CO CO CO Ct3 CO Ol CO CO DO u CO CO ON in rsi in tf CN r- cn p- tf ro tf rt en ro CO in r - cn r o rtCO o r» CM •-» •— cn O vO r- rn o i n o o ro cn CO vO ro cn f * CO tf r- tfr-l vO tf tf «— CO /n »— cn ro ro r- ro fN rt tf CN CN r-o *— ro vO tf tf vO m m tf •n o m tf vO oo cn vO in cn P* ro vO vO ro en i n O **- in «~ vO I-* in CN «— tr- o CO P* CO r» r o cn tf CO m r* fN o CN r- fN o ro ro tf in rO cn \0 fN ro r» r- o vO cn cr. C* CO vO tf fN cn cn vO p» rn tf *~ ro *~ in *~~ P- vO r o in o o o O o O O O O o o O o © O o o o o o o O o o o *~ CN f n in vO vO l A r» in vO vO in -* tf tf ro O o O O o o o O o o o o o o o o o Cu CO CO CO w CO CO CO CO CO CO CO CO CO DO CO 1 CO ro CO CN fN w-O cn o vO CO in r - p- cn fN CO CO o r - CO vO vO «— p- r— co tf tf *— tf CN o vO. cn p» CN vO. tf- «— CO CO r— in tf tf p*- tf rt o rt vO vO cn CN fN p* m «— C0 in cn fN CN r o CO CO C* o "X» <N vO CO fN m tr- P* ro P* in in p* ~— r* ro vO CO o p» p» tf p- CM in m co tf p- r- «— «— o fN ro tf ro CO o r* CO CO tf cn tf tf cn tf r-» vO o ro tf tf tf o a o vO • • • r- tf vo -— r- ro in vO CN vo ro tf CN tf r r ro CO CN o o O o O o O o o o O O o o O O o o O o O  * T- CN tf in vO m P- in m m uo in m tf tf 1 tf ro • o o o o o o o o o o o O o o o o o O * w b3 1 CO 1 CO W CO u CO l CO to w 1 CO CO CO 1 CO 1 CO 1 CO 1 CO 1*4 CO «— tf CO ro r> r»"> in tf vO tf cn «— f CO vO in «— cn r - fM Q in LO CT* cn r o o fN r* in vO •— ro cn p*> r - fN O r- o tf ro cn ro »— r i •~ fN »— vO vO IN r» r- vO tf tf •» tf «— r~ CO tf CO r** rvi u r» in tf o CO vO vO O CN r- co in in in CO fN *— tf o CN cn ro M «  o in *— CO fN vO fN o *— p- ^« in ro CO r- cn cn a tf CO f- vO CO p- ro p* en •n vO rt in o vO rn p- tf o cn o ro o in o os • • m fN vO ^* cn CN p- CO (N «— ro ro tf CN »~ *— (T» fN a o fN > o O o O o o O o o o O o o o o O c o O o o w-<N m m VO vO vO in vO in in in in tfin tf 1 tf 1 m o o O o o o o o o O o o o o o o o o o (0 to W w w CO CO w w w CO H i CO w CO CO 1 CO I ro i CO co fN vo p» fN ^» f vO CO cr tf vO CO CO r- *— in in in fN i » cn in vO :=> p- r- r- tf \o fN r-i tf fN tf »— r> fN vo O" o cr r— o •J tf vO tf ro CO i— r- vO cn CO cn fN ro «— vO fN o o in o t— ro «< fN O in tf tf vO r- r-* CN vO tf m fN tf ro CO in in in ro o O CN > «~ CO VO fN r— vO en «—i fN p* CO O vO r* vO vO  tf tf ro P- vOiO m *-* ro O •o vO rr tf vo vO cr- vO »— V— fN in ro ro  en •J • • CO m tF— fN CN in CN r— CO -~ P- CN tf ro in fN cn ca ro o << CM M o o o O O o o o o o O o O o o o o Q -O o o o M «- fN tf vO vO vO vO vO vO vO m in tf tf in 1 in CO CO 2 o O o o o O o o O O o o o o o o o o o *~* CO co CO CO CO w w i CO w CO CO CO CO CO CO i CO 1 CO 1 CO • CO CO m •— tf vo m CN CO tf fN p* tf p* fN r - r - CO vO ro p» ro• 00 cn r- tf o r- cn r- o CO o o r* rsi «o tf ro o tf tf ro Q • ^» vo cn r- CO cn co r» tf m »— fN QO in fN CO o o in in vO no fN •— r- «— tf m fN CO CO CO tf o tf »— ro vO in f— tf cn tf r o cn 00 vO tf *— »— VD m ro fN m o cn tf O CN vO O vO *— tf p» r— r- vO fN C-i tf o fN in o v0 ro fN o »— cn O • cn tf *- ro vO m CN vO fN ro in p* vO in tf cn — — vO tf ro o o o o O o o O o O O o o O o o o O O a O O o o oooo.oooooooooooooooooooooooooooooooooooooooooooo O O O O O O O O O r t O O O O O O O O O O O O O O O O r t O O O O r t O O O r t O O O r t O O O O r t O O O »»• IHITIAl VALUES OF STREAM FUHCTIOS •*• D.O 0.0 0.0 0.0 0.0 0.0 0.0 0. 3166351E-01 0.5157673E-01 0.5363615E-01 0.40S791DB 0.0 0.0 ft ft 0. 1210178 0.2322458 0.2190959 0. 1748646 U • u 0.0 rt rt 0.2585750 0. 442981 8 0.5000356 0.42158S9 u • u 3.0 0.4361131 0.7630147 0.8940217 0.7952866 0.0 0.0 ft ft 0.6505539 1.156322 1. 396379 1. 303639 u • u 3.0 ft ft 0.9053867 1. 626942 2.008596 1.950569 U . U 0.0 ft ft 1.209157 2. 188885 2.744302 2.744241 0 • U 0.0 ft ft 1. 573606 2.863231 3.627474 3.704163 U . U 0.0ft ft 2.013128 3. 676451 4.692855 4.862879 U . u 0.0 ft ft 2.545129 4.660694 5.982316 6.264517 U • U 0.0 ft ft 3.190701 5.854926 7. 545331 7.964046 U • U 0.0 ft ft 3.9754B0 7. 306573 9.446457 10.02845 U • U 0.0 ft ft 4.930648 9.073275 11.75913 12. 53952 U . U 0.0 ft ft 6.094239 11.22533 14.57577 15.59700 u. (J 0.0 n ft 7. 512609 1 3. 84847 13.00858 19.32245 u • u 0.0 ft n 9.242293 17.04715 22. 19426 23.86438 U • U 0.0 ft A 11.35221 20.94890 27. 29553 29. 40381 U . u 0.0 ft ft 13.92657 25.70908 33.52794 36. 16150 u • u 0.0 ft ft 17. 06801 31.51790 41.12737 44.40619 u • u 0.0 ft ft 20.90201 38. 60753 50. 40373 54. 46637 U . u 0.0 25.58179 47.25945 61.71883 66.74300 0.0 3.0 ft A 31.29436 57.81926 75. 53226 81.72527 U • 0 0.0 38.26833 70.71066 92. 38792 100.0000 0.0 0.0 0.0 0.0 0.21889358-01 0.70071262 -02 0. 2433537E-03 0.1011599 0. 37371 45! -01 0.5312692B-02 0.2605929 0.1065531 0.2157733E-01 0.5248731 0.2328833 0.5593926E-01 0.9176478 0.4382318 0. 1 153362 1.453294 0.7475468 0.2144723 2.161308 1. 187733 0. 36 19673 3.040434 1.785275 0.5756383 4.116013 2.564778 0.8754433 5.420938 3.543873 1.285153 7.002493 4.767262 1.832363 8.321789 6.252353 2.549453 11.25446 8.057783 3.463511 14.09279 10.25026 4.631682 17. 54367 12.91733 5.0821 13 21.76305 16. 16533 7.870827 26.9012D 20. 12453 10.05304 33. 16959 24.95505 12.72333 40.81833 30.85486 15.96343 50. 15288 38.06493 19.91283 61.54584 46.37323 24.75475 75.44626 57.61794 30.739 32 92. 38800 70.71074 38.26843 ro CO ITER MAX STREAK MAX VORT/8 MAX VORT/S 5001 0 . 1525879E- 03 0 . 4673004E- 04 0 .4377365E- 03 (16.90) (20 , 2) (10, 1) 5002 0 . 1371291E- 03 0. 4863739E- 04 0. 533 1039E- 03 (21,90) (22, 2) (10, 1) 5003 0 . 1220703E- 03 0 . 4321337E- 04 0 .4538894E- 03 (21.89) (24, 2) (24, 1) 5004 0 . 19 8364 3E- 03 0 . 3165007E- 04 3 .4521608S- 03 (15,88) (22, 3) (21, 1) 5005 0 . 1831055E- 03 0 . 4088879E- 04 3 .3767014E- 03 (15.07) (21 , 2) (19, 1) 5006 0 . 1 3 732 9 1 E-03 0 . 4577637E- 04 0. 3805 16 1E- 03 (15,86) (19, 2) (19. D 5007 0 . 1 373291 E-03 0 . 35405 16E- 04 0. 57697 30E- 03 (10.90) (21 , 2) (11. 1) 5008 0 . 1 6784 67 E-03 0 . 429 1534 E- 04 0.4 196 167E- 03 (20, 92) (17 , 2) (11. 1) 5009 0 . 1 3 7 3 2 9 1 E-03 0. 4005432E- 04 D.4491806E- 03 (20,92) (16. 2) (16, 1) 5010 0 . 1525879E- 03 0 . 4005432E-•04 0 .4758835E- 03 (20,89) (19 , 2) (18, 1) son 0 . 1220703 E- 03 0 . 4386902E- 04 3 .8459091E- 03 (20,90) (1R, 2) (15, 1) 5012 0 . 1525879E- 03 0 . 4959106E- 04 0 .46 15784E- 03 (19,89) (16, 2) (14, 1) 5013 0 . 1831055 E- 03 0 . 4863739E-•04 0 .4978180E- 03 (17,92) (18 . 2) (11. D 501ft 0 . 1525879R- 03 0 . 4196167E-•04 0. 3032584E- 0 3 (18.92) (16 . 2) (12, 1) S015 0 . 1 631 055 E-03 0 . 371 9.330E- 04 3. 388 1454E- 03 ( i a , 9 i ) (12, 2) (20. 1) 5016 0 . 1 678467E-.03 0 . 333 7H60E-•04 0. 3967285E- 03 (21,92) (14, 2) ( 7, 1) 5017 0 . 1 3 732 9 1 E-03 0 . 2777576E-•04 0 .2908707E-•03 (18,89) (21 , 3) ( 6 , 1 ) 5018 0 . 1220703 E- 03 0. 2652407E- 04 0 .2765656E- 03 (18,B9) (21 , 3) (11. D 5019 0. 1 525879 E-03 0. 2634525E-•04 0 .5540848E- 03 (17,BU) (21 , 3) (14, 1) 5020 0 . 1525879E- 03 0 . 324 2493E-•04 0 .5598068E- 03 (15,86) (14, 2) (11. D 5021 0 . 183I055E- 03 0. 3147125E-•04 0.54 16870E- 03 ( U . 85) (11 . 2) (10, 1) 5022 0 . 1525879E- 03 0 . 333 7860E-•04 0.4863739R- •03 (21,90) (10, 2) ( 9 , 1) 5023 0 . 1678467E- 03 0 . 343322AE-•04 0 .406 26538-•03 (18.91) ( 9, 2) (14, 1) 5024 0 . 1220703E- 03 0 . 2956390E-.04 0 .3795624E-•03 (21,90) (1H. 2) ( 8 , 1) 5025 0 . 136811SE-•03 0 . 2479SS3E-.04 0 .4873276E-03 (19,87) (13 , 2) (10, 1) XAVG STREAM %KVG V3RT/B XAV3 V09T/S 0 . 30 32535E- 34 0 . 11 39359E- 02 0 . 1 2234U2E-01 3 6 9 6 5 . 0 3 4 39 . 281 5 40 .0 3 783 0 .3367694E- 04 0 .1120930B- 02 0 . 1 085493E^ 01 3 6 9 6 5 . 0 2 4 3 9 . 2 8 ) 8 40 .03783 0 . 3 2 7 3 2 6 6 E - 04 0 . 1121744E- 02 0 . 1 336374E-01 3 6 9 6 5 . 0 3 4 3 9 . 2798 40 .03 767 0 .3162512E- 04 0 . 1071175B- 02 0 . 1 4 6 5 3 9 3 E - 01 36965 .02 4 39 . 2798 4 0 . 0 3 7 9 8 0. 302482BB- 04 0 .10 3 3273E- 02 0 . 3954 730E- 02 3 6 9 6 5 . 0 3 4 3 9 . 2 7 9 5 40 .03764 0.2B 15023E- 04 0 . 1037479E- 02 0 . 7656839E- 02 36965 .0 3 4 3 9 . 2791 40 .0174 1 0 .2850003E- 04 0 . 1078373E- 02 0 . 1 1 5 2 4 3 5 E - 01 3 6 9 6 5 . 0 2 4 3 9 . 2 7 3 6 40 .03754 0.28B7 346E- 04 0 . 1067379B- 02 0 . 112 742 8B- 01 3 6 9 6 5 . 0 2 4 3 9 . 2783 4 0 . 0 3 7 4 8 0. 2693971E- 04 0 . 1D69725B- 02 0 . 123 1 3 91E-01 3 6 9 6 5 . 0 2 4 3 9 . 2 7 7 6 4 0 . 0 3 7 2 9 0 .2723134E- 04 0 . 1 3 7 5 3 3 2 E - 02 0 .9395043E- 02 36965 .01 4 3 9 . 2776 40 .03712 0. 28251 18E- 34 0 .106H965S- 02 0 . 1143550E- 01 3 6 9 6 5 . 0 2 4 3 9 . 2 7 6 3 40.0.3 706 0. 3000229E- 04 0 .13U0786E- 02 0 . 121 1420E- 01 36965 .01 439 .2764 4 0 . 0 3 6 0 0 0 .3119644E- 34 0 . 1067272E- 02 0 . 13581 79E- 01 36965 .01 439 .2764 40 .03694 0. 3 127920B- 04 0 . 105B124B- 02 0 . 9 2 3 6 6 7 8 E - 02 3 6 9 6 5 . 0 2 4 3 9 . 2 7 5 6 40 .036H5 0. 3000027B- 04 0 . 10634B0B- 02 0. 3 133H50E- 02 36965 .00 4 39. 2754 40 .03671 0. 3043893E- 04 0 . 1 3 3 5 3 6 6 E - 02 0 . 3 533 6 3 4 E-02 36965.01 4 19. 2751 4 0 . 0 3 6 8 5 0. 3040 301 E- 34 0 . 131 6366E- 02 0 .3453 326E- 02 36964. 99 4 3 9 . 2 7 4 7 4 0 . 0 3 6 9 0 0.2874 531E- 34 0 . 1D09549B- 02 0 . 9 4 8 1 8 1 4 E - 02 36965 .0 1 4 39 . 2 74 4 4 0 . 0 3 6 8 0 0. 3009409E- 34 0 . 1 0 2 0 8 1 8 E - 02 0 . 1 1 3 7 3 1 5 E - 01 36965 .01 4 39 . 2744 40 .03656 0. 2859 565E- 04 0 . 1038332B- 02 0 . 1 1 34327E-01 36965.00 4 39 . 2742 40 .0164 8 0 .3125262E- 34 0 . 101 1196E- 02 0 . 1250598E- 01 36965.00 4 3 9 . 2739 4 0 . 0 3 6 3 8 0 .2982428B- 04 0 . 1 3 4 6 2 5 4 E - 02 0. 1134442E- 01 36965 .01 4 3 9 . 2 7 3 4 40 .33592 0 .3103838B- 04 0 . 1338796E- 02 0 . 1339615E- 01 36965 .01 439 .2734 40 .0361 8 0 .2994195E-•04 0 .1014166E- 02 0 . 8 3 6 3 8 7 9 E - 02 36964 .99 4 3 9 . 2 7 3 4 4 0 . 3 3 5 9 6 0 .3046263E- 04 0 . 1033248B- 02 0 . 1 1 5 7 3 1 6 E - 01 3 6 9 6 4 . 9 8 4 3 9 . 2 7 3 2 40 .03593 ITER 11 AX STBEAH MAX VORT/B MAX VORT/S 5026 0. 18 310 5 5 E-03 - 0. 3242493E-•04 0. 5909075E-•03 (14,92) (10, 2) (10, 1) 5027 0. 1 9 8 3 6 4 3 E-0 0. 2312660E-•04 3. 4 9209S9E- 03 (14,92) (22, 4) ( 9, 1) 5028 0. 1373291E- 03 0. 2861023E- 04 0. 37 19.330E- 03 (14,91) (21. 2) <lt. 1) 5029 0. 1 525H79E-03 0. 3099442E-•04 0. 485 1222E-•03 (16,92) (25, 2) (21, 1) 5030 0. 1 678467E-03 0. 324 2493E-•04 0. 4043579E- 03 (14,92) (21. 2) (15, 1) 5031 0. 1 678467E-03 0. 314 7125 E- 04 0. 533 10 39E- 03 (17.92) (19, 2) (12, 1) 5032 0. 1 C78467E-03 0. 257492 1E-•04 3. 8764267E- 03 (14,92) (16, 2) (11, 1) 5033 0. 137129 1 E-03 0. 3433228E-•04 3 - 5846024E- 03 (14.92) (13. 2) (10, 1) 5034 0. 1 83 1055E-03 0. 286 1023E-•04 0. 4 148483E- 03 (18,92) (10, 2) (10, 1) 5035 0. 1670467E- 03 0. 257492 1E- 04 0. S006790E- 03 (14.90) (19. 2) (11. 1) 5036 0. 1 368 1 15 E-03 °- 22292 14E- 04 0. 4 167557E-03 (20. 90) (24, 4) (19, 1) 5037 0. 1220703E- 03 0. 2384 186E- 04 0. 428 199BE- 03 (18.92) (20, 2) (18. 1) 5038 0. 1220703 E- 03 0. 3051758E- 04 0. 406 1460E- 03 (21,88) (18, 2) (23, 1) 5039 0. 1220703E- 03 0. 2777576E- 04 0. 367 1646E- 03 (17.86) (23, 2) (20, 1) 5040 0. 1220703 E- 03 0. 290B707E-•04 0. 387 19 18E- 03 (22,92) (21, 2) (10, 1) 504 1 0. 1525U79E- 03 0. 2670288E- 04 0. 408 1726E- 03 (14,92) (19. 2) (19, 1) 5042 0. 1 525879E-03 0. 2145767E- 04 0. 3 166 199E-03 (20, B9) (22, 4) (18, 1) 5043 0. 122O703E- 03 0. 2133H46E- 04 0. 2899 170E- 03 (20,92) (27, 2) ( 9, 1) 5044 0. 1 3 732 9 1 E-03 0. 21 15965E-•04 3. 3 9 196 0 1E-03 ( 1 8. 90) (24, 4) ( 8, 1) 5045 0. 1 3 732 9 1 E-03 0. 2 3 78225 E-•04 3. 5779266E- 03 (11.92) (24, 2) ..(10, 1) 5046 0. 1220 703 E- 03 0. 3051758E- 04 0. 4 100800E-03 (21.92) (18, 2) (12, 1) 5047 0. 1 678467 E-03 0. 231H621E-•04 0. S865097E- 0 3 (17.91) (21. 2) (16, 1) 5048 0. 16784B7E- 03 0. 3147125E-•04 5. 4577S37E- 03 (14.92) (16, 2) D 5049 0. 1678467E- 03 0. 2765656E- 04 0. 3919601E- 03 (14,89) (18, 2) (18, 1) 5050 0. 1220703E- 03 0. 2098083E- 04 0. 3B14697E- 03 (18, 88) (17. 2) (19, 1) SAVG STREAM fAVG VORT/B * A V 3 VORT/S 0. 3 166374E- 04 0.13189832- 02 0. 1368764E- 01 36964.99 4 39. 2727 40.03569 0. 2983 30BE- 34 0.9363840B- 03 0. 3222530E- 02 36964.98 439.2722 40.03587 0.3153236E- 04 0.9735744E- 03 0. 9 636290E-02 36964.97 4 39. 2725 40.035B7 0.3109127E- 34 0.9687142E- 03 0.1368631E- 01 36964. 96 4 39. 2720 40.0 3595 0.3102815E- 04 0.9737389E- 03 0.1262989E- 01 36964.96 439.271 5 40.03581 0. 30 10007E- 04 0.9307673B- 03 0. 350568OE- 02 36964.98 419. 