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A study of circular couette flow by laser doppler measurement techniques Geach, Nigel Douglas Sinclair 1974

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A STUDY OF CIRCULAR COUETTE FLOW BY LASER DOPPLER MEASUREMENT TECHNIQUES by NIGEL D.S. GEACH B.A. Sc . , University of Br i t ish Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA FEBRUARY 1974 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study,. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f / ^ ^ A ^ L A / A - ^ * ^ t - L y -The U n i v e r s i t y o f B r i t i s h C o l umbia Vancouver 8, Canada Date /hrif l£ { /<? 74 i i ABSTRACT A laser Doppler velocimeter is constructed and used to make flow measurements in c i rcular Couette flow. The flow is created between concentric cylinders with a small gap-to-radius rat io , and measurements of the velocity profi les are made in both laminar and turbulent flow regimes. Distortion due to end effects is noted in the laminar case, but the turbulent case is shown to conform well to a three region model. A study of the mean velocity profiles allows estimates of skin f r i c t ion and Reynolds stresses. Turbulent velocity fluctuations are also estimated from the laser Doppler technique, and their intensity compared with existing results for plane Couette flow. i i i TABLE OF CONTENTS Chapter Page 1. INTRODUCTION 1 2. THEORY 3 2.1 Laminar Couette Flow . 3 2.2 Turbulent Couette Flow 4 2.3 Reynolds Stresses 6 3. INSTRUMENTATION 8 3.1 Background 8 3.2 Components . . . . . 8 3.3 Calibration . . . . 9 3.4 Signal Broadening 10 4. MEASUREMENTS AND RESULTS 13 4.1 The Flow Apparatus 13 4.2 Procedures 13 4.3 Analysis 15 4.4 Laminar Profi les . . 16 4.5 Turbulent Profi les 16 4.6 Spectral Broadening 17 4.7 Measurements of/ u'2 . . . . . . . 17 4.8 Measurement of Reynolds Stresses . . . . . . . 18 4.9 .Measurements of ^ w ' ^ 19 5. DISCUSSION . . . . . . 20 6. CONCLUSIONS 24 . REFERENCES . . . . . . . . . . . . . . . . . . . . . . 26 i v Chapter Page APPENDIX I - Exact Solution of the Navier-Stokes Equations for Laminar Circular Couette Flow • • 2 8 APPENDIX II - A Three Region Model for Turbulent Couette Flow . 32 APPENDIX III - Measurements of Reynolds Stresses by Laser Doppler Velocimetry . 37 APPENDIX IV - Theoretical Description of the Laser Doppler Velocimeter . . . . . 40 APPENDIX V - Tracker Calibration 43 APPENDIX VI - Data , . 45 V LIST OF FIGURES Figure Page 1.1 Typical LDV signals from particles of approximately uniform size . . . . . . . . . . . . 53 2.1 Theoretical velocity profi les for plane Couette flow 54 3.1 Reference beam operation 55 3.2 Formation of LDV fringe pattern through interference of intersecting laser beams . . . 56 3.3(a) Schematic i l lus t rat ion of the laser-Doppler system used . 57 3.3(b) LDV and Couette flow apparatus 58 3.4 Calibration curves for DISA tracker 59 3.5 Calibration of ambiguous broadening . 60 4.1 Schematic of Couette flow apparatus . 61 4.2 Laminar flow profi les . . . . 62 4.3 Laminar flow profi les . . 63 4.4 Turbulent flow profi les 64 4.5 Turbulent flow profi les . . . . . . . 65 4.6 Core region slope as.a function of Reynolds number . 66 4.7 Typical laminar flow spectrum 67 2 4.8 u turbulence intensities vs normalized position . . 68 4.9 Measurement of Reynolds Stresses 69 2 4.10 w1 turbulence intensities vs normalized position . . 70 5.1 Determination of C- from the Clauser curves 71 vi Figure Page 5.2 Skin f r i c t ion coefficient vs Reynolds number by various workers 72 •5.3 Representative semi-Tog plot showing the logarithmic wall region 73 v i i NOMENCLATURE A^ turbulent mixing coefficient a,b real and imaginary constants in the equation for Hamel spiral motion b distance to the midpoint of the flow 0^ coefficient of f r ic t ion E mean voltage output of LDV tracker e1 fluctuating voltage output of LDV tracker f 5 f 0 » f - j frequency components of LDV signal H curvil inear coordinate for spiral motion h 2b, distance between inner and outer cylinder K conversion constant of optical geometry I turbulence length scale, after von Karman cIr>clQ radial and circumferential velocity components R Reynolds number based on cylinder velocity and gap width, v R, Reynolds number based on midstream velocity and half gap, U c b • - ~ (R/4) r^,r^ radi i of inner and outer cylinders, respectively c . , b 9U S core region slope, -g- T^T c y y=b U circumferential velocity of Couette flow U£ center!ine flow velocity U outer cylinder velocity, 2U o J J c U ,u ' components of velocity normal to the clockwise rotated fringe pattern fluctuating velocity components of flow in Cartesian coordinates f r i c t ion velocity velocity components in Navier-Stokes equations components of velocity normal to the clockwise rotated fringe pattern analytic function in Hamel '.s solution Cartesian coordinate system x + iy constants of order unity in mixing length theory apparent or "eddy" viscosity half angle between the l ight beams von Karman constant, 0.4 wavelength of l ight viscosity (absolute) viscosity (kinematic) density shear stress, shear stress at wall laminar and turbulent contributions to shearing stress curvil inear coordinate in Hamel spiral motion frequency of turbulent fluctuations angular velocity of inner, outer cylinder respectively components of vort ic i ty stream function of flow ACKNOWLEDGEMENTS The author would l ike to thank Drs. I.S. Gartshore and E.G. Hauptmann for their advice and guidance in the course of this research. The computing f a c i l i t i e s of the Computing Centre of the University of Br i t ish Columbia were used for the reduction of data contained herein. Columbia This research was supported by the University and the National Research Council of Canada. of Br i t ish •1 1. INTRODUCTION Plane Couette flow is the simplest form of shear flow to treat mathematically, but is very d i f f i c u l t to create physically because of the d i f f i cu l t ies involved in avoiding boundary effects. It is for this reason that rotating concentric cylinders with a small gap-to-radius ratio are often used to approximate the flow because of their physical s implicity . The shear flow between rotating concentric cylinders is also interesting in i ts own right because of the application to journal bearing design, or indeed any lubricated rotating system. The number of workers who have made measurements in c ircular Couette flow since i t was i n i t i a l l y studied by Couette1 [1870] is small. Some of the work includes the studies of Sir G.I. Tay lo r 2 ' 3 [1923 and 1936] S . I . P a i 4 [1939], and D.C. McPhail [1941]. Further attempts have been made to measure plane Couette flow using immersed rod techniques by H. 5 Reichardt [1955], and with pi tot tubes and hot wire anemometry by fi 7 Robertson [1959]. More recently the work of Coles and Van Atta [1965] 8 and of Coles [1966] has produced information on spiral turbulence and accurate measurements of laminar c ircular Couette flow with end effects. 