IV1£3I_J RETURNING MATERIALS: PIace in book drop to LIBRARIES remove this checkout from ——. your record. FINES win be charged if book is returned after the date stamped below. FLO! EAST A.D-SECTION 8w! “FIRED 3! mm 'ALLS 3! Richard Gregory Iataon A WIS Schnitted to Iichigan State University in partial fulfill-ant of the requirements for the degree of mm (B SCIDlG Depart-cut of Mechanical Engineering 1985 33—173 :5 7‘1/ ABSTRACT FLO! PAST A D-SECTION BODY CDNFINED BY CHANNEL 'ALLS By Richard Gregory Iatson The channel flov past a D-section has been investigated for five different channel widths for a given obstruction dinension. The flow of main interest. aft of the D-section. was quantified in terns of its tine nean pressure and velocity distributions. The prine focus of the study. the oscillatory features of the flow field. were sinilarly quantified in terns of fluctuation intensities. Also. a flow nodel featuring a shifting streanline pattern. dependent on the alternate vortex shedding. vas deveIOped based on the salient observations. Finally. an appropriate characteristic velocity uas identified such that a nodified universal Strouhal number could be defined. This for- of the universal Strouhal nunher was approximately constant for all gee-etric configurations. AINOILEDGIENTS I would like to thank Ir. I. Ielkerson and Ir. I. Rose for their contributions to the design and construction of the experinental test section. Also. I an pleased to aknowledge the work of Ir. 8. Burns on the drafting of the figures and the helpful assistance of the entire Free Shear Flows Laboratory. A special thanks goes to the I.8.0. Rugby Club for providing an exciting diversion to the daily grind of college and for allowing us to keep a sound body while attenpting to retain a sound nind. I would like to express uy gratitude to qy thesis adviser. Dr. John F. Foss. for his thoughtful insights into ny work and for his guidance during ny graduate progran. Finally. I wish to express ny deepest appreciation to ny parents for their continual support during ny entire education and to ny fiance. Jill. for her constant love and encourage-ent e TIBLE OF CONTENTS LIST OF TABLES LIST OF FIGURES .. UBTOFSDBGJ..u.u.n.”.u.u.n.u.u.u.n.u.u.u.u.. CHAPTER 1 _ INTRODUCTION ......................................... 2 EXPERIIBNTIL EQUIPIENT'AND PROCBDDRB ................. 2.1 The Bxperinental Facility ....................... 2.2 Bxperinsntal Procedure .......................... 2.2.1 Pressure Ieasureuents ..................... 2.2.2 Velocity leasurenents ..................... 2.2.3 Signal Analysis ........................... 3 nlscussmuormnuum mums 3.1 Introduction to Discussion of Results ........... 3J nehnme”nuuunuuuuunununu. 3.2.1 pressure Distributions .................... 3.2.2 Velocity Distributions .................... 3.3 The Oscillatory Flow ............................ 3.3.1 Fluctuation Intensity leasurenents ........ 3.3.2 Spanwise Correlations of the Fluctuating Velocity and a Flow Iodel for the 0.0i11‘tm'ot10‘ OOOOOOOOOOOOOOCO.IOO0.... vi PAGE 10 11 11 13 13 16 20 23 3.3.3 Streanwise Correlations of the Fluctuating Velocity ...................... 25 3.3.4 The Strouhal Nunber 27 3.4 The Universal Strouhal Nunber ................... 28 3.4.1 st‘ Based on Roshko's Criteria ............ 29 3.4.2 St. Based on Bear-en's Criteria ........... 31 3.4.3 The Effect of Confining 1.11. on St. ...... 33 3.4.4 a New sc‘ ................................. 34 4 SUM! AND mamasmus. 38 APPENDIX A PRESSURE DATA ............................. 110 APPENDIXB DmmATIONW'io“ 133 mam 0.0.000...OO0......OOOOOCOOOOOOOOOOOOOOOOOO...0...... 137 3.1 3.2 3.3 3.1 3.2 3.3 3.4 LIST OF TAB]... Variation of St‘ with r ............................. Influence of r on 0.. h(lp). and uh ................. Variation of 8t;o with r ............................ Pressure Values for r - 1.0 ... Pressure Values for r - 0.8 ... Pressure Values for «r - 0.6 .. h...".V‘1u.tux-OO‘ .........OOOOOOOOOOOOOOOO PAGE 34 35 37 110 116 122 127 FIGURE 10 11 12 13 14 15 16 ' LIST OF FIGURES The D-section Obstruction .............................. Schenatic Representation of the Flow Facility .......... Schenatic Drawing of the Tcst Section .................. Scale Drawing of the Test Section ...................... Scale Drawing of Variable Channel Vail Settings ........ Scale Drawing of the Sliding Upper Surface of the Test Section ............................ Schenatic Representation of the Data Aoqni'it£on sy‘t- 0.00.0.0.........OOOOOOOOOOOOO0.00... Schenatic Representation of the Tho-Vire hp.ri..nt. ......0.0......OOOOOOOIOOOOOOOO00.000000...O Isobar Contours: r I 1.0 .............................. Isobar Contours: r - 0.8 .............................. Isobar Contours: r - 0.6 .............................. Isobar Contours: r - 0.4 .............................. Variation of Average Base Pressure with r .............. Base Pressure Distribution on the Aft Bd‘. at “. D-'.°tion 00.00..........OOOIOOOOOOOOOOO.... Pressure Distribution Along the Rake-Axis (y - 0) ...... Pressure Distribution on the Surface 0‘ &. n-‘.°t£°n: r.2.s 0.0.0.000.........OOOOOOOOOO. vi PAGE 41 42 43 44 43 46 47 49 50 51 52 53 34 55 56 17 18 20 21 22 23 24 23 26 27 8 30 31 32 33 34 33 36 37 38 39 40 Pressure Distribution on the Surface at tho ”-0.0t10n: r-IOO 00000000000000000000000000000 Pressure Distribution on the Surface 0‘ a. D-..°tion: r-008 00000000000000000000000000000 Pressure Distribution on the Surface 0‘ a. ”-0.0tion: t.006 00000000000000000000000000000 Pressure Distribution on the Surface 0‘ a. n-..°ti°‘: t.004 00000000000000000000000000000 lean Velocity Profile at x I 3.3d: r I 2.3 ............ lean Velocity Profile at x I 3.3d: r I 1.0 ............ lean Velocity Profile at x I 3.3d: r I 0.8 ............ lean Velocity Profile at x I 3.3d: r I 0.6 ............ lean Velocity Profile at x I 3.3d: r I 0.4 ............ lean Velocity Profile at x I 2.23d: r I 2.3 ........... lean Velocity Profile at x I 2.23d: r I 1.0 ........... lean Velocity Profile at x I 2.23d: r I 0.8 lean Velocity Profile at x I 2.23d: r I 0.6 ........... lean Velocity Profile at x I 2.23d: r I 0.4 ........... Location at Vhich a Vool Tuft indicated 30! Rack Flow - 30$ Foruard Flow as a Function of r ............. h“ v.1“1ty “0‘11. .t x-d: t-205 000000000000000 I.“ V.lo°1” hotil. .t x - d: r - 10o 000000000000000 lban Velocity Profile at x d: r I 0.8 ............... I.“ V.1°.it’ PtOIil. .t x d: r - 00‘ 000000000000000 I.“ v.loc£ty hotil. .t ‘ d: t - 004 000000000000000 Boundary Leyer Profiles at the Aft Edge of the D-section (x I d) .......................... Fluctuation Intensities at x I 3.3d: r I 2.3 .......... Fluctuation Intensities at x I 3.3d: r I 1.0 .......... Fluctuation Intensities at x I 3.3d: r I 0.8 .......... V“ 37 38 39 60 61 62 63 64 63 66 67 68 69 70 71 72 73 74 73 76 77 78 79 80 41 42 43 44 43 46 47 49 30 31 32 33 34 33 36 37 39 60 61 62 63 64 Fluctuation Intensities at x I 3.3d: r I 0.6 .......... Fluctuation Intensities at x I 3.3d: r I 0.4 .......... Fluctuation Intensities at x 2.23d: r I 2.3 ......... Fluctuation Intensities at 2.23d: r I 1.0 ......... 2025‘: t-o08 000000000 Fluctuation Intensities at x Fluctuation Intensities at x 2.23d: r I 0.6 ......... Fluctuation Intensities at x I 2.23d: r I 0.4 ......... Sand Pass Fluctuation Intensities at x I 3.3d .......... Dand Pass Fluctuation Intensities at x I 2.23d ......... Band Pass Fluctuation Intensities at 13(P—nin) Norn- alised by the laxinnn Intensity Along the Span ......... Fluctuation Intensities at y I d: r I 2.3 ............. Fluctuation Intensities at y I (d+g)/2: r I 1.0 ....... Fluctuation Intensities at y I (d+g)/2: r I 0.8 ....... Fluctuation Intensities at y I (d+g)12: r I 0.6 ....... Fluctuation Intensities at y I (d+g)l2: r I 0.4 ....... Rand Pass Fluctuation Intensities at y. (&.)I2 [y-‘ tut-205] 00000000000000.00000000 Fluctuation Intensities and Correlation co.’t1°t.‘t' .tx-zIzSd: t‘10o 00000000000000.00000 Schenetié Representation of the Flow “.1 fat-1.0 0......0.................0..0......... Phase Shift at the Shedding Frequency Illm08201028 0000000000000000000000000000000000000 Variation of Convection Speed with r ................... Variation of Strouhal nnnber with r .................... v-xiatton oth .1.» r .............................. Variation of Foruatiou.Length with r ................... 'ithr 00000000000000000000000000000000 Variation of St; vfii 81 82 83 84 83 86 87 88 89 90 91 92 93 94 93 96 97 98 99 100 101 102 103 104 63 66 67 69 3.1 3.2 Varietion of 05 with t ................................. Variation of St; with r ................................ Fluctuation Intensities It 13 .......................... Variation of 01, with r ................................ v.ti‘t‘ou 0‘ st:° 'ithr 0000000000000000000000000000000 A Typical Energy Spectrun Aft of the D-section ......... A Typical Sand Pass Energy Spectrun .................... 103 106 107 108 109 133 136 ‘0 St St. “5 0 St“ 8:; LIST OF sumac Lateral spacing between vortices of opposite signs Base pressure coefficient ‘ Iidth of obstruction Distance between free streanlincs Error Gap width between the side wall and the D-section Span between the naxina of velocity fluctuations at the shedding frequency Length Fornation length Plenun pressure (Bernoulli constant) Pressure on obstruction surface 0.123d upstrean of separation Ratio of gap width to obstruction width (g/d) Reynolds nunber Root nean square Strouhal nunber Universal Strouhal nunber Searnan's (1967) universal Strouhal nunber Universal Strouhal nunber based on U10 Roshkc's (1934b) universal Strouhal nunber Velocity approach Velocity of approach flow Inviscid velocity calculated fron base pressure Free streanline at separation Inviscid core velocity Inviscid velocity outside the boundary layer 0.123d upstrean of separation Freestrean approach Velocity Cross correlation between wires A and B lean square of velocity fluctuations RlS of fluctuations (full pass) . [3..2 + 32..2]l/2 RlS of fluctuations (band pass - 00) RB of fluctuations (band pass - 2a.) Streanwise coordinate Transverse coordinate Vertical coordinate ' Greek Synbols Boundary layer thickness Phase shift lavelength of passing structures Density Tine delay of cross correlation Frequency Shedding frequency Second harnonic of the shedding frequency CHAPTER 1 INTRODUCTION The present study of flow past a D-section (Figure 1) confined by channel walls is one elenent of a large category of oscillatory flows of concern in engineering design. The