2713 40.3 3 55 8 0.3032B72E- 04 0.9785J36E- 03 0. 13524 75E- 01 36964.98 4 39. 27)8 40.03551 0. 2829218E- 04 0.3323B72B- 03 0.1131617B- 01 36964.97 4 39. 2.73 3 40.31S39 0. 2905953B- 04 0.9650185E- 03 0. 1 1 40799E-01 36964. 96 4 39. 2730 40.03 54 3 0. 29 39766B- 04 0.9637957B- 03 0.11H6327E- 01 36964. 96 439. 2738 40.0 3 52 3 0. 29 17 7 35E-34 0.9234827S- 03 0. 9 53 bl71E- 02 36964.96 4 39. 273 5 40. 0 3554 0.2H 15462E- 04 0.9341282B- 03 0. 1 1 1 524 9E-01 36964.96 4 39. 2 63 8 40.03546 0.2B93035E- 34 • 0.9432957B- 03 0. 1394H29E- 01 36964.96 4 39. 269 5 40.03531 0. 303186HE- 04 0.9412340E- 03 0.1339926E- 01 36964.97 4 39.2 60 8 40.01525 0. 2897 4 54B-34 0.3498513E- 03 0.3902410E- 02 36964.96 4 39 . 2 63 5 40.03 513 0. 2923 1 352-04 0.9427743E- 03 0.3619651E- 02 36964. 96 4 39. 2698 40.0351 7 0. 30 10957E- 04 0.9269328E- 03 0.34 15470E- 02 36964.96 4 39. 2733 40.31519 0. 211723 3BE- 04 0.9210431E- 03 0.6712113E- 02 36964.96 4 39. 2733 40.0 3514 0. 2993793E- 04 0.321U376S- 03 0. 13646 75E- 01 36964.95 4 39 . 2 69 0 40.31516 0.3023274E- 04 0.9330443B- 03 .0.1185300E- 01 36964.96 4 39. 2733 40.0 3493 0. 30.374B1E- 04 0. 94.35383E- 03 0.382H378E- 02 36964.95 4 39. 269 5 40.03464 0. 3054943E- 04 0.9215283E- 03 0.113B471E- 01 36964.9 5 439.2695 40. 0 3482 0. 3020713E- 04 0.9069396B- 03 0. 3428153E- 02 36964. 9S 4 39.2636 40.03482 0.2907751E- 04 0.83325978- 03 0.1318804E- 01 36964.94 439.2633 40.03491 0.2811367E- 04 0.301 1132B- 03 0.1332757B- 01 36964.95 439.2633 40.03467 ro on O I T E R MAX STREAM MAX VORT/B MAX VORT/S XAVG STREAM KAV3 VDRT/B XAV3 VORT/S 5051 0. 1220 703 E- 03- 0. 2384 186E-•04 0. 5960464E-•03 0. 289 1931 E-•04 0.8888422E- 03 0. 1275934E- 01 (17, 85) (19, 2) (12, 1) 36964.94 439.2696 40.03477 5052 0. 13681 15E- 03 0. 2574921E-•04 0. 3021955E- 03 0. 2752632E- 04 0.8B68647B- 03 0.3844007E- 02 (19,92) ( 1 » . 2) (22, 1) 369S4.95 4 39.2636 40.03465 5053 0. 13 732 9 1E- 03 0. 2765656E-•04 0. 3633499E- 03 0. 2805340E- 04 O.9011313E- 03 0.3235261E- 02 (19.92) (13, 2) ( 5, 1) 36964.93 4 39 . 2 63 6 40.03444 5051 0. 1 37329 1 E-03 0. 2765656E-•04 0. 327 1 103E-03 0. 2882739B-•04 0.9090332E- 03 0. 3 1 8754 9E- 02 (17,91) (11, 2) 111. 1) 36964.93 439. 2678 40.03412 5055 0. 1 525879E-03 0. 2574921E-•04 3. 4 196 167E- 03 0.2811955E- 04 0.3969586E- 03 0. 10634 87E- 01 (17,92) (13. 2) (15, 1) 36964.93 4 39.2 67 6 40.03429 5056 0. 1220703E- 03 0. 286 1023E-•04 3. 285 1486E-•03 0. 2729899B- 04 0.9123427B- 03 0. 73 361 87E- 02 (13.89) (15, 2) (14. 1) 36964.9 3 4 39.2676 40.03409 5057 0. 1 83 1055 E-03 0. 352 U595E- 04 0. 3385544E- 03 0. 2846 39 1E- 04 0.9 1 104 51E-03 0. 1 242 2 82 E-01 (18,88) (14, 2) (17, 1) 36964.93 439.2676 40.0340 1 5058 0. 1 831 055 E- 03 0. 3242493E- 04 0. 3633499E- 03 0. 2941 256E- 34 0.9 19 1297B- 03 0.3130733E- 02 (15,90) (13, 2) (10. 1) 36964.93 4 39. 2673 40.0 3 3 81 5059 0. 167G467E- 03 0. 314 7 125E-• 04 0. 3757477E- 03 0.2735080E- 04 0.9 13 34 1 3B-03 0.3719305E- 02 (15,92) (10, 2) ( 8, 1) 36964.9 3 439.2671 40.03371 S060 0. 1 37329 1 E-03 0. 2986193E-•04 3. 3728867E- 03 0. 2634993E-•04 0.929 1554E- 03 0. 124364 1 B- 01 (15,92) (21, 2) (12, 1) 36964.93 4 39. 2661 40.03349 5061 0. 122O703E- 03 0. 3433228E-•04 0. 5598068E- 03 0.2697967E- 04 0.946551 8e- 03 0. 1 106507E- 01 (15,91) (20. 2) (11. 1) 36964.92 439.2661 40.0 331 9 5062 0. 1 678467E-03 0. 3814697E-•0 4 0. 7324219E-•03 0. 27 30533E- 34 0.9499732E- 03 0. 1410455E- 01 (14,92) (18, 2) (11. 1) 36964.91 4 39.2656 40.03105 5063 0. 1 3 732 9 1 E-03 0. 3B14697E-•04 0. 5435944E- 03 0. 2778 31 1E- 34 0.93 327318- 03 0.1339839E- 01 (19,92) (18, 2) (11, D 36964. 9 1 4 39.2654 40.0 3258 5064 0. 1 3732 9 1 E-03 0. 5054474E-•04 0. 614 1663E-•03 0.2713141S- •04 0.9938037B- 03 0. 1312166E- 01 (17,87) (17. 2) (11. 1) 36964.9 1 439.2651 40.03230 5065 0. 1 525879E-03 0. 4959106 E-•04 0. 4272461E- 03 0. 2835759E- 04 0. 1036987E- 02 0. 1 1 76592E- 01 (21,92) (14, 2) (17, 1) 36964.89 4 39. 2651 40.01198 5066 0. 1 83 1 055 E-03 0. 4577637E-•04 0. 6275177E-•03 0. 2799286E-•34 0. 10 1 7252B- 02 0.1312425E- 01 (15,89) (17, 2) (12, 1) 36964.90 439. 2654 40.01175 5067 0. 1B3 1055E- 03 0. 5054474E-•04 3. 8449554E- 03 0. 2819599E- 04 0.1029234B- 02 0.1742905E- 01 (15,91) (12, 2) (11. D 36964.89 439. 2644 40.03146 5068 0. 1 678467E-03 0. 581 74 13E-•04 3. 64659 12E- 03 0.2R29 373E- 34 0. 1D70256E- 02 0. 1 42644 5E- 01 (17,87) (17, 2) ( 9. 1) 36964.90 4 39. 2 64 6 40.03091 5069 0. 1 525B79E-03 0. 5435944 E- 04 0. 5664825E- 03 0.2602144E- 04 0. 1D75376E- 02 0.1471501E- 01 (18,86) (14, 2) (11. D 36964.89 4 39. 2619 40.03067 5070 0. 1220703 E- 03 0. 5054474E-• 04 0. 49 1 14 23E-03 0. 2796727E- 04 0. 1070946E- 02 0. 1374620E- 01 (18,92) (11. 2) (16. 1) 36964.89 4 39. 2634 40.01046 5071 0. 1373291E- 03 0. 4673004E-•04 3. 4062653E- 03 0. 2796 174E- 04 0. 1074377E- 02 0.9953976E- 02 (16,86) (20, 2) (15, 1) 36964.87 4 39. 2629 40.03023 5072 0. 1220703 E- 03 0. 4386902E- 04 3. 4940033E- 03 0.2736473E- 04 0. 1365191E- 02 0. 1 382572E- 01 (19,89) (16, 2) ( 8, 1) 36964.88 4 39. 2634 40.03009 5073 0. 1525879E- 03 0. 4196 167E- 04 0. 4825592E- 03 0.2706643E- 04 0.1063399B- 02 0. 1262376E- 01 (19,85) (1H. 2) (1"». 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CO CO ^  CO M CO t-> cr ro r- ro vO ro CN zr O zr co zr ro zr r - CN CO CN CO CN O CN u i CN O CN cc r i ro CN uo fN uo CN cr CN cr CN ro CN uO rsi uo CN ro CN OC CN co CN DS o cr cr «—i r- CO CO co in in ro cr vC CN cr rt r - c cr vO cr cr CC «n o rsi r- te r— * ro »in * O w O cr * r - % p" *• O »m * CC » zr *C » vO *C * o V cc » zr » o * in zr * CM »r* > CN *— f " «— O -^ r— •— UO *- r - cc ro sO »— «^ O •— cr r-i cc f- r- in •— co cr o ro cr CN zr «— CN sO O in cc rr CN in o cr w-00 fN ro rsi cr CN «  rsi r-rsi in CN cr rsi sO rsi m CN in vO CN rsi ro o CN sO r- in CN fN O rsi cr CN cr ^- »— *— rsi zr r» *- in X sO in —TO *™* CN .—^ o CO o o w ' zr — LT ro —*x CC wr* w o ~— O " CN v cr —• in CN "—so o << CN CN CN na CN CM CN rsi ro ro ro ro ro CN rsi CN ro rt ro ro ro ro ro CN ro 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 I I cu ca^.to w cu ^-.CJ—.D: .— co — a: ^-.cu ^-.cu ^-.bJ — CJ ^~.CJ — cu ^.cu —.tJ ^ . t : ^-.co ^-.cu — CJ ^cu ^_co —^co ^to ^  ^ N ^ ^,~ o ^ O i n N ' " ^ f N i - » - ^ r i ^ c ^ ^ ' - o c o ^ r ^ c ^ f N ^ o j w O ( ? O L ^ N LO C L n c r c r c r c r c r t n c r c r c r s c c i ^ c r c r c c ^ r ^ c r c r c ^ 3) rr o* ^  o r* o » o % rvj *rs] * o *vO *zr «CN >r-> »cc »CN %co ^rsi »co •» cc *so * co » o »w »co «rr *>o »<o »vo »cO « ^ c c ^ ~ r t ^ r t v o » — c o r t c c ^ c o r t c c o z r i n z r r t c c i n z r r t rtr—rt»—r^rsir^^rtr-cOf—r^r—r^f— r s i ^ o i ^ r ^ co "~'co •—• rt —'rt co c* "—»o —'ro *—"rsi w i n ro —'in w r o '—'in in *-*cr " i n *co ~-*in —• vn —*vo co cr cr <n r- r* p* CONVERGENCE TEST ON GRID AFTER 5100 ITERATIONS: • MAXIMUM CHANGES OVER 3RID: STBSAH FUNCTION' 0.2136230E-03 BULK VORTICITY- 0.2765656E-04 AVERAGE X CHANGE OVER GRID: STREAM FUNCTION" 0.3553228E-04 BULK VORTICITY- 0. 8203302E-03 SURFACE VDRTICirY- 0.3671646E-03 SURFI:E vsaricirx- o.IO231«IE-OI MITH RELAXATION: MAXIMUM CHANGES OVER SRID: STREAM FUNCTION" 0. 3845200E-03 BULK VORTICITY- 0. 5531310B-05 AVERAGE X CHANGE OVER GRID: STREAM FUNCTION- 0. 6395788E-04 BULK VORTICITY- 0. 164O760E-03 SURFACE VOBTICirY- 0. 36716152- 04 SURFACE VORTICITY- 0.1023140E-02 AVERAGING THE STEP TO STEP CHANGES OVER THE LAST 100 ITE2ATIONS: OVERALL X CHANGES, (HO RELAXATION): STREAM FUNCTION" 0.30021498-04 BULK VORTICITJ- 0. 9792182E-03 SURFACE V3BTICirx» 0.1076943E-01 WITH RELAXATION: OVERALL X CHANGES, (NO RELAXATION): STREAM FUNCTION- 0.5403848E-0U BULK VORTICITY- 0.1958196E-03 SURFACE VORTICITY- 0.1076942E-02 RINF- 100.000000 SIZE OF GRID- 33 X 93 NUMBER OF ITERATIONS-RE- 10.000000 5100 SURFACE VORTICITY DISTRIBUTION THETAG 1) 0.0 VORTICITY ( 1,1 3 0.0 THETAG 2) i . 0.9817475E-01 VORTICITY ( 2,1 3 0.4397055 THETAG 3) o 0. 1963495 VORTICITY ( 3,1 = 0.8673537 THETAG «) a 0.2915213 VORTICITY ( 4.1 = 1.271414 THETAG 5) 3 0.3926990 VORTICITY ( 5,1 1.6410 84 THETAG t 6) B 0.1908738 VORTICITY t 6,1 n 1.966905 THETAG 7) 3 0.5390485 VORTICITY ( 7,1 = 2.241029 THETAG 8) a 0.6H72233 VORTICITY ( 8,1 3 2.457419 THETAG 9) a 0. 7853980 VORTICITY ( 9,1 a 2.612244 THETAG 10) B 0. 8835728 VORTICITY (10,1 IS 2.703779 THETAG 11) 3 0. 9817475 VORTICITY (11,1 •s 2.732823 THETAG 12) 3 1. 079922 VORTICITY (12, 1 3 2.702162 THETAG 1.1) B 1. 178097 VORTICITY (13.1 3) 2.616772 THETAG (IH) B 1. 276271 VORTICITY (14,1 *t 2.483301 THETAG 15) B 1. 374446 VORTICITY (15,1 2.309817 THETAG 16) B 1.472621 VORTICITY (16,1 = 2. 105314 THETAG 17) a 1.570796 VORTICITY (17,1 -a 1.879213 THETAG 18) S 1.668970 VORTICITY (18, 1 3 1.640(127 THETAG 19) a 1.707145 VORTICITY (19,1 s 1.398988 THETAG 20) a 1.865320 VORTICITY (20, 1 1.161667 THETAG 21) 3 1. 963494 VORTICITY (21.1 3 0.9356664 THETAG 22) a 2.06 1669 VORTICITY (22, 1 a 0.7265244 THETAG 23) a 2. 159844 VORTICITY (23,1 X3 0.5384942 THETAG 2 D = 2.258018 VORTICITY (24, 1 = 0.3746592 THETAG 2 5) B 2. 356194 VORTICITY (25,1 s 0.2370090 THETAG 26) 3 2.454369 VORTICITY (26, 1 tj 0.1265936 THETAG 27) B 2.552543 VORTICITY (27,1 <S 0.4355459E- 01 THETAG 28) a 2.650718 VORTICITY (28,1) s - 0. 1290640E-01 THETAG 29) 3 2.748893 VORTICITY (29,1 a- 0.4462009B- 01 TH STAG 30) V 2.847067 VORTICITY (30, 1) B - 0.5456545E- 01 THETAG 3 D a 2.945242 VORTICITY (31,1 3 - 0.4684433E- 01 THETAG 32) a 3.043417 VORTICITY (32,1) *>«• 0.2665148E- 01 THETAG 33) B 3.141592 VORTICITY (33.1 n 0.0 ro i n SURFACE PRESSURE DISTRIBUTION THETAG ( 3) = 0. 1963495 SURFACE PRKSSURE( 3) B 1.393917 THETAG ( S)» 0. 3926990 SURFACE PRESSURE ( 5) a 1 .138027 THETAG ( 7) = 0.5890485 SURFACE PRP.S5URE( 7) 3 0. 7583042 THETAG ( 9)» 0.7BS3980 SURFACE PRESSURE! 9) 3 0. 3139000 THETAG (1 1) « 0. 98 17475 SURFACE PRESSURE(11) 3 ' -0. 1310129 TH ETAG(13)» 1.178097 SUHFACE PRESSUHE(13) 3 -0. 5205097 THETAG (15)" 1.37114 46 SURFACE PRESSURE (15) 3 ' -0. 8168278 THETAG (17) » 1.570796 SURFACE PRESSURB(17) a -1 .004923 THETAG (19) » 1. 767 115 SURFACE PRESS(JRE(19) 3 -1 .090684 THETAG (21) • 1. 963494 SURFACE PRESSURE(21) 3 -1 .0940 18 THETAG (23) « 2. 159841 SURFACE P3ESSHRE (23) 3 -1 .040699 THETAG (25) « 2. 356 191 SURFACE PRESSURE (25) 3 ' -0. 9556675 THETAG (27) • 2.552513 'SURFACE PRESSURE(27) S --0. 8605127 TH ETAG (29)" 2.718893 SURFACE PRESSURE(29) B < -0. 77417 85 THETAG (31) " 2.915212 SURFACE PRESSURE(31) =-0. 7136545 THETAG (33) • 3. 111592 SURFACE PRESSURE(33) • < -0. 6918373 DRAG PASAHETERS AFTER 5100 ITERATIONS: R-INFINITY" 100.0000 REYNOLDS NO.» 10.00000 SIZE OF GRID" 33X93 FRONTAL STAGNATION PRESSURE" 1.484224 REAR STAGNATION PRESSURE=-0.6918373 THE SKIN DR A3 COEFFICIENT" 1. 223155 THE FORM DR AS COEFFICIENT" 1.559752 CDSKIN/CDFORH" 0.7841986 THE TOTAL DRAG COEFFICIENT • 2.782907 ro cn cn STOKES { 1)- 0.5000000 STOKESJ 2)- 1.000000 XKAP< 1)- 0.9999999E-03 XKAP{ 2)« 0.2000000 INITIAL VARIABLES RINF= 100.000 REYNOLD'S NO.- 10.000 XP (1 )« -100.0000 SIZE OF GRID- 33 X 93 NUMBER OF STEPS PER GRID CELL" 3.000000 »*••• NUMERICALLY CALCULATED FLOW FIELD USED ••**»«* •••*•* EFFICIENCY WITH INTERCEPTION **••**• STOKES* NO.- 0.500 EFFICIENCY- 0. 379538908S0E-02 STOKES' SO. • 1.000 EFFICIENCY- 0.1041456142171 •»•••• EFFICIENCY WITH INTERCEPTION •••••*« STOKES' NO.• 0.500 EFFICIENCY- 0.10357707739 STOKES' NO.• 1.000 EFFICIENCY- 0.224134 8624 2 DP/DF- 0.001000 I- 0.3791599S345E-02 DP/DP- 0.001000 1-0.13435211658 DP/DF" 0.200000 I- 0.86314201355E-01 DP/DP" 0.200000 I » 0.18677902222 ro I T E R MAX STREAM MAX V O R T / B MAX V O R T / S 5101 0. 1 5 2 5 8 7 9 E - 03 • 0. 2 8 6 1 0 2 3 E - 04 0 . 4 4 7 2 7 3 3 E - 0 3 ( 2 1 , 9 0 ) ( 1 5 , 2) ( 0 , 1) 5 1 0 2 0. 1 8 3 1 0 5 5 E - 03 0 . 2 0 3 8 4 7 9 E - 04 0 . 3 6 1 4 4 2 6 E - 0 3 ( 2 1 , 8 9 ) ( 2 2 , 4) ( 2 0 , 1) 5 1 0 3 0. 1 9 8 3 G 4 3 E - 03 0 . 2 0 9 0 0 8 3 E - •04 0 . 4 0 1 4 9 6 9 E - 0 3 ( 1 8 . 8 9 ) ( 2 0 , 2) ( 1 1 , D 510H 0 . 1 5 2 5 8 7 9 E - 03 0 . 2 0 0 2 7 1 6 E - 04 0 . 5 3 4 0 5 7 6 E - 0 3 ( 1 9 . 8 7 ) ( 1 1 . 2) (11, 1) 5 1 0 5 0. 1 5 2 5 8 7 0 E - 03 0. 1 9 6 6 9 5 3 E - 04 0 . 2 7 3 7 0 4 5 E - 0 3 ( 1 5 . 8 9 ) ( 2 2 , 4) ( 1 0 , 1) 5 1 0 6 0. 1 3 7 3 2 9 1 E - 03 0. 193 1 1 9 0 E -•04 0 . 5 1 0 2 1 5 3 E - 0 3 ( 1 6 . 9 0 ) ( 2 3 , 4) ( 1 2 , 1) 5 1 0 7 0. 