9 Robertson and Johnson [1970] have made measurements of the turbulence structure in plane Couette flow using conventional techniques. With the advent of the laser, i t became possible for the . f i r s t time to employ optical techniques for flow velocity measurements, and this was demonstrated in 1964 by Yeh and Cummins1^ with their "laser-Doppler" velocimeter. This type of measuring technique lends i t s e l f to velocity measurement in c ircular Couette flow because the probe is simply an e l l ipsoid of l i gh t , with no potentially disturbing intrusions into the flow. For this reason, and because there are few known published measurements of turbulent c ircular Couette flow, i t was decided a laser Doppler system should be developed and velocity measurements taken. The system, which w i l l be described in detail in a later chapter, essentially consists of two beams of laser l ight which cross. The small volume where they cross is the point of measurement with small particles which move with the f lu id generating a frequency pro-portional to velocity. Typical laser Doppler signals are shown in Figure 1.1. The measurement of any mean velocity merely requires the ab i l i t y to measure the mean frequency; while to measure a fluctuating velocity requires an ab i l i t y to follow the changes in frequency. In the report which follows is a description of circular Couette flow, both laminar and turbulent, and measurements which have been made in water contained in a circular Couette flow apparatus. Mean velocit ies have been measured, as well as some representative measurements of turbulence intensities and core region prof i le slopes. 3 2. THEORY 2.1 Laminar Couette Flow The study of the laminar regime in c i rcular Couette flow is of interest in that the theory is well developed and allows for accurate prediction of the velocity profiles of the flow between in f in i te cylinders. Laminar flow is also free of turbulent velocity f luctuations, so measurements can be made of the spectral or ambiguous broadening of the s ignal , an effect which wi l l be discussed later in the text. This is created by an in f in i te upper plate moving with a velocity U0 with respect to an in f in i te stationary lower plate. The intervening f l u i d , which is incompressible, shears in such a way that the velocity at any height y is given by the relat ion: where h is the distance between plates. Furthermore, for laminar flow of a Newtonian f lu id the shearing stress is proportional to the slope of the velocity profi le i . e . : The ideal plane Couette flow profi le is shown in Figure 2.1(a). U = 2.1 h T = y dU dy 2.2 4 This shearing stress increases rapidly upon transition from laminar to turbulent flow. The exact profi le of Couette flow between in f in i te concentric rotating cylinders can be predicted by solving the Navier-Stokes equations for incompressible flow (see Appendix I). The tangential velocity component is given by: i 2 2 l 2 2 r 1 ^ 2 U = ~2 2 C r ( u 2 r 2 " w l r l ^ ^ w 2 ~ w l ^ 2 - 3 r 2 " r l r where r-j and are the radii of the inner and outer cylinders respectively, which rotate with angular velocit ies co-j and u ^ . A l l measurements reported in this study have been made with the inner cylinder f ixed, i . e . , = 0 , so that Equation 2 .3 reduces to: 2 2 1 2 r l r 2 U = —p p [roj ?r ? u)p ] 2 .4 r 2 " r 1 Equation 2.4 is the basis of the theoretical curves plotted with measured laminar results. 2 .2 Turbulent Couette Flow Robertson [1959] has observed that his measurements of plane Couette flow in a i r l ine up well with Couette's concentric cylinder results. Thus, for the purposes of this study, the turbulent Couette flow between the cylinders is approximated by plane Couette flow because of the small gap to radius ratio of the apparatus ( 1 : 2 1 ) . That the effect of curvature is minimal is borne out by the experimental pro-f i l es described in Chapter 4. Turbulent plane Couette flow is approximated by three regions, as shown in Figure 2.1(b) after Reynolds^ [1963]. These are the so-called "viscous sublayers" at either wal l , a log-law region further from the wal l , and a linear region in the core. The viscous sublayers are assumed to have a Reynolds number so small that the Reynolds stresses are negligible, and that their 19 thickness is of the order lOv/u* [Tennekes and Lumley, 1972]. Further 1 3 more, experimental evidence from pipe flow [Hinze, 1959] suggests the prof i le is more accurately approximated by assuming the eddy viscosity is nowhere larger than 0.07 bu*. The viscous sublayers are assumed to change abruptly to a log region, which extends well into the gap before merging into a linear region in the core. At the matching point, the core region velocity U and slope are equal to the velocity and slope of the log region. The composite velocity profiles have been worked out both with and without Hinze's restr ict ion and can be found in Appendix II. These curves are plotted in conjunction with measured values, as described in Chapter 4. The shearing stress T remains constant across the gap (to a f i r s t approximation), and is equal to that at the wall (T q ) . This stress consists of the laminar contribution given by Equation 2.2 plus the turbulent contribution T , where 6 with y measured from the stationary wal l . The total shearing stress is then given by: where A^ is a mixing coefficient for the Reynolds stress in turbulent flow. 2.3 Reynolds Stresses In addition to measurements of the mean velocity prof i les , estimates of shear stresses are reported in Chapter 4. These shear stresses, or Reynolds stresses, arise from the interaction between the u' and v1 components of the turbulence, as long as a shear layer exists , and can be demonstrated as the mechanism by which the wall stress is imparted to the opposite wal l . For turbulent flow far from the wal1, r t » T ^ , hence . 9U 3U T = Axay = P £W 2.7 where p is density, and e is eddy viscosity. From mixing length con-siderations, the following equalities are va l id : (see reference 12) 8U „ 2 , dU 2 •p u v = pe = pA ( ^ ) 2.8 where -u 'v ' is a Reynold's stress, and I is a mixing length. Von Karman made the assumption that turbulent fluctuations are similar at a l l points in the f i e ld of flow. The mixing length £ can be chosen as the characteristic linear dimension for the fluctuation, 7 A f r i c t ion velocity, u*, which is characteristic of the turbulent motion, can be defined in terms of the shear stress as follows: Thus T also sat isf ies the following: x = pu; = -p u'v' 2 .10 As seen from Appendix II , the value of u* can be arrived at through the measurement of the velocity prof i les . From these measurements, the Reynolds stress is estimated for turbulent circular Couette flow, and reported in Appendix VI. 8 3. INSTRUMENTATION 3.1 Background The fact that the Doppler shi f t of laser l ight could be used to measure flow velocit ies was f i r s t demonstrated by Yeh and Cummins^ [1964], and subsequent investigations by Goldstein and K r e i d ^ [1967], Rudd1 5 [1969], Durst and Whitelaw1 6 [1970], and Greated 1 7 [1971] have a l l served to extend the technique. It is now commonly accepted that there are two separate and distinguishable modes of optical velocimeter operation, these being the reference beam technique (optical heterodyning) and the dual scatter mode (fringe pattern)(see Figures 3.1 and 3.2). The theory governing: these different points of view is described in Appendix IV. The measurements performed during the course of the investigation reported herein were made with a dual-scatter system. 