doninant feature of this flow was the quasi-periodic nature of the fluctuations of the wake region which occur as the result of the alternate shedding of vortices fro- each side of the body. If the resonant frequency of the D-section could be excited by the flow field oscillations. then structural dan- age nay occur. As sunnarixed by Roahkc [l]. nan has been aware of the periodic nature of flows puat bluff bodies for centuries (e.g.. the sounds pro- duced by wind flowing past cables) and this general subject has been the topic of nunerous investigations. The excellent review articles are presented by lair and laull [2] and by Bear-an and Grahan [3]. Roshkc [I] noted that sone of the earliest work in the area was perforled by Strouhal [4] and Rayleigh [3] late in the 19th Century and by lArnhn [6] in the early 20th Century. These initial research- ers first characterized the flow past circular cylinders in terns of the now faniliar paraneters: Reynolds nunber and Strouhal number. The Strouhal nunber is a dinensionless frequency defined as St I uL/U (1) where L and U are characteristic length and velocity. respectively. lore recently. attenpts have been nade to unify all such vortexrshedding flows in terns of a universal Strouhal nunber. St‘, based on the flow characteristics in the wake'of the body. This tern (St‘) was first introduced by Roshko [7] to describe the flow past a variety of bluff objects. The St. of his work‘was based upon the use of a Notched-Eodograph nethod. Roshko [7] hypothesized that the prop- er ncrnalixing constants should be the velocity on the free streanline It separation. “1.. and the distance. d'. between these streanlines at - the downstreau location where the vortices are first turned. 3c nain- taincd that this downstrean "fornation length." lF' corro.pond.d to the point on the wake centerline at which the pressure reached a nini- nun value. This analysis has been extended by others to include "enpirical equivalents" of the Roshko [7] analytical nethod. Brief sunnaries of 80-0 of this work.nay be found in articles by Griffin and Banberg [8] and by Griffin [9]. Bear-an [10. 11] presented an enpirical nethod to describe the ‘ .. .iaé'lnw ”Ol'mumll‘lfl «a ' ...° - tries-a s use mi": 7- ., {-3, e any can I a he he. . ... fig uc-uainoh OCT .ngseeo ._.. .9335'59339312 edr io eraser ' ~..~;rt;;i~.:zza eiscxeele ed: to 11 .31.: tumors: ed: 1)" .t.‘ . _:-_~..'::a.'?.iuao “In." wo' I. Ices-r! u"! can .[l] aided: 00.1322. .6; n 3. 9~ ' \ 7 ’a he} “I .0333 18"..aa. 0 -" ‘0: , .-'.i'.‘£. .e.nuf..‘ht :35 i“ '. 0‘ or” has {3? {It '2 35.. unu ed: to «not Jarfi daieltcfi be; it; t L use!) H303 virus a... : :ceq r013. ed: harm": 3 flow past a body (roughly half an ellipsoid) nuch like the D-section. For that flow. and for the D-section. both the separation point and separation angle are fixed. 3c calculated a characteristic velocity. “t. fro. the obstruction's base pressure using Bernoulli's equation. The characteristic length was assuned to be the span-wise distance. h. between the points of naxinen velocity fluctuation at the fornation length behind the obstruction. An analytical nethod of deternining the span. b. between the vortices was docunented by Rronauer [12] and shown. by Bear-an [11]. to yield a constant St' for a variety of bluff bodies. The influence of confining channel walls on flows 'past bluff bodies has been the subject of several investigations. Shair. Grove. Peterson and Acrivos [13] concluded that. at low Re. instabilities in vortex fornation occurred at severe confine-out ratios. Tbxkas [14] and Chen [l3] exanined St‘ for flow past cylinders and noted increas- ing irregularities in the fluctuations as the channel walls were noved closer to the obstruction. Additional related studies were docunented by Shaw [16] and Thebes [l7]. Richter and Naudasoher [18] exuuined the Strouhal umber for the afar-entioned studies. as well as their 0'3. 130! concluded that St. is approximately constant (for all con- finenent ratios tested) for flow past a circular cylinder. The intent of the present study is to apply sole of the preceding ideas to the specific problen concerning channel flow past a D-section. Five different channel wall settings have been exanined 4 including a "control” case where the valls are far away fro. the obstruction. The najority of data were obtained through the use of pressure neasurenents. a linearised hot-wire anenoneter. and conputer aided signal processing. Since the scope of this investigation does not require extreue precision nor a naxinnn signal to noise ratio. a linearized anenoneter was used to expedite the processing of the ana- log data. Flow visualization techniques. including surface streaking (with a leap-black and kerosene nixture) and wool tuft observations were also utilized. It is inportant to note that all the data were taken with the inviscid flow velocity. just upstrean of separation. nainr tained as a constant. This was acconplished via 7 a pressure neasureuent at a tap near the trailing edge of the D-section (Figure 1). This pressure is torn-d P, and the corresponding velocity is 301-0‘ U: 'h.:. 0. - [2(Ppl-P,)Ip]1’2. (2) (N0t03 Pi; is the stagnation or plenun pressure.) The present study will establish a thorough data base. for the specific flow fields investigated. and interpret the data in the con- text of previous vork where applicable. The nean velocity and pressure distributions are subject to an in-depth analysis in order to identify the gross features of the flow. Also included is a conplete 5 study of the different aspects of St‘. However. the prinary purpose of the work is to deternine the effect of the location of confining walls on several characteristics of the flow including St‘. pressure and velocity (lean and fluctuating) distributions. vortex fornation. and convection speed of coherent structures. CHAPTER 2 EIPERIIENTAL EQUIPIENT AND PROCEDURE ‘ 2.1 The Experinental Facility A.schenatic representation of the experinental flow systen is shown in Figure 2. This facility is the principle flow systen of the lichigan State University Depart-ent of lechanical Engineering's undergraduate Fluid lechanics Laboratory. Albient air is drawn through a contoured inlet by a Buffalo 3-1/2 E fan. The fluid then passes through a radial diffuser into a plenun. and then through the test section (Figures 3-3). For the present study. the desired flow rates through this channel were regulated by adjusting the bleed flow through a slit-jet (also attached to the plenun). It is inportant to note that the free strean turbulence was quite high (I 5;) gt the inlet to the test section. The test section entrance consisted of a snooth rounded contrac- tion fro. the plenun into a 10.16 on x 60.96 on inlet plane. Four different contoured blocks were available; when inserted into the channel. these blocks provided four different widths for the renainder of the flow passage. The paraneter used to describe these different geonetries is the ratio r; it is defined as r I g/d (3) where g is the gap width betneen the side wall and the D-section and d is the width of the obstruction (Figure 4). Experinents were per- forned for channel settings of r I 0.4. 0.6. 0.8 and 1.0. A case for which the contour blocks and the following side walls were renoved oonpletely (r I 2.3) was exanined in order to best approxinate an infinite channel width (Figure 3). A channel extension of length: 30.80 on (3 obstruction widths) was utilized to deternine if the inpo- sition of the atnospheric boundary condition at the channel exit (2.3d downstrean of the aft edge of the D-section) affected the experinental results. The present D-section (Figure 1) can best be described as having a cylindrical half-round nose. of radius dIZ. connected to a cubical section of dinension d. This obstruction.vas positioned in the center of the span of the channel. The nose of the D-section was located 1d downstrean of the end of the contraction. 2.2 Erperinental Procedure 2.2.1 Pressure leasurenents The pressure data were obtained using a Validyne DP13-22 pressure transducer in conjunction with a C313 carrier dencdulater (recording instrunentation is listed in section 2.2.3). All the neasurenents vere teken eith the velne of Ppl-P. (Figure 1) equal to a fixed value. This quantity was held constant at 80.2 Nlnz which corresponded to the naxinun flow rate available for the r I 2.3 case. Extensive data were obtained fron a previous study (Lorry and lcsaian.[l9]) concerning the pressure neasured at the upper surface of the channel. A sliding plate with three rows of pressure taps (Figure 6) afforded the experinentalist access to pressure neasurenents at 1.27 cn.intervals in the strernwise direction and at any desired span- wise location. This infornation was entered into conputer data files and processed into contour naps of the isobars. Surveys of the pressure distribution around the perineter of the D-section were carried out with special care taken to provide accurate neasurenent of the base pressure (on the aft edge of the obstruction). Finally. a series of pressure neasurenents were taken in the wake region aft of the obstruction. An L-head static probe was traversed along the wake centerline fron as close as possible to the aft-edge. to the channel exit. 2.2.2 Velocity leasurenents Several velocity surveys were inplenented for this study at all r values. A schenatic representation of the data acquisition systen is presented in Figure 7. leasur-ents were taken with a two channel TSI A hot-wire anenoneter: nodel 1031-2 power supply. 1036 variable decade resistance and 10343 constant tenperature linearized anenoneter. Bot-wire traverses were perforned at the aft edge of the obstruction as well as downstrean of the D-section at the channel exit and at an internediate x-location halfway between the previous two positions. Additional traverses were nade at x-locations deternined by the point of vortex fornation. Data were also recorded along the wake center- line in order to deternine the fornation point. Detailed boundary layer surveys were perforned at the aft edge of the D-section. Care was taken to not danage the hot-wire through con- tact with the surface of the body. This hazard was neat pronounced when the probe was in close proxinity to the obstruction and the probe support was in a region of large anplitude oscillation. A series of two-wire studies were nade to discern certain veloci- ty correlations in the flow field (Figure 8). In order to deternine the convection speed of the coherent structures. one wire was held fixed at the fornation point (where there exists a very strong oscil- latory signal). while the other wire was traversed fron a point approxinately 0.4d upstrean of the fixed wire. to the channel exit (y IO and z renaining constant). A siniliar traverse was perforned across the y-span (r I 1.0) with one wire fixed outside the shear layer. as the other was noved across the entire channel width. 