10681 1 5 E - 03 0 . 1 9 1 9 2 7 0 E - 04 0 . 6 8 6 6 1 5 5 E - 0 3 ( 1 9 , 9 0 ) ( 2 2 , 4) ( 1 0 , 1) 5 1 0 8 0 . 1 0 6 8 1 1 5 E - 03 0 . 2 3 8 4 186 E -•04 0 . 7 3 2 4 2 1 9 E - 0 3 ( 1 7 , 9 2 ) ( 1 0 , 2) ( 2 0 , 1) 5 1 0 9 0. 1 3732 0 1 E - 03 0. 2 5 7 4 9 2 1 E - 0 4 0 . 5 8 6 5 0 9 7 E - 0 3 ( 1 6 , 8 6 ) ( 2 0 , 2) ( 7 , 1) 5 1 1 0 0. 1 3 7 3 2 9 1 E - 03 0 . 314 7 125 fi-0 4 0 . 6 6 2 8 0 3 6 E - 0 3 ( 1 7 , 8 7 ) de, 2) ( 6 , 1) 5111 0. 167II467E- 03 0. 3 8 0 2 7 7 6 E - 04 0 . 7 1 9 0 7 0 4 E - 0 3 ( 2 1 . 9 2 ) ( 2 1 , 2) ( 1 2 , 1) 5 1 1 2 0 . 1 5 2 5 8 7 9 B - 03 0 . 4 5 7 7 6 3 7 E - 04 0 . 5 5 5 9 4 5 B E - 0 3 ( 2 1 , 9 1 ) ( 1 2 , 2) ( 1 0 , 1) 5 1 1 3 0 . 1 3 7 3 2 9 1 E - 03 0 . 4 6 7 3 0 0 4 E - •04 0 . 7 5 3 4 0 2 7 E - 0 3 ( 2 1 . 9 2 ) ( 1 7 , 2) ( 0 , 1) 511U 0 . 1 8 3 1 0 5 5 E - 03 0 . 1 6 7 3 0 0 4 E -•04 0 . 6 4 3 7 3 0 2 E - 0 3 ( 1 8 . 9 1 ) ( 1 2 , 2) ( 8 . 1) 5 1 1 5 0. 1 8 3 1 0 5 5 E - 03 0. 1 2 9 1 5 3 1 E - •04 0 . 5 4 2 6 4 0 7 E - 0 3 ( 1 8 . 9 0 ) ( 1 3 , 2) ( 1 2 . 1) 5 1 1 6 0 . 1 2 2 0 7 0 1 E - 03 0 . 1 3 8 6 9 0 2 E - •04 0 . 4 0 5 3 1 1 6 E - 0 3 ( 2 0 , 8 8 ) ( 1 2 , 2) ( 7 . 1) 5 1 1 7 0 . 1 6 7 8 4 6 7 E - 03 0 . 37 1 93 30 E -•04 0 . 4 0 0 5 4 3 2 E - 0 3 ( 1 7 , 9 1 ) ( 1 0 , 2) ( 1 0 , 1) 5 1 1 8 0. 2 1 3 6 2 3 0 E - 03 0 . 3 3 3 7 8 6 0 E - 04 0 . 6 0 0 3 9 7 9 E - 0 3 ( 1 9 , 0 2 ) ( 1 3 , 2) ( 1 2 , 1) 5 1 1 9 0 . 1 6 7 f l ' l b 7 E - 03 0 . 36 2 3 96 2 E-04 0 . 5 1 0 2 1 5 8 E - 0 3 ( 1 0 , 9 2 ) ( 1 6 , 2) ( 1 1 . D 5 1 2 0 0 . 1 5 2 5 8 7 9 E - 03 0 . 3 6 2 3 9 6 2 E - •04 0 . 3 8 8 1 4 5 4 E - 0 3 ( 1 1 . 9 2 ) ( 1 8 , 2) ( 1 6 , 1) 5121 0 . 1 6 7 8 4 6 7 E - 03 0 . 343 3 2 2 8 E -•04 0 . 3 7 0 0 2 5 6 E - 0 3 ( 1 4 , 9 2 ) ( 1 6 , 2) ( 1 8 , 1) 5 1 2 2 0 . 1 3 7 3 2 9 1 E - 03 0 . 324 2 4 9 3 E -•04 0 . 4 6 4 4 3 9 4 E - 0 3 ( 1 4 . 9 1 ) ( 1 « . 2) ( 1 0 , 1) 5 1 2 3 0. 1 S 2 5 8 7 9 E - 03 0 . 3 1 4 7 1 2 5 E - •04 0 . 4 3 2 9 6 B 1 E - 0 3 ( 1 8 , 9 0 ) ( 1 6 , 2) ( I t , D 5 1 2 0 0. 1 3 7 3 2 9 1 E - 03 0 . 2 7 6 5 6 5 6 E -•04 0 . 4 2 0 5 7 0 4 E - 0 3 ( 1 7 , 9 0 ) ( 1 4 , 2) ( 1 1 . D 5 1 2 5 0. 1 S 2 5 B 7 9 E - 03 0. 3147125fi- •04 0 . 6 4 6 5 9 1 2 E - 0 3 ( 1 8 , 9 2 ) dS, 2) ( 1 8 , 1) XAVG STREAM 3 U V G VDRT/B U V G VORT/S 0 . 3 2 9 3 0 8 5 B - 0 4 0 . 9 2 0 2 2 8 9 E - 03 0 . 1 0 1 7 8 1 7 E - 01 3 6 9 6 4 . 8 0 4 3 9 . 2 5 6 6 4 0 . 3 2 85 8 0 . 3 1 5 4 2 3 2 B - 04 0 . 9 0 1 7 9 6 6 E - 03 0 . 9 5 3 7 6 6 7 E - 0 2 3 6 9 6 4 . 7 9 4 3 9 . 2 5 6 6 4 0 . 32 8 5 8 0 . 3 2 4 3 2 1 2 E - 04 0 . R 8 3 7 7 5 U E - 03 0 . 9 4 0 3 1 0 2 E - 0 2 3 6 9 6 4 . 8 0 4 3 0 . 2561 4 3 . 3 2 8 7 3 0 . 33 1 4 6 0 9 E - 04 0 . 8 5 8 1 3 3 7 E - 0 3 0 . B 9 1 0 2 3 5 E - 0 2 3 6 9 6 1 . 7 9 4 3 9 . 2561 4 0 . 3 2 0 9 5 0 . 3 2 9 5 9 8 1 E - 34 0 . 8 4 1 5 5 5 0 E - 03 0 . 79 762 9 7 E - 0 2 3 6 9 6 4 . 7 9 4 3 9 . 2561 4 0 . 3 2 9 0 7 0 . 3377 3 8 3 E - 04 0 . 8 3 8 S 3 1 9 E - 03 0 . 9 4 3 9 1 0 0 E - 0 2 3 6 9 6 4 . 7 8 4 3 9 . 2 5 6 3 4 0 . 3 2 0 3 2 0 . 3 3 7 9 7 8 3 E - 04 0 . 8 3 7 2 6 3 4 E - 03 0 . 1 4 5 2 9 1 4 E - 01 3 6 9 6 4 . 7 9 4 3 9 . 2 5 5 9 1 0 . 3 2 U 9 0 0 . 3 4 5 7 6 0 2 E - 04 0 . 8 U 5 8 6 2 8 E - 03 0 . 180 53 3 9 E - 01 3 6 9 6 4 . 7 8 4 3 9 . 2 5 5 6 4 0 . 3 2 885 0 . 3 4 0 3 9 1 9 E - 04 0 . 8 9 2 7 1 7 1 E - 03 0 . 1 7 1 5 7 5 7 E - 01 3 6 9 6 4 . 76 1 3 9 . 2 5 5 4 4 3 . 3 2 8 3 1 0 . 3 3 9 7 4 7 3 E - 0 4 0 . 9 3 3 1 2 8 1 E - 03 0 . 1254 8 4 0 E - 01 3 6 9 6 4 . 7 6 4 3 9 . 2 5U9 4 0 . 3 2 7 7 7 0 . 3 2 4 2 0 9 8 E - 04 0 . 9 S 6 6 3 6 7 E - 03 0 . 1 5 1 3 1 7 2 E - 01 3 6 9 6 4 . 7 5 4 3 9 . 2 5 4 6 4 3 . 0 2 7 4 4 0 . 3 3 7 5 5 B 4 E - 04 0 . 9 7 0 3 6 7 6 E - 03 0 . 1 3 7 6 2 2 4 E - 01 3 6 9 6 4 . 7 5 4 3 9 . 2 51*2 4 3 . 3 2 7 1 5 0 . 3 3 9 0 8 2 8 E - 0 4 0 . 9 9 0 63 3 4 E - 03 0 . 1 2 2 2 9 5 5 E - 01 3 6 9 5 4 . 7 4 4 3 9 . 2 5 3 7 1 3 . 3 2 6 7 9 0 . 3 4 H 4 R 2 9 B - 0 4 0 . 1 0 0 2 1 2 2 E - 02 0 . 1 1 7 6 0 0 1 E - 01 3 6 9 6 1 . 7 5 4 3 0 . 2 5 2 9 1 0 . 3 2 6'* 9 0 . 3 1 9 0 2 1 8 E - 01 0 . 9 9 4 9 3 4 0 E - 03 0 . 9 7 0 S 1 3 0 E - 0 2 3 6 9 6 1 . 7 5 4 3 9 . 2 52 9 1 0 . 3 2 6 3 7 0 . 3 2 3 9 0 0 6 E - 0 4 0 . 9 9 7 6 8 1 2 E - 03 0 . 13781 9 9 E - 01 3 6 9 6 4 . 7 3 4 3 9 . 2 5 2 7 4 0 . 32 620 0 . 3 3 2 7 4 4 8 E - 01 0 . 9 9 5 1 4 0 8 E - 03 0 . 1 0 5 8 6 8 5 E - 01 3 6 0 6 4 . 7 3 4 3 9 . 251 7 40 . 3 2 6 3 9 0 . 3 3 3 2 3 3 B E - 04 0 . 9 7 1 8 B 7 3 B - 03 0 . 1 3 3 1 5 7 0 E - 01 3 6 9 6 4 . 7 3 4 3 9 . 251 2 4 3 . 3261 2 0 . 3 2 2 0 8 0 5 E - 04 0 . 9 6 3 8 4 4 4 E - 03 0 . 1 50 1 2 6 9 E - 01 3 6 0 6 4 . 7 3 4 3 9 . 2 51 2 4 3 . 32 61 5 0 . 3 3 9 4 6 9 7 E - 04 0 . 9 4 9 B 4 8 7 E - 03 0 . 9 3 2 2 4 7 6 E - 0 2 3 6 9 6 4 . 7 4 4 3 9 . 251 3 4 3 . 3261 1 0 . 3 3 9 0 7 4 2 E - 0 4 0 . 9 4 0 5 2 6 4 E - 03 0 . 8 8 0 0 7 U 2 E - 0 2 3 6 9 6 4 . 7 2 4 3 9 . 2 5 ) 2 4 0 . 0 2 6 3 B 0 . 3 3 5 2 1 B 4 E - 0 4 O . 9 2 0 O 1 8 5 E - 03 0 . 1 1 5 3 5 4 8 E - 01 3 6 9 6 4 . 7 3 4 3 9 . 2 5 3 2 4 3 . 0 2 6 3 5 0 . 3 2 8 2 5 7 0 E - 04 0 . 9 2 2 8 6 1 5 E - 03 0 . 8 9 9 3 8 0 8 E - 0 2 3 6 9 6 4 . 7 3 4 3 9 . 2 4 9 8 1 3 . 3 2 6 3 5 0 . 3 4 2 2 9 1 8 E - 0 4 0 . 9 1 1 7 7 0 9 E - 03 0. 1 2 2 1 8 5 0 E - 01 3 6 9 6 4 . 7 2 4 3 9 . 2 4 9 3 4 3 . 0 2 6 0 2 0 . 3 2 0 3 6 0 7 E - 0 4 0 . 9 2 4 0 9 9 9 B - 03 0 . 1 2 7 1 3 5 9 E - 01 3 6 9 6 4 . 7 3 4 3 9 . 2 4 8 8 4 0 . 3 2 5 8 0 ro cn ITF.B MAX STREAM MAX VORT/B MAX VORT/S 5126 0 . 1525879E- 03 - 0. 2765656E- 04 0 . 44727 33E--03 (14,92) (16, 2) (17. 1) 5127 0 . 1831055E- 03 0 . 209U083E- 04 0 . 5178452E--0 3 (18 ,91 ) (14, 2) (10. 1) 5128 0 . 167R4G7E- 03 0 . 2479553E- 04 0 . 3204346E--03 (18.90) (13 , 2) d 1), 1) 5129 0 . 1 220703E- 03 0 . 1919270E- 04 0 . SO540 1 1 E--0 3 (20.91) (21 , 4) (17, 1) 5130 0 . 1220703E- 03 0 . 191 1309 S-•04 0 . 2622S04E--0 3 (15 .86 ) (22 , 4) (14. D 5131 0 . 16781b7E- 03 0 . 1907149E- 04 0 . 2727509E--03 (11.92) (21 , 4) (14, 1) 5132 0 . 1 3 73291 E -03 0 . 1 84 1784 E-04 0 . 4854202E--0 3 (18.91) (21 . 4) ( 11 , D 5133 0 . 1 220 703E-03 0 . 1907349E-•04 0 . 185 37 39 E--0 3 (18 .91 ) (11 . 2) (11- D 5131 0 . 1068115E- 03 0 . 20027 16E- 04 0 . 5051171E--03 (19.86) (10 , 2) (18, 1) 5135 0 . 1220 701E- 03 0 . 209fl0n3K- 04 0 . 4 1 10336 E--03 (21.92) ( 8 , 2) ( 8 , 1) 5136 0 . 1525879B- 03 0 . 1907349E- 04 0 . 4 100800E--0 3 (18,92) (17 , 2) (16. 1) 5137 0 . 1173291E- 03 0 . 20027 16E- 04 0 . 6S42206E--0 3 (17.91) (16, 2) (16, 1) 5138 0- 1678467E- 03 0 . 2056160E- 04 0 . 4 053 1 16E--0 3 (16,90) ( 2 2 , 2) ( 8 , 1) 5139 0 . 1678U67K- 03 0 . 2050400E- 04 0 . 4 362454E--0 3 (14.91) (21 , 2) (24, 1) 5110 0 .1220703E- 03 0 . 1907349E-•04 0 . 46 15781E--03 (17.92) (12 , 2) (12, 1) 5141 0 . 1220703E- 03 0 . 168085 1E- 04 0 . 34 11151E--03 (16.91) ( 21 , 5) (11 . 1) 5112 0 . 1220703E- 03 0 . 17166 14 E-•04 0 . 3 175735E--0 3 (15.90) ( 9, 2) ( 5 , 1) 5143 0 . 2136230E- 03 0 . 165 104 9 E -•04 0 . 4989505E--0 3 (21.92) (23 , 5) (23 , 1) 5141 0 . 1220 701R - 03 0 . 2479553E-•04 0 . 6 188154E--0 3 (21.88) (15, 2) (21 , 1) 5145 0 . 167RU67E- 03 0 . 3099442E-•04 0 . S99B511E--0 3 (18,90) (2 1, 2) (18, 1) 5146 0 . 1220701E- 03 0 . 3137860E-•04 0 . 632286 1 E--0 3 (21,92) (18, 2) (16, 1) 5147 0 . 1525879E- 03 0 . 40054 32E- 04 0 . 4 768372E--0 3 (16 ,86 ) (16, 2) (17. 1) 5148 0 . 13 732 91B- 03 0 . 3147125E-•04 0 . 6237030E--03 (20 . 87) (15. 2) (17, 1) 5149 0 . 1373291E- 03 0 . 3623962E- 04 0 . 5369186E--03 (17 .92 ) (17 , 2) (10 , 1) 5150 0 . 1525879E- 03 0 . 4673004 E-•04 0 . 4959106E--03 (14,89) ( 11 . 2) ( 9 , 1) XAVG STREAM *AVG VDRT/B XAVG VORT/S 0.3309744E- 04 0 .8955339E- 03 0 . 1 145375E-01 36961.71 4 3 9 . 2 4 3 3 10 . 32600 0 .3297126E- 04 0 .8802931E- 03 0 .9504206E- 02 36964 .72 4 3 9 . 2 4 7 6 13 . 32 639 0.3 182217E- 34 0 .8966681E- 03 0 . 11 60000E- 01 36961 .72 4 39 .2 471 10 . 32 585 0.3380 323E- 04 0 .8545044E- 03 0 .9834450E- 02 36964.71 4 3 9 . 2 4 6 6 40 . 3261 7 0 .3253 327E- 04 0 .8509515B- 03 0 . 9271096E- 02 36964 .70 4 3 9 . 2463 43 .32621 0 .3274276E- 04 0 .8401666E- 03 0 .9040929E- 02 36964 .70 4 3 9 . 2 4 5 4 40 . 3263 8 0.3 180435E- 04 0 .8 108938E- 03 0 .9410653E- 02 36964 .70 4 39 . 2456 40 . 3 2 6'4 3 0. 3036794E- 04 0 .8227886E- 03 0 .9809572E- 02 36964. 69 4 3 9 . 2 4 5 8 4 3 . 0 2 6 5 0 0.3 1357 35B- 04 0 .8237415E- 03 0 . 1 150986E-01 36964 .69 4 39 . 2456 40. 3 2 64 3 0 .3361104E- 04 0.83OH909E- 03 0 .86123 12E- 02 36964 .70 4 3 9 . 2451 40 . 3261 5 0.32626 11E- 04 0 .8300620E- 03 0 .9977695E- 02 36964 .69 4 3 9 . 2 1 4 9 4 3 . 3 2 6 2 0 0. 3228027E- 04 0 .8315756E- 03 0 . 1 308735E-01 36964. 69 4 39 . 2449 40 . 32625 0. 3 1410 37E-04 0 .8158921E- 03 0.951 7547E- 02 36964 .69 4 39 . 2449 43 . 326H6 0.3 130041E- 04 0 .8100693E- 03 0 . 1 192757E-01 36964 .69 4 39 . 2444 43 . 32658 0 . 33144J8E- 04 0 .79 3 51 10 E-03 0.9UG0393E- 02 36964 .70 439 .2441 40.326R1 0 .3221093E- 04 0 .7851452E- 03 0 .9391584E- 02 36964 .68 4 3 9 . 2 4 4 6 43 .32673 0 . 329674BE- 04 0 .7952419E- 03 0.100U053 E- 01 369f,4. 69 4 39 . 2439 43. 32 660 0. 3324 146E- 04 0 .7861641E- 03 0 . 12582 H6E-01 36964. 68 4 3 9 . 2 4 3 4 40. 32658 0. 3296672E- 04 0 .8 125771E- 03 0 . 1 2 6 74 14 E-01 36964 .68 4 39 . 24 34 43. 32625 0. 32626 43E- 04 0 .8 123573E- 03 0 . 104864 8E- 01 36964. 66 4 39 . 2432 43 .32635 0. 3021062E-•04 0 .8279241E- 03 0 . 123 8891E- 01 36964 .68 4 39 . 2129 1 3 . 3 2 5 8 0 0 .3066452E- 04 0 .8388185B- 03 0 . 102 80 15E- 01 36961 . 67 4 3 9 . 2 4 1 9 10 .32562 0. 2901626E- 34 0 .8367D79E- 03 0 . 1 1 60427E-01 36961 .67 4 3 9 . 241 7 40 . 32545 0 .3011217E- 04 0 .8565348E- 03 0 . 1274044E- 01 36964 .63 4 3 9 . 241 7 43 . 32516 0 .31B5873E- 04 0 .8573621B- 03 0 .1159762E- 01 36964 .67 4 3 9 . 241 5 4 0 . 3 2 5 3 5 CO ITER MAX STREAM MAX VORT/B MAX VORT/S 5151 0. 1220701E- 03 - 0. 3337060E- 04 0. 3 166 199E-•03 (19.92) ( 9, 2) (16. 1) 5152 0. 1220703E- 03 0. 2956390E- 04 0. 35905B4E-•0 3 (19.91) (11, 2) (23, 1) 5153 0. 1371291E- 03 0. 3337H60E- 04 0. 3267527 E-•0 3 (18, 88) (10, 2) (23, 1) 5154 0 . 1 525B79E-03 0. 3337B60E- 04 0. 4 06 2553E-•0 3 (21,91) ( 9, 2) (18, 1) 5155 0. 1525879E- 03 0. 295"6390E- 04 0. 5526543E--0 3 (11.90) C 8 „ 2) (21, 1) 5156 0. 1373291E- 03 0. 2574921E- 04 0. 3919601 E--0 3 (19,88) (10, 2) (18. 1) 5157 0. 1220703 E- 03 0. 2765656E- 04 0. 4901886E--0 3 (19,88) ( 9, 2) (11. D 5158 0. 1373291E- 03 0. 219345 1E- 04 0. 2689362E--0 3 (20, 87) (20, 2) (13, 1) 5159 0. 1525879E- 03 0. 209B083E-•04 0. 45B7173E--0 3 (17,87) (18, 2) (16. 1) 5160 0. 1.171291E- 03 0. 26702H8E- 04 0. 46 157B1E--03 (21,90) (16. 2) (14, 1) 5161 0. 1373291E- 03 0. 286 1023E- 04 0. 3099412E--0 3 (21,91) (15, 2) (11, D S162 0. 11732 91E- 03 0. 2574921E- 04 0. 4 06 25 53E--03 (16,83) (18, 2) ( 9, 1) 5163 0. 1670467E- 03 0. 26702H0E- 04 0. 387 19 18E--0 3 (18,92) (12, 2) (12. 1) 5164 0. 1371291E- 03 . 0. 210401ME-•04 0. 5 187988E--0 3 (18,92) (2 3, 2) (20, 1) 5165 0. 1373291E- 03 0. 286698.3E- 04 0. 1281998E--0 3 (20,89) (2 1, 2) d f l . D 5166 0. 1 525879E-03 0. 314 7125 E- 04 0. 43 100 12E--0 3 (18,88) (19, 2) (24, 1) 5167 0. 1373291E- 03 0. 2539158E-•04 0. 34 33228E--0 3 (18,1)9) (25, 2) (16, 1) 5168 0. 1525879E- 03 . 0. 2825260E-•04 0. 4024506E--0 3 (18,90) (22. 2) (20, 1) 5169 0. 1525079E- 03 0. 34 1.3 2 28 r.-•04 0. 3 39500 1 E-0 3 (10,89) (21, 2) (11, D 5170 0. 1525U79E- 03 0. 2676249E-•04 0. 5264282E--0 3 (18, 88) (22. 2) ( 1, U 5171 0. 19 8 3 643 E-03 0. 333 7860E-•04 0. 3519058E--0 3 (13,90) (19, 2) ( S, D 5172 0. 1.3712 9 1 E-03 0. 333 7U60E-•04 0. 4730225E--0 3 (22,91) (10, 2) (18, 1) 5173 0. 1220703E- 03 0. 1907149E-•04 0. 3433223E--0 3 (22,92) ( 8, 2) ( 1 t , D 5171 0. 1220703E- 03 0. 1752377 E-•04 0. 4644394E--03 (19,89) (21, 3) (IS, 1) 5175 0. 1373291E- 03 0. 2098083E-.04 0. 4558563E--03 (19,91) (16, 2) ( 7, 1) *AVG STREAM %AVG VORT/B SAVS VORT/S 0.3117419E- 04 0.8100362E- 03 0.859 7154 E- 02 36964.68 439.2412 43. 3251 0 0. 3 163521 E- 04 0.8208160E- 03 0. 1066765E- 01 36964.66 439.2413 43. 3251 9 0.3 19.15B5E- 34 0.O2 3913 8E- 03 0.1052067E- 01 36954.66 4 39 . 241 3 43. 3251 5 0.3045026E- 04 0.B20B133E- 03 0.9871166E- 02 36964. 66 4 39. 2432 43.32496 0. 3021 384E- 34 0.8236620E- 03 0. 1 54432BE- 01 36964. 66 439.2435 40.32498 0.2868772E- 04 0.B40671 8B- 03 0.1031031E- 01 36964.64 4 39. 24)5 43.32469 0. 2942474E- 04 0.8470346E- 03 0.96521 34E- 02 36964.64 4 39 . 2 3)7 43.3 2154 0. 