3.2 Components Shown in Figure 3.3(a) is a block diagram of the dual scatter system used, while Figure 3.3(b) shows a photo of the experimental set-up. The beam source was a 15 mil l iwatt Spectra Physics Helium-Neon laser operating in the TEM-00 mode. The l ight wavelength was 6328 Angstroms and the beam diameter at point of sp l i t t ing was 1.2 millimeters. The sp l i t t ing was accomplished using a f i f t y percent beam sp l i t ter which gave two beams at an angle of 90 degrees. They were realigned parallel to within 0.1 percent using a front silvered mirror. Individual beam 9 intensities measured between 5 and 6 mi l l iwatts , indicating a certain amount of loss from the reflecting surfaces. The gap between the beams was measured as 11.68 mill imeters. An off - the-shelf 100 millimeter focal length lens was used to focus the l ight beams into a focal volume of approximately 0.07 millimeters in diameter and 0.64 millimeters in length. The resulting set of interference fringes was then imaged to the detecting surface of a Motorola PIN photodiode in an amplifying c i rcu i t by a 50 millimeter focal length PHYWE lens. The time varying signal frequency (whose mean covered a range of 2 to 200 Khz) was caused by foreward scattering of l ight from particles passing through the bright fringes at varying speeds. It was then band pass f i l tered to remove low and high frequency noise by a pair of Krohn-Hite model 3202 R f i l t e r s before being fed into a DISA type 55L30 preamplifier. The DISA type 55L35 frequency tracker was then used to convert the frequency to a voltage, and this voltage was measured by DC and true RMS voltmeters (DISA type 55D30 and type 55D35 respectively). Visual monitoring of the signal was maintained throughout the experiments by a Tektronics model 502A dual-beam oscilloscope. 3.3 Calibration The calibration of the DISA tracker was carried out as follows. In order to ascertain the accuracy of frequency to voltage conversion, a sinusoidal signal of known frequency was fed into the tracker unit from a signal generator, and the analogue output was measured by digital 10 voltmeter. In a l l ranges tested, the tracker performed to manufacturer's specifications of 1 percent accuracy. Calibration curves appear in Figure 3.4(a). The AC capabil i t ies of the DISA system were measured by triggering the signal generator with a second signal generator such that an a r t i f i c i a l frequency modulation (slew rate) of the sinusoidal signal was created. The capture bandwidth, i .e . that region centred on the centre frequency (selected manually) was kept at i t s maximum of 8 percent, and the range of frequency fluctuations was varied up to 50 percent of the DC frequency. These curves appear in Figure 3.4(b). (See also Appendix V). 3.4 Signal Broadening The signal being tracked is of the form f = f sinwt + f, 3.1 o 1 where f-| is the DC component, f the range of f luctuation, and w the frequency of f luctuation. Ideally i f the probe volume were in f in i te l y small and i f the particles were in a continuous stream, the frequency f-j would be given by the following: = -f sin e 3 ' 2 where U is velocity, A is wavelength of the laser l ight , and 9 the half angle of intersection. The frequency f^  is directly proportional to the DC voltage from the frequency tracker. S imi lar ly , with f the average amplitude of velocity fluctuations and w their frequency (the majority less 11 than 100 hz), the RMS voltage from the frequency tracker should be directly proportional to the RMS of the frequency f , hence also the velocity fluctuations Ju'^ . However, there exists in a l l optical anemometers an ambiguous broadening of the s ignal , which adds an uncertain-ty to any measured RMS values of voltage. Physically, this effect arises from the fact that 0 is indeed a range of angles dependent on the beam diameter and lens focal length. An ideal representation of this broadening is obtained by differentiating Equation 3.2 with respect to 6, giving d f l = f 1 ctn 9 3.3 In practice, however, i t is often more advisable to measure the broaden-ing directly from a known laminar flow where fluctuations of velocity (hence frequency) do not exist . The broadening is then corrected for directly by subtraction of the mean square voltages from the turbulent and laminar contributions as follows: - [ < f ) 2 - ] 1 / 2 3.4 U T l turb T l lam where Af^ is the measured broadening in laminar flow, (see Reference 19) It must be noted that the use of Equation 3.4 as shown above represents a simplified approach to the problem of broadening. Generally in turbulent flow there exist the following effects: broadening due to variations in velocity across the scattering (probe) volume, Af^; and broadening due to the fluctuations of volume averaged velocity, Afu Q . 12 Other factors which contribute to the broadening of the Doppler spectrum are gradients of mean velocity acroos the scattering volume Afg; Brownian motion of scattering part ic les , Afg; and the non-monochromaticity of the laser l ight source, Af<.. Assuming these effects to be Gaussian, the bandwidth observed would be given as follows: 2 ? 2 2 2 2 ? Af* = A f V + Aff. + Af„ + Af* + Af* + Af* 3.5 0 T Jc G B S At present, nothing can be said of the contributions of the last three terms, except that they are small with respect to the f i r s t three. We are le f t with: Af* - Af: = Af* + Af* 3.6 J6 u o • 1 2 The existence of Af-j. is the factor which introduces the uncertainties into the turbulence measurements. For this reason, the results obtained using Equation 3.4 w i l l be greater than the true values by an amount Af ( -f— ). The just i f i cat ion for not attempting to compensate for this T l factor is that uncertainty of beam position (as described in the next section) is of the order of three percent. It also varies as the cylinder rotates because of i ts eccentricity, although refractive effects of the wall are negligible. The error introduced by Afj is small when compared with this effect. Shown in Figure 3.5 is a calibration curve of the laminar broadening, which indicates a sl ight variation of the percentage with the output voltage of the tracker. Correction of turbulence measurements was carried out u t i l i z ing this curve, i . e . , the value of ambiguous broadening was chosen depending on the mean D.C. voltage at the measuring point. 13 4. MEASUREMENTS AND RESULTS 4.1 The Flow Apparatus The Couette flow under investigation was set up using water contained between two concentric plexiglas cylinders 24 inches in height and of radi i 20.95 and 22.08 inches respectively (see Figure 4.1). The inner cylinder remained fixed at a l l times, the outer cylinder rotating at various speeds, governed by a VARIAC controlled 1 1/2 horse power electr ic motor which drove a reduction gear system, which in turn drove the cylinder via a belt drive. Due to the large inertia of the cylinder (wall plus base weighed over 100 l b s . ) , high frequency velocity fluctuations were eliminated. Long term dr i f t in rotational speed was observed, but did not exceed 2 percent. Since measurements were made with the motor well warmed from running, d r i f t was not expected to be a major factor. 