2.2.3 Signal Analysis lean voltages. corresponding to both pressure and velocity sig- nals. were deternined using a Disa type 32330 true integrator. The output was displayed on a liethley 179 TRlS digital nultineter. Qualitative observations of the output signals were nade with the use of a Tektronix 3111 storage oscilloscope. The bulk of the signal processing was done by the Genrad 2307 Structural Analysis Syst-A. lith the aid of the Interactive Signal Analysis Progr- software package. the following functions were evalu- ated: 1. Root nean sqaure of the fluctuations 2. Energy spectra 3. Cross correlation 4. Cross spectra and phase shift Another inportant facet of the Genrad's softvare was the capability to serve as a band-pass filtering device (see Section 3.3.1 for details); this enabled the user to deternine the fluctuation intensity over a specific frequency range. CHAPTER 3 DISCIISSIM 0F EXPERIUITAL RESULTS 3.1 Introduction to Discussion of Results The nain thrust of the present study was to deternine the effect of confining walls on the flow past a D-section obstruction. A dis- cussion of the significant aspects of the nean flow is used to introduce the general features of the flow field. Subsequent analyses of the oscillatory phenonena associated with the flow provide the nain results of this investigation. The initial experinents. docunented in Section 3.2, were carried out in order to gain sone insight into how the the velocity and pres- sure distributions changed as a result of repositioning the channel side walls. It was evident that the doninant feature of the flow fields was the periodic vortex shedding and its effect on the near wake region of the obstruction. Therefore. a detailed data base of acne of the oscillatory characteristics of the flow was established. A series of experinents were executed (Section 3.3) with an enphasis on spectral analysis of hot-wire signals. 1] 12 The final phase of the study dealt with the characterization of the oscillations in terns of a universal Strouhal nunber. Section 3.4 is devoted to an in-depth investigation of several aspects of St‘. luch of this work was atinulated by previous studies of related topics. Prior to the presentation of the results. it is inportant to recognize a few significant features of the particular experinental facility utilized for this study. Of prinary concern is that the test section was one of a "squat" geonetry: the channel height was equal to the width of the D-section. As could be readily anticipated. this geonetric condition leads to three-dinensionality in the flow fields and hence in the observed results. Another peculiar feature of this flow systen is that the channel exit (the atnosphere) was located 2.3d downstrean of the D-section's aft edge. Since an atnospherie pressure boundary condition is inposed at the periphery of the exiting channel flow. this relatively short distance to the channel exit can play a significant role in the dynan- ics of the flow problnn. The results. which nay have been influenced by this specific flow geonetry. are discussed in the subsequent sec- tions. Finally. to aid in the presentation of the results. a rectangular Cartesian coordinate systne has been established. The origin of the coordinate cysts-5 (Figure 4) was located at the center of the 13 half-round section; It was situated at the nid-plane of the channel's ' height. 1.3d downstrean of the end of the contraction and 3.3d upstrean of the end of test section. 3.2 The lean Flow 0f prinnry concern in this phase of the study was the establish- nent of a conplete data base of the tine averaged velocity and pressure distributions in the flow fieldr exenined. This infornation serves to establish the reference base for the oscillatory phenonena studied in the subsequent phases of the study. 3.2.1 Pressure Distributions The results of a detailed series of pressure neasurenents. taken by Lorry and Rosaian [19] at the upper surface of the test section. are shown in Figures 9-12. A conplete list of this pressure data is presented in Appendix A. The contours represent the nean isobars. [P(x.y)-P.JIIP,1-r,l. for r values of 1.0. 0.3. 0.6 and 0.4 (r - 2.5 was not included in this study). extending fron the end of the cone traction (x I -1.3d) to the channel exit (x I 3.3d). Data were processed by the Surface II software package on the lichigan State University Cyber 730 conputer. If the flow is essentially planar (i.e.. if the streanline curva- ture is negligible except in the x-y plane). then it is safe to assune 14 that these pressure distributions are representive of the flow at all z-planes of the channel. It is inportant to note that the isobars were developed by the construction of a grid of pressure values. based on the neasured values at nearby locations. and a snoothing of con? nected curves. Therefore. these plots should be used. nainly. to identify trends in the data rather than to deternine the value of the pressure at any particular location. The plots show the sane general pressure distribution for all r values:' i.e.. low pressure at the shoulder [where the half-round nose joins the square cross section (x n.0, y. a. d/2)] and .1: of the obstruction. relatively high pressure upstrean of the nose and an area of fairly constant pressure between the obstruction and the side wall of the channel. As the r value decreases. the area of high pressure. forward of the D-section. increases as a result of tunnel blockage. Also. as expected. the distance between sinilar isobars decreases. indicating greater fluid acceleration through the snaller gaps. Another trend. which was evident in the plots. was an increase in the nagnitude of pressure in the region aft of the obstruction (in the recirculation zone) for the snaller r values. Detailed pressure neasurenents on the aft surface of the D-section showed sinilar results. The average base pressure (Figure 13) increased as r decreased. Note that as the side walls were noved closer to the obstruction. the average base pressure approached the 'Ilfl. 0‘ P,. As shown in Figure 14. the distribution across the aft 15 edge is less variable for snall r while the pressure is lower near the nid-plane (y I 0) for larger r. Additional pressure surveys along the wake axis (Figure 13) also revealed a distinct trend as the r values were changed. The nagnitude of the nininnn pressure increased as r decreased and the location of the nininun pressure noved farther downstrean. This suggests that the extent of the recirculation zone increased with decreasing r. The contour plots also suggest that the greatest variation in the 'pressure distributions occurs on the perineter of the D-section. Consequently. the fluid passing close to the obstruction will experi- ence the largest acceleration near the shoulder of the body. Figures 16-20 show the pressure distributions on the surface of the D-section (exclusive of the aft edge). The neasurenents were taken at the nid-plane of the channel. The results indicate that the channel wall position has a signi- ficant effect on the nininmn pressure experienced at the shoulder and thus the severity of the pressure gradient will vary with r in this region. Surface streaking experinents showed that there was no narked effect on the location of the separation point on the shoulder of the D-section. (The data showed a slight variation with r; however. these differences nay be attributable to experinental uncertainty.) The average location of the separation point was approxinately 86-degrees fron the nose. This visualization technique indicated that the spa- )6 tial extent of the separation bubble (near the shoulder) was increased as the r value was decreased. 3.2.2 Velocity Distributions The results of hot-wire surveys at r I 3.3d (channel exit) and x I 2.23d are shown in Figures 21-30. A significant feature of the velocity profiles at the channel exit is the velocity defect in the wake of the obstruction. This defect is nuoh larger for the snall r cases. supporting the assunption that the size of the recirculation zone increases as r decreases. Closer to the D-aection (x I 2.23d). however. there are addition- al conplications in the flow field which influence the nean data. Since this is approxinately the region where vortices are being forned (see Section 3.4). a strong periodic transverse conponent to the velo- city vector exists in this region (Roahko [7]). This effect. in addition to any reverse flow (as suggested by wool tuft visualiza- tions). will bias the nean value registered by the hot-wire anenoneter. The neasured nean voltage cannot be accepted. prina facie. as being indicative of the nean streanvise velocity. Hence. the dashed lines in Figures 26-30 (representing plausible velocity distributions) deviate fron the data points recorded directly aft of the obstruction. These profiles were based upon the best available evidence: the )7 integration of the results of velocity surveys at x I d (0.3d i y i g+d/2) provided approxinate values of flow rate per unit depth. The velocity profiles at x I 2.23d were then "adjusted". over the wake region. in order to provide natching integral (0 I y f g+d/2) vgjngg (and to provide good qualitative agrenuent with tuft observations in these regions). Since three-dinensional effects were particularly significant in the wake of the obstruction. this two-dinensional con- putation is of linited accuracy; however. it does allow a welcone and an independent evaluation of the hot-wire data. Relatively sophisticated nethods for the deternination of rest- tachnent or the closure of free strean separation donains are available. For exanple. the thernal tuft technique used by Eaton and Johnston [20] would provide a quantitative basis for the assessnent of the recirculating zone length in the present study. However. since only an approxinate value was sought. the nore elenentary and less accurate technique of using a wool tuft was adopted. By observing the notion of wool tufts at various locations along the wake axis. the direction of the flow could be deternined. The approxinate