3025726E- 04 0.0215629E- 03 0.B.3 72 54 9E- 02 36964.63 4 39. 2333 10.32466 0. 30700 58E- 04 0.8000338B- 03 0. 1 250453E-01 36964. 64 4.39. 2383 10.321H0 0. 304 13 44E-04 0.82C0032E- 03 0.7676132E- 02 36964.63 439. 2333 10.32452 0. 3052149E- 04 0.8318427E- 03 0.78643 19E- 02 36964.64 4 39. 2333 10.32435 0. 2993536E-•04 0.8396704E- 03 0. 1 1 1.1588E-01 36964.64 4 39 . 2333 13.32432 0. 29900 59 E-• 01 0.B3221H4E- 03 0.925B516E- 02 36964.64 4 39. 2 378 10.32123 0.3 123007E- 01 0.8 18B190E- 03 0.10B1 04 7E- 01 36964.63 4 39. 2 378 13.32426 0.3045906E-•01 0.8239231E- 03 0. 1 1 1 6202E-01 36964.63 4 39. 2 375 40.32435 0. 3006024E- 31 0.B 1 243 67E-03 0. 9 532 U35E- 02 36964.63 4.39. 2 375 40.32415 0. 30930 23E-•04 0.8274384E- 03 0.9090556E- 02 36964.62 4 39. 2 371 40.32 306 0.3 1 1B8 49 E-01 0.8238351E- 03 0. 0 50 0 3 64 E-02 36964.63 4 39. 2360 40.32376 0. 32 17961E-•04 0.8246330E- 03 0.U 60434 0E- 02 36964.62 4 39 . 2 373 43.32376 0. 3 19 36 43E-•01 0.0219311E- 03 0.1173250E- 01 36964.6 1 4 39. 2371 40.32373 0. 3282427E- 04 0.83513 66E- 03 0.9790957E- 02 36961. 6 1 439.2368 40.3 2 354 0. 32537 14E-•01 0.839 13 31E- 03 0.105U556E- 01 36964.61 439.2363 43. 32 144 0.32B1455E- •04 O.B056690E- 03 0.1114503E- 01 36964.62 439.2358 40. 32373 0.3 187274E-04 0.7870390E- 03 0.1074778E- 01 36964.61 439.2361 40.32393 O.3170675E-04 0.7801993E- 03 0.76.39598B- 02 36964.61 439.2358 43.32390 ro tn 260 r - fN »— »— rsi fN «— «~ fN fN fN fN »— «— fN fN fN CO 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 $-* Cut to CO CO CO to CO CO u to CO CO CO CO CO CO CO CO CO CO CO CO CO 1 CO 1 CO oa r- X »- tf »— o tf <3s p» VO OS X fN SO r- u0 ro o in ro cn X X p* Q in CO X tf in *- 00 ro »— CT* r-i r- o in — X fN ro in ^ p» f^-p* cn X tf rsi r— X r— p- »— rsi —f in r- p* rg •— vO r- o ro cn X rs r— r  > t-O cc r-j r- CO r- X C 'JO r— in r— ro tf X rsi X »— uo r- ro *o ro CT X r— sO »— UO rsi r-. r— rs cr OS X CT X in w~ rsi rvi o rO CT rs sO x r - cr ro ro ro CS ro fN ro X ro o ro tf r-i c- ro o ro p» rsi 01 fN tf fN tf o r-i ro sC ro tf ro r * fN X rsi ; 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CN OS ( ' CO ^ i CN in CN i cr r • .co »*• - o X vO i • fN fN •— ; to ^ co . •JO fN \c t » 04 * fN ; ro O tf O i — P-r^ rsi cn r-. cn » — tf O tf ^ r- «- x CN «~ » o »cn » cn x ^ ^j-cn T-. to A CO A C O n i tf CN o rsi ro rsi *- cn o> X »ro * zr •» ro vO rsi O tf rsi in CN& fN 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 mz i i i i i i i i i i i i i i i i i i i i t i t i i *< to—»oj^— co^-W^-.co — tu—co^— co — t o ^ - t o ^ c o ^ . c o ^ . t o ^— to co .—. to — tiz — t o ^ t o « c o ^ . t o -^.co ^ .co —«ro co^ to • - r ^ P * r N ^ ^ ^ O ' " C ^ r ^ O ' / ! r s o o c ^ c ^ f - a 3 l ^ x ^ > ' - r • o ^ . ' ^ J D u , ' " L ' , r ~ , — r s i r ~ - '— ^ r^i •— r~sir~"i«— r o o c n « — c ^ r s i c n cn os c n c n v o c n c n c n p ^ c n c r x o s c n i n x i n x p ^ x c n a s u ^ C-« rg % t f . r s j » X *CN % . o » o t,x «r-i * o » ^ va- » o ««o * o » r g » . r s i .CN »r* * r - . co »co * X « IT. r o r s i X r ^ r o r ^ u o ^ r ^ x X X < — 0 > — o;uOXr^P*ir-P-'XLO r--rgP,'»— r * - r s i r s i «— l r-r*^-r~^-r*Sr— r s i ^ - r ^ r - r ^ ^ — r~«— i •— r n ^ r ^ ^ — r - > r g r - ~ r s i r > - »— r^-r— — (~grNrsi«~rgrs)rsiCNCNCN W ro sO - ^ r o —' LT? •—C~ —sD—'CO'—'X w JO —* ro ^ CC v sO W \ C — X CO '—' CC "r-> —• v£ « r O —' ro psl ^ - r g — uo —,inw»n^' r: « t t • • « 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 sO X cn o fN no tf in sO p» X cn o •— fN ro uo sO p* CO cn o P* r- p* p» X X X CO CO X X X X X cn cn Cn cn cn cn cn cn cn cn o in vn in m uo in m in m in in in m in uo m m in m in m m in in CN m CONVERGENCE TEST ON GRID AFTER 5200 ITERATIONS: • MAXIMUM CHANGES OVER 3RID: STREAM FUNCTION- 0. 1525879E-03 BULK VORTICITY- 0. 37 19330E-04 AVERAGE X CHANGE OVER GRID: STREAM FUNCTION" 0.3176701E-04 BULK VORTICITY- 0.7528174E-03 SURFACE VORTICITY" 0.5149841B-03 SURFACE VDBTICITY= 0.1294620E-01 WITH RELAXATION: MAXIMUM CHANGES OVER GRID: STREAM FUNCTION- 0.2746570E-03 BULK VORTICITY- 0.7138659E-05 AVERAGE X CHANGE OVER GRID: STREAM FUNCTION- 0.5718040E-04 BULK VORTICITY- 0.1505635E-03 SURFACE VORTICITY- 0. 5149838E- 04 SURFACE VORTICITY- 0.1234619E-02 AVERAGING THE STEP TO STEP CHANGES OVER THE LAST 100 ITERATIONS: OVERALL X CHANGES,(NO RELAXATION): STREAM FUNCTION- 0. 3259513E-04 BULK VORTICITY- 0. 8388790E-03 SURFACE VORTICITY- 0. 1075931E-0 1 VITH RELAXATION: OVERALL X CHANGES, (NO RELAXATION): STREAM FUNCTION- 0.5867101E-04 BULK VORTICITY- 0.1677758E-03 SUBFACE VORTICITY- 0.10759318-02 RINF- 100.000000 BE- 10.000000 SIZE OF GRID- 33 X 93 DUMBER OF ITERATIONS- 5200 rso cn SURPACE VORTICITY DISTRIBUTION THETAG 1) • 0.0 V O R T I C I T Y ( 1, 1 a 0.0 THETAG 2) 3 0. 9 8 1 7 4 7 5 E - 0 1 V O R T I C I T Y ( 2 . 1 3 0 . 4 3 9 6 6 5 1 THETAG ! 3) a 0 . 1 9 6 3 4 9 5 V O R T I C I T Y ( 3, 1 3 0 . 8 6 7 2 9 5 4 THETAG 4) a 0. 2 9 4 5 2 4 3 V O R T I C I T Y ( 4, 1 = 1.2712H2 THETAG 5) a 0 . 3 9 2 6 9 9 0 V O R T I C I T Y ( 5, 1 a 1.640920 THETAG ( 6) a 0 . 4 9 0 8 7 3 8 V O R T I C I T Y ( 6. 1 a 1.9 6 674 3 THETAG 1) a 0 . 5 8 9 0 4 8 5 V O R T I C I T Y ( 7, 1 s 2 . 2 4 0 8 0 9 THETAG 8) a 0. 6 8 7 2 2 3 3 V O R T I C I T Y ( 8, 1 ss 2 . 4 5 7 1 6 1 THETAG 0) B 0 . 7 8 5 3 9 8 0 V O R T I C I T Y ( 9, 1 a 2 . 6 1 1 9 5 9 THETAG 10) a 0 . 8 8 3 5 7 2 8 V O R T i : i T Y ( 1 3 . 1 a 2 . 7 0 3 5 1 3 THETAG 11) a 0 . 9 8 1 7 4 7 5 V O R T I C I T Y ( 1 1 . 1 3 2 . 7 3 2 4 7 1 THETAG [12) m 1. 0 7 9 9 2 2 V O R T I C I T Y ( 1 2 . 1 = 2 . 7 0 1 7 9 5 THETAG 11) a 1. 1 7 8 0 9 7 V O R T I C I T Y ( 1 3 , 1 3 2 . 6 1 6 4 1 9 THETAG 14) a 1. 2 7 6 2 7 1 V O R T I C I T Y ( 1 4 , 1 a 2 . 4 8 2 9 4 1 THETAG 15) a 1. 3 7 4 4 4 6 V O R T I C I T Y ( 1 5 , 1 3 2. 3 0 9 4 5 6 THETAG 16) a 1 . 4 7 2 6 2 1 V O R T I C I T Y ( 1 6 , 1 m 2. 1 0 4 9 6 9 THETAG 17) a 1 . 5 7 0 7 9 6 V O R T I C I T Y ( 1 7 . 1 a 1 . 8 7 8 8 2 7 THETAG 18) a 1 . 6 6 8 9 7 0 V O R T I C I T Y ( 1 8 , 1 A 1.640441 THETAG 19) a 1.767 145 V O R T I C I T Y ( 1 9 . 1 a 1. 1 9 8 6 3 2 THETAG 20) a 1 . 8 6 5 3 2 0 V O R T I C I T Y ( 2 0 , 1 3 1.161340 THETAG 21) a 1. 96 3494 V O R T I C I T Y ( 2 1 , 1 a 0.9.15.1606 THETAG 22) 3 2.06 1669 V O R T I C I T Y ( 2 2 , 1 a 0 . 7 2 6 2 3 9 3 THETAG 23) a 2. 1 5 9 8 4 4 V O R T I C I T Y ( 2 3 , 1 a 0 . 5 1 B 2 3 2 9 THETAG 24) a 2 . 2 5 8 0 1 8 V O R T I C I T Y ( 2 4 . 1 3 0. 3 7 4 4 1 6 3 THETAG 2 5) a 2. 356 194 V O R T I C I T Y ( 2 5 , 1 a 0 . 2 3 6 7 8 7 9 THETAG 26) 3 2.454 369 V O R T I C I T Y ( 2 6 , 1 3 0. 1 2 6 3 9 3 3 THETAG 27) a 2 . 5 5 2 5 4 3 V O R T I C I T Y ( 2 7 , 1 3 0 . 4 3 3 7 7 5 8 E THETAG 28) a 2 . 6 5 0 7 1 8 V O R T I C I T Y ( 2 8 , 1 3 - 0 . 1 3 0 6 0 2 3 8 THETAG 29) a 2. 7 4 8 8 9 3 V O R T I C I T Y ( 2 9 , 1 3 — 0 . 4 4 7 5 3 3 9 3 THETAG 30) a 2 . 8 4 7 0 6 7 V O R T I C I T Y ( 3 0 , 1 3 - 0 . 5 4 6 6 8 7 0 2 THETAG 31) a 2 . 9 4 5 2 4 2 V O R T I C I T Y ( 3 1 . 1 a - 0 - 1 4 5 9 2 0 1 0 8 THETAG 32) a 3 . 0 4 3 4 1 7 V O R T I C I T Y 132.1) s- 0 . 2 6 6 9 5 8 4 2 THETAG 33) a 3. 1 4 1 5 9 2 V O R T I C I T Y ( 3 3 , 1 a 0 . 0 ro cn ro S U R F A C E PRESSURE D I S T R I B U T I O N T H E T A G ( 3)" 0. 1963495 S U R F A C E TH ETAG ( 5)» 0. 3926990 S U R F A C E T HETAG ( 7) = 0.5890405 SU R F A C E TH E T A G ( 9)» 0. 7H53980 S U R F A C E T H E T A G (11) « 0.98 17475 S U R F A C E T H E T A G ( 1 3) » 1. 178097 S U R F A C E T H E T A G (1 5) = 1. 374446 su R F A : E TH ETAG (1 7) • 1.570796 S U R F A C E T H E T A G (19)* 1.767 145 S U R F A C E T H E T A G (21) » 1. 963494 S U R F A C E T H E T A G (23) » 2. 159844 S U R F A C E T H E T A G (25) •» 2.356194 S U R F A C E T H E T A G (27) » 2.552543 S U R F A C E T HETAG (29) » 2.748893 S U R F A C E T H E T A G ( 3 1 ) » 2.945242 S U R F A C E T H E T A G ( 3 3 ) » 3 . 141592 S U R F A C E P R E S S U R E ! 3) « P R E S S U R E ( 5)-= P R E S S U R E ! 7) * P R E S S U R E ! 9) » P S E S S U R E ( 1 1) =-P R E S S U R E ( 1 3 ) P S E S S U R E (15) =-P R C S S U K E ( 1 7 ) * P R S S S U R E ( 1 9 ) = P R E S S U R E (21) " P R E S S U R E (23) » P R E S S U R E (25) "• P R E S S U R E (27) »• P R E S S U R E (29) • -P S E S S U R E (31) »-P R E S S U R E (33) »• 1. 393908 1. 138104 0.7584244 0.3141203 0. 1307583 0.5201273 0.8163815 -1.004440 -1.090083 -1.09337 1 -1.040020 0.9549694 0.8598185 0.7735138 0.7130127 0.6911907 DRAG P A R A M E T E R S A F T E R 5200 I T E R A T I O N S : R - I N F I N I T Y » 100.0000 R E Y N O L D S KO.» 10.00000 S I Z E O F G R I D - 33X93 F R O N T A L S T A G N A T I O N P R E S S U R E " 1.484200 REAR S T A G N A T I O N PRESSURE--0.6911907 T H E S K I N DRA3 C O E F F I C I E N T " 1.222929 T H E FORM DRAG C O E F F I C I E N T " 1.559216 C D S K I N / C D F O R H " 0.7843232 T H E T O T A L DRAG C O E F F I C I E N T • 2.782145 ro cn O J STORES ( 1)» 0.5000000 STOKESf 2 ) ° 1.000000 XKAP( 1)» 0. 99999998-03 XKAP( 2)• 0.2000000 INITIAL VARIABLES RINF" 100. 000 REYNOLD'S NO.* 10.000 XP (1 )» -100.0000 SIZE OF GRID' 33 X 93 NUMBER OF STEPS PER GRID CELL" 3.000003 •••*• NUMERICALLY CALCULATED FLOWFIELD USED •*«**«* * * * * * * EFFICIENCY WITH INTERCEPTION * * * * * * * STOKES' NO." 0.500 EFFICIENCY" 0. 37953890860E-02 STOKES' NO.» 1.000 EFFICIENCY" 0.1044564247 1 * * * * * * EFFICIENCY WITH INTERCEPTION *•**••• STOKES' UO.» 0.500 EFFICIENCY" 0.10357707739 STOKES' NO." 1.000 EFFICIENCY" 0.22409653664 DP/DP" 0.001000 DP/DF" 0.001000 I" 0.37915995345B-02 I" 0. 1343521 1658 DP/DP" 0.200000 I « 0.86314201355E-01 DP/DP" 0.200000 I" 0.13674713373 rsi 265 Computer Programme used to Calculate Stagnation Pressures and Drag Coefficients 1 PEAL S ( 3 3 , 93) ,V (33,93) ,PSIN (33) ,PCOS (33) 2 BEAL P (33) , PT (33) 3 REAL RG (93) ,THETAG(33) 4 READ(U,290O) RIH P, RS, RLXS, RL IV , RLI ? S , A, B 5 READ (<t,3000) N R , N A , I T E R , I I C Y C L , m j H 8 A 6 8RITE(6,3100) RINF, RE, NA, NR, NUH8A 7 DO 10 J-1 .HB 8 READ(4,2910) (V ( I , J) , 1= 1, BA) 9 10 CONTINUE 10 DO 20 J=1,NB 11 READ(M,2910) (S ( I , J ) ,I=1,SA) 12 20 CONTINUE 13 PI=3.141592653589793 14 B = PI/(NA-1.0) 15 DO 40 1=1,SA 16 A ! I G L E = P I / ( N A - 1 . 0 ) » ( I - 1 . 0 ) 17 FSIN (I) =SIN (ANGLE) 18 FCOS (I) =COS (ANGLE) 19 THETAG (I) =ANGLB 20 40 CONTIHOE 21 DO 80 1=1,NA 22 80 CONTINUE 23 HA2=SA-2 24 NH2=SB-2 25 CDSKIN=0.0 26 DO 30 I=1.NA2,2 27 CDSKIN=FSIN (I) *V (1,1) +4. *FSIN (1+1) *? (I>1 , 1) 28 1 •PSIN (1+2) *V (1*2, 1) •CDSKIH 29 30 CONTINUE 30 CDSKIN=CDSKIH*4.0*B/3.0/BB 31 PO=0.0 32 DO 53 J=1,NB2,2 33 PO = P 3 » ( - 3 . 0 * V ( 1 , J ) • 4 . * V ( 2 , J ) - V ( 3 , J ) > 34 1 *4 . » (-3. *V ( 1 , J M | + 4.*V (2, J * 1 ) - V ( 3 , J * 1 ) ) 35 2 + ("3.*Y ( 1 , J*2) *4.*V ( 2 , J * 2 ) - V ( 3 , J » 2 ) ) 36 50 CONTINUE 37 PO = PD«4.*A/3-/RE/B/2. M.O 38 PZER0=P3 39 P (1) =0-0 40 DO 63 I=1,HA2,2 41 KK=I*2 92 P (KK) = ( (-3.*V ( I , 1) »4.*V ( I , 2 ) - V (1,3)) 43 1 » 4 . » (-3. *V (1*1, 1) »4.*V (1*1,2)-V ( t » 1 ,3) ) 44 2 • ("3. *V (1*2, 1) »4.«V ( I » 2 , 2 ) - V (1*2, 3) ) ) / 2 . / A * P ( I ) 45 PT(KK) =PO»P(KK) *4.*B/3./RB 46 60 CONTINUE 47 PT ( 1) =PO 48 PREA R= PT (N A) 49 CDFORH=0.0 50 DO 73 1=1,NA2.4 51 CDFORf1 = CDFORM»PT (I) *FCOS (I) *4 .0*PT ( I » 2 ) *FCOS (1*2) 52 1 *PT (1*4) »FCOS ( I » 4 ) 53 70 CONTINUE 54 THSEP=3.0 55 D1 = 1./(NA-1.0) 56 DO 130 1=1,NA 57 I F ( I . E O . I ) GO TO 100 58 I F ( I . E a . NA) GO TO 100 266 59 IF(V (I,1).GT.O.O) GO TO 100 60 THSEP=180.* (1.0-D1* (1-2. - V (I - 1 ,0) / (V (I, 1) -V (I- 1, 1) ) ) ) 61 GO T3 90 62 100 CONTINUE 63 90 CONTINUE 6a CDF0RH=CDFORM*B/1.5 65 CDTOT=CDSKIS*CDFOHH 66 CRAT=CDSKIH/CDFOBB 67 WRITE<6,4200) 68 VRITE(6,4000) RE, RIH P, N A, HR 69 W 8 I T £ ( 6 , 4 1 0 0 ) PZERO,PREAR,THSEP,CDSKIN,CDPORH,CRAT,CDTOT 70 VJRITE(1 ,43 00) RINF,RE,HR,HA,NU NBA,PZ ERO, PR EAR ,THS EP, CDSKIB , CDFO BM, 71 1 CRAT.COTOT 72 SRITE(6,4200) 73 WRITE(6,4200) 74 1000 FO RM A T (1 0 X ,•THE SKIN DRAG COEFFICIENT=•, GH.7/) 75 1 100 FORMAT(5P12.5) 76 1200 FORM AT (1 OX, ' TH E FORM DRAG COEFFICIENT = «,G14.7/) 77 1300 FORNAT(6I5) 78 1400 FORMAT(10X,'THE TOTAL DRAG COEFFICIENT = ' , G 1 4 . 7 / / ) 79 1500 FORHAT(10X,'FRONTAL STAGNATION PRESSURE* • , 3 14 .7/) 80 1600 FORMAT(10X, 'THETAG(' , 12,')=• ,G14 .7 , 5X, 'SURFACE PRESSURE^, 81 1 12, •) = ' ,G14 .7 / / ) 82 1700 FORMAT(10X, • TH BT A G (• , 12 , •) =• » G14 . 