4.2 Procedures Early measurements consisted of traverses across the test section in order to get representative laminar and turbulent velocity prof i les , while in later measurements turbulence intensities and Reynolds stresses were also attempted. The measurements were accomplished by mounting the optical components of the LDV on a moveable lathe bed. The large mass of the lathe bed reduced vibration to a minimum; and by moving the optics in the horizontal direction (normal to the cylinder walls) the beam intersection could traverse the gap. Displacement was 14 measured to 0.001 inches by a micrometer fixed to the stationary part of the lathe bed. The receiving optics were mounted on a 0.5 meter optical bench, which in turn rested on a f la t 0.5 inch thick base plate with rubber mat supports as vibration isolat ion. Profi les were taken at varying heights above the base of the cylinders in an attempt to find a region of the flow which was relat ively free from end effects. Unfortunately, since the height was of the same order as mean cylinder radius, end effects appeared in the laminar flow regime. Most traverses were made in the region between 2 and 4 inches below the free surface. The water between the cylinders was seeded with small , approximately neutrally buoyant (density 1.05 gm/cc) polystyrene spheres of mean radius 0.372 microns in a concentration of about 1:100,000 by volume so as to increase the scattering of l ight to the detector. Drop-out (loss of signal) due to insuff icient numbers of scattering centres was thus eliminated. However, refractive effects of the moving plexiglas caused the beams to misalign momentarily, placing an uncertainty on measurements which wi l l be discussed later in the text. Traverses were carried out in approximate steps of 0.05 inches, some to within 0.15 inches of the inner (stationary) cylinder wal l . Closer proximity resulted in a D.C. flow frequency below the lower l imi t (2 Khz) of the DISA tracker, and therefore loss of tracking. The resulting profiles were then corrected for mean refractive effects on mean beam intersection position, normalized, and plotted. In a l l , twenty-two traverses were carried out successfully, 7 in the laminar regime, 12 turbulent, and 2 in a regime which was 15 assumed to be part ia l ly turbulent (transit ion). Flow visualization was attempted using dye. It was noted in the case of transition that the streaks exhibited laminar s tab i l i t y for much of the circumference, then rapidly broke into turbulent eddies and became well mixed. This phenomenon has been studied by Coles and van Atta [1966], and would lend i t s e l f readily to investigation by LDV methods. In each traverse, care was taken to make readings at the same point on the outer cylinder circumference in order to minimize the sl ight effect of eccentricity, which was measured to be 3 percent of the gap width. 4.3 Analysis The parameters measured were as follows: the position of the probe volume; the voltage (DC and true RMS) output of the frequency tracker; the mean frequency (as displayed on the tracker meter unit ) ; and the percentage signal drop-out. The rotational speed of the outer cylinder was timed so as to give an independent measure of the mean velocity. Throughout the experiments, i t was discovered that the instantaneous mean velocity fluctuated up to 4 percent around the circum-ference. This was attributed to the eccentricity of the cylinder as previously mentioned, i . e . that the probe volume did not remain at a constant position in the flow. However, since measurements were taken on a damped voltmeter, and at the same circumferential posit ion, this effect has been minimized. 16 4.4 Laminar Profi les A total of seven laminar profiles were taken, at depths ranging from mid-height to within 0.5 inches of the free surface of the water. In a l l cases, consistent behavior was noted, with curvature markedly greater than predicted, probably as a result of end effects. This phenomenon has been noted by Coles [1966], in which laminar flow was maintained for Reynolds numbers up to 9,000. During the course of the present investigation, transition to turbulence was complete at Reynolds numbers of the order of 5,000. Comparison of two of the present results with those of Coles are shown in Figures 4.2 and 4.3. As well as mean velocity measurements, RMS voltages were also taken as a measure of the spectral broadening of the system. It was found that these values did not remain constant as expected, but appeared as a sl ight dependency on the mean voltage output of the tracker. Corrections to turbulent RMS voltages have been applied accordingly. Complete data from the laminar measurements are shown in Appendix VI. 4.5 Turbulent Profiles Turbulent circular Couette flow as observed during the course of this study has exhibited reasonable agreement with the three-region theoretical model as described in Appendix II. The turbulent prof i le is highly dependent on the value of the f r i c t ion velocity u*, as well as assumptions made about the eddy viscosity e. Appendix VI shows calculated parameters as a function of Reynolds number, while representative 17 turbulent profiles are shown plotted in Figures 4.4 and 4.5. The measurements were made in the region between 2 and 4 inches below the free surface. Measurements were made successfully up to Reynolds numbers of the order of 16,000; beyond this point the tracker could not follow the flow due to distortion of the probe volume as a result of the rapidly rotating cylinder. Turbulent flow data is also contained in Appendix VI and a plot of core region slope against Reynolds number is shown in Figure 4.6. 4.6 Spectral Broadening audio frequency range, measurements were made of the laminar flow spectrum in order to observe the ambiguous broadening of the LDV signal . Shown in Figure 4.7 is a typical spectrum which corresponds to a velocity of about 4.8 cm/sec at the 17 Khz peak. The existence of the secondary peak at 12 Khz is puzzling and unexpected, and i t has been interpreted as a function of the moving plexiglas. Band pass f i l t e r i n g of the signal was used to reduce this effect, but this may s t i l l be a source of uncertainty in the calibration of the ambiguous (spectral) broadening. Measurements of - ~ — were performed by correcting the measured RMS voltage for spectral broadening, then arriving at a percentage value by dividing by the mean DC voltage. Shown in Figure 4.8 are the values Since the analyzing equipment was readily available in the 4.7 Measurements of 18 for Reynolds numbers of 6,256, 10,820, and 15,700. Robertson and Johnson [1970] report similar percentage values for measurements of u' in a i r . There appears to be a sl ight Reynolds number dependency evident from Figure 4.8, and this is contrary to the observations of Robertson and Johnson, which indicated that turbulence intensit ies were independent of flow Reynolds number. 