location of the closure of the tine nean separation bubble was estinated by noting the postion at which the tuft indicated that the flow was noving equally fore and aft. As shown in Figure 31. the location of closure. or "aft stagnation point." noved farther downr strean. away fron the body. as r decreased. This infornation was useful in deternining an estinate of the nagnitude (and direction) of IS the velocity at y I 0 for the x I 2.23d velocity profiles. Sone noteworthy features of the nean velocity distributions are present at x I 2.23d. For all r values. an inviscid core of non-sheared fluid existed between the sidewall boundary layer and the shear layer growing fron the step at the aft edge of the D-section. These data also showed that the nagnitude of the velocity in these core regions increased nonotonically as r decreased (for constant U‘), At the channel exit the shear layer had grown such that the inviscid core was no longer evident except for the r I 2.3 case. Velocity profiles were also constructed for the span extending fron the side wall of the D-section to the channel wall at the aft edge of the obstruction. x I d (Figures 32-36). At large r values. and especially for r I 2.3. there was a pronounced "bulge" in the pro- file near the obstruction. For snall r. the profiles were relatively ”flat" outside the boundary layers. This observation correlates well with the pressure data which showed that the fluid passing near the shoulder region of the D-section experiences a analler and snaller favorable pressure gradient as r decreases. Thus. for snaller r values. the nagnitude of the acceleration of the fluid near the body is closer to that of the overall flow in the gap and the downstrean profile becones nore flat. The aft edge data included a nunber of data points inside the boundary] layers on the obstruction. Figure 37 shows that. for a wide 19 range of r. the boundary layer velocity profiles are quite sinilar. Error nay have resulted due to inaccurate initial positioning of the hOt'ViIO PIObO (0/5 .’7!) causing a slight shift in the profile. It was assuned that since the fluid separated fron the body upstrean. effectively tripping it at the shoulder. a turbulent boundary layer existed for all r values. Oscilloscope traces of the anenoneter sig- nals fron a wire positioned in the boundary layer supported this hypothesis. 3.3 The Oscillatory Flow Initial observations of the oscillatory flow were nade with hot-wire probes. One of the distinctive features of these observa- tions was that an oscillatory conponent of velocity fluctuation was present in the flow virtually everywhere downstrean of the obstruction (except for the r I 2.3 case where the signal dininished at large y). These oscillations occured at a constant frequency throughout the flow (10! I particular 3 I‘d U.) and were l80-degrees out-of-phase on either side of the obstruction. The latter result is apparently a consequence of alternate vortex shedding. Directly behind the D-section. the doninant frequency was twice that of the shedding fre- quenoy. This result suggests that the causal factors for the oscillatory flow on either side of the D-section are jointly experi- enced at this location. 20 3.3.1 Fluctuation Intensity leasuruents In order to quantify the fluctuating conponent of the flow. a series of root nean sqare neasuruents were taken at the downstrean locations: x I 3.3d and x I 2.23d [Figures 38-47 (triangles)]. The developing shear layers. which originate fron the boundary layers at the trailing edge of the D-section. are revealed by these data. Energy spectra of these signals showed that a large percentage of the fluctuating intensity was present over a snall frequency band cor- responding to the shedding frequency of the vortices. The data points. shown as squares in Figures 38-47. represent to“; this quantity 'is defined as ‘osc " “0.2 + 320.2]1I2 (4) “It. 1.. is the RlS of the signal over a band of frequencies inclu- aive of the peak in the spectra corresponding to the shedding 1‘09""! “d ‘3... is the RlS over a band centered about the second harnonic of this frequency (if one existed.) This operation was per- forned utilizing the band pass capabilities of .the Genrad conputer and "operator selected" linits for the frequency bands. A typical fre- quency spectra is shown in Figure 3.1 and a detailed explanation of the steps of this operation is presented in Appendix B. The results show that the distributions of the Ti and ‘30“ exhibit 2] the sane trends. The peaks of the curves are located. noninally. at A the sane y-locations. This result was also noted by Bearnan [10] and suggests that. at these points in the flow. nost of the energy is present at the shedding frequency. However. there is an increase in the intensity at frequencies outside the selected band 5735.11 P888 - 773.6] in the center of the shear layer. It is interesting to note that there is still oscillatory notion present in the flow across the entire span of the channel (except at r I 2.3). at 2.23d. That is. the oscillatory notion extends well beyond the vortical region of the developing shear layer. Figures 48 and 49 show the oscillatory conponent of the RB sig- nal for 4 all r values. At x I 3.3d. the distributions of ‘osc are sinilar for all r. However. at x I 2.23d there is a disinct grouping of the curves. At r I 2.3 and 1.0 the RS distributions are approxi- nately the sane as are the curves at 0.8 and 0.6. ne RlS distribution at r I 0.4 is altogether different fron either of the other two "fanilies" of curves. It is obvious fron these results that the location of the side walls has sone effect on the anplitude of the oscillations at this particular x-location. “91“9‘“ evaluations 0‘ P... were nade at the x-location cor- responding to the point of nininun pressure along the wake axis. Figure 30 shows the results nornalized by the naxinun 11.. "1“ .10” the span. These nornalized distributions show two distinctive charac- teristics. The relative intensity fron y I 0 to the naxinun value 22 exhibits a sinilar distribution for all r values. Conversely. the distributions. fron the y location of the naxina. to the y location of the boundary wall. are quite different for the separate conditions. As r is decreased. the ratio of 11.. to 3..(nax) was 81““: at each point along the span. This result suggests that the closer proxinity of the channel walls to the D-section enhances this effect. in the channel downwind of the obstruction. causing a higher relative anpli- tude of oscillation (for y > 0.3d) at the shedding frequency. Figures 31-37 illustrate how 3 and ‘30” 830' ii. a. streanwise direction. These data were taken at the nid-plane between the side wall of the obstruction and the channel well except for the r I 2.3 data case where data were taken at y I d). The results show that. aside fron the initial free strean turbulence. the najority of the fluctuation intensity is at the shedding frequency for nuch of the even- "nee month” of 1|“. for all r values (Figure 36). it is evident that the growth of the oscillation is suppressed until the fluid has noved farther downstrean for the r I 0.4 case. It is strik- ing to note that the nagnitude of to“ 1. quite “.11 in a. ”pro.“ channel. 8 3 1d. This indicates that there is virtually no upstrean feedback. fron the oscillatory notion to the approach flow. 23 3.3.2 Spanwise Correlations of the Fluctuating Velocity and a Flow lodel for the Oscillatory lotion. A cross correlation. at an x - location of 2.23d. provided sus- tantial insight into the oscillatory notion. These data were acquired for an r value of 1.0. One wire. 'A'. was held fixed outside the shear layer (y I 1.23d) and the other wire. '3'. was traversed across the channel (x r-aining constant). The value of the cross correla- tiOI- W. at 1: I 0 (zero tine delay) was recorded at each staion of the traverse. The results (Figure 37) show that there was. in general. a nega- tive correlation when the wires were on opposite sides of the channel. It is inferred that large vortices are being alternately shed fron the wake region and that the 180-degree phase shift. in the oscillatory flow of the two sides. is a result of this alternate shedding. The results also show an opposite trend. for the sign of the correlation function in the neighborhood of the centerline and aft of the D-section. These observations are represented in the conprehensive flow nodel that is presented below. Figure 38 presents a flow nodel. for one instant in tine. that is conpatable with the present observations. This sketch depicts a stre-line pattern sinilar to those shown by Roshko [1] and by Griffin and Rsnberg [8]. Although their sketches illustrate flow around a cylinder. it is asst-ed that the vortex fornation would be sinilar in 24 the wake of a D-section. Note that on the side of the D-section fron which the vortex is being shed (Figure 38). the separated streanline would be closer to the side well than its counterpart on the other side of the obstruc- tion. Hence. the velocity at '1' will be relatively high and the velocity at '4' will be relatively low. For reference purposes. the flow at '1' is considered to be in a "nozzle” while that at '4' is considered to be in a "diffuser." The specific features of the vortex fornation. shown in Figure 38 for the sane conditions as the cross correlations of Figure 37. are conpatable with the reversed sign of the cross correlation in the neighborhood of the wake-centerline. Specifically. the velocity at '2' is lower and the velocity at '3' is higher. than their respective tine nean values. (A qualitaive sense of their tine nean values can be gained by reversing the streanline pattern of Figure 38 and averaging the inplied velocity values.) This flow nodel can also be used to describe the distribution of the relative intensity of the oscillations: l../;..(..,), ‘3 ‘30., in Figure 30. It is plausible to assert that the relative strength of the nozzle-diffuser effect is increased as the r value is decreased. That is. the oscillatory flows in regions '1' and '4'. are nore uni- fornly distributed across the span for the snaller r values. Conversely. it is also reasonable to infer. fron the flow nodel. that the confining wall restrains the notion of the separation streanlines. In the sense. the streanlinea are not as ”free" to alter their posi- 25 tions as the channel walls approach the obstruction. Hence. the larger oscillatory intensity values for the larger r values (Figure 49) are conpatable with this flow nodel. Another aspect of the flow nodel is the observation that the tine dependent streanline pattern is only considered to exist downstrean of the obstruction's aft edge. Hence. a relatively steady flow exists ‘9! 