7 , 5X, 'VORTICITY (', 12, ' , 1 > =• , 83 1 G14.7/ / ) 84 1800 FORMAT(10X. • CDSKIN/CDFOR M= • , G 14 .7/) 85 2900 FORMAT(8F10.6) 86 2910 PORMAT(1 7G14.7) 87 3000 FORNAT(5I5) 88 3100 F O R M A T ( • 1 ' , 1 0 X , ' R I N P = ' , P 1 2 . 6 , 5 X , » R B = ' , F 1 2 . 6 / / 1 0 X , ' S I Z E OF G R I D " ' , 89 1 1 X . I 2 , ' X •.I2/ /10S, 'NUMBER OF ITERATIONS- ' ,15 / ) 90 3200 FORMAT(•1',15X,'SURFACE VORTICITY DISTRIBUTION"//) 91 3300 FORMATf•1' ,15X. 'SURFACE PRESSURE DISTRIBUTION'/ / ) 92 3400 F O R M A T C1 « , 1 5 X , ' DRAG PARAMETERS'//) 93 3500 FORHAT(10X,'REAR STAGNATION PRESSUBE=',314.7/) 94 3600 FORM AT (• 1 ' , 10X) 95 3700 FO3MAT(10X, 'AFTER', 1X, 14, 1X, 'ITERATIONS: •///) 96 3800 FORMAT(10X, 'R-INFINITY=' ,G14.7,5X, 'REYNOLDS NO.= ' , G 1 4 . 7 / 97 ' 1 l O X . ' S I Z E OF GRID= ' , IX ,12, 'X•,12/) 98 3900 FORMAT! 10X, ' ANGLE OF HAKE SEPARATION* • , F 8 . 3 , " DEGREES'/ / ) 99 4000 FORMAT! 1 3X, • R EYNOLD ' 'S NO. = • , F5. 2, 18X , • OU T ER BOUNDARY RADIUS = ', 100 1 F5. 1 / /50X, 'GRID DIMENSIONS: ',12,' X ' , I 2 / / / 2 4 X , 101 2 'CALCULATED V A L U E S ' / / ) 102 4100 FOSMAT(13X,'FRONTAL STAGNATION PRESSURE=•,1X,Fl0 . 6 / / 103 1 13X,'REAR STAGNATION PRESSORE=' ,3X,Fl1 .6 / /13X 104 • 2 'ANGLE OF WAKE SEPA RATIO 3= • , 4X , Fl 0. 6//1 3 X, 105 3 'SKIN DRAG COEFFICIENT* ' , 7X . F ID . 6//1 3X, • FORM DRUG COEFFICIENTS ', 106 1 7X ,F10 .6 / /13X,•RATIO: CDS KIS/CDFD RM=',8X,F10.6//13X, 107 5 'TOTAL DRAG COEFFIC IE :)T= • , 6 X , F 10. 6) 108 4200 FORMAT('I ' ) 109 4300 FORMAT(12G14.7) 110 STOP 111 END R EI HOLD'S NO." 0.20 OUTER BOUNDARY RADIUS'100.0 GRID DIMENSIONS: 33 X 93 CALCULATED VALUES REAR STAGNATION PRESSURE' ANGLE OF HAKE SEPARATION" SKIN DRAG COEFFICIENT" FORK DRAG COEFFICIENT" RATIO: CDSKIN/CDFOBM" TOTAL DRAG COEFFICIENT" 12.U7U579 CO 1 1. 292297 am 0.0 i fD 18. 632019 O 18.669302 GO c= C+ cr 0.998001 OJ -a CQ c 37.301361 a -s na <-+• Cl> CT -+> ca -s o o o rs 3 o ro T3 -h -s -s -b fD o —i. 10 ca o CO -s —it c a> n> -5 3 3 fD 3 C+ CO fD to Dl O 3 O) CL —1 o c —1 r + 3 ca ro cn B Et HOLD' S NO. = 10.00 OUTER BOUNDARY RADIUS-100.0 GRID DIMENSIONS: 33 X 93 CALCULATED VALUES FRONTAL STAGNATION PRESSURE" 1.484242 REAR STAGNATION PRESSURE" -0.691041 ANGLE OF HAKE SEPARATION" 33.753765 SKIN DRAG COEFFICIENT" 1. 223443 FORB DRAG COEFFICIENT- 1.559248 RATIO: CDSKIN/CDFORM" 0.784637 TOTAL DRAG COEFFICIENT" 2.782691 ro cn CO REYNOLD'S NO.-40.00 OUTER BOUNDARY RADIUS-100 GRID DIMENSIONS: 33 X 93 CALCULATED VALUES FRONTAL STAGNATION PRESSURE- 1. 1390U1 REAR STAGNATION PRESSURE" -0.573UB3 ANGLE OF WAKE SEPARATION" 55.672012 SKIN DRAG COEFFICIENT" 0.525U63 PORM DRAG COEFFICIENT" 1.036173 RATIO: CDSKIN/CDFORH" 0.507116 TOTAL DRAG COEFFICIENT" 1. 561601 270 Computer Programme used to Calculate Impaction E f f i c i e n c i e s i 2 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 31 35 36 37 38 39 QO a i «2 U3 aa a s U6 a? a 8 49 50 51 52 53 54 55 56 57 58 C C c c c c c c c c c c c c c c c c c BE A t PS I S ( 5 1 ) , FCOS ( 5 1 ) ,RG (9 3) ,TKET AG (5 1) ,XP ( 2 0 0 0 ) . I P ( 2 0 0 0 ) T H I S I S THE MAIN PROGRAMME FOR C A L C U L A T I N G THE P A R T I C L E C O L L E C T I O N E F F I C I E N C I E S , FOR A L L C A S E S , E I C E P T T H i T WHERE STOKES* H03BER I S EQUAL TO ZERO. BT PROPER S E L E C T I O N OF THE V A R I A B L E " J S U B " ANY OF THE FOLLOWING THREE F L O W F I E L D S CAN BE USED: P O T E N T I A L , D A V I E S ' B E S S E L E Q U A T I O N ONE AT R E = 0 . 2 , OR THE G R I D OF N U M E R I C A L V A L U E S FOR THE G I V E N REYNOLD'S NUMBER. I T SHOULD B E NOTED THAT THE CO-ORDINATES FOR THE P A R T I C L E T R A J E C T O R Y HAVE 3EES MOVED TO THE CENTRE OF THE C Y L I S D E R . T H I S HAS HAD E F F E C T OH C E R T A I N EQUATIONS I N THE H/P AND S/R 'LOCATE'. >»»»»»»»«»*•*»*»»»*»•< R E A L VGX ( 5 1 ,93) , VGT (5 1,93) ,VPX (200 0) , VPY ( 2 0 0 0 ) R E A L S ( 5 1 , 9 3 ) , V ( 5 1 , 9 3 ) R E A L E P S I L ( 2 0 ) , P ( 2 0 ) , K S I Z E ( 2 0 ) , E P F ( 2 0 ) R E A L X P S T , R E , R I N F , A , B . G H U B I N T E G E R NR.NA, J S U B , J I N T COMMON X P S T , R E . R I N F , A , B , H R , J A COMMON R G , T H E T A G , V G X , V G Y , P , K S I Z B COMMON S N U M , J S U B , J I N T I P A = 0 READ ( 8 , 2 9 0 0 ) S T A R T , 3HUH WHITE ( 6 , 9 9 1 0 ) R E A D ( 7 , 9 1 1 0 ) J S U B , J BOUND, J U T READ ( 5 , 2 3 0 0 ) MSTK1 , K S T K 2 ,MKAP1 , HKAP2 , H A N I DO 2 2 0 1=1,HANI R E A D ( 5 , 2 1 0 0 ) K S I Z E ( I ) , P ( I ) 220 C O N T I N U E REWIND 5 READ ( 4 , 2 9 0 0 ) R I N F , R E , R LXS , RLX V, R L X V S READ ( 4 , 3 0 0 0 ) N R , N A , I T E R , S C Y C L , S U M B A X P S T = - R I N F ' S T A H T DO 3 5 0 I = M S T K 1 , M S T K 2 WRITE ( 5 , 9 7 3 0 ) I , P ( I ) 3 5 0 C O N T I N U E DO 3 5 0 I = H K A P 1 , H K A P 2 WRITE ( 6 , 9 7 4 0 ) I , K S I Z B ( I ) 360 C O N T I N U E I F ( J S . U B . S E . 1) GO TO 6 3 0 DO 20 J = 1 , N H R E A D ( 4 , 2 9 1 0 ) ( 7 ( 1 , J ) , 1 = 1 , MA) 20 C O N T I N U E DO 30 J = 1 , N 8 READ ( 4 , 2 9 1 0 ) 30 C O N T I N U E 630 C O N T I N U E REWIND 4 WRITE ( 6 , 3 3 0 0 ) ( S ( I , J ) , I = 1 . N A ) 271 5 9 I P ( J S U B . EQ- 3) GO TO 6 4 0 60 6 6 0 WRITE ( 6 , 1 1 0 0 ) B M P , B E 61 GO TO 6 5 0 62 6 4 0 WRITE ( 6 , 1 1 1 0 ) BE 63 6 5 0 C O N T I N U E 64 WRITE ( 6 , 2 5 0 0 ) X P S T 6 5 WRITE ( 6 , 1 8 0 0 ) 3 1 , HR 6 6 WRITE ( 6 , 3 6 0 0 ) GH0H 6 7 I F (JBOUND. EQ-1) GO TO 5 0 0 6 8 I P ( J B O U N D . E Q . 2 ) GO TO 5 1 0 69 WRITE ( 6 , 5 2 0 1 ) 70 GO TO 2 5 1 71 5 0 0 WRITE ( 6 , 5 2 0 0 ) 72 GO TO 5 2 0 73 510 WRITE ( 6 , 5 3 0 0 ) 74 5 2 0 C O N T I N U E 7 5 I P ( J I N T . E Q . 1) GO TO 5 5 0 76 I F ( J I H T . E Q . 2 ) GO TO 5 6 0 77 WRITE ( 6 , 6 2 0 1 ) 7 8 GO TO 2 5 1 79 550 WRITE ( 6 , 6 2 0 0 ) 80 GO TO 5 7 0 81 5 6 0 WRITE ( 6 , 6 3 0 0 ) 82 5 7 0 C O N T I N U E 83 Z I N F = A L O G ( R I S P ) 84 A = Z I N P / (NR-1) 8 5 P I = 3 . 1'4 1 5 9 2 6 5 3 5 8 9 7 9 3 86 B = P I / (NA-1) 87 DO 10 1 = 1 , H i 8 8 A N G L E = P I / ( N A - 1 . ) * ( 1 - 1 . ) 8 9 F S I N ( I ) = S I N ( A N G L E ) 9 0 FCOS ( I ) =COS (A N G L E ) 91 THETAG ( I ) =AHGLE 92 10 C O N T I N U E 93 DO 11 J = 1 , N B 94 RG ( J ) = E X P ( Z I N F / ( N R - 1 . ) * ( J - 1 . ) ) 9 5 11 C O N T I N U E 9 6 I F ( J S U B . EQ. 1) GO TO 4 0 0 9 7 I P ( J S U B . EQ.2) GO TO 2 6 0 9 8 I F ( J S U B . EQ.3) GO TO 2 7 0 99 WRITE ( 6 , 9 2 0 1 ) 1 0 0 GO TO 25 1 10 J 4 0 0 WRITE ( 6 , 9 2 0 0 ) 1 0 2 NA1=NA-t 103 NB1=NR-1 104 DO 4 0 1=2,NA1 105 DO 4 0 J = 2 , N H 1 106 VTH = (S ( I , J « 1 ) - S ( I . J - 1 ) ) / 2 . / A / B G ( J ) 107 VR= (3 ( I - 1, J ) -S ( I H , J ) (/2./B/RG ( J ) 1 0 8 V G X ( I , J ) = V T H * P S I N ( I ) - V R * F C O S ( I ) 1 0 9 VGT ( I , J ) =VTH*PCOS ( I ) •yR»FSIJ ( I ) 1 1 0 40 C O N T I N U E 111 00 5 0 1=1,NA 112 V G X ( I , 1 ) = 0 . 0 113 VGT ( I , 1) =0.0 114 V G X ( I , N R ) = 1 . 0 1 1 5 V G T ( I , N R ) = 0 . 0 1 1 6 50 C O N T I N U E 272 1 1 7 DO 1 2 0 J = 2 , H H 1 1 1 8 78= (-3.*S ( 1 # J ) »4. *S ( 2 . J ) - S ( 3 , J ) ) / 2 . / B / R G ( J ) 1 1 9 VGX ( 1 , J ) =V8 1 2 0 YGY ( 1 , J ) = 0 . 0 121 VR1= ( 3 . *S ( S i , J ) - 4 . * S ( N A - 1 . J ) »S (H A-2 , J ) ) / 2 . / B / H G ( J ) 1 2 2 VGX ( N A , J ) =—VB1 1 2 3 VGY ( N A , J ) = 0 . 0 1 2 4 120 C O N T I N U E 1 2 5 GO TO 2 8 0 1 2 6 260 WRITE ( 6 , 9 3 0 0 ) 1 2 7 GO TO 2 8 0 1 2 8 2 7 0 WRITE ( 6 , 9 4 0 0 ) 1 2 9 2 3 0 C O N T I N U E 1 3 0 DO 2 5 0 KR=.1KAP1,HKAP2 131 DO 2 3 0 I = S S T K 1 , H S T K 2 132 Y H A X ' K S I Z E ( K R ) »1.0 1 3 3 I F ( S T A R T . G T . 0 . 9 8 ) X P S T = - S Q R T ( R I N F » R I N F - Y J A Y * Y H A X ) 134 E P S = 1 . 0 E - 4 1 3 5 F 1 9 = 5 7 6 5 . 1 3 6 F 2 0 = 1 0 9 4 6 . 1 3 7 S T A R = P 1 9 / F 2 0 + E P S / F 2 0 1 3 8 YP1 = 0.0 1 3 9 Y P 2 = 1 . 0 - S T A B 1 9 0 Y P 3 = S T A B 141 Y P 4 = 1 . 0 142 N F I B = 1 9 143 I F (P ( I ) .GT. 1.0) GO TO 6 2 0 144 H F I B = 2 9 1 4 5 F 2 8 = 5 1 4 2 2 9 . 1 4 6 P 2 9 = 8 3 2 0 4 0 . 1 4 7 S T A R = F 2 8 / P 2 9 1 4 8 I P 2 = 1 . 0 - S T A B 1 4 9 Y P 3 = 3 T A B 150 6 2 0 WRITE ( 6 , 9 9 30) N F I B , P ( I ) , K S I Z E (KR) , XPST 151 WRITE ( 6 , 9 9 2 0 ) 152 T ? 1 0 = Y P 1 * Y N A X 153 Y P 2 D = Y P 2 * Y " A X 154 Y P 3 D = Y P 3 * Y H A X 1 5 5 X P 4 D = Y P 4 * Y N A X 1 5 6 C A L L F I T R A J ( Y P 1 D , I , K R , H 1 ) 1 5 7 C A L L F I T R A J ( Y P 2 D , I , KR,H2) 1 5 8 C A L L F I T R A J ( Y P 3 D , I , K R , H 3 ) 1 5 9 C A L L F I T R A J ( Y P 4 D , I , KR.H4) 160 D12 = ABS (H1-H2) 161 D23 = ABS (H2-H3) 1 6 2 D34=ABS (H3-H4) 163 I F (ABS ( D 1 2 - 1 . 0 J . L E . 1 . 0 E - 4 ) GO TO 6 0 164 I F (ABS ( D 2 3 - 1 . 0 ) . L E . 1.0E-4) GO TO 7 0 1 6 5 I F (ABS ( D 3 4 - 1 . 0 ) . L E . 1 . 0 E - 4 ) GO TO 7 0 1 6 6 60 I P 4 = X P 3 1 6 7 H4 = H3 1 6 8 Y I = Y P 1 » Y P 4 - Y P 2 1 6 9 Y TES T= Y l * Y M A X 1 7 0 C A L L F I T R A J ( Y T E S T , I , K R . R T E S T ) 171 I F ( Y I . G T . Y P 2 ) I P 3 = Y I 1 7 2 I F ( Y l . L T . YP2) GO TO 160 1 7 3 GO TO 170 174 160 Y P 3 = Y P 2 273 1 7 5 H3 = H2 176 H2 = H TEST 1 7 7 IP2=YI 1 7 8 GO TO 80 1 7 9 170 R3=HTEST 1 8 0 GO TO 8 0 181 7 0 Y P 1 = Y P 2 182 H1*H2 183 Y I = Y P 1 » Y P » - Y P 3 184 YTES r = Y I * Y H A I 1 8 5 C A L L F I T 9 A J ( T T S S T . I , KB.HTBST) 1 8 6 I F ( Y I . L T . Y P 3 ) Y P 2 = YI 1 8 7 I F ( Y I . G I . Y P 3 ) GO TO 140 1 8 8 GO TO ISO 1 8 9 140 YP2=YP3 1 9 0 H2=B1 191 H3=HTEST 1 9 2 YP3=YI 193 GO TO 8 0 194 150 H2=HTESr 1 9 5 8 0 C O N T I N U E 1 9 6 DO 90 J = 1 , B F I 8 1 9 7 D12 = A8S (H1-H2) 1 9 8 D23 = ADS (H2-H3) 1 9 9 D34 = ABS (H3-H4) 2 0 0 I F (ABS ( D 1 2 - 1 . 0 ) . L E . 1.0E-4) GO TD 1 0 0 2 0 1 I F (ABS ( D 2 3 - 1 . 0 J • L E . 1 . 0 E - 4 ) GO TO 1 1 0 2 0 2 I F (ABS ( D 3 4 — 1 . 0 ) • L E . 1.0E-4) GD TD 1 1 0 2 0 3 100 Y P 4 = Y P 3 2 0 4 H4 = H3 2 0 5 Y I = Y P 1 * Y P 4 - Y P 2 2 0 6 Y T E S T = Y I * Y « A X 2 0 7 C A L L F I I B A J ( Y T E S T , I , K B , B T E S T ) 2 0 8 I F ( Y I . 3 T . Y P 2 ) Y P 3 = YI 2 0 9 I P ( Y I . L T . Y P 2) GO TO 180 2 1 0 GO TO 190 2 1 1 180 Y P 3 = Y P 2 2 1 2 H3=B2 2 1 3 H2=HTEST 2 1 4 Y P 2 = Y I 2 1 5 GO TO 1 3 0 2 1 6 190 H3=HTEST 2 1 7 GO TO 1 3 0 2 1 8 110 Y P 1 = Y P 2 2 1 9 H1=H2 2 2 0 Y I = Y P U Y P 4 - Y P 3 2 2 1 Y T 2 S T = Y I * Y H A X 2 2 2 C A L L P I T B A J ( Y T E S T . I , K a.HTEST) 2 2 3 I F ( Y I . L T . Y P 3 ) I P 2 = Y I 2 2 4 I P ( Y I . S T . Y P 3 ) GO TO 2 0 0 2 2 5 GO TO 2 1 0 2 2 6 200 Y P 2 = Y P 3 2 2 7 H2=H3 2 2 8 H3=HrESI 2 2 9 Y P 3 = Y I 2 3 0 GO TO 130 2 3 1 210 H2=HrEST 2 3 2 130 COHTINOB 274 233 9 0 C O N T I N U E 23ft D 1 2 = A B S ( H 1 - H 2 ) 2 3 5 D23=ABS (H2-H3) 2 3 6 D34=ABS (H3-H4) 2 3 7 E P S I L ( I ) =0.0 2 3 8 I F (ABS ( D 1 2 - 1 . 0 ) . L T . 1. OE-5) E P S I L ( I ) = T P 1 H P 2 2 3 9 I F (ABS (D2 3-1.0) . LT. 1. OE-5) E P S I L ( I ) = Y P 2 * Y P 3 2 4 0 I F (ABS ( D 3 4 - 1 .0) . LT. 1 . 0 E - 5 ) E P S I L ( I ) = Y P 3 » Y P 4 2 4 1 E P S I L ( I ) = E P S I L ( I ) *YMAX/2.0 2 4 2 2 3 0 C O N T I N U E 2 4 3 WHITE ( 6 , 3 3 0 0 ) 2 4 4 WHITE ( 6 , 1 1 0 0 ) R I N F . B B 2 4 5 WRITE ( 5 , 2 5 0 0 ) X P S T 2 4 6 WRITE ( 6 , 1 8 0 0 ) NA, HR 2 4 7 WRITE ( 6 , 3 7 0 0 ) SHUM,JSUB.NOHBA 2 4 8 WRITE ( 6 , 3 9 0 0 ) 2 4 9 I F (JBOUND. EQ. 2) GO TO S 3 0 2 5 0 WRITE ( 6 , 5 2 0 0 ) 2 5 1 GO TO 5 4 0 2 5 2 5 3 0 WRITE ( 6 , 5 3 0 0 ) 2 5 3 540 C O N T I N U E 2 5 4 I F ( J S U B . E Q . 2 ) GO TO 4 10 2 5 5 I F ( J S U B . EQ. 3) GO TO 4 2 0 2 5 6 WHITE ( 6 , 9 2 0 0 ) 2 5 7 GO TO 4 3 0 2 5 8 4 1 0 WRITE ( 6 , 9 3 0 0 ) 2 5 9 GO TO 4 30 2 60 4 2 0 WRITE ( 6 , 9 4 0 0 ) 2 6 1 430 C O N T I N U E 2 6 2 I F ( J I N T . EQ.2) GO TO 5 8 0 2 6 3 WRITE ( 6 , 6 2 0 0 ) 2 6 4 GO TO 5 9 0 2 6 5 580 WRITE (6 , 6 3 0 0 ) 2 6 6 5 9 0 C O N T I N U E 2 6 7 DO 2 4 0 I X = H S T K 1 . N S T K 2 2 6 8 E F F ( I X ) = E P S I L (IX) / (1 . » K S I Z S (KB) ) 2 6 9 WRITE ( 6 , 2 4 0 0 ) P ( I X ) , E P S I L ( I I ) , K S I Z E (KR) , E F F ( I X ) 2 70 2 4 0 C O N T I N U E 2 7 1 WRITE ( 1 , 3 0 0 0 ) J S U B , J B O U N D , J I N T 2 72 W R I T E ( 1 , 2 9 1 0 ) R I N F , R E , S T A R T , G N U B 2 7 3 WRITS ( 1 , 3 0 0 0 ) N R , N A , I T E R , N C I C L , N U H B A 2 74 DO 3 7 0 I X = M S T K 1 , H S T K 2 2 75 WRITE ( 1 , 9 8 0 0 ) P ( I X ) , K S I Z E (KR) , E P S I L ( I X ) , E F F ( I X ) 2 7 6 370 C O N T I N U E 2 7 7 250 C O N T I N U E 2 7 8 251 C O N T I N U E 2 7 9 1000 FORK AT ( 2 P 1 2 . 