4.8 Measurement of Reynolds Stresses As described in Appendix I I I , the values of the Reynolds stress u'v1 can be measured by taking the difference between the RMS voltages measured from each configuration (Figure 4.9). In order to simplify data reduction, the angle of fringe pattern rotation should be plus and minus 45°. Sample measurements of u'w' were made with the LDV probe volume directed normal to the cylinder wal l . These results had large scatter, but were distributed about zero as expected due to the negligible shear in the z direction. Attempts were then made to probe the flow from an angle different from the normal in an effort to get a component of u'v 1 . These were unsuccessful due to the increased reflective loss of l ight intensity caused by an increased angle of incidence, combined with d i f f i cu l t i es involved in the location of the l ight receiving optics. 4.9 Measurements of As described in Appendix I I I , the slant fringe technique permits the measurement of thev w component of turbulence. Representative values for the case of Re = 10820 are shown in Figure 4.10. It w i l l be noted that these values are substantially smaller than the values, and that they tend to approach zero further from the wall than the u1 values. The large scatter encountered in measuring the rms voltages in the slant configurations make the accuracy of the w1 measurements open to question, however i t is probable that the indicated trend is accurate. 20 5. DISCUSSION From the values obtained for mean flow veloc i t ies , i t is seen that circular Couette flow in water is consistent and predictable. Using the mean prof i les , and law-of-the-wall assumptions, i t is possible to arrive at estimates of the f r i c t ion coefficient C^, the f r i c t ion velocity u*, and the shear stress T . These values can then be compared with previous results and appropriate conclusions drawn. 1 o Clauser [1954] made extensive boundary layer measurements in a wind tunnel, and from these results was able to obtain a family of universal curves with C^  as a parameter. Experimental points taken near the wall are plotted, and C^  is determined by selecting the appropriate curve which f i t s the points. Shown in Figure 5.1 is the determination of C f for the turbulent profiles reported, with U/U yU c plotted against log-jQ — ~ . As can be seen, allowing for scatter yields a f r i c t ion coefficient in the order of 0.0035. The previous results of Couette in water and Robertson in a i r showed a dependency of the C^  value on the Reynolds number given by the relation 0.072/(1 ogR-j) . This relation is not apparent in the values reported in this study, although the values f a l l within a range shown in Figure 5.2, after Robertson and Johnson. It is f e l t that more accurate determination of C^  might result from torque measurements, rather than from log law inference as reported. The f r i c t ion velocity u* is related to the shear stress T by 5.1 while the coefficient of f r i c t ion is 5.2 Thus u* can be estimated directly from the C f value by Alternate values of u* are arrived at by solving the equation for the velocity prof i le as given in Appendix II. Measured and calculated values of u* appear in Appendix VI. Further manipulation of the above relationships, combined with the core region slope ^ - y i e l d s a measure of the eddy viscosity e. From this the turbulent Reynolds number can also be found. This should remain approximately the same for the range of Reynolds numbers measured. The pertinent equations are as follows: 22 The normalized core region slope S is given by 5.5 so U b c S 5.6 e These values also appear in Appendix VI, accompanied by some representative results from previous work. plots have been made of the normalized profiles on semi log paper, and the linear region becomes evident, as shown by the representative profile in Figure 5.3. the core as expected, although the apparent slight Reynolds number dependency is surprising. The core region of the flow stays relatively constant in intensity, with a slight increase in the vicinity of the moving wall. This has been observed in previous work (Robertson and Johnson) as a more pronounced effect, and was also constrained to a thin layer closer to the wall. Of course, in the very near wall region, u' is expected to approach zero as a result of the dominant viscous effects, and this justifies the plot in Figure 4.8 being extended to the moving wall. As justification of log law relationships in the wall region, The RMS values of the turbulent velocity fluctuations in the circumferential (x) direction (i.e. / u ' 2 / u ) displayed a consis 23 The most probable explanation for the rather broad region of increased turbulence intensity near the outer wall is that the wall is fluctuating some 3% of the gap width in position. This is due to the eccentricity effects cited earlier. Consequently, due to the fact that fairly long (up to 30 seconds) integration times were used in the RMS voltage measurements, a broad portion of the wall region has been sampled. The stationary wall region has the higher turbulence intensities, and had i t been possible to make measurements in this region, the higher intensity would have been correspondingly more narrow because of the better spatial resolution. However, the low mean velocities in this region give rise to frequencies below the lower limit of the tracker, and thus measurement is impossible. One of Johnson's 1970 values for plane Couette flow in air has been included in Figure 4.8 as an indication of general agreement. 6. CONCLUSIONS 24 Laser Doppler velocimetry has been successfully used to make measurements of velocity profiles in both laminar and turbulent c i rcular Couette flow. Physical l imitations inherent in the apparatus have introduced an uncertainty to measured values close to the moving wal l , while the natural limitations of laser Doppler systems have prevented measurements from being taken in the low velocity region close to the stationary wal l . These l imitations are the f in i te dimensions of the focal volume (0.64 mm in length) and the lower l imi t of velocity resolution (about 0.6 cm/sec). However, accurate measurement of core region slopes for varying Reynolds numbers has allowed the determination of the skin f r i c t ion coefficient for the plexiglas cylinder, and subsequent estimates of shear stress and Reynolds stress. Furthermore, the complete turbulent prof i le across the gap has been shown to approximate a three region model as f i r s t proposed by Reynolds [1963] in studies of bearing turbulence, while laminar measurements have confirmed the existence of profi le d istort ion, probably due to end effects, as observed by Coles. Laser Doppler methods as applied to the measurement of tur-bulence intensities produced results which had somewhat greater scatter than those observed by conventional techniques in a i r . Also, turbulence intensities showed a sl ight Reynolds number