8 $11.3d as shown by the 730.c(x/d) results of Figures 31-36. 3.3.3 Stre-wise Correlations of the Fluctuating Velocity A series of experinents were perforned to deternine the convec- tion speed of the' structures being shed fron the D-section. One hot-wire probe was held fixed at a location such that: 1.) the x-coordinate was the position at which ‘2‘. reached . naxinun on a. wake axis (y I 0) 2.) the y-coordinate was the position at which I. 0 '48 I “81- [T " l“lid/2]. (Note that the x-position. as given by 1.). is the 11? value defined by Floor and Gerrard [21].) The other wire was traversed fron an x-location corresponding to the nininun pressure along the wake centerline. to the channel exit (the y-coordinate of the second wire r-ained the sane as that of the first wire). Figure 39 shows the phase shift (deternined fron a cross spectra analysis on the Genrad) as a function of x/d. It is interesting to note that the phase shift does not quite go to zero when the wires are 26 at the sane xrlocation. Since the wires were separated by a distance of approxinately 0.2d in the x-direction. it appears that three-dinensional effects are inportant in that the vortex structure nay not be vertically aligned in the channel. "Least squares" fits to the data points. for each r value. are shown in Figure 39. The fits are based on the data for x 2 0,24 downr strean of the fixed wire. Since only one data point not this criterion for r I 0.4. no straight line is indicated. It is consi- dered to be instructive that the x 2 0.24 condition was necessary: there is. apparently. an adjust-ant tine as the flow develops fron its fornation location. It is possible to calculate a convection velocity of the passing structures fron the wavelengh. 1. (derived fron the slope of the best fit straight line) and the shedding frequency. o., uconv ' 1's (3) The results of these calculations are shown in Figure 60. The convec- tion speed is relatively constant (Uhon’ a.o.5n’) over the range of r: 00‘ s t ‘ 2.5. Shuilar experinents were executed with two wires which were nainr tained at a fixed distance apart (in the stresnwise direction) and for which both wires were traversed in the stresnwise direction. The results of this experinent also showed that the convection speed was noninally constant for all r. The addition of a channel extension had 27 no effect on the results of these experinents. indicating that the inposition of atnospheric pressure at the channel exit did not have any effect on the convection speed. 3.3.4 The Strouhal Nunber The previous sections have dealt with the anplitude of the oscil- lations. This section will address the equally significant aspect of the frequency of the oscillations. The shedding frequency was deter- nined by perforning a spectral analysis of the hot-wire signal. Since the spectral intensities in the frequency range which define ‘osc (.. described in section 3.3.1) exhibited a narrow and a synnetric distri- bution. "the" shedding frequency. “e. was selected as the one which corresponded to the naxinnn point in the spectrun. A Strouhal nnnber (St) was calculated for each r value: at . ..d’ufippr000h° (6) Since the velocity was not constant over the span of the channel at the end of the contraction. the characterisic velocity. "approach: ... obtained by averaging the velocity over the span. By nornalizing the frequency by the average approach velocity in the channel and by the obstruction width. the non-dinensional St was deternined. Figure 61 shows that the Strouhal nunber is not constant for all r but decreases nonotonically as the r values is increased. Again. the addition of the channel extension had little. or no. effect on the results. Sinilar trends for St(r) have been docunented for a variety of bluff cylinders (Tozkas [14]. Chen [13]. Shaw [l6]. Toebes [17] and Richter and Naudascher [18]). 3.4 The Universal Strouhal Nunber The conventional Strouhal nnnber (Figure 61 for the present study) is an appropriate nornalized frequency for the boundary value problen of flow past a cylindrical obstruction. It has been well docunented (e.g.. Rayleigh [3]) that the shedding frequency is depen- dent on the geonetry of the flow and on the Reynolds nunber. As sunnarized by Bearnan and Grahan [3]. it has been hypothesized that: "all vortex-streets possess a sinilar structure and that a universal Strouhal nunber can be defined which is independent of the body generating the wake." The results presented in the following sub-sections concern the characterization of the oscillatory flow in terns of universal Strouhal nunbers as defined by Roshkc [7] and Bear- nan [11]. The effect of the confining walls on the parsneters of the universal Strouhal nunber is discussed and. as a result of the present investigation. a new definition of St‘ is proposed. 29 3.4.1 St‘ Based on Roshko's Criteria As outlined in Chapter 1. Roshkc [7] proposed an analytical solu- tion for the deternination of a universal Strouhal nunber. Recall that Roshko's [7] universal Strouhal nunber was defined as 8t; - “wag, (7) where d' is the lateral distance between the free streanlinea (deter- nined using a Notched-Hodograph technique) at the location where the vortices are being forned and U1, 1. the velocity .10., the 13.. streanline at the separation point. Roshkc [7] applied the concept of a universal Strouhal nunber to flow past a variety of cylindrical (circular and others) obstructions. The separation point is often not well defined' for nany geonetries (e.g.. circular cylinders) and therefore. it nay be difficult to neas- ure the velocity accurately in the neighborhood of the separation point. Therefore. Roshkc [7] adopted a plausible alternative for the velocity scale. The velocity U‘, 'hich 1. aggingd g. 05 - u.[1 - c,,)1’2 (8) where C9. is apparently derived fron a single pressure neasurenent on tho back Of ‘30 0711140! I“ “a is the freestrean approach velocity. was used in his evaluation of St‘. 30 For the present study. the separation point was well defined and th' P‘°"“‘° (P3). neasured at a tap 0.123d upstrean of the trailing edge of the D-section. allows a good approxination to be nade of the of, value. Nanely. U5 I’Uf. and U8 has previously been defined; see Equation 2- Th. VII“. 0‘ U. was held constant (11.64 n/sec) for all r values. The length scale for the present study. is derived fron neasure- nents that conforn to an extension of Roshko's [7] original fornulation by Calvert [22]. Calvert [22] hypothesized that d' was equivalent to the span between the naxina of velocity fluctuations at th‘ 103-Iti0fi IOIIth- 3(13). For the present investigation. a hot-wire annenoneter was used to deternine but?) for a. “V. r values [Figure 62 (squares)]. Hence. h(lF) ... gggd g. the nor-glizin‘, length for the Roshkc [7] Strouhal nunber: St; .. ‘ppligd to the present study. The fornation length was defined using the suggestion. 0‘ 303330 [7] th't 13 is the location at which the pressure reaches a nininnn value along the wake-centerline. Figure 63 (squares) shows the variation of 1F with gggpggg to r. The universal Strouhal nunber was deternined for each confinnent ratio using the Roshkc [7] criteria; see Figure 64. It is interesting ‘9 30*. th‘t ‘59 "1‘9 0‘ 3t; I 0.179 for r I 2.3 agrees quite well with the results published by Roshkc [7] for a variety of cylinders in a relatively wide approach flow. However. the present data also indi- cate tilt 3t; increased as the value of r was decreased. 3] 3.4.2 St‘ Based on Bearnan's Criteria Bearnan [11] proposed a nodification to the Roshkc forn of the universal Strouhal nunber; nanely. he suggested that it could be defined as 8t; - ...bmb (9) 'hItO 0; is as above and b is the lateral spacing between vortices (Rronauer [12]). He applied this concept to the flow past a variety of cylinders including a body with a blunt trailing edge (Bearnan [10]: see Chapter 1). that was sinilar to the D-section. 13' 3‘; value was conputed for the present study by using the spatially averaged base pressure (Figure 13) to calculate "b using the following forn of Bernoulli's equation: - 1 2 The 308i1t8 0‘ u5(r) are presented in Figure 63. It is noteworthy 33“ Us aaynptotically approaches a constant value (05 I1.06U.) as r is increased. while for snall r. 05 approaches 0,. Th. '31“. 0‘ llup) [Figure 62 (triangles)] was used to 'approxi- nate b (Bearnan 1967) and was deternined in the sane nanner as in Section 3.4.1. However. for the present study. the fornation length 32 was defined as: the point along the wake-axis where 12.. ro‘ch.‘ ‘ naxinun value (Floor and Gerrard [21]). Bearnan's [10] definition of 13 was slightly different (i.e.. 13 was the point where h reaches a nininnn) but cursory experinents indicated that these two definitions yielded. noninally. the sane value of 1F [Fi‘nro 53 (trign31.3)1. These data also revealed. for r > 0.4. that the naxinun fluctua- tion intensity in the entire flow field was found at the location y I Mlp)/2) for a given r value. This observation is in agreenent with the results of investigations by Bearnan [10] and Schaefer and Eskina- zi [23]. Figure 66 shows the variation of St; 'ith r. The .g.. ‘gngr.l trend for St; was observed for St; (i.e.. St. increased on r 400:0!804)3 however. 3‘; was noninally 13$ snaller than Sti. The snaller values are attributed to two factors: "b ... .t..t.r ‘3‘, 0. (Figure 63) and h was snaller at 13(32..-nax) thgn .g 1F(P-.in) (31‘- ‘10 ‘2). 13' 1038th 1p(P-nin) nay have been influenced by the inposition of atnospheric pressure at the channel exit since the nini- nun pressure along the wake-axis nay not occur at the sane position in a longer channel (i.e.. without this boundary condition). Once again. ‘30 Q‘IItit'ti'O '31“. 0‘ 3t; I 0.137. for the r I 2.3 case. is in agreenent with previously published results. 