6 . E 1 2 . 7 ) 2 8 0 1 100 F O R M A T ( 1 0 X , ' R - I N F I N I T Y = ' , F 1 0 . 3 , 5 X , 1R E Y N O L D 1 ' S SO. = • , F 1 0 . 3 2 8 1 1110 FORMAT ( 1 0 X , ' R - I N F I N I T Y = I N F I S I T E ' , 7 X , ' R E Y N O L D 1 • S NO.=•,P10, 2 8 2 1200 F O R M A T ( 8 X , P 1 0 . 6 , 7 X , F 1 0 . 6 , 7 X , F 8 . 3 , 1 1 X , I 4 ) 2 8 3 1 3 0 0 F O R H A r ( ' 1 ' , 1 0 X , ' X - V A L U E ' , 1 0 X , ' Y - V A L U E ' , 1 0 X , ' T I M E ' , 1 0 X , 284 1 ' I T E R A T I O N S ' / / ) 2 8 5 1 4 0 0 FORMAT ( 2 F 1 0 . 6 ) 2 8 6 1500 F O R M A T ( ' 1«,5X, •••* IMPACT AT TIME= • F 8 . 3 , 1 X . ' • * * • / / 2 87 1 10X,'BETWEEN I P * ' , F 1 0. 6 , 2 1 , • Y ? = ' , F 1 0 . 6// 2 8 8 2 1 0 X , ' A N D ' , 5 X , ' X P = ' , P 1 0 . 6 , 2 X , ' Y ? = ' , P 1 0 . 6 / / ) 2 8 9 1600 F O R M A T ( 1 0 X , ' V I S C O S I T Y OF F L U I D * ' , E 1 2 . 6 / / , 1 0 1 , 2 9 0 1 'RADIUS OF P A R T I C L E * ' , F 1 0 . 6 , / / 1 0 X ' D E N S I T Y OF P AR T I C L E = ' 275 2 9 1 1 7 0 0 FORMAT ( P 1 2 . 6 ) 2 9 2 1 8 0 0 F O R M A T { 1 0 X , ' S I Z E OF G R I D = ' , 11,12, • X * , I 2 / ) 2 9 3 1 9 0 0 F O R M A T ( / / / 1 7 X • I N I T I A L P A R T I C L E P O S I T I O N ' / / / , 1 0 1 , ' X - I N I T I A L * * 2 9 4 1 P 1 0 . 6 , / / 1 0 X ' Y - I N I T I A L = « , P 1 0 . 6 / / ) 2 9 5 2 0 0 0 FORMAT ( 1 0 X , • S T D K E S " NO. = • , F 1 0 . 3, 5 X , • DP/DP= •, F 1 0 . 4//) 2 9 6 2 1 0 0 FORMAT ( 2 P 8 . 3 ) 2 9 7 2 2 0 0 F O R M A T ( / / / / 1 8 I * * * * * * * E P S I L I E N C Y WITH I N T E R C E P T I O N ******* V//| 2 9 8 2 3 0 0 FORMAT (511) 2 9 9 2 1 0 0 FORMAT ( 5 X . ' S T O K E S " SO. = • , F 1 0 . 3, 5 X , » E P S I L I E N C Y = •, 3 14 . 7 , 5 1 3 0 0 1 ' D P / D P = ' , P 1 0 . 6 , 5 X , • 1=',G14.7/) 3 0 1 2 5 0 0 PORMAT ( 1 0 X , ' X P (1) =',G 1 5 . 7 / ) 3 0 2 2 9 0 0 FORMAT ( 8 F 1 0 . 6 ) 3 0 3 2 9 1 0 F O R M A T ( 1 7 G 1 1 . 7 ) 304 3 0 0 0 FORMAT ( 5 1 5 ) 3 0 5 3 2 0 0 F O R M A T ( 1 0 X , ' A F T E R ' . 1 3 , ' I T E R A T I O N S . TIM E= ' , F 6 . 2 , 3 0 6 1 2 X . « X - V A L U E = ' , F 1 0 . 6 , 2 X , • Y-VA LUE= ' . F 1 0 . 6 / ) 3 0 7 3 3 0 0 F O R M A T ( • 1 • , 2 0 X , ' I N I T I A L V A R I A B L E S ' / ) 3 0 8 3 4 0 0 F O R M A T ( / / ' * * * * * * * * * * * * * * * F I B O N A C C I R E S 3 L T S * * * * * * * * * * * * * * * * * * » / 3 0 9 1 / 1 0 X , • YP1 = ' , F 1 0 . 6.2X, 'H1= ', F 1 0 . 6 / 1 0 X , 'YP2 = «,F10. 6, • H 2 = « , 3 1 0 2 F 1 0 . 6 / 1 0 X , *YP3= ' . F 1 0 . 6 . • H 3= •, F1 0. 6/ 1 0 X , «YP4 = •, F 10.6 , 2X , 3 1 1 3 'H4 = ' , P 1 0 . 6 / / ) 3 1 2 3 5 0 0 P O R M A T ( / / 1 0 X , • E P S I L I E N C Y = • , P 1 0 . 6) 3 1 3 3 6 0 0 F O R M A T ( 1 0 X , ' N U M B E R OF S T E P S PER G R I C C E L L = • , F l 3 . 6 / / ) 3 1 4 C T H I S I S TO T E S T THE LENGTH DP THE L I N E ENTERED ******************* 3 1 5 3 7 0 0 F O R M A T ( 1 0 X , ' G S U 3 = ' , 3 1 4 . 7 , 5 X , ' J S U B = ' , 1 2 , 5 X , ' N U M B E R OP I T E R A T I O H S IH 3 1 6 1 G R I D = ' . G 1 4 . 7 / / J 3 1 7 3 8 0 0 FOR MAT (' 1' , 1 OX) 3 1 8 3 9 0 0 FORMAT ( 2 2 X , ' K L Y A C H K O " S FORMULA FDR DRAG C O E P S I L I E N T USED AT A L L V 3 1 9 1 A L U E S OF RE'//) 3 2 0 4 0 0 0 FORMAT (/////10X, • AT YMAX= • ,G 1 4. 7, 2 X, • AND RIN F=', 3 14 . 7 , 321 1 2 X , ' X P S T = ' , G 1 4 . 7 ) 3 2 2 5 2 0 0 F O R M A T ( 3 I X , * * * * * * * * KUWABARA''S ZERO V O R T I C I T Y MODEL * * * * * * * « / / ) 3 2 3 5 2 0 1 FORMAT (• 1','WRDHG S P E C I F I C A T I O N DF BOUNDARY C O N D I T I O N S ' / / ) 324 5 3 0 0 F O R M A T ( 3 0 X , • • • • * * * * * * H A P P E L ' ' S ZERO SHEAR STaSSS MODEL *****•/) 3 2 5 6 2 0 0 FORMAT ( 4 0 X , ' E P S I L I E N C Y WITH I N T E R C E P T I O N '//) 3 2 6 6 2 0 1 FORMAT (• 1«, 'WRONG S P E C I F I C A T I O N FDR I N T E R C E P T I O N E F F E C T ' / / ) 3 2 7 6 3 0 0 FORMAT ( 3 8 X , ' N O I N T E R C E P T I O N : POINT P A R T I C L E S ' / / ) 3 2 8 9 0 0 0 FORMAT (• 1',20X,•****»*»* NSW S T D K E S ' ' AND DP/DP * * * * * » * • • / 3 2 9 1 1 0 X , ' S T O K E S (',12, ') =' , F 1 0 . 6 , 5X, ' D P / D F f , 1 2 , •) = ' . F 1 0 . 5 / / ) 3 3 0 9 1 1 0 FORMAT ( 3 1 2 ) 331 9 2 0 0 FORMAT ( 3 5 X , ****** NUMERICAL F L O W F I E L D USED * * » * * * * • / / ) 3 3 2 9 2 0 1 FORMAT (• 1* ,'WRDNG S P E C I F I C A T I O N DF F L O W F I E L D ' / / ) 3 3 3 9 3 0 0 FORMAT (3 I X , ' ******** D A V I E S * • B E S S E L EQUATIONS USED ********•//) 334 9 4 0 0 F O R M A T ( 3 3 X , * * * * * * * * * * * P O T E N T I A L F L O W F I E L D USED ****•**•//) 3 3 5 9 7 3 0 FORMAT (//10X,'STOKES (',12, • ) = ' , P 7 . 2) 3 3 6 9 7 4 0 FORMAT (// 10 X , ' K S I Z S ( * , I 2 , *) = * , F 7 . 4 ) 3 3 7 9 8 0 0 F O S S A T ( 2 F 1 2 . 5 , 2 G 1 4 . 7 ) 3 3 8 9 9 3 0 F D R K A T ( / / 1 0 X , * * * * * * * * NUMBER DF F I B O N A C C I C Y C L E S : * , 1 3 ) 339- 9 9 1 0 FORMAT ( I X , ' E N T E R VALUES OF J S U B , J B C U N D , AND J I N T ' / / , 11, 3 4 0 1 ' 1 = N U M E R I C A L / K U W A . / i : i T . «*• 2=DA V I E S / H A P P E L / S D I N T . ** 3=PDT') 3 4 1 99 20 FORMAT (1X,<ITE3',6X,'riME',11X , ' r ? (1) ', 13X, 'XP', 1 4 1 , MP', 131, 3 4 2 1 • V P X ' , 1 5 X , ' V P Y ' , 1 1 X , • V M A G ' / / 4 X , ' R P ' , 1 2 X , ' M I N R P ' , 9 X , 3 4 3 2 ' R A D I A N S • , 9 1 , ' D E G R E E S • , 9 X , 'CDS (ASG) ' , 9 X , 'MAX R E ' , 9 X , 3 4 4 3 • MIN CD'//) 3 4 5 9 9 3 0 FORMAT (' 1',//5X, 'NO. F I B O N A C C I C Y C L E S = • , 1 1 , 1 2 , 5 X , •3TOXES S O . * 1 3 4 6 1, P 1 0 . 5 . 5 X , 'DP/DF= ' , P 8 . 5 , 5 X , •XPS T=',G 14 . 7 / / / ) 3 4 7 STOP 3 4 8 END 276 309 SUBROUTINE P I T ? A J (TP N, I , K H , B) 350 IBP L I C I T REAL*8 (A-H, O-Z) ,INTEGER *4 (I-N) 351 REAL YPN . H 352 REAL RG (93) ,THETAG ( S I ) 353 REAL ABS.ALOG 350 8EAL*8 XP (2000) .TP (2000) 355 REAL*8 XTES (5) ,ITES (5) 356 REAL VGX (5 1,93) , VGT (51,93) 357 REAL*8 VPX (2000) , VPY (2000) , S U B T ( 2 0 0 0 ) ,VTOT(2000) 358 REAL P (20) ,KSIZE (20) 359 REAL*8 R E? (2000) , DIS XP (2000) 360 REAL* 8 DELXP (2000) , DELXP (2000) , RAT XT (2000) 361 REAL GSPAC (2000) 3 62 RSAL*8 VBA3X (2000) , V BAR Y (2000) ,TSTEP (2000) ,VFX (2000) , VFY (2000) 363 REAL XPST,BE,RINP,A, B.GNUfl 360 INTEGER N8,HA, JSUB, JINT 3 65 COBBOH XPST,RE,RINP,A,B,NR,NA 366 COBHON RG,THETAG,VGX,VGI,P,K SIZS 367 COBBOH GKUB,JSUB,JINT 368 XP(1)=XPST 369 ZIHF = ALOG (RIHF) 370 VPX (1) =1.0 371 VPY (1) =0.0 372 SPANS=5.0 373 IF (JSUB. EQ.2) SPANS = 0.0 3 70 IF (RIHP.LE. 10. 0) SPANS = ABS (XPST) / I . 5 3 75 YP (1) =YPN 3 76 SOHT (1)=0.0 377 KITER=0 378 KDU»=0 3 79 RPBIN=1.0E6 380 REBAX=0.0 381 COBIN=1.0E6 3 82 DO 60 K=1,2000 383 XPK=0. 0 3 84 IPK=0.0 385 VXK=0.0 3 86 VYK=0.0 387 VTK=0.0 388 NOLD=K 389 HEW=K*1 390 KITE3=KITEB*1 391 CALL LOCATE(XP,IP,K ,BP,TBETAP,NT,NZ) 392 IP (NZ. EQ.N8) G3 TO 10 393 I B I G = N Z » 1 394 ISBL=NZ 395 GO TO 20 396 10 IBIG=N3 397 ISBL=NR-1 398 20 CONTINUE 399 DISX? (K) =XP (K) -RINP 400 GSPAC (K)=RG(IBIG)-RG (ISHL) 401 IF (JSUB. EQ. 1) GO TO 21 402 IF (JSUB. EQ. 2) 33 TO 22 003 CALL POT EN (X?,YP,K,VELY,7ELY) 400 GO TO 23 005 21 CALL CLOSE(RP,THETAP,NT,NZ,VELI,VELY) 0 06 GO TO 23 277 107 22 CALL EQOAT {X P, I P , K, Et P, V ELX, 1ELY) 108 23 CONTINUE 4 09 VTOT (K)=DSQRT(?PX (K) *VPX (K) » VPY (KJ *V PI (A") ) . 010 TSTEP (K) =GSPAC (K)/VTOT (KJ/GNOS ttll HB1=N8-1 112 IF (SZ.GE.NR1) TSTEP (K) =GSPAC (K)/VTOT (K) /GSUH/3. 0 413 IF (T3TEP (K) . GE. 10.0) TSTEP(K) =10 .0 414 VFX(K)=VELX 115 VPY(K)=VELY 416 I F ( K . G T . I ) GO TO 100 417 JKL=1 418 XTES ( l ) - X P ( l ) + GSPAC (K)/GNOH/3.0 419 TTES (1) =YP (1) 420 CALL LOCATE(XTES,YTES,JKL,RP,THETAP,NT,NZ) 421 IF (J3UB. EQ. 1) GO TO 200 422 IF (JSUB.EQ.2) GO TO 210 423 CALL POTEN ( X T E S . I T E S , JKL,VXTES, Y I I E S ) 424 GO TO 220 425 200 CALL CLOS E(RP,THETAP,NT,HZ,V X TES,VITES) 426 GO TO 220 427 210 CALL E3U AT (XTES, YTES , JKL , RP, VXTSS, 7YTES) 428 220 CONTINUE 429 VBARX (1) = (YFX (1) +VXTES)/2.0 430 VBARY (1) = (VPI (1) •VYTES)/2.0 431 GO TO 110 432 100 YBARX (K) =V FX ( K) • ( VF X (K) - VFX (K-1) ) *TSTEP (K) / 2 . /TSTEP (K- 1) 433 VBARY (K)=VFY (K) • (VFY (K)-VFY (K-1) ) »TSTEP(K) / 2 . /TSTEP (K- 1) • 434 110 CONTINUE 435 SUST (K* 1) =S0HT(K) •TSTEP(K) 436 RELX = VBARX (K)-7PX (K) 437 RELT = VBARY (K)-VPY (K) 438 SURD= (R ELX*RSLX •RELY*B ELY) 439 UREL=DSQRT (SUBD) 440 RE? (K) =RE*ORSL*KSIZ8 (KR) 441 IF (REP (K) .GT.REHAX) HEHAX = REP(K) 442 BBTA=1./P(I) 443 IF (REP (K) .GT.0.0) BETA= (1.0* (RE? (K) **0.6666666) / 6 . 6) /P (I) 444 IF (HEP (K) . EQ.O.O) GO TO 221 445 CDRAJ = BETA*24.0*P (I )/aBP(K) 496 IF(CDSAG.LT.CDHIH) CDMIN=CDRAG 447 221 CONTINOE 498 C5=1.0/BET\ 449 ARGE = -BETA*TSTBP (K) 450 C4 = DEXP (ARGS) 451 XP(K*1)=XP(K) *VBARX (K) * (C4*C5-C5*TSTEP (X) ) »VPX (K) * 452 1 (C5-C4»C5) 453 Y P ( K » 1 ) = Y P ( K ) » V B A R Y ( K ) » (C4*C5-C5*TSTEP (X) ) *VPT (K) * 4 54 1 ( C 5 - C 4 » C 5 ) 455 I K 1 = K » 1 456 IK2=K*2 457 DISXP ( K H ) =RIHP-XP (K*1) 458 DSLXP <K» 1) =XP ( K » 1 )- I P (K) 459 DELYP (K* 1) = YP ( K » 1 ) - Y P (K ) 460 VPX (K*1) =V3ABX (X) * (1.-C4) »VPX (K) *C4 461 VPY (K* 1) =VBA3Y (K) » (1.-C4) f/pr (K) »C4 462 X?K=XP(K) 463 YPK"YP(K) 464 YXK=VPX(K) 278 4 65 VYK=VPY(K) 466 . VTK=VTOT (X) 467 VTOT (K*1) =DSQRT(VPI (K» 1) *VPX (K » l ) + VPY (K» 1) *VPY (K»1) ) 468 CALL LOCATE (XP.YP,SEW,HP,TRETAP, X T ,H Z ) 469 IP (RP. LT. RPMIN) RPHIH=BP 4 7 0 FINSIZ=KSIZE (KR) 471 IF (JINT. EQ.2) FISSIZ = 0 . 0 472 CLEA3 = RP-1.0-FINSIZ 473 IF (CLEAR.LE.0.0) GO TO 140 474 IF (XP (K) .GT. SPANS) GO TO 133 475 60 CONTINUE 4 76 140 H=0 . 0 477 GO TO 7 0 478 130 H=1 . 0 479 70 COHTIHOE 480 XCOSS=DCOS(THETAP) 481 THDE3 = THETAP*180./3.14 159 2653 589793 4 82 WRITE (6,979 2) NOLD, SUHT (NOLD) ,YP(1), IPK,YPK,VXK» 483 1 VYK, VTK.R?.RPMIN,THETAP,THDEG,XC3SS,R EM AX, COB IH 4 84 1000 FORMAT (2F12.6,E12.7) 485 1700 FORMAT(F12.6) 486 2000 FORMAT(1OX,'STOKES'• HO.=',F10.6//) 487 2100 FORMAT (2F12.6) 488 2900 FORMAT (8F10.6) 489 2910 FORMAT (17G14.7) 490 9000 FORMAT(/15X,•*****»« FITRAJ: CYCLE COM PL ETEO **»***'/ 491 1 10X, 'SUMT(',I4, •) =• ,P10.6 ,5X, 'SUNT (• ,14 ,') = ', P10.6) 492 9001 FORMAT(10X,'CURRENT LARGE TIME STSP=•,F10.6) 493 9200 FORMAT (10X, 'DISXP (',I4,') = ',F10.6,5X,'DISXP(',I4,')='F10.6/) 494 9300 FORMAT (/I OX, 'STOKES (',I2,') = ',F13.6,5X, 'DP/DP (',I2,')=',F10. 6///) 495 9400 FORMAT(/10X,'STARTING Y= •,F10.6,5X,'FINAL H=•,F10.6,5X, 496 1 • AFTER* , 1X.I4, IX, ' IT ERATIO J S •/) 497 9600 FORMAT (10X, 'NUMBER OF STEPS*', 1X,I4,5X, 'VALUE OP K = '.I4/) 498 9705 FORMAT(///IX, >******************** ******************************* 499 I***************************************'/) 500 9710 FORMAT (1 OX, "XP (' ,14, ') =' , F13. 6.5X, • XP (• ,14, •) * *, F10.6.5X 501 1 ,'YP(',I4,') = ',F10.6,5X,'YP(',I4, ') = ',F10.6/J 502 9720 FORMAT(10X,'VPX(',I4,') = ',F10.6,4X,'VPX(',I4,') = ',F10.6,4I, 503 1 • VPY (• ,14, •) * ', F10. 6,4X, 'VPY ( ', 14, •) = • , F10. 6/) 50 4 9730 FORMAT(10X.'HT=',12,4 X, ' SZ= ', 12 , 5X,« RG {', 14 , ' ) * ' , F 1 0.6 ,5X, 505 1 'RG (• ,14 ,') = ', F10. 6.5X, 'GSPAC (', 14, •)=• ,F10. 6/) 506 9740 FORMAT(10X,'VFX(',14,•)=•,F10.6,5X,•VFY(•,14,•)=',P10.6,SX, 507 1 'DISXP (',14, ') = ',F 10.6,51, 'RP=',P10.6/) 508 9750 FORMAT (10X, • RELATIVE VELOCITY WRT FLOI D= ' , E1 2. 5, 81, 509 1 'REYNOLD' ' S H3. ('.14,') = ',E12.5/) 510 9760 FORMAT (IX, • !»#»! TS T EP (• , 14, •) = •, F 1 2. 6 , 2X , • 11111 • , 5X, 511 1 •TOTAL TIME TO XP (•,14, ) = ' , F l2.6/)512 9770 FORMAT (1 OX, 'DELXP («,I4,')=',F10.6, 2X,'DBLYP(',I4,«)=', 513' 1 F10.6/) 514 9780 FORMAT(IX,'ZER3 EPSILIENCY CONDITION: VPX(•,14, •) =» , 515 1 E12.5,5X,'DELXP(',I4,') = ',E12.5,5X, 'XP(',I4,')=«,B12.5, 516 2 5X, • YP (• ,14, •) * ' , E12.5//) 517 9790 FORMAT (10X, ' STARTING VALUE: T P (1) = ' , E1 2. 5/) 518 9791 FORMAT (10X , 'VBARX (', 14 , • ) = ', F10. 