dependency, which is contrary to findings in plane Couette flow in a i r . The a i r measurements were in a higher Reynolds number range, but further work is indicated in this area. 25 Estimates of the Reynold's stress u 'w' in the core from slant fringe methods were found to exhibit scatter about zero as expected. The technique was also applied in an effort to measure u'v 1 , the Reynold's stress which dominates because of the non-isotropy of the flow, but this was unsuccessful for reasons discussed in the text. The w' component of turbulence reported is lower than the u' component, and does not exhibit an increase near the moving wal l . Its measurement comes about from the slant measurements, and is subject to large scatter. Due to the small number of points, no con-clusions can be drawn other than that the intensity is low. It is f e l t that LDV methods can be signif icant in taking measurements in d i f f i c u l t situations. Further work in Couette flows is feasible; of special interest would be the transition regime. The obvious advantages of the LDV system, i . e . the absence of flow-disturbing probes combined with a linear response, make i t the most practical tool available for this type of measurement. The versat i l -i ty of the system w i l l make i t the logical choice for many future applications. The significance of this work has been to provide measure-ments of circular Couette flow which have not been affected by the presence of a probe. Whether or not a probe does produce a sub-stantial disturbing effect in measurements of such a flow has not been investigated in this study, but the possibi l i ty has been removed by the use of the LDV technique. 26 REFERENCES 1. Couette, M. "Etudes sur le frottement des l i q u i d s , " Ann. de Chemie et de Physique, Ser. 6, Vo l . 21, 1890, pp. 433-510. 2. Taylor , G.I . " S t a b i l i t y of a viscous l i q u i d contained between two rotat ing c y l i n d e r s , " P h i l . Trans. 1923, A 223, 289. 3. Taylor , G.I . "F lu id F r i c t i o n Between Rotating Cy l inders , " Proc. Roy. S o c , A 157, pp. 546-564. 4. P a i , S . I . "Turbulent Flow Between Rotating Cy l inders , " NACA TN 892, March 1943. 5. Reichardt, H. "Uber die geschwindigkeitsvertei lung i n einer geradl inigen turbulenten Couette stromung," ZAMM Sonderheft, Vo l . 36, 1956, pp. S26-29. 6. Robertson, J .M . "On Turbulent Plane Couette Flow," Proceedings of the S ixth Midwest Conference on F lu id Mechanics, 1959, pp. 169-182. 7. Van A t t a , C , "Exploratory measurements in sp i ra l turbulence," J . F lu id Mech. (1966), Vo l . 25, Part 3 , pp. 495-512. 8. Coles, D; and Van A t t a , C. "Measured d i s t o r t i o n of a laminar c i r c u l a r Couette flow by end e f f e c t s , " J . F luid Mech. (1966), Vo l . 25, Part 3 , pp. 513-521. 9. Robertson, J . M . , and Johnson, H.F. "Turbulence Structure in Plane Couette Flow," Journal of the Engineering Mechanics D i v i s i o n , ASCE, Vo l . 96, No. EM6, Proc. Paper 7754, Dec. 1970, pp. 1171-1182. 10. Yeh, H., and Cummins, H.Z., "Local ized f l u i d flow measurements with He-Ne laser spectrometer," Appl . Phys. Le t t . 4 , 176, 1964. 11. Reynolds, A . J . "Analysis of Turbulent Bearing F i lms . " Journal Mechanical Engineering Science, Vo l . 5, No. 3 , 1963, pp. 258-272. 12. Tennekes, H. , and Lumley, J . L . "A F i r s t Course in Turbulence." The MIT Press , 1972. 13. Hinze, J .O. "Turbulence, An Introduction to Its Mechanism and Theory." McGraw-Hi l l , 1959. 27 Goldstein, R . J . , and Kreid, D.K. "Measurement of Laminar Flow Development in a Square Duct using a Laser-Doppler Flow-meter." Journal of Applied Mechanics, E, Vol. 34, pp. 813-818, 1967. Rudd, M.J. "A New Theoretical Model for the Laser Dopplermeter." J . Sc i . Instruments, 2, pp. 55-58, 1969. Durst, F., and Whitelaw, J .H. "Optimization of Optical Anemometers. Imperial College, Mech. Eng. Dept., ET/TN/A/1. Greated, C A . "Resolution and back scattering optical geometry of laser Doppler systems." Journal of Physics E: Sc ient i f ic Instruments, Vol. 4, pp. 585-588, 1971. Clauser, F.H. "Turbulent Boundary Layers in Adverse Pressure Gradients." Journal of the Aeronautical Sciences, Feb. 1954, pp. 91-108. George, W.K., and Lumley, J .L . "The laser-Doppler velocimeter and i t s application to the measurement of turbulence." J . Fluid Mech. (1973), Vol. 60, Part 2, pp. 321-363. APPENDIX I EXACT SOLUTION OF THE NAVIER-STOKES EQUATIONS FOR LAMINAR CIRCULAR COUETTE FLOW We have: 3u. 3u. 1 3p 3 2u. — - + u. — - = + v 1 3t 3xk p 3x. 3x£ 3u. = 0 3xk S i n c e t h e f l o w i s p a r a l l e l , u 1 = u 1 ( x 2 , t ) u 2 = u^ = 0 D e f i n i n g v o r t i c i t y , i . e . : 3u. 3u. a - - a . - A 3xk 3x, l e a d s t o t h e v o r t i c i t y e q u a t i o n : Deo. 3u. 3o). Dt * 3xk 3t 2 3io. 3u. 3d). + u J 3xk 3xk 3x \ 29 The f i r s t term on the le f t represents total variation of vort ic i ty with time, while the second term represents deformation of a vortex tube. The right represents diffusion of vort ic i ty due to viscosity. In two dimensional flow, deformation terms vanish, and 4 becomes: 2 3u 9o) 3 a) — + u, = v - y k = 1,2 3t 3xk 3xk Introducing the stream function ty, where dty dty U l ' 7 7 = • U 2 and 3x2 3x1 2 2 3 ty 3 ty = - ( — £ + — 2 ) = ~ hp 3x^  3x£ gives dAty 3i|> 3A^ 3^ 3ATJJ + = vAAty 3t 3x2 3x-j 3x1 3x2 This is the vort ic i ty transport equation. In steady flow (no time variation) this becomes: dty dAty dty dAty 3x7 3 ^ - 3 ^ 3 ^ = 9 Hamel found solutions of 9 such that A cJ) = 0 * = f(*) , A }\i f 0 Introducing an analytic function W(Z) = W(x-j + i x 2) such that W(z) = <j> + iH 10 Hamel found that i f the analytic function W satisf ied the following condition, 2 d W l = a + i b = c o n s t > n (dW/dz)2 the function f(<j>) sat isf ies the following: f " f b = v [ f 1 v + f " (a2+b2)+ 2f'"a] 12 (primes refer to differentiation w.r . t . <J)) Integration of 11 gives 2 * = ~?—9 (a log r + b 9 ) + (j) . 13 a^+tT 0 where <j>Q is a constant of integration, and the polar coordinates r and 6 are defined by 31 i 9 z - z Q = re with zQ = const. 14 The streamline <j> = constant is a logarithmic s p i r a l , i . e . a log r + b 9 = constant. When b = 0, the streamlines are concentric c i rc les r = constant. The velocity components are: a = -ab f 2a f a +b r For b = 0, Equation 12 gives f = C + r 2 (A + B log r) 17 with q r = 0 ' % = \ < 7 + A r + B r 1°9 r) 18 For the case of two concentric rotating cylinders, constant B must be zero because the pressures at 9 = 9 and 9 = 9 + 2TT are the same. Thus, q9 = I £ +A^ > \ = 0 19 32 Applying appropriate boundary conditions leads to the velocity com-ponents as stated in the text. Since the gap is small with respect to the radius, we may let r - r ] + A , where r ] is the inner cylinder radius, A is variable, and erywh< Then, t r - « 1 everywhere. r l C l % = T + C 2 r C 20 r l ( 1 + ^ + C 2 h + A ) Expanding by the binomial theorem gives q e * C 2 r ] + £ i + C 2 A + C, ( - ^ ) 1 r l A' + B 'A 21 C 1 C T where A' = C ?r + -1 , B' = C9 + - L 1 r 1 Since qQ = 0 when A = 0, A' = 0, and q Q = B 'A = B'Cr-r-j) So to a f i r s t approximation, q Q is a linear d istr