33 3.4.3 The Effect of Confining Valle on St‘ The results of the investigation of the universal Strouhal nunber indicate that the confinenent ratio has a dranatic effect on two of the inportant paraneters of the flow field. The fornation length (13). either calculated ala Roshkc [7] or Bloor and Gerrard [21]. increased as r decreased (Figure 62). The anplitude of the oscilla- tions at the fornation location are also observed to be a function of r. The decrease in this anplitude as r decreased is evident fron the hot-wire data traverses inplenented to deternine h(lF) (Section 3.4.2): see Figure 66. Recall that these values of‘hb. .3. the larg- est fluctuation intensities in each particular flow field. (This result was inconclusive at r I 0.4 since the peaks of the RlS distri- butions were relatively flat.) It is conjectured that these effects are interrelated; nanely. the delay in the fornation of the vortices (1F increasing .. r 1. decreased) nay be responsible for the decreased naxinnn anplitude of the oscillations. This conjecture is based on the possibility that the deflections in the separated streanlinea (proposed in the physical nodel of Section 3.3.2: see Figure 38) are linited by the increased fornation length. Chan [13] noted that the oscillatory flow was strongly influenced when the r value was sufficiently snall; be con- cluded that: "the regular shedding nechanisn giving rise to large vortices. which involves an interaction of the two shear layers. is greatly inhibited by the restraint of the conduit walls." 34 A universal Strouhal nunber has been deternined for flow past a circular cylinder (sub-critical Re) for a variety of confinenent ratios by Richter and Naudascher [18]. They concluded that St‘ I 0.16 for the range of confinenent ratios tested. The St. for their data. presented in Thble 3.1. was calculated using an analytical approach derived fron Roshko's [7] study. TABLE 3.1 Variation of St. with r d/2g r St. 1" 2.3 0.130 1/4 1.3 0.133 1/3 1.0 0.137 1/2 0.3 0.162 Vhile it is clear that 0.16 provides a noninal value for their neas- wants. it appears that there is a systenatic trend in the data which is not unlike that found in the present study for St; .,4 gt; (i.e.. increasing St. with decreasing r). 3.4.4 A New st‘ As discussed in Sections 3.4.1 and 3.4.2. the conventional neans of calculating St. for r I 2.3 yielded results that were in agreenent with prior. unbounded flow studies. However. the sane calculations failed to provide a constant value of St. for r 1 1,0, The paraneters which define St. are presented. in Table 3.2. as a function of r for 33 the present experinent. The data in this table show why St‘ was increased as the channel walls were noved closer to the obstruction. (Note: th. '11“. 01 U. was naintained as a constant for all r.) TABLE 3.2 Influence of r on 0., h(lp). gnu “b 1’ e.uir) h(lp)/d rib/u, 2.3 23.3 0.82 1.06 1.0 24.0 0.82 1.03 0.3 24.0 0.34 1.04 0.6 24.6 0.84. 1.03 0.4 26.7 0.86 1.02 If ti. VII“. 01 511p) [neasured where 32.. reaches a naxinun (y I 0)] is assuned to be the correct characteristic length. then it would be inappropriate to nornalize the frequency by U' in the present experinent. The constant value of U. would pgovid. . value of St. thtt 18 proportional to I.h(lp) and this product is clearly increasing as r decreases. Fron.Thble 3.2 it is obvious that nornalizing by 0‘ would tend to cause an even greater deviation in St‘. A characteris- tic velocity which would yield a constant St. nust increase as r is decreased. he inviscid core of fluid. between the shear layer and the side wall boundary layer at x I 2.23d for all r values. was identified in Section 3.2.2. Upon further investigation (evaluation of the pressure neasurenents on the upper surface). it was discovered that a sinilar 36 core of inviscid flow existed at the fornation length; the velocity in th. 0010 30810! It 1 ' 1p is terned “10' (Note that there was no dis- tinct flat region to the pressure distribution for r I 0.4; hence. the naxinun velocity across the span represented U1c in this c....) Fi‘ur. 68 shows that “1.6. increased as r decreased. Thus. the inviscid core velocity at the fornation length. displays the ”proper trend" required to nornalize ‘30 product. ueh(lp). such that a relatively constant St. is obtained. The use of the inviscid core velocity is also supported by its role in the physical process depicted in Figure 38. Specifically. this velocity. apparently. provides an appropriate scale velocity for the physical process of collecting the shed boundary layer vorticity into the large scale vortex notion. This collection process would appear to define the rate of vortex shedding since the tine between releases. of the large scale vortex.notions. would depend on the col- lection fit. (proportional t0 01¢) and the naxinun size [proportional to hu3)] of the vortex. A "new" universal Strouhal nunber: 8t:c - n.h(lF)/Uic (11) '30:. 13 was neasured at the point where 2n. reached a naxinun value (I ' 0). 'II calculated (Figure 69). The results indicate that 8t;c was relatively constant for all r; st;c . 0.202 e.0.003. As shown by 37 the data in Table 3.3 (sane as Figure 69). there is no suggested sys- tenatic trend of Stzc. Thble 3.3 Variation of St:c with t r 0.4 0.6 0.8 1.0 2.3 8t;c 0.206 0.200 0.204 0.197 0.203 If 01° is the proper nornalizing velocity for this particular flow. then this night explain why the vortex fornation was so severely linited at r I 0.4. Shair. et. al.. [13] and Chen [13] both noted that instabilities in the vortex street exist for severely confined flows. If the interaction of the shear layers were to be interrupted by the preacence of a side wall boundary layer. than there would be no 13318014 00" II 1p. Also. for sufficiently snall r values. the wake dynnnics would have to be characterized by a velocity other than nic- CHAPTER 4 SUlMIR! AND CONCLUSIONS A.systenatic series of experinents were carried out on the flow past a D-section geonetry. 0f prinary concern was the effect of the proxinity of the channel side walls on several aspects of the flow field. The nean flow was docunented by deternining several pressure and velocity distributions. The following conclusions are supported by the results of the present investigation of the nean flow: 1) As r is decreased. the pressure at the shoulder of the D- section increased (becane less negative). ii) The location of the separation point. near the shoulder. was not influenced by the r value. The length of this lo- calized separation bubble increased as r was decreased. iii) The size of the recirculatory zone. aft of the obstruction. increased as r is decreased. 38 39 iv) An inviscid core of fluid existed (between the shear layer and the outer wall boundary layer) for acne downstrean extent. A series of RlS neasurenents. spectral neasurenents. and velocity correlations were executed. Thus. the oscillatory content of the flow was well quantified aft of the D-section. A qualitative flow nodel of a tine dependent streanline pattern. based on RlS and velocity corre- lation data. is presented. The following conclusions are supported by the results of the present investigation of the oscillatory flow: i) The naxinun oscillation anplitude increased as the r value was increased. This increase. however. occurred irregularly eeeh thet "pleteene' in the ‘0“(nax) - f(r) distribution were observed. ii) As the value of r was decreased. the unifornity of the "dif- fuser-nozzle” effect was increased such that ‘3 [3. (nnx) e e fpwas increased across the span of the channel: 0.3d I y i (d/2 + g). iii) The convective speed of the coherent structures (vortices) renained constant (“conv I 0.30.) for the r values of the present study. iv) The conventional Strouhal nunber ['ed/napprcach] incgggggd 4O dranatically as r was decreased. Finally. the factors involved in "the” universal Strouhal nunber were nade the subject of a detailed investigation. This investigation relied heavily upon the earlier work of Roshkc [7] and Bearnan [11]. The following conclusions are supported by the results of the present investigation of St‘: i) The universal Strouhal nunber. based on either the criteria of Roshkc [7] or Bearnan [11]. increased as r was decreased. ii) The several choices. for the definition of the length scale. Mlp). to be used in the universal Strouhal nunber did not provide different trends of St. I St.(r). Hence. a new velocity scale. that would provide a constant universal Strouhal nunber. was sought. iii) A universal Strouhal nunber based on the inviscid core vel- ocity (at the fornation lenghth) was found to be approx- inately constant for r 2 0,4; its vglue was S‘Ie .. 0.202. 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O 4 > a VI 4 4 > a o H o I i ‘ F a O Ilsa.8 4 I Ilsadad 3.0 HANSEN 36 I 06$ 1 ,0. 99.4 I 3-. has? 100 u Aunt wanna acauoo>uou «a coda-«uu> am GEN ; _ Sax" .oe ouaumm 101 u new: Hoalua uaaaouum no douucmu¢> .nw ouunum Gmé 102 u gums Andy: «a no««-«u.> .«e ounaam Gaé 9:. 2.2.. «a... -4 35-5“.— .0 88.8 I smd II $4.8 II smé III sm.& I 1.... ..I. 84 103 a dud: gunned acme-luau mo acauamua> .mo ounuqm Gad EQN as...” 88.8 _ _ F _ _ _ _ _ _ _ _ _ asé. n— .u a II n. 4 4 sad .1 4 I ,II b a X 13:... .3 — -4 4 35-..} -o I r: 104 0’ I... O —- . h— ‘0 ‘0 FIII|IIIIIIIIIIIIIIIIII a s s 0! '1 8. 8 fl, a a 0.00 Variation of Stu with t Figure 64. 105 u an“: a: we aa«».«u-> .3 93.3 as... 1.... NSA fl .vsé 106 u gum: man «a acme-«u¢> .wo ounudm 8. m a N 9. v-l 3. & 9 [VI I I I I I I I I I I I I I I I I I I I I I I I 88.8 107 an an unaumanouum acmuanuonam .he ouuaam ‘ x Smd 84.8 smé ENG _ . . r _ . . . . _ _ . _ r _ _ . _ _ 3.8 § I 84:8 . § * § . I .9 9 § I . I b u .3... N N . n r290 m b N N N 4 I H . . a I as 2-. m m m m . I 2..» H 3.. to-.. WI I08 a An“. om: no acme-«u¢> .ao ouuuum as...” . 3a .33 8.... _____. _ _ _ _ I. a. I 8N8 I 84.8 .I 8m.8 o .I 88.8 Um: ,I 88..“ 109 oi . _— 8 —-8 01 fig 0 “—8. v1 9 .— ‘0 {O I.— IIIII IIIIIIIIIITIIIIIII 8° 9 8 a a. G! a 3.2 Q Q a. «0 Variation of St;c with r Figure 69. APPBRDICBS Tabla 3.1 Praaaur x/d -1.5000 -1.3750 -1.2500 "'1 o 1250 -1.0000 *0.8750 -0.7500 -0.6250 *0.5000 -0.4830 ‘0 o 4330 ‘093536 ’00 2500 ‘00 1294 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.0000 1.0000 -1.5000 -1.3750 ‘1 o 2500 “'1 0 1250 -1.0000 -0 o 8750‘ -0.7500 “‘0 0 625° ‘0 0 500° APHDHHX.A PRESSMHIDNLA 71d 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1294 0.2500 0.3536 0.4330 0.4830 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.3125 0.1875 0.0625 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875 ‘HO 0 valua £0: 1' " 1'0 (P-P.) I (2,14,) 0.7006 0.7006 0.7267 0.7535 0.7825 0.8368 0.8795 0.8849 1.0022 0.9389 0.5645 0.0916 -0.3385 -0 05724 -0.0785 -0.0047 0.0149 0.0185 0.0214 0.0026 -0.0336 -0.1509 -0 o 1 546 -0.1582 0.6876 0.7035 0.7137 0.7383 0.7586 0.7948 0.8237 0.7795 0.7823 “'1 05000 -1.3750 ~1.2500 “‘1 0 1250 ~1.0000 -0 08750 -0.7500 -0 06250 “005000 ‘003750 —1.5000 -1.3750 -1.2500 ‘1 0 1250 -1.0000 -0.8750 ‘0 07500 -0.6250 -0.5000 -0.3750 -0.2500 -0 0 1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 -1.5000 -1.3750 -1.2500 -1.1250 ‘1 0 0000 ~0.8750 -0 07500 ~0.6250 -0.5000 -0.3750 “'0 0 2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 -1.5000 ‘1 03750 "I 0.3750 0.3750 0.3750‘ 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.7500 0.7500 0.6695 0.6782 0.6905 0.6992 0.7072 0.7086 0.6992 0.6151 0.4875 0.2011 0.6615 0.6601 0.6652 0.6623 0.6579 0.6507 0.6232 0.5222 0.3939 0.1872 ”000638 -0.2539 ‘002518 -0.0222 0.0395 0.0693 0.0554 0.0631 0.0492 0.0326 -0.0361 0.6572 0.6550 0.6557 0.6514 0.6434 0.6275 0.6101 0.5056 0.4008 0.2025 -0.0285 ~0.1873 —0.1727 ~0.0195 0.0381 0.0525 0.0596 0.0603 0.0541 0.0263 ”000167 0.6304 0.6232 -1.2500 ‘1 0 1250 -1.0000 ‘008750 -0.7500 -0.5000 -0.3750 -0.2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 . 