6, 3X, 'VBKRY (', 14, • ) = • ,F10.6/) 519 9792 FORMAT (15,7 (4X,G12.5)/G12.5,6 (4X,3 12.5) ) 520 RETURN 521 END 522 SUBROUTINE LOCATE (IP; YP, K , BP, THETAP, ST , N Z ) 279 523 IMPLICIT REAL*9 ( A-H, 0-Z) , IHTEGEa*4 ( l - H ) 52U REAL SKGL 525 REAL'S XP(2000) ,YP (2000) 526 REAL RG (93) .THETAG (5 1) . VGX (51,93) , 7G Y (5 1,9 3) . P ( 2 0 ) , K S I Z E (20) 527 REAL XPST.RE,RIHF, A, B,GH0» 528 IHTE3ER N R , H A , J S U B , J I H T 529 C3.1.103 XPST, BE, RINF, A,B,»R,»A 530 COMMON RG,THETAG,VG X, VGY , P,KSIZE 531 COMMON 3N0H,JSUB,JINT 532 R = -XP(K) 533 RP=D33RT (YP(K) *YP (K)+a*H) 534 PI=3. 14 1592653589793 535 IF (DABS (B) . L T . 1.0E-5) GO TO 10 536 THETAP = DATAN (YP (K) /R) 537 IF(R.LT.O.O) THETAP=PI-DATAS (YP (K) /DABS (R) ) 538 GO TO 20 539 10 THETAP=PI/2.0 540 20 CONTINOE 541 NZ=IPIX (SNGL (DLOG (9PJ/A)) »1 542 IF (RP.GE.RINF) NZ = HB 543 NT=I FIX (SNGL (THETAP/B) )• 1 544 RETURN 545 END 54 6 SUBROUTINE CLOSE (BP, TH ET A P, NT, H! , 7 ELX, V ELY) 547 IMPLICIT BEAL*8 (A-H,0-Z) .INTEGERS (I-N) 548 REAL RG (93) ,THETAG (51) 549 REAL VG X (5 1 ,9 3) , VGY (5 1 ,9 3) ,TH (51 . 93 ) , W (5 1 ,93) 550 REAL P (20) ,KSIZE (20) 551 REAL XPST,RE,RINF,A,B,GHOH 552 INTE3ER NR,NA,JSUB,JINT 553 COMMON XPST,RE.RINF,A,B,NR,HA 554 COMMON RG,THETA3,V3X,VGY,P,KSIZ8 555 COMMON GNUM,JSUB,JINT 556 VELX'0.0 557 VELY=0.0 558 IF (RP. GE.RG(NR) ) GO TO 400 559 NZ1=NZ*1 560 ARGW = RP-RG(NZ) 561 XL=AR3H/(BG(NZ1)-BG (NZ)) 562 GO TO 4 10 563 400 XL=0.0 564 NZ1=NZ 565 410 CONTINUE 566 NT1=BT*1 567 ARGT=DABS(THBTAP-THETAG(ST)) 568 ALPHA=ABGT/B 569 VX1 = YSX(NT.NZ) 570 VX2=VGX (NT,NZ1) 571 VX3* V3X (NT 1, NZ) 572 VX4 = VGX (NT 1 ,HZ1) 573 VY 1 = VG I (NT, NZ) 574 VY2*VGY (NT.SZ1) 575 VY 3 = V3 Y (NT 1, NZ) 576 VY4 = V3Y (NT1,NZ1) 577 VELX= (1.0-ALP HA) * (V X 1 * ( 1 . 0-X L) »VX2 *XL) • A LPHA * (VX3 » (1 . O-XL) • 578 1 VX4»XL) 579 V E L Y = ( 1 . 0 - A L P H A ) * ( 7 T 1 « ( 1 . 0 - I L ) + Y Y 2 * X L ) * A L P H A * ( V r 3 * ( 1 . 0 - X L ) » 580 1 VY4*XL) 280 581 R E T O B H 582 BHD 5 8 3 S U B R O U T I N E E Q U A T ( I S , T S . K , R P , V E L X , V E L T ) 5 8 0 I M P L I C I T R E A L * 8 ( A - Z ) 5 8 5 I N T B 3 E R * 4 K 5 8 6 R E A L B E S S K 0 , B E S S X 1 5 8 7 R E A L * 8 XS ( 2 0 0 0 ) , Y S ( 2 0 0 0 ) 5 8 8 R E A L R3 (9 3) , T H E T A G (51) , YGX ( 5 1 , 9 3 ) , V 3 1 (5 1 , 9 3 ) , P ( 2 0 ) , K S C Z E ( 2 0 ) 5 8 9 R E A L J , K 1 , L , M , S 5 9 0 R E A L X P S T , B E , R I N P , A , B.GHOH 591 I N T E 3 S R N R , N A , J S U B , J I N T 5 9 2 COMMON X P S T , R E , R I N F , A , 8 , H B , S A 5 9 3 COMMON R G , T H E T A G , V G X . V G T . P , K S I Z E 5 9 4 COMMON G N U n , J S O B , J I N T 5 9 5 X = XS (K) 5 9 6 Y = IS (K) 5 9 7 X2 = XS (K) *XS(K) 5 9 8 X3 = X 2 * X S (K) 5 9 9 B 2 = R P * B P 6 0 0 R 4 = R 2 * B 2 601 R 6 = R 4 * R 2 6 0 2 A R G 1 = R E / 4 . 0 60 3 A R G 2 = 8 S * R P / 4 . 0 604 B 1 = B E S S K 1 ( R E / 4 . 0 ) 6 0 5 B 2 = 3 E S S K 0 ( R E / 4 . 0 ) 606 B 3 = B F . S S K 0 ( R E * R P / 4 . 0 ) 60 7 B 4 = B E S S K 1 ( R E * R P / 4 . 0 ) 6 0 8 C = D E X P ( B E » X / 4 . ) 6 0 9 A R G K = R E * R P / 4 . 6 1 0 C1 = D E X P ( B E * B P / 4 . 0 ) 611 C K 0 = B 3 * C 1 6 1 2 C K 1 = B 4 * C 1 6 1 3 D = - 1 . 0 / 2 . 0 / B 2 6 1 4 F = X / R 2 6 1 5 G = R E / 1 6 . * B 2 6 1 6 H = 1 . 0 - 5 . / R 1 6 1 7 G 1 = G * H 6 1 8 J = B E / 6 » . 6 1 9 K 1 = 1 . 0 - 6 . / B 2 6 2 0 L = K 1 » 5 . / B 4 621 M = X 2 / B 4 622 N = X 3 / B 6 * R E / 8 . 623 P P = I / H P * B » 624 Q = P P » B 3 6 2 S R B = C * Q 6 2 6 D E H = B 2 * 0 . 5 6 2 7 Y A = Y / 8 2 6 2 8 I B = R E / 1 6 . » B 2 62 9 Y C = 1 . 0 - 1 . / R 4 6 3 0 Y D = Y B»IC 631 Y E = R E / 6 4 . * ( 1 . - 2 . / R 2 H . / R 4 ) 6 3 2 Y F = X * Y / B 4 6 3 3 Y G = X 2 * Y / R 6 « R E / 8 . 6 3 4 Y H= D E X P ( R E » X / 4 . ) 6 3 5 Y K = Y / B P « B 4 6 3 6 V E L Y = ( Y A » ( B 1 - Y D * Y E ) » Y F » Y G - r H * Y K ) / DBS 6 3 7 V E L X = 1. 0* ( D » F » ( B 1 - G 1 » J » L ) *M • » - R B ) / DEN 6 3 8 B E T U R N 281 639 END 690 SUBROUTINE POTES(XP,TP,K,VELX,VELY) 641 IMPLICIT REAL*8 (A-H, 0-Z) ,INTEGEa»4 (I-H) 642 REAL*8 XP (2000) , YP (2 000) 643 X2 = XP (K) *XP(K) 644 I2 = YP (K) *YP(K) 645 VELX=1.- ( X 2 - Y 2 ) / ( X 2 * Y 2 ) / ( X 2 » Y 2 ) 646 VELY = -2.*XP(K) *YP (K) / (X2H2) / (X2+12) 647 RETURN 648 END INITIAL VARIABLES B-IHFINITY" 100.000 REYNOLD'S NO." 10.000 XP(1)- -100.0000 SIZE OF GRID- .13 X 93 NUMBER OF STEPS PER GRID CELL" 3.000000 * * * * * * * KUUABftRA'S ZERO VORTICITY MODEL ••••»»* EFFICIENCY KITH INTERCEPTION ** * * * NUMERICAL FLOWFIELD USED *»••»•« C O OJ 3 -a —~j CD O c r t •a c el-'s o 3 m -h -h o fD 3 o ~i o - J OJ 3 3 fD rsi co ro NO. FIBONACCI CYCLES" 29 STOKES NO." 1.00000 DP/DF" 0. 10000 XPSI" -99.99394 ITEH TIME YP{1) XP YP VPX VPY VII AG RP KIN RP RADIANS DEIBEES COS (ANG) MAX RE NI N CD 277 102.30 0.0 -1.1092 -0.0 0. 15476 0. 0 0.15476 1. 0913 1. 0913 0.0 0.0 1.0000 0. 2021 1 125.56 430 111.85 0.42016 5.0404 1.6841 3. 48185 -0. 57837D-01 0. 4853 1 5. 3949 1. 4456 2.8262 16 1.93 -0.95067 0. 15285 164.50 404 109.29 0.67984 5.3116 2.2342 0. 64390 -0. 50505D-01 0. 64588 5. 5694 1. 7426 2.7303 156.43 -0.91659 0. 13859 180.90 376 107.71 1.1000 5.3257 2.8355 3. 81044 -0. 27932D-01 0.81092 5. 8546 2. 1899 2.6367 15 1.07 -0.87522 0. 114 4 4 217.95 453 116.21 0.25967 5.3542 1.1404 0. 34176 -0. 49804D-01 0. 34537 5.2605 1. 2452 2.9254 157.62 -0.97673 0. 17150 147. 14 4 74 124.82 0. 16049 5.0022 0.60195 3. 24303 -0. 33153D-01 0. 24490 5. 1211 1. 1103 3.0258 173.37 -0.99330 0. 18525 136.57 282 102.58 0.99187D-01 -1.0361 0.37639 3. 15746 0. 11848 3. 19706 1. 0930 1. 0930 0.36209 20.746 0.935 16 0. 19367 130.83 465 120.33 0. 19837 5.3786 0.83047 3. 28013 -0. 39573D-01 0. 28291 5. 2274 1. 1631 2.9843 170.99 -0.98766 0. 18024 140.24 288 102.87 0. 13707 -0.95011 0.56273 3. 16836 0. 16181 0.23351 1.0997 1. 0997 0.55037 31.534 0.85233 0. 19005 133.24 471 122.86 0. 17496 5.3532 0.69089 0. 25803 -0. 33791D-01 0.26024 5. 1825 1. 1307 3.0100 172.46 -0.99135 0. 18300 138. 19 296 103.37 0. 15154 -0.84120 0.70881 3. 16585 0. 17226 0.23913 1. 0990 1. 0990 0.71645 41.050 0.75414 0. 18694 135.38 473 124.06 0. 16601 5.3338 0.63470 3. 24907 -0. 31440D-01 0.25104 5. 1563 1. 1181 3.0202 173.05 -0.99265 0. 18434 137.22 475 125.49 0.15706 5.0222 0.57647 0. 24022 -0. 28935D-01 0.24195 5. 1382 1. 105S 3.03 11 173.67 -0.99390 0. 18586 135. 14 476 125.90 0.15496 5.0277 0.56125 3. 23827 -0. 2 82 4 OD-01 0.23994 5. 1421 1. 1025 3.0342 173.84 -0.99423 0. 18626 135.86 477 126.29 0.15364 5.3 586 0.54833 3. 23772 -0. 27540D-01 3. 2393 1 5. 1714 1. 1007 3.0372 174.02 -0.99456 0. 18652 135.68 298 103.51 0.15286 -0.81555 0.73834 3. 16547 0. 17172 0. 23847 1. 0997 1. 0997 0.75204 43.089 0.73029 0. 18667 135.57 477 126.26 0.15419 5.3782 0. 54992 3. 23876 -3. 27526D-01 0. 24034 5. 1912 1. 1014 3.0373 174.03 -0.99457 0. 18641 135.75 477 126.32 0.15341 5.3558 0.54698 3. 23741 -0. 27492D-01 0. 23900 5. 16 86 1. 1003 3.0374 174.03 -0.99458 0. 18656 135.64 299 103.58 0.15310 -0.83308 0.75176 0. 16494 0. 17082 0. 2 3 746 1.0999 1. 0999 0.7687 1 44.044 0.71881 0. 18663 135.60 477 126.34 0.15333 5.0547 0. 54653 0. 23730 -0. 27477D-01 3. 23888 5. 1674 1. 1002 3.0375 174.04 -0.99459 0. 18658 135.63 477 126.31 0.15318 5.3421 0.54692 3. 23683 -0. 27547D-01 0.23843 5. 1549 1. 1000 3.0372 174.02 -0.99455 0. 18661 135.61 477 126.30 0.15325 5.3431 0.54731 3. 23692 -0. 27560D-01 0.23852 5. 1S59 1. 1001 3.0371 174.01 -0.99455 0. 18660 135.62 299 103.58 0. 15317 -0.80307 0.75191 3. 16509 0. 17094 0.23765 1.1000 1. 1000 0.76881 44.050 0.71874 0. 18661 135.61 477 126.30 0. 15324 5.0430 0.S4 723 3. 23691 -3 . 27S57D-01 0.23851 5. 1559 1. 1001 3.0371 174.02 -0.9945S 0. 18660 135.62 477 126.31 0. 15323 5.3430 0.54716 0. 23690 -0 . 27554D-01 0.23850 5. 1559 1.1001 3.0372 174.02 -0.99455 0. 18660 135.62 477 126.31 0. 15322 5.0430 0.54708 0. 23689 -0.27550D-01 0. 23849 5- 1558 1. 1001 3.0372 174.02 -0.99455 0. 18660 135.62 H77 126.31 0. 15320 5.3430 0.54700 0. 23688 -0.27547D-01 0.23848 5. 1558 1.1000 3.0372 174.02 -0.99455 0. 18660 135.61 477 126.31 0. 15319 5.3430 0.54693 3. 23687 -0.27544D-01 0. 23 845 5. 15S8 1.1000 3.0372 174.02 -0.99456 0. 18661 135.61 U77 126.31 0. 15318 5.3430 0.54685 0. 23686 -0. 27540D-01 0.238'45 5. 1558 1.1000 3.0372 174.02 -0.994S6 0. 18661 135.61 299 103.58 0. 15317 -0.80307 0.75192 0. 16510 0,17095 0. 23766 1.1000 1. 1000 0.76882 44.050 0.71873 0.18661 135.61 477 126.31 0. 15318 5.3421 0.54693 0. 23683 -0.27548D-01 0.23843 5. 1549 1.1000 3.0372 174.02 -0.99455 0. 18661 135.61 299 103.58 0. 15317 -0.80307 0.75193 3. 16511 0.17095 0. 23766 1.1000 1.1000 0.76882 44.050 0.71873 0. 18661 135.61 299 103.58 0. 15317 -0.80307 0.75193 0. 16511 0.17095 0.23766 1, 1000 1.1000 0.76883 44.050 0.71873 0. 18661 135.61 299 103.58 0. 15317 -0.80307 0.75193 0. 16511 0.17095 0.23767 1.1000 1.1000 0.76883 44.050 0.71873 0. 18661 135.61 HO. FIBONACCI CYCLES" 19 STOKES HO.» 2.00000 DP/DF" 0.10000 XPST" -99.99394 CTEB TIME » P ( D XP YP VPX VPY VHAG BP MIN BP 8ADIANS DEJBEES COS (lNG) MAX BE NIN CO 277 101.24 0.0 -1.1057 -0.0 3. 39307 0.0 0.39307 1.0870 1.0870 0.0 0.0 1.0000 0.38413 67.981 451 115.49 0.42016 5.D432 1.1680 3. 35577 -0. 52 946D-01 0. 35959 5. 2557 1.1827 2.9199 167.30 -0.97553 0.27622 93.030 1418 110.07 0.67984 5.D851 2.0831 3. 61375 -0.43062D-01 0.61526 5.5792 1.5166 2.7602 158. 15 -0.92817 0.22633 112.60 3 81 107.86 1. 1000 5.3043 2.7897 3. 81567 -0.29062D-02 0.81567 5. 8111 1. 9926 2.6110 151.32 -0.87727 0. 17944 140.84 2 81 101.57 0.25967 -0.94037 0.57519 3. 40267 0. 15100 0.43005 1. 0915 1.0915 1 D.56187 32. 193 0. 84626 0.32291 80.154 136 112.23 0.51935 5.3591 1. 6261 3. 47967 -0. 54836D-01 0. 48279 5. 3952 1.3174 2.8371 162.57 -0.95409 0.25360 100.96 300 102.18 0.35886 -0.62971 0.90307 3. 39316 0.19687 0. 43970 1.0985 1.0985 0.97801 56.036 0.55867 0.29199 88.223 445 113.93 0.45805 5.0766 1.3668 3. 40717 -0.55567D-01 0.41095 5. 33 96 1.2357 2.8851 165.30 -0.96728 0.26823 95.677 456 117.02 0. 39675 5.3588 1.0142 3. 31897 -0.492590-01 0.32276 5. 2392 1.1486 2.9493 168.98 -0.98157 0.28161 91.326 459 118.23 0.38228 5.3519 0.90709 3. 29512 -0. 46030D-01 3. 29869 5.2131 1. 1268 2.9692 170. 12 -0.98518 0. 28533 90. 190 461 119.24 0. 37333 5.3 750 0.83096 3. 28112 -0. 43036D-01 0. 2 844 0 5. 2235 1. 1130 2.9843 170.99 -0.98765 0.28778 89.455 462 119.76 0.36781 5. 3 336 0.79016 3. 27226 -0.416250-01 0. 27542 5. 1764 1.1044 2.9908 171.36 -0.98866 0.28935 88.991 305 102.38 0.36438 -0.54442 0.95602 0. 37903 0. 19287 0.42528 1.0995 1.0995 1.0694 61.272 0.48066 0.29035 66.698 461 119.39 0.36991 5.3039 0.81291 0. 27531 -0. 42747D-01 0. 2785 1 5. 1504 1. 1077 2.9856 171.06 -0.98786 0.28876 89. 167 463 120.10 0.36618 5.3816 0.77131 3. 27069 -0. 40537D-01 0. 27371 5.2212 1. 1023 2.9957 171.64 -0.98938 0. 28974 88.878 463 120.16 0.36571 5.3 683 0.76638 3. 26935 -0. 40408D-0 1 0. 27236 5.2073 1.1011 2.9963 171.68 -0.98946 0.28997 88.812 463 120.14 0.36516 5.3451 0.76496 3. 26824 -0. 404 87D-01 0.27128 5. 184 1 1.1002 2.9959 171.65 -0.98941 0.29013 88.765 306 102.43 0.36193 -0. 52834 0.96513 0. 37599 0. 19159 0.42199 1. 0999 1.0999 1.0862 62.236 0.46583 0.29019 88 .745 463 120.14 0.36518 5.3556 0.76620 3. 26885 -0.40477D-01 0.27188 5. 194 7 1.1007 2.9960 171.66 -0.98942 0.29003 88.792 463 120.17 0.36521 5.3 555 0.76413 3. 26849 -0. 10388D-01 0.27151 5. 1943 1.1004 2.9964 17 1.68 -0.98947 0.29010 88.772 307 102.47 0.36501 -0.51239 0.97346 3. 37236 0.18992 0.41799 1. 1000 1.1000 1.1025 63. 171 0.45132 0.29017 88.752 463 120.14 0.36509 5.3412 0.76492 0. 26809 -0.43509D-01 0.27113 5. 1802 1. 1001 2.9958 171.65 -0.98940 0.29015 88.759 463 120.14 0.36508 5.3412 0.76482 3. 26807 -0.40505D-01 0.27112 5. 1802 1. 1001 2.9959 171.65 -0.98940 0.29015 88.758 463 120.15 0.36502 5.0412 0.76429 3. 26798 -0.404 82D-01 0.27102 5. 1801 1.1000 2.9960 171.66 -0.98941 0.29017 88.753 HO. FIBONACCI CYCLES' 19 STOKES MO.• 3. 00000 DP/DP- 0. 10000 XPST- - 9 9 . 9 9 3 9 4 ITER TIME YP(D XP RP KIN RP RADIANS DEGREES 277 100.81 0.0 -1.1050 1.0871 1. 087 1 0.0 0.0 289 101.32 0.42016 -0.79006 1. 0991 1.0991 0.79203 45.380 429 110.96 0.67984 5.0314 5. 4328 1.3529 2.8007 160.47 392 108.14 1.1000 5.0667 5.8026 1.8492 2.6666 152.79 D13 109.38 0.84032 5.0467 5.S961 1.5498 2.7352 156.72 442 112.91 0.58065 5.0195 5. 1121 1.2226 2. 8635 164.06 453 11S.40 0.51935 5.0778 5. 2852 1. 1364 2.9250 167.59 300 101.70 0.48146 -0.56044 1. 0981. 1.0984 1.0532 60. 346 una 114.19 0.54276 5.0249 5. 2658 1. 1700 2.8968 165.98 U55 116.10 0.50487 5.0307 5.2215 1. 