ibut ion, as in the plane case. This result indicates that the small gap-to-radius ratio just i f ied the use of a plane model in the turbulent flow. 33 APPENDIX II A THREE REGION MODEL FOR TURBULENT COUETTE FLOW (a) No modification Starting from the a pr ior i assumption of <_ 10 ( i .e . that 10 V the viscous sublayer thickness is - 7 ^ ) t n e velocity can be matched to the log law in the wall layer. Region 1. Viscous sublayer: 2 9 U  T0 = PU* = y — s y Integrating gives: 2 u* y u = v Region 2. Log law region 9U u* 9y icy where K is von Kantian's constant. Integrating gives: U 1 = log y + C Matching velocities at y = — gives C,, and 4 becomes: u* 1 II i u * y i uT K 1 0 9 ( — ) - ^ o g l 0 + 10 Region 3. Linear Region From the scale relation between vorticity of the turbulence and vorticity of the mean flow, we have a 3 U = 1 — 3y where a-j is a coefficient of order 1 and £ is a length scale of the turbulence. Assuming £ « b in the core, 6 above integrates to u* c^b L2 We know U = Uc at y = b, thus: Matching derivatives between log and linear regions at y = ym: 3 U ay = b^ T = ^ ' s o b a i = « y y=y 1 171 m ba and y = — -35 Matching the velocit ies at y m , we obtain a relationship between — and a, , as follows: l l * I _y_ 1 U r 1 1 b a i U * 1 K u+ a, y =y U U bU bU a , or K -— + log — = l o g ( — ) + log(—~) + ^oq(•~-) + (10ic-l) + — u* 3 u* 3 V v ' 3 V v ' y v 1 0 K ' x ' bU U For any given Reynolds number (—-) this equation relates a, to (—). v U u * If a, is chosen by means of a best f i t to a measured prof i le , — can be estimated from this equation; alternatively, i f — is found using a u * best f i t to the log law region (Clauser technique) then a-j can be estimated using this equation (all profiles showed an in the range of 0.05 to 0.1). In either case, the equation above assumes a known viscous sublayer thickness, which is bui l t into the derivation above. (b) The Three Region Model with Hinze's Modification In the three region model, the effective kinematic viscosity (e = t/Pgy ) is constant in the core region and varies l inearly in the log region, since T is constant to a f i r s t approximation. Thus, e reaches a maximum in the core region, and i f this value is given by e £ 0.07 bu*, then the value of a-j is equal to 0.07, i . e . : 36 2 from the definit ion of e, and the fact that T = pu* = constant. 3U u * 1 However, — = -g- — in the core, so that 2 0.07 bu* = , or a, = 0.07 . 11 b a . If e reaches a maximum less than .07 u* in the core region, then the previous model, described in (a) is applicable. Using a, = .07, and Equation 9 the value of TT - can be found, U b 1 u c for any {—^ -) within the assumptions of the three region model with the assumed viscous sublayer thickness. The f r ic t ion coefficient u* 2 U b (Cf = 2(TJ~) ) c a n D e found as a function of - ^ - w i t h i n Hinze's assumption and is plotted in Figure 5.2 for comparison with other data. 37 APPENDIX III MEASUREMENTS OF REYNOLDS STRESSES BY LASER DOPPLER VELOCIMETRY Consider the simplist ic approach as shown schematically in Figure 4.9. We have measured voltages which are directly related to velocity by a constant K, both mean and fluctuating thus: KE = [)± = — Ke' - u; For the two configurations shown, we have the following equations: u x + ui = — (U + u' + w') JT 1 V, + v' = — (U + u' - w') Taking root mean squares of Equation 2 yields the following: (UA + u;) 2 = ul + 2U xu x' + u x ' 2 = K2 (E2 + 2E ] e^ + e] 2) e ' 2 J (UA + UJL' = KE-,0 + \ - g - + higher order terms) 4 E l Operating now on the right hand side of Equation 3 y ields: ^(G+u'+w1)2 = l(U2+u ,2+w,2+2U(u ,+w')+ 2 u V ) + higher order terms) ,2 U /, A 1 u"" ^ 1 w " , u'w' 9 - u / l . ' U . I W , Similarly Equation 3 yields Q (VA + v^ ' ) 2 = KE2 (1 + \^r+ higher order terms) ,2 and / (U + u'-wT = ^ U T 2 D2 T 2 r 2 — D2 + higher order terms) Subtracting Equations 6 and 7 from Equations 4 and 5 results in 4 E * where E• = E ] = E2 . Recalling Equation 1, this simplif ies to 3 9 u V = \- (e j 2 - e 2 2 ) where K is the calibration constant which is a function of the laser Doppler system. By adding equations 5 and 7 , and equating their sum to the sum of Equations 6 and 8 , the following expression arises: u , 2 + w . 2 I " l - 2=2 , -p -IT U* -2 u ' 2 If a value of is known, then estimates of the w' component D* of turbulence can be found. APPENDIX IV THEORETICAL DESCRIPTION OF THE LASER DOPPLER VELOCIMETER (a) Reference Beam Operation Figure 3.1 shows geometrically the Doppler shi f t of laser l ight incident on a particle moving in a f l u i d . The number of wave-fronts incident on the part icle per unit time i s : c - v k . After scattering, an observer in the direction k s c would observe an apparent wavelength of: c - v k c - v k SC v V ' v -> . ' p c - v k i The frequency of this scattered radiation is given by: -»• r v k . c c ~ v * k i c c • v = c ( ) = [ ] S C A i c - v k A i v k sc 1 sc v*k sc and the frequency shi f t is given by: 1 - v k v o = v sc - v i = XT C TJ— ] - v I v k 1 1 _ sc 4 This frequency difference can be measured when the scattered l ight is heterodyned with an unscattered reference beam on the face of a square law optical detector such as a photomultiplier tube or a photodiode. (b) Dual Scatter Operation The dual scatter system requires the formation of a focal volume containing a fringe pattern, as shown in Figure 3.2. This is accomplished by the intersection of two equal intensity laser l ight beams which set up fringes of known geometry. If the angle between the beams is 20, the fringe spacing is given by 2 sin 0 where both \. and 9 are measured in the f l u i d . As the scattering centre traverses the focal volume (interference pattern) with a velocity v, l ight is emitted with a frequency which corresponds to the rate at which the bright fringes are cut. This frequency is given by: v 2v v = -x = T1- s i n 6 d x. Equation 4 in (a) reduces to Equation 6 above i f the angle between incident and reference beams is 20. The two different governing principles result in identical equations for velocity. 43 APPENDIX V Tracker Calibration (a) Frequency Tracking The DISA type 55L35 frequency tracker was used to process the laser Doppler signal from the flow. Although the instruction manual presented data on the tracking performance of the unit, an independent study was also undertaken. I n i t i a l l y , a pure sine wave input was fed to the tracker, which was tested in each range. The frequency to voltage conversion was within one percent for a l l ranges used, i . e . 