1.0000 ‘1 05000 -1.3750 -1.2500 “'1 0 1250 -1.0000 -0.8750 -0.7500 -0.6250 -0.5000 -0.3750 -0.2500 '0 0 1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 -1.5000 ”'1 03750 ”'1 02500 -1.1250 "1.0000 -0.8750 -0.7500 -0.6250 ‘0 05000 ‘0 03750 -0.2500 ‘00 1250 0.0000 0.1250 "2 0.7500 0.6210 0.7500 0.6007 0.7500 0.5884 0.7500 0.5645 0.7500 0.5319 0.7500 0.4515 0.7500 0.3773 0.7500 0.2864 0.7500 0.1713 0.7500 0.0825 0.7500 0.0437 0.7500 0.0478 0.7500 0.0631 0.7500 0.0679 0.7500 0.0624 0.7500 0.0658 0.7500 0.0658 0.7500 0.0513 0.7500 0.0457 0.9375 0.6000 0.9375 0.5862 0.9375 0.5812 0.9375 0.5667 0.9375 0.5667 0.9375 0.5232 0.9375 0.4899 0.9375 0.4320 0.9375 0.3731 0.9375 0.3128 0.9375 0.2469 0.9375 0.1935 0.9375 0.1512 0.9375 0.1297 0.9375 0.1172 0.9375 0.1123 0.9375 0.0957 0.9375 0.0984 0.9375 0.0894 0.9375 0.0860 0.9375 0.0832 1.1250 0.5775 1.1250 0.5710 1.1250 0.5696 1.1250 0.5457 1.1250 0.5203 1.1250 0.4993 1.1250 0.4660 1.1250 0.4147 1.1250 0.3766 1.1250 0.3301 1.1250 0.2802 1.1250 0.2427 1.1250 0.2087 1.1250 0.1844 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 -1.5000 -1.3750 -L.2500 -1.1250 -1.0000 *0.8750 -0.7500 “'0 0 6250 ‘0 0 5000 -0 0 3750 ‘0 0 2500 ~O.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 "1 0 500° ‘1 0 3750 -1.2500 -1.1250 -1.0000 “'0 0 875° ‘0 0 7500 “'0 0 625° “‘0 0 5000 -0.3750 -0.2500 “'0 0 1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 2.2500 2.2500 2.2500 2.2500 2.2500 ”3 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 1.4687 0.0000 0.1250 0.2500 0.3750 0.5000 0.1616 0.1408 0.1255 0.1276 0.1193 0.1172 0.1144 0.5674 0.5573 0.5515 0.5261 0.5080 0.4885 0.4631 0.4085 0.3759 0.3377 0.3003 0.2670 0.2372 0.2122 0.1782 0.1671 0.1491 0.1428 0.1401 0.1290 0.1297 0.5667 0.5536 0.5428 0.5254 0.4979 0.4848 0.4639 0.4126 0.3890 0.3564 0.3121 0.2857 0.2691 ' 0.2510 0.2205 0.1851 0.1588 0.1387 0.1109 0.1082 0.1227 0.1311 0.1435 0.1741 0.2136 0.2503 2.2500 .2.2500 2.2500 2.2500 2.2500 2.5000 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 3.0000 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 fl“ 0.6250 0.8750,”. 1.0000 1.1250 1.2500 0.0000 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 0.0000 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875 0.1875 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.2753 V0.3010. 0.2975 0.2913 0.2788 0.2032 0.2351 0.2323 0.2483 0.2600 0.2864 0.3065 0.3197 0.3259 0.3301 0.3301 0.3336 0.3343 0.2649 0.2926 0.2906 0.2871 0.2933 0.3051 0.3155 0.3232 0.3245 0.3266 0.3308 0.3349 0.3377 ‘0 0 1248 -0.1560 "002275 -0.2393 -0.2074 -0.1422 -0.0680 0.0187 -0.1144 -0 0 1782 ‘002455 ‘002462 -0.1956 -0.1186 -0.0222 0.0409 "0 0 1 1 17 -0.1491 -0.1623 -0.1214 -0.0562 0.0250 0.0964 2.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 1.1250 1.2500 1.3750 1.5000 1.6250 “5 0.3750 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5312 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.5625 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.9375 0.9375 0.9375 0.9375 0.9375 0.9375 0.9375 0.9375 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.3125 1.4687 1.4687 1.4687 1.4687 1.4687 0.1352 -0.0499 “000610 -0.0319 0.0055 0.0742 0.1234 0.1720 0.2101 ~0.0319 -0.0326 ~0.0097 0.0465 0.0825 0.1470 0.1852 0.2143 0.0388 0.0548 0.0818 0.1089 0.1422 0.1810 0.2157 0.2496 0.0888 0.1019 0.1179 0.1428 0.1761 0.1976 0.2372 0.2531 0.1207 0.1324 0.1505 0.1643 0.1817 0.2060 0.2344 0.2600 0.1366 0.1345 0.1401 0.1595 0.1734 0.1983 0.2240 0.2406 0.1283 0.1373 0.1567 0.1754 0.1956 1.7500 1.8750 2.0000 116 1.4687 1.4687 1.4687 0.2302 0.2379 0.2600 Thblo 3.2 Pressure values for r - 0.8 :16 -0.5000 “004830 -0.4330 -0.3536 “002500 -0.1294 ‘ 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.0000 1.0000 “105000 -1.3750 “102500 “101250 -1.0000 -0.8750 -0.7500 -1.5000 -1.3750 “102500 “101250 -1.0000 -0.8750 -0.7500 “105000 “103750 “102500 “101250 -1.0000 -0.8750 -0.7500 -1.5000 “103750 -1.2500 “101250 -1.0000 “008750 yld 0.0000 0.1294 0.2500 0.3536 0.4330 0.4830 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 ‘ 0.3125 0.1875 0.0625 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.4062 0.4062 0.4062 0.4062 0.4062 0.4062 (P-P.) /(Pp1-P.’ 1.0191 0.9130 0.5834 0.1496 “002475 ~0.4920 “000464 0.0196 0.0289 0.0305 0.0356 0.0191 “000108 “000954 -0.1067 -0.1052 0.7108 0.7237 0.7335 0.7629 0.7866 0.8294 0.8748 0.7159 0.7159 0.7299 0.7484 0.7665 0.8072 0.8433 0.7051 0.7067 0.7087 0.7201 0.7319 0.7505 0.7531 0.7046 0.6974 0.7056 0.7087 0.7129 0.7170 “007500 -1.5000 -1.3750 -1.2500 “1 0 1250 -1.0000 -0.8750 -0.7500 -1.5000 -1.3750 -1.2500 -1.1250 “1 00000 -0.8750 “007500 “1 05000 -1.3750 “1.2500 “1 0 1250 -1.0000 ~O.8750 “007500 “1 05000 -1.3750 “1 0 2500 ~1.1250 -1.0000 -0.8750 *0.7500 -1.5000 -1.3750 “1 0 2500 -1.1250 -1.0000 -0.8750 -0.7500 -1.5000 “1 0 3750 -1.2500 -1.1250 “1 00000 -0.8750 “00 7500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.5000 117 0.4062 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.8750 1.0000 1.1250 1.2500 0.0000 0.7129 0.6876 0.6850 0.6896 0.6860 0.6783 0.6680 0.6473 0.6659 0.6633 0.6571 0.6520 0.6350 0.6231 0.5870 0.6504 0.6443 0.6329 0.6138 0.6035 0.5829 0.5463 0.6303 "" 0.6236 0.6076 0.5911 0.5751 0.5530 0.5169 0.6185 0.6107 0.5989 0.5793 0.5648 0.5447 0.5081 0.6092 0.6180 0.5782 0.5684 0.5473 0.5323 0.5029 0.1362 0.1386 0.1639 0.2004 0.2510 0.2695 0.2850 0.2879 0.2855 0.2904 0.2174 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 12.7500 3.0000 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 “0 0 6250 -0.5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 “0 0 5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 “8 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 0.0000 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.4062 0.2612 0.2617 0.2641 0.2889 0.2894 0.3035 0.3142 0.3273 0.3448 0.3502 0.3502 0.2894 0.3127 0.3098 0.3050 0.3069 0.3127 0.3264 0.3317 0.3380 0.3482 0.3541 0.3604 0.8964 “000817 “00 1138 -0.1751 -0.1970 “00 1834 -0.1357 -0.0564 “000024 0.8317 0.8517 -0.0817 “00 1231 -0.1498 -0.1946 -0.1688 “00 1367 “000613 0.0146 0.7043 0.5948 -0.0652 -0.1133 -0.1605 -0.1352 —O.1085 -0.0501 0.0156 0.0910 0.6411 -0.5000 -0.3750 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 “0 0 6250 -0.5000 -0.3750 -0.2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 “0 0 6250 -0.5000 -0.3750 “0 0 2500 “0 0 1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 ‘H9 0.4062 0.4062 0.4062 0.4062 0.4062 0.4062 0.4062 0.4062 0.4062 0.4062 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.5078 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.6562 0.5097 0.2821 -0.0589 “000914 -0.1031 -0.0851 “000413 0.0282 0.0910 0.1396 0.5837 0.4441 0.2680 0.0054 -0.2208 -0.2179 0.0199 0.0686 0.0798 0.0734 0.0764 0.0666 0.0613 -0.0302 -0.0287 -0.0457 -0.0306 -0.0073 0.0404 0.0963 0.1391 0.1960 0.5292 0.4470 0.3225 0.1673 0.0516 0.0107 0.0462 0.0725 0.0871 0.0778 0.0764 0.0691 0.0477 0.0272 0.0165 0.0272 0.0418 0.0710 0.1021 0.1518 0.1955 0.2335 “0 0 6250 “0 0 5000 -0.3750 -0.2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 “00 6250 -0.5000 -0.3750 “0 0 2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 “0 0 3750 -0.2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 120 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.8125 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 0.9687 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 0.4956 0.4309 0.3536 0.1863 0.1425 0.1206 0.1051 0.1085 0.0900 0.0885 0.0783 0.0681 0.0647 0.0608 0.0696 0.0730 0.0851 0.1119 0.1474 0.1761 0.1994 0.2286 0.4839 0.4339 0.3677 0.3020 0.2456 0.1994 0.1741 0.1469 0.1362 0.1094 0.1046 0.0939 0.0851 0.0890 0.0866 0.0987 0.1119 0.1221 0.1522 0.1790 0.2111 0.2310 0.4737 0.4280 0.3813 0.3264 0.2748 0.2403 0.2077 0.1800 0.1605 0.1289 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 “'0 o 3750 -0.2500 ~0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 121 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 1.2813 0.1138 0.1041 0.0900 0.0895 0.0968 0.0997 0.1021 0.1133 0.1386 0.1712 0.2038 0.2378 0.4669 0.4348 0.3872 0.3434 0.3064 0.2709 0.2554 0.2101 0.1469 0.1187 0.1065 0.0968 0.0885 0.1036 0.1104 0.1196 0.1459 0.1678 0.1834 0.2164 0.2218 0.2597 122 Table 3.3 haunt-0 values for r 8 0.6 xld -0.5000 -0.4830 -0.4330 ‘003536 -0.2500 -0.1294 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.0000 1.0000 -1.5000 ‘1 03750 ‘1 02500 "1 0 1250 -1.0000 -0.8750 -0.7500 -1.5000 ‘1 03750 -1.2500 “'1 0 1250 -1.0000 -0.8750 -0.7500 -1.5000 ~1.3750 -1.2500 ‘1 0 1250 -1.0000 “008750 “007500 “1 05000 ‘1 03750 -1.2500 -1.1250 ‘1 00000 -0.8750 yld 0.0000 0.1294 0.2500 0.3536 0.4330 0.4830 0.5000 0.5000 0.5000 . 0.5000 0.5000 0.5000 0.5000 0.3125 0.1875 0.0625 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1250 0.1250 0.1250 0.1250 0.1250 0.1250 0.1250 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 1.0021 0.9091 0.6333 0.2514 ‘00 1081 .