1151 2. 9389 168. 38 157 116.74 0.49594 5.0452 5. 2233 1. 1017 2.95 12 169.09 304 101.83 0.49039 -0.48273 1. 099 1 1. 0991 1. 1331 64.925 456 116.51 0.49932 5.0460 5. 2285 1.1068 2.9469 168.84 307 101.93 0.49377 -0.42924 1.0997 1.0997 1.1865 67.980 H57 116.71 0.49714 5.0637 5. 2426 1.1035 2.9508 169.07 457 116.72 0.49497 5.0144 5.1926 1. 1002 2.9505 169.05 309 102.00 0.49 4 74 -0.39452 1.1000 1.1000 1.2204 69.923 457 116.73 0.49572 5.0358 5.2141 1. 1014 2.95 09 169.07 457 116.72 0.49549 5.0287 5. 2070 1. 1010 2.9507 169.06 457 116.73 0.49527 5.0266 5. 2047 1.1007 2.9509 169.07 457 116.75 0.49504 5.0266 5. 2044 1.1003 2.95 12 169.09 457 116.73 0.49482 S.0143 5.1924 1.1000 2.9507 169.06 457 116.72 0.49489 5.0144 5. 1925 1. 1001 2.9506 169.06 310 102.03 0.49481 -0.37760 1.1000 1.1000 1.2367 70.859 IP ?PI VPY VBA3 COS<ANG) HAX RE -IN CD •0.0 0. 52363 0. 0 0.52363 1.0000 0.50716 52.338 0.77692 0. 53 872 0. 16061 0.56216 O.7O240 0.37491 59.563 1.8229 0. 53 796 -0 . 40343D-01 0. 53947 -0.94247 0. 29835 85.428 2.6524 0. 77814 0. 10063D-01 0.77821 -0.88932 0.23103 110.40 2.2150 0. 65328 -0 . 19478D-01 0.65357 -0.91856 0. 25357 95.561 1.4688 0. 43721 -0. 50734D-01 0.44014 -0.95157 0. 3229 1 80.154 1.1482 0. 35533 -0 . 51501D-01 0.35904 -0.97663 0. 33932 76.464 0.94879 0. 52668 0. 17889 0.55624 0.49477 0.35203 73.842 1.2873 0. 38910 -0. 52282D-01 0. 39259 -0.97020 0.33267 77.917 1.0640 0. 33384 -0 . 50658D-01 0. 33766 -0.979S2 0. 34390 75.498 1.0013 0. 31927 -0. 49561D-01 0.32309 -0.98193 0. 34684 74.383 0.98960 0. 51759 0. 17910 0.54770 0.4 2381 0. 34373 74.505 1.0245 0. 32480 -0. 49969D-01 0. 32662 -0.98110 0. 34572 75.119 1.0135 0. 50 94 7 0. 17764 0. 53955 0.37494 0. 34758 74 . 7 39 1.0068 0. 32103 -0. 495850-01 0. 32484 -0.98186 0.34544 74.372 0.99928 0. 31802 -0. 49652D-01 0.32187 -0.98180 0.347 17 7 4.3 22 1.0272 0. 50 321 0. 17595 0. 53309 0. 34 328 0. 34725 74.306 1.0012 0. 31901 -Oz 49598D-01 0. 32284 -0.98187 0.34592 74.873 1.0007 0. 31872 -0. 49619D-01 0.32256 -0.98184 0. 34599 74.853 0.99948 0. 31837 -0 . 49605D-01 0.32222 -0.98187 0. 34707 74.842 0.99791 0. 31800 -0. 49576D-01 0.32184 -0.98193 0. 347 14 74.827 0.99825 0. 31777 -0 . 49634D-01 0.32163 -0.98183 0.34722 74.812 0.99873 0. 31789 -0 . 496420-01 0.32174 -0.98182 0. 34720 74.816 1.0333 0. 49976 0. 174 88 0.52 94 8 0.32790 0.34722 74.811 NO. FIBONACCI CYCLES" 19 STOKES NO.» 5.00000 DP/DP" 0. 10000 XPST" -99.99394 F.TEB TIME YP(U XP KP VPX VPY VMAS BP MIN RP RADIANS DECREES COS (ANS) MAX RE WIN CO 277 100.36 0.0 -1.1040 -0.0 0. 66039 0.0 0. 66039 1.0 861 1.0861 0.0 0.0 1.0000 0.64410 4 1.893 283 100.66 0.42016 -0.90892 0. 62426 0. 66648 0. 9630OD-O1 0. 67340 1. 0895 1.0895 0.61300 35.122 0.81793 0.55412 48. 182 8 112.36 0.67984 5.0093 1. 4287 0. 42496 -0. 35730D-01 0.42646 5.2884 1. 1455 2.8694 164.41 -0.95319 0.39955 65.498 403 108.33 1.1000 5.3382 2.3579 0. 70 763 0.21O44D-01 0.70794 5. 6488 1.6609 2.7105 155.30 -0.90849 0.30273 85.236 428 110.07 0.84032 5.0862 1.873B 0. 55776 -0. 13922D-01 0.55793 5. 5033 1.3518 2.7946 160. 12 -0.94040 0.35570 73. 118 293 101.01 0.58065 -0.65061 0.89498 3. 67074 0. 13171 0.68355 1.0987 1.0987 0. 95779 54.877 0.57533 0.47388 55.776 1 4 «0 111.32 0.74114 5.0 786 1.6185 3. 4 8051 -0.280620-01 0.48133 5.H36 1.2263 2.8390 162.66 -0.95457 0.38071 68.559 307 101.43 0.64195 -0.35338 1.0424 3. 65253 0. 14164 3.66772 1.0988 1.0988 1.2602 72.203 0.30564 0.44892 58.686 415 111.94 0.70324 5.3480 1. 5042 3. 44686 -0.33O23D-01 0. 44808 5. 3508 1. 1767 2.8579 163.75 -0.95004 0.39193 66.702 450 112.71 0.66535 5.0047 1.3 774 3. 41062 -0. 37459D-01 0.41232 5. 2702 1.1259 2.8787 164.94 -0.96S64 0.4 1164 63.681 452 113.12 0.65643 5.3715 1.3378 3. 40093 -0. 38886D-01 0. 40281 5. 3287 1. 1137 2.8895 165.S6 -0.96839 0.42705 6 1.512 452 113.13 0. 65087 5.3130 1.3222 3. 39577 -0. 39123D-01 0. 39770 5. 2640 1. 1060 2.8894 155.55 -0.96836 0.43667 60.234 453 113.26 0.64751 5.0238 1. 3080 3. 39219 -0. 39530D-O1 0. 39418 5. 2709 1. 1014 2.8925 165.73 -0.95913 0.44323 59. 394 310 101.52 0.64530 -0.29618 1.0595 0. 64589 0. 14105 0.66 111 1.0993 1.0993 1.3145 75.317 0. 25348 0.44697 58.927 453 113.26 0. 64866 5.0 367 1.3112 3. 39327 -0. 39484D-01 0.39525 5. 2842 1.1030 2.8925 165.73 -0.96914 0.44086 59.694 312 101.57 0.64645 -0. 26023 1.0690 3. 64121 0.14035 0.65639 1.1000 1. 1000 1.3483 77.253 0. 22064 0.44539 59. 123 453 113.25 0.64760 5.0238 1.30 84 3. 39229 -0. 39520D-01 3. 39427 5.2710 1.1015 2.8924 165.72 -0. 95912 0.44304 59.418 453 113.23 0.64655 5.0006 1.3066 3. 39140 -0. 39497D-0 1 3. 39338 5. 2480 1.1001 2.8917 165.68 -0.96893 0.44521 59. 146 453 113.26 0.64742 5.0238 1.3077 3. 39210 -0.39539D-01 3. 39408 5. 2708 1. 1013 2.8926 165.73 -0.95915 0.44342 59. 371 453 113.26 0.64733 5.3238 1.30 73 3. 39200 -0. 39548D-O1 3. 39399 5.2707 1.1012 2.8926 165.73 -0.95917 0.44361 59.347 453 113.25 0.64723 5.3166 1.3077 0. 39197 -0.39518D-01 0. 39395 5.2639 1.1010 2.8922 165.71 -0.96907 0.44380 59.322 453 113.25 0.64714 5.3166 1.3073 3. 39187 -0. 39527D-01 3.39386 5. 2638 1.1009 2.8923 165.72 -0.95908 0.44399 59.298 453 113.23 0.64705 5.3060 1.3080 3. 39187 -0.39476D-01 3. 39386 5.2537 1.1008 2.8917 165.68 -0. 95893 0.44418 59.274 453 113.23 0.64696 5.3051 1.3078 3. 39178 -0. 39481D-01 0. 39377 5.2527 1.1007 2.8917 165.68 -0.95893 0.44437 59.251 NO. FIBONACCI CYCLES' 19 STOKES NO.« 7.50000 DP/DP» 0.10000 XPST" -99.99394 ITEB TINE YP(1) XP RP UIN RP RADIANS DECREES 277 100. 05 0.0 -1.1034 1.085U 1. 0854 0.0 0.0 281 100. 29 0.42016 -0.95219 1.0892 1. 0892 0.53987 30.932 295 100. 76 0.67984 -0.57758 1.0934 1. 0934 1.0334 59.207 U 1 3 108. 22 1. 1000 5.0204 5. 5197 1.5236 2.75 37 157.77 tuo 109. 83 0.84032 5.3145 5.3566 1.2135 2.8330 162.32 U29 109. 07 0.93951 5.3309 5.D337 1. 3345 2.8002 160.44 l)ll 9 110. 65 0.77902 5.3884 5. 3880 1. 1366 2.86 11 163.93 306 101. 08 0.74114 -0.32818 1. 0986 1.0986 1.2843 73.587 HI 5 110. 28 0.80244 5.3458 5. 1623 1. 1662 2.8493 163.25 450 110. 73 0.76455 5.3356 5.1285 1. 1182 2.8642 164. 10 D51 110. 85 0.75561 5.0320 5. 3 192 1. 1067 2.8678 164.31 312 101. 2D 0.~75008 -0.21144 1.1000 1.1000 1. 3938 79.859 1)51 110. 84 0.75901 5.3536 5. 3D20 1.1110 2.8675 164.30 U52 110. 97 0.75348 5.0732 5. 3572 1.1039 2.8710 164 .50 1)52 110.92 0.75221 5.3457 5. 3 301 1. 1023 2.8700 164.44 D52 110. 92 0.75134 5.3403 5. 321)14 1.1012 2.8701 164.44 D52 110. 90 0.75095 5.3279 5. 3123 1.1007 2.86 96 164.41 U52 110. 90 0.75047 5.3256 5. 3098 1.1001 2.8697 164.42 D52 110. 90 0.75055 5.3256 5.3098 1. 1002 2.8696 164.42 313 101. 26 0.750 16 -0.13368 1.0997 1.0997 1.4101 80.795 313 101. 26 0.75024 -0 .19368 1.0998 1.0998 1.4102 80.796 313 101. 26 0.75032 -0. 19239 1. 0999 1.0999 1.4114 80.865 319 101. 29 0.75040 -0.17462 1.1000 1.1000 1.4277 81.802 1)52 110.90 0.75048 5.3256 S.3098 1.1001 2.8696 164.42 IP VP! VPY VHAS COS (AN G) HAX RE HIN CD 0.0 0. 74602 0. 0 0. 74602 1.0000 0.72987 37.325 0.55835 0. 74 889 0. 63610D-01 0. 75159 0.85778 0.65607 4 1. 185 0.93686 3. 75145 0. 10217 0. 75836 0.51194 0.55578 48.048 2.0847 3. 67008 0. 25344D-01 0. 67056 -0.92569 0.35961 72. 364 1. 6274 3. 52166 -0. 338O3D-02 0. 52157 -0.95277 0.42500 6 1.790 1.8182 3. 50476 0. 79910D-02 0. 5 8481 -0.94228 0.39566 66.106 1.4938 3. 47604 -0. 1 1 723D-01 0. 47618 -0.96092 0.48976 54.078 1.0512 3. 74197 0. 10976 3. 75004 0.28256 0.53496 49.791 1.5469 3. 49419 -0. 83682D-02 0. 49426 -0.95758 0.45381 58.091 1.4619 0. 46589 -0. 12808D-01 0. 46606 -0.96177 0.5 1370 5 1.714 1.4411 3. 45902 -0. 1 3 826D-01 0. 45923 -0.96275 0.52878 50.334 1.0 801 3. 73262 0. 10983 0. 74081 0. 17607 0.53589 49.710 1.4484 3. 46131 -0. 13678D-01 0. 46152 -0.96 268 0.52303 50.852 1.4348 3. 45671 -0. V4552D-0 1 3. 45694 -0. 96362 0.53255 50.001 1.4326 3. 45615 -0. 1 4 3 84 D-01 0. 45638 -0. 96335 0.53500 49.787 1.4307 0. 45556 -0. 14 4 2 3 0- 0 1 0. 45579 -0.96337 0.53668 49.641 1.4302 3. 45545 -0. 14327D-01 0. 45567 -0.96323 0. 53745 49.575 1. 4291 0. 45511 -0. 14356D-01 0. 45534 -0.96325 0.53837 49.496 1.4293 3. 45517 -0. 14 34 60- 0 1 0. 4554 0 -0. 96324 0. 53821 49.510 1.0829 0. 73075 0. 1096 1 0. 73892 0. 15996 0.53759 49.563 1.0830 3. 73078 0. 10962 0. 73895 0. 15995 0.53742 49.577 1.0833 0. 73067 0. 13962 0. 73 885 0. 15876 0.53735 49.584 1.0861 3. 72 878 0. 13939 0. 73695 0. 14260 0.53812 49.517 1.4292 3. 45512 -0. 14355D-01 0. 45534 > -0.95325 0.53835 49.498 HO. FIBONACCI CYCLES" 19 STOKES HO." 10.00000 DP/DP" 0.10000 XPST" -99.99394 CTER TIME XP YP VPX VPX VMAS RP KIN RP RADIANS DBSR EES COS (ANG) MAX BE MIN CD 277 99.860 0.0 -1.1031 -0.0 0. 79554 0. 0 0.79554 1.0851 1. 085 1 0.0 0.0 1.0000 0.77945 35.137 280 100. 06 0.42016 -0.97580 0.52595 0. 79 751 0. 47304 D-01 0.79892 1.0925 1. 0925 0.50344 28.845 0.87593 0.7 1193 38. 191 290 100. 42 0.67984 -0.68341 0.86860 3. 83081 0. 764 10D-0 1 0.80444 1.0956 1. 0956 0.91796 52.595 0.60744 0.61684 4 3.607 420 108. 03 1.1000 5.3729 1.9032 0. 66869 0. 2644 1D-01 0.66922 5.5029 1.4397 2.7878 159.73 -0.93805 0.39727 65.854 449 109. 38 0.84032 5.0770 1.4824 0. 53457 0. 43605D-02 0.53459 5. 37H7 1. 1327 2.8620 163.98 -0.95118 0.55137 48.406 437 108. 75 0.93951 5.3836 1.6533 0. 59107 0. 13040D-01 0.59122 5.4311 1. 2521 2.83 19 162.26 -0.95243 0.43822 60.033 300 100. 72 0.77902 -0.43397 1.0156 0. 79 756 0. 86988D-0 1 0.80229 1. 0993 1. 0993 1. 1826 67.759 0.37850 0.58284 45.967 mm 109. 11 0.87821 5.0731 1.5492 0. 55719 0. 77795 D-02 0.55724 5. 3899 1. 1787 2.8498 163.28 -0.95774 0.49712 53. 327 951 109. 44 0.81691 5.0103 1.4395 3. 52167 0. 303 0 ID-02 0.52168 5. 29K6 1. 1040 2.8662 164.22 -0.95231 0.58884 45.530 306 100. 88 0.80244 -0.33618 1.0584 0. 79225 0. 89106 D-01 0.79724 1.0990 1. 0990 1.3053 74.790 0. 25236 0.58479 45.324 450 109. 39 0.82585 5.0236 1.4559 0. 52693 0. 36770D-02 0.52694 5. 3161 1. 1150 2.864 1 164. 10 -0.96 174 0.57450 46.587 31 1 101. 00 0.81138 -0.23537 1.0809 0. 78612 0. 89526D-01 0.79121 1. 0990 1. 0990 1.3993 80.176 0. 17062 0.58943 45.438 151 109. 51 0.82032 5.3615 1.4460 0. 52261 0. 27588D-02 0. 5226 1 5. 3498 1. 1082 2.8678 164.31 -0.95276 0.58343 45.923 152 109. 54 0.81479 5.3529 1.4359 0. 51934 0. 23645D-02 0.51935 5. 3 388 1. 1014 2. 8692 164.39 -0.95314 0.5927 1 45.254 31 0 101. 07 0.81350 -0.14978 1.0898 0. 78204 0. 89357D-01 0.78713 1. 0998 1. 0990 1. 4506 03. 110 0. 11996 0.59234 45.280 452 109. 56 0.81562 5.3 6 84 1.4374 0. 51951 0. 22659D-02 0.51951 5. 3542 1. 1024 2. 8697 164.42 -0.96327 0.59118 45. 363 452 109. 54 0.81433 5.3518 1.4350 3. 51908 0. 23372D-02 0.51909 5. 3 371* 1. 1008 2.8693 164.40 -0.95316 0.59354 45. 195 452 109. 51 0.81396 5.3355 1.4343 0. 51923 0. 24853D-02 0.51923 5. 3216 1. 1004 2. 8686 164.36 -0.95298' 0.59424 45.146 452 109. 51 0.81388 5.3355 1.4341 0. 51918 0. 24785D-02 0.51918 5. 3215 1. 1003 2.8687 164.36 -0.95299 0.59438 45. 136 311 101. 07 0.81358 -0.14978 1.0899 0. 78206 0. 89365D-01 0.78715 1.0999 1. 0999 1.4506 83. 111 0. 11995 0.59219 45.291 314 101. 07 0.81365 -0.14978 1.0900 0. 78209 0. 89373D-01 0.78718 1.1000 1. 1000 1.4506 83.112 0. 11994 0.59204 45. 301 452 109. 51 0.81373 5.3355 1.4338 3. 51908 0. 24654 D-02 0.51909 5. 3214 1. 1001 2.8687 164.36 -0.96 300 0.59465 45.117 452 109. 51 0.81380 5.3355 1.4340 0. 51913 0. 24717D-02 0.51913 5.3215 1. 1002 2.8687 164.36 -0.95299 0.59452 45.126 452 109. 51 0.81373 5.0355 1.4338 0. 51908 0. 24649D-02 0.51909 5.3214 1. 1001 2.8687 164.37 -0.95300 0.59466 45.116 INITIAL VARIABLES R-INFIHITY- 100.000 REYNOLD'S NO.- 10.000 XP<1)» -99.99394 SIZE OP GRID- 33 X 93 GNUH- 3.000000 OSUB= 1 NUMBER OP ITERATIONS IN GRID- 5000 KLYACHKO'S FORMULA FOR DRAG COEFFICIENT USED AI ALL VALUES OF RE •«***»* KUWABARA'S ZERO VORTICITY MODEL »»•*»*• ***•• NUMERICAL FLOW FIELD USED »•***•* EFFICIENCY WITH INTERCEPTION STOKES' NO.- 1.000 EPSIL- 0. 1S31760 OP/DP- 0. 100000 EFF- 0. 1192509 STOKES' NO.- 2.000 EPSIL- 0.36 50161 DP/DF- 0. 100000 EFF" 0.3318329 STOKES' NO. - 3.000 EPSIL- 0.1948146 DP/DF- 0. 100000 EPF- 0.4498317 STOKES' NO.- 5.000 EPSIL- 0.6464987 DP/DP- 0. 100000 EFP- 0.5877264 STOKES* NO. - 7.SO0 EPSIL-. 0.7504339 DP/DF- 0. 100000 EFP- 0.6822130 STOKES' NO.- 10.000 EPSIL- 0.8136897 DP/DF- 0. 100000 EPF- 0.7397183 STOP 0 EXECUTION TERMINATED JSIGNOPP 

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