—0.3731 “000003 0.0442 0.0414 0.0396 0.0351 0.0218 -0.0076 -0 0 0531 -0.0545 -0.0528 0.7635 0.7652 0.7740 .0.7848 0.8111 0.8451 0.8885 0.7726 0.7747 0.7803 0.8045 0.8213 0.8514 0.8871 0.7677 007645 0.7733 0.7925 0.7922 0.8104 0.8297 0.7579 0.7551 0.7582 0.7565 0.7638 0.7701 -0.7500 -1.5000 -1.3750 "1 0 2500 -1.1250 -1.0000 -0.8750 -0.7500 “1 05000 I “'1 03750 -1.2500 ~1.1250 -1.0000 ~0.8750 -O.7500 -1.5000 -1.3750 "'1 02500 -1.1250 -1.0000 -0 0 8750 -0.7500 -1.5000 -1.3750 -1.2500 ~1.1250 -1.0000 -0.8750 -0.7500 -1.5000 -1.3750 “'1 02500 -1.1250 ~1.0000 -0 08750 -0.7500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.5000 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 123 0.3750 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.8750 1.0000 0.0000 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 0.7666 0.7407 '0.7379 0.7341 0.7334 0.7246 0.7292 0.7005 0.7295 0.7239 0.7211 0.7103 0.7008 0.6872 0.6620 0.7159 0.7099 0.7015 0.7005 0.6770 0.6567 0.6273 0.7068 0.7005 0.6917 0.6844 0.6623 0.6424 0.6095 0.7036 0.7022 0.6910 0.6784 0.6595 0.6340 0.5972 0.0372 0.0397 0.0555 0.1114 0.1616 0.1746 0.1943 0.2062 0.1605 0.2530 0.2509 0.2406 0.2733 0.2551 0.2691 0.2902 0.3064 0.3183 3.0000 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 ~0.6250 -0 0 5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0 0 6250 -0.5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 ~0.6250 -0.5000 -0.3750 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 -0.3750 124 0.0000 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1250 0.1250 0.1250 0.1250 0.1250 0.1250 0.1250 0.1250 0.1250 0.1250 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.3750 0.5156 0.5156 0.5156 0.2997 0.3334 0.3264 0.3148 0.3099 0.3173 0.3306 0.3426 0.3489 0.3623 0.9301 0.9951 -0.0229 -0.0299 -0.0355 -0.0457 “0.0840 -0.0907 ‘000886 -0.0671 0.8844 0.9174 -0.0176 -0.0299 “000334 -0.0679 “000928 -0 0 1026 -0.1097 -0.0728 0.8001 ~ 0.7414 ~0.0215 -0.0292 -0.0506 -0.0517 -0.0904 -0.0967 ‘001016 _ “'0 0 0759 0.7115 0.6012 0.3598 -0.0179 ‘000169 -0.0296 “000408 -0.0524 -0.0591 -0.0299 0.0021 0.6353 0.5372 0.3475 -0.2500 “001250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 -0.3750 -0.2500 -0.1250 I 000000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 “006250 “005000 -0.3750 -0.2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 125 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.5156 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.0804 “001501 -0.1522 0.0467 0.0822 0.0857 0.0716 0.0710 0.0594 0.0491 -0.0035 -0.0029 -0.0050 ~0.0106 -0.0102 “000060 0.0172 0.0485 0.0896 0.5987 0.5116 0.3787 0.2024 0.0594 0.0116 0.0594 0.0766 0.0861 0.0664 0.0660 0.0565 0.0344 0.0196 0.0119 0.0130 0.0165 0.0242 0.0404 0.0660 0.0941 0.1240 0.5682 0.4961 0.3991 0.2842 0.1886 0.1296 0.1159 0.1015 0.1022 0.0752 0.0752 0.0639 0.0505 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -O.6250 “0 05000 “0 0 3750 ‘0 0 2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 “00 6250 -0.5000 -0.3750 “0 02500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 126 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 0.8750 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0432 0.0369 0.0376 0.0432 0.0463 0.0586 0.0815 0.0892 0.1282 0.5538 0.4912 0.4135 0.3299 0.2523 0.1971 0.1655 0.1384 0.1233 0.0941 0.0853 0.0741 0.0597 0.0488 0.0397 0.0319 0.0323 0.0351 0.0439 0.0695 0.0920 0.1226 0.5481 0.4919 0.4237 0.3499 0.2842 0.2326 0.2027 0.1778 0.1430 0.0924 0.0847 0.0600 0.0281 0.0109 0.0105 0.0175 0.0298 0.0439 0.0639 0.0948 0.1293 0.1528 127 Ihble 3.4 Pressure vslnes for r - 0.4 xld -0.5000 “004830 -0.4330 -0.3536 -0.2500 “00 1294 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.0000 1.0000 “1 0 5000 -1.3750 -1.2500 “1 0 1250 -1.0000 -0.8750 -0.7500 “1 05000 -1.3750 “1 02500 “1 0 1250 ~1.0000 -0.8750 “007500 -1.5000 “1 03750 -1.2500 -1.1250 -1.0000 “008750 yld 0.0000 0.1294 0.2500 0.3536 0.4330 0.4830 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.3125 0.1875 0.0625 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0781 0.0781 0.0781 0.0781 0.0781 0.0781 0.0781 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 (24,) I (291-11,) 1.0263 0.9747 0.7582 0.4453 0.0918 “002085 0.0631 0.0733 0.1038 0.0858 0.0383 0.0360 “0 0 0194 “000235 -0.0368 -0.0201 0.8703 0.8677 0.8730 0.8912 0.8964 0.9225 0.9564 0.8703 0.8651 0.8703 0.9017 0.9147 0.9277 0.9486 0.8625 0.8677 0.8677 0.8782 0.8834 0.9069 “007500 “105000 ~1.3750 “102500 “101250 *1.0000 -0.8750 ~0.7500 “105000 “103750 -1.2500 “101250 -1.0000 -0.8750 -0.7500 “105000 “103750 -1.2500 -1.1250 -1.0000 -O.875O “007500 -1.5000 ~1.3750 “102500 “101250 -1.0000 “008750 -O.7500 “105000 -1.3750 -1.2500 “101250 -1.0000 -0.8750 ~0.7500 -1.5000 ~1.3750 -1.2500 “101250 -1.0000 ~0.8750 -0.7500 “1.5000 -1.3750 -1.2500 “101250 -1.0000 “008750 ~0.7500 “105000 “103750‘ “102500 “101250 128 0.1562 0.2344 0.2344 0.2344 0.2344 0.2344 0.2344 0.2344 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3906 0.3906 0.3906 0.3906 0.3906 ,0.3906 0.3906 0.4687 0.4687 0.4687 0.4687 0.4687 0.4687 0.4687 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.7031 0.7031 0.7031 0.7031 0.7031 0.7031 0.7031 0.7812 0.7812 0.7812 0.7812 0.9304 0.8651 0.8625 0.8651 0.8730 0.8730 0.8912 0.9069 0.8651 0.8651 0.8677 0.8703 0.8860 0.8834 0.8886 0.8625 0.8625 0.8599 0.8625 0.8625 0.8599 0.8521 0.8495 0.8469 0.8469 0.8469 0.8416 0.8390 0.8247 0.8390 0.8390 0.8312 0.8286 0.8234 0.8182 0.7921 0.8390 0.8390 0.8260 0.8260 0.8260 0.8077 0.7764 0.8286 0.8312 0.8234 0.8051 0.8077 0.7947 0.7712 0.8260 0.8260 0.8208 0.8129 ~1.0000 -0.8750 -O.7500 "'1 0 5000 -1.3750 -1.2500 ‘1 o 1250 -1.0000 -0.8750 ‘0 o 7500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.2500 2.5000 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 2.7500 3.0000 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 3.2500 -0.6250 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 ‘0 o 5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 129 0.7812 0.7812 0.7812 0.8594 0.8594 0.8594 0.8594 0.8594 0.8594 0.8594 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.8750 0.0000 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 '0.0000 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0781 0.0781 0.0781 0.0781 0.0781 0.0781 0.0781 0.0781 0.0781 0.0781 0.8077 0.7895 0.7660 0.8077 0.8129 0.8156 0.8025 0.7947 0.7816 0.7529 -0.0133 -0.0211 "0.0290 0.0074 0.0432 0.0555 0.0926 0.0189 0.0909 0.0921 0.1020 0.1322 0.1356 0.1518 0.1683 0.1929 0.1774 0.2479 0.2396 0.2285 0.2268 0.2415 0.2634 0.2828 0.2948 0.9565 0.0305 0.0324 0.0278 0.0256 0.0226 0.0292 *0.0098 -0.0081 0.9356 0.9634 0.0273 0.0258 0.0322 0.0243 0.0229 0.0172 -0.0034 -0.0179 -0.6250 -0.5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 ”‘00 6250 -0.5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0 o 6250 -0.5000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 “'0 03750 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 “'0 06250 -0.5000 ‘0 o 3750 -0 o 2500 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 130 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 0.1562 -0.1562 0.1562 0.1562 0.2344 0.2344 0.2344 0.2344 0.2344 0.2344 0.2344 0.2344 0.2344 0.2344 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3125 0.3906 0.3906 0.3906 0.3906 0.3906 0.3906 0.3906 0.3906 0.3906 0.3906 0.3906 0.4687 0.4687 0.4687 0.4687 0.4687 0.4687 0.4687 0.4687 0.4687 0.4687 0.4687 0.4687 0.5469 0.9113 0.9138 0.0373 0.0248 0.0258 0.0219 0.0248 0.0219 0.0027 ‘0 00128 0.8717 0.8334 0.0428 0.0278 0.0322 0.0209 0.0342 0.0179 -0.0029 -0.0243 0.8275 0.7595 0.0317 0.2844 0.0231 0.0572 0.0270 0.0064 “0.0054 -0.0113 0.7823 0.7091 0.5145 0.0351 0.0319 0.0312 0.0197 0.0270 0.0172 0.0150 0.0091 0.7499 0.6695 0.5005 0.2231 0.0302 0.0302 0.0297 0.0283 0.0344 0.0236 0.0268 0.0160 0.7236 -0.5000 -0.3750 ~0.2500 ‘0 o 1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 -0.3750 -0.2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 ~0.6250 -0.5000 -0 o 3750 "0 o 2500 -0.1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 131 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.5469 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.6250 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.6243 0.5047 0.2894 0.0759 0.0494 0.1351 0.1482 0.1531 0.1216 0.1187 0.1066 0.0860 0.0455 0.0450 0.0332 0.0489 0.0334 0.0351 0.0300 0.0241 0.0383 0.7052 0.6324 0.5147 0.3575 0.2037 0.1366 0.1509 0.1474 0.1543 0.1462 0.1224 0.1042 0.0818 0.0649 0.0543 0.0641 0.0526 0.0381 0.0314 0.0369 0.0327 0.0383 0.6921 0.6246 0.5201 0.3914 0.2725 0.2000 0.1769 0.1622 0.1634 0.1337 0.1219 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 -0.3750 -0.2500 -0 0 1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250" 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 2.0000 -0.6250 -0.5000 -0.3750 ’0 0 2500 “'0 0 1250 0.0000 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750 1.0000 1.1250 1.2500 1.3750 1.5000 1.6250 1.7500 1.8750 132 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7037 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.7312 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.3594 0.1031 0.0377 0.0713 0.0550 0.0459 0.0371 0.0243 0.0256 0.0233 0.0295 0.0263 0.6333 0.6134 0.5263 0.4206 0.3174 0.2467 0.2074 0.1311 0.1794 0.1541 0.1265 0.1079 0.0762 0.0636 0.0366 0.0157 0.0074 0.0115 0.0133 0.0221 0.0300 0.0334 0.6779 0.6140 0.5322 0.4344 0.3396 0.2636 0.2310 0.2061 0.2033 0.1676 0.1057 0.0590 0.0333 0.0364 0.0233 0.0290 0.0364 0.0346 0.0337 0.0452 0.0457 APPENDIX B DBMATION 0F ‘osc For nany positions aft of tho D-soction. a largo porcontago of tho fluctuation intonsity was prosont in a snall troqnoncy hand inclu- ‘1'0 01 ‘5' 3404413! froqnoncy. 0.. Figaro 3.1 roprosonts a typical onorgy spoctrun at a location downstroan of tho obstruction. By using tho Conrad.conpntor it was possihlo to obtain a graphic display of tho spoctra and to aocwratoly dotornino tho contor of tho posh froqnoncy '84 th": 4 '41“. 0‘ 0.. Tho Conrad was also nsod to calcalato tho oworall BIS (full pass) which is oqniwalont to tho aroa undo: tho spoctral cnrvo. Noto that. in addition to tho vory largo poah at 0., a“. ... . socond poak corrosponding to tho socond harnonic of tho shodding fro- qnoncy. 30.. This post was not ovidont in tho spoctra for y > 0.511. llowowor. as tho hot-wiro was novod fro- y a 0.54 toward a“ wako-oontorlino. tho nagnitndo of this posh incroasod until. at y - 0. tho socond poak was as largo or largor than tho ono at 11., 133 134 Tho Intoractivo Signal Analysis Progran of tho Gonrad systo- allowod tho nsor to soloct a spocific bandwidth of froqnoncios to bo displayod; soo Fignro 3.2. By choosing tho propor linits so as to inclndo tho ontiro poah. tho BIB of tho hot-wiro signal 913; this hgndlidth can thon ho calculatod. Tho RIB walno of tho first post (at th. 83044138 troqnoncy) '48 tOIIOdi ‘3‘. A sinilar oporation yioldod tho RIB waluo at tho socond posh. 32... if it oxistod. In ordor to roprosont tho flnctation intonsitios of tho oscilla- tory notion. both of thoso BIS walnos nnst ho inolndod. Tho sun of thoso walnos was tornod‘lo‘o .35 1. dgfingd .. 3°“ _ [71...” + 32.31112. (Noto: for y > 0.5d. ‘osc';o.°) 135 603031: on» «o and .330on hnuonm moot—ha. 1 «d can»; . 533 M: .33 . _1. . 1— 18. .3 _ _ _ _ _ _ _ _ _ _ _ G . A '51:? I36 4? 8.3025 :3ch ans.— uaon door—ma. < «.n 0.3.:— 1h0§>oz