15, 50, 150, and 500 khz. The curves normalized to produce the composite shown in Figure 3.4(a). As an independent test on the rate at which the tracker would follow frequency fluctuations, one signal generator was connected so as to vary the frequency of a second signal generator. This produced a signal of the form f = f-j + f sin tot where to was the rate at which the frequency was varied. Various amplitudes of fluctuation about the mean frequency f-j were tested, resulting in the curves shown in Figure 3.4(b). Since the ratio of frequency fluctuation to mean frequency seldom exceeded 10 percent, i t can be seen that the tracker was consistently following fluctuations of 200 hz and below, and often following fluctuations up to 1000 hz. 44 (b) Ambiguous Broadening Measurements of the RMS voltage output of the frequency tracker taken in laminar flow produced a spectral broadening in the order of 3 percent. However, this value had a sl ight dependence on the mean DC voltage output of the tracker, and is shown graphically in Figure 3.5. This effect was f e l t to be a function of the tracker rather than a physical effect in the flow, because theory predicts that the broadening is a function of the optics alone. Figure 3.5 results from measurements made in the 15 and 50 Khz ranges, and corrections for the 150 Khz range have been assumed to be the same, as there is no reason to suspect otherwise. Since 150 Khz corresponds to a velocity of over 40 cm/sec, flow in this range was turbulent and thus i t could not be checked for inconsistencies. Corrections were applied to the RMS voltage by subtracting the ambiguous value which corresponded to the voltage produced by the mean velocity, as per Figure 3.5. Thus turbulence levels have been corrected for the sl ight non-linearity of the tracker. 45 A P P E N D I X I V CALCULATED FLOW PARAMETERS R U c m c sec S u c m * sec e bU c £ 5186 9.04 .543 .39 - .43 .045 - .054 291,239 0.0035 5338 9.30 .432 .39 - .41 .054 - .060 245,222 0.0034 6027 10.50 .461 .44 - .47 .057 - .065 262,230 0.0030 6055 10.55 .462 .44 - .47 .057 - .065 265,233 0.0035 6200 10.80 .519 .45 - .49 .056 - .062 298,252 0.0030 10,820 13,800 ] 18.85 .425 .79 - .80 .112 - .115 242,236 0.0035 24.04 .360 1.1 .19 181 -15,700 27.35 .393 1.14 - 1.13 1.74 226 0.0035 18,4002 - .240 - - - 0.0054 23,2002 - .195 - - - 0.0052 27,OOO2 - .217 ! - - 0.0048 Murguly (1971, unpublished). 2Robertson (1959). 47 •s". 1ST PRHFT-t-10 1 1 -tf -13 14 -+5-16 17 -f8-10 ?0 R=IO2O PuTpp C Y L I N D E R \ ' F L r ir J T Y 1 . 0000 0.9426 0.8971 0 .8578 0 .8030 0.7646 •O.72P0 0 .6878 0 . 6 2 U ? 0 .5938 0.5455 - O v W W ) — 0.461 1 0.U158 8.7000 8.3000 8.0400 7.790(1 7,a000 7.0700 6.9000 6.6000 6.3600 5. fl 4 n o 5.3700 -a-j-ft-fr'HV-a. a o o o 3.8300 0,0874 O.0R7O O.0R75 0 . 0876 0.OR76 O.0R77 -O.0R77 0, 0P78 0.0878 0 . 0P7" 0.0879 0 . ORRO -fi-j-fVRR-f)-0.0881 0.OPR1 13,0500 -i-?;-905-fV-12.4500 12,060 o •1 1 .6850 11 .1000 10.6050 -H>T3irfra-9.9000 9,5400 9.0900 fl.7600 P.0550 —7-rVi?ft-ft-6.6000 5.7450 = 3,56 0, O048 —fhr6-l fl-7-1 .9196 3. '1318 a.7422 6.5625 7.P361 —9-.~frttft9-10.3796 ! i . a 4 a 3 12.481? 13.4R42 15.0767 -i-<>T3i-ftP— 17.R533 19,34 0 1 c M / s r c 35.5771 —35.2813 33.9130 32.8289 31.7899 30.1743 28.8127 -?R-rl-fra-9-26.8673 25.87R2 ?a.6a63 23.7411 21 .R152 -t^i-fH-3-9-17.8528 15.5299 22 23 2a 25 26 -?7-2P 29 -3 ft-31 32 -43-3 a 35 -36-37 38 -^9-ao ai -42-a3 aa ft.3950 0 .3605 0 .3225 0.2965 0 ,26'IP n.23P5 0.299 1-0.3a0fl 0.3962 -o-iWWfr-0 .52R0 0 .5R00 o . h a c n 0 , 6R65 0.75^5 -ft-i-7 5-/4-5-0.821 0 0.P930 0 .05 3+-3.a7"0 3,1300 2.6200 2.2RO0 1.R50O 1.500A ? . 2 a 0 ft 2.7600 3.4600 rt—pJtfio... a . R ! 0 0 5.4100 h . 0 a ft 0 • 6.3700 6 . O 0 o o -—6-.-«2ftft— 7.4400 7.9500 8.3700 0 . Aflft ] 0 . 0PR2 0 . 0HR2 0 . 0«83 0 . ftPR3 0 , 0 R P 3 >ft5ft 20.022'J 1 a .0660 0 . 0 P fl 2 0.Opp t -ft-j-ftH-R-ft-0.OPRO 0.0R79 ft . Of 78 4 ,6950 3.9300 3.4200 2.7750 2. ?500 • 3. 36 "0 4.1400 5,1900 —6-,a?ftft-7.2150 8.1150 Q . 0 I, 0 "i 21.1535 22.3071 23.2477 24.2837 25.1424 23.1630 21 . 1 9 . 9 8 % 4 -1 7.5A72 -15.6526 1 3. "395 1 1 . 6-9^— 0 , 0P7R 0.0877 0 . 0P76 A . 0 P 7 5 0 .9932 1 . 0 0 0 o •1 .OOOft-0.OP74 • P.6200 0.0P7a 8,9000 0.0P7U -R-j-5-ft-fl-fl <».5550 10.3500 -1 4-i-3flft-ft— 11,1600 1 1 .9250 1 2.5550 12,9300 13.3500 -+ -? - , -7 Jjftfl— 1 0.4226 P. 1 71 0 —R-H-7~!-ft-5.9647 3.5713 —Hr5-6-*2-12.6814 1 0.6093 0.2291 7.4P51 6.0667 —0, 0f,7'j 11.1 792 14.0257 H-7-T-36-8 2-19.5353 21.9RR7 -2-4-^ 5^ 3-5-25.9306 28. 1157 -2R-I-! 9-7-2-30.3454 32„a595 •34.2 0-4-2-0.2315 0.004P -o -yooaf t - -35.2a64 36.3950 -34-J-7-5-92-P=1ISO nilTFR CYl.INnFR Vfc ' i n r i T Y = 4.02 CM/SEC 46 47 -ftft-49 50 -54-52 53 - 54-55 56 58 5° 60-61 6? POS'NdKO HC Vm.TR THFTA FPFOIIFMCY POP ' M ( N'M ) VELOCITY (""/SEC) - 4 - j - O - A A O -0 .951 0 0 . R 9 4 0 0.838ft-0.7690 0.7030 ,6«-.ftft 0.5770 * . 5 1 4 0 0.461ft-•~2v°5fift-2.7300 2.4400 2.2400 -~ft-,-ft-P-7-«-0 . 0 R 7 4 0.np75 0 . 0 8 7 6 -1-4-1 3 1 2 6.3100 5.6200 il-.-HAA-O-. 3 . 9 7 0 ft 3 . 1 a 0 0 2.500,-1 o.a OPO 0 . 4 P a ft -n.A.fcO/y. 0.7660 0.9390 2.26^0 -4-.-/(-2.1ft.-6 . 1 2 o R . 3 a A 0 o . 0A77 0.0P7R _n.--fift.7-su 0.0079 ft , 0 P p ,1 0 . r (i.«1 -H-9 p —7 5 /1 —3 -.-75ftft-.6500 . 2000 .4650 .4300 -j 2 0 ft ft • .9550 ,710ft T^fftft-O - i - f t f t V I R -1.6393 3.53P0 —5-i-4ftft"i— 7.6904 O.P765 -H-.620 4-) a , n 3 R 5 16,113* -H7-I-P5-62--«-0-.-2HJ7-37.1862 33.20P5 0.oaa j 0 . 0 - <5 0 A...nA7R — 0 . 0 P 7 7 r . OP75 3 <, — 6-9 1 2 . 3 o 0 0 . o ft 0 -.6 3 0 0 -. 1 8 0 0 . 5 1 0 0 1 9 , 5 -' * 2 1 6 , Q < M -4 1 .2"HP 7 . 7 M O - J ?. 0 39.< ?5.7171 22.RR30 -1 9.-529 1-1 6. 1 352 12.750 2 1 0 . 1 a 3-7-«. 1629 1 0 . 5 S 3 7 ,-QR5 /-, 0 « 1 6 , 0 7 4 5 —! 7 24. 3/! 6 a 6 5 6 7 6 8 6 0 ••Pf l^ - f^ . ) ^(i-vflfc-T-S- - T - « f - T - 4 F P f - n i l f - v ^ . y . — o r v g f „ u ^. 48 7 0 71 -?2-7? 7a -7-5-76 77 ••78-79 80 l.oooo n . o y t r, i . < » a 7 0 0 . 8 8 7 - 6 -0 . 8 0 7 0 0 . 7 6 1 5 - 0 . 7 1 2 6 0 . 6 4 7 1 0 . 5 9 0 5 - l . ^ a - H • 0 . 5 0 4 0 0 . 5 0 4 0 ' i . 2 n o r. 4 . 2 1 n o-4 . 0 6 0 0 3 . 9 « o n 3 . 5 4 0 0 3 . 3 0 O 0 3 . 0 6 0 0 2 . 7 0 0 0 2 . 5 7 0 0 - 2 - ^ 3 2 1 0 -2 . 1 6 0 0 6 . 9 5 0 0 o . o n 7 a 0.0R74 0 . 0 8 7 5 -n . Of!75-o. O R R O 0 , ORflO 21 , a -i n o . nf, o o 2 0 . 3 0 0 0 1 9 , 7 0 0 0 -4-fl-r^O-frft-0 . O R 7 6 1 7 . 7 0 0 0 0 . O R 7 7 16.5000 o .on77 i - g . 3 o o p O . 0 R 7 B 1 3 . 5 0 0 0 0 . 0 R 7 9 1 2 . P 5 0 0 I O . R O O O 1 0 , 4 2 5 0 o . o o a i n . Q r; q.R-1 . 7 7 2 6 2 . 6 6 5 5 6 . 4 3 0 0 7 . 9 3 9 2 9 . 5 'j 9 0 • 1 1 . 7 2 5 0 1 3 . 5 9 3 5 - + 5 - r 2 2 1 - 7 -1 6 . 4 4 2 P 1 6 . 4 4 2 8 8 3 C»VSFC -vt t nr T Tv-f«*vse-e-j-5*.341 0 5 7 . 3 6 2 9 • 5 5 . 2 9 9 4 5 3 . 6 4 3 9 r) 1 . 4-4-1-7-4 8 . 1 1 8 6 4 4 . 8 2 6 7 • ^ 1 . 5 3 7 1 • 3 6 . 6 1 5 7 3 4 . 8 2 4 1 -*t- . -4-i -a- i -2 9 . 2 3 1 9 2 8 . 2 1 6 9 8 2 A 3 R a RS 8 6 - 6 ^ 7 -Rfi R9 9 0 91 9 2 - 9 5 -94 9 5 9 6 -9 7 98 o<j-1 0 0 101 1 0 2 -1 0 3 104 H 5 -1 Of. 1 0 7 -j n R J 09 1 1 0 -+++-1 1 2 1 1 3 - H u -ns 1. 1 6 - H - 7 — 1 1 R 1 1 9 1 2 0 ~ 121 12? 1 2 4 125 1 26 1 27 1?R -}-?<*-1 3 0 0 . 4 6 9 P 0 . 4 3 2 2 0 . 4 0 3 0 0 . 3 7 7 5 0 . 3 2 9 5 0 . 2 8 9 5 0 . 2 4 9 5 0 .2165 6 . 4 5 0 0 5 . 9 4 0 0 5.5000 4.7300 3 . 9 3 0 0 3.0600 1 . 5 5 0 0 0 . o 0 n i-0 . P R R 1 0 . O R R 1 0 . O R R 2 0 , 0 R R 2 0 . 0 R R 3 •0.0003 0 . P R R 4 - 0 . 6 7 5 P -8 . 9 1 PP R . 2 5 0 P 7 . 0 9 5 0 5 . R O 5 0 4 . 5 9 0 ( 1 3 . 3 7 5 0 -- 1 7 . 5 9 3 5 1 8 . 8 0 1 9 1 9 . 7 6 0 3 2 0 . 5 9 6 1 2 2 . 1 6 « 2 2 3 . 4 7 6 3 - 2 4 . 7 f l 3 3 2 , 3 2 5 0 2 5 . R 6 0 4 2 6 . 1 7 3 7 2 2 . 2 9 7 4 1 9 . 1 6 8 7 l r ' . 9 1 S 6 1 2 . 3 8 5 1 " . 1 0 1 S -6 . 2 6 6 9 - » = i - B 1 1 - - fHt -TFR-Cr Y1 1 "WE P -Vfc-|_ OC~J T V - 6 . ^ 5- e«/ 9EC P " S ' M ( T V , n C V O L T S T H F T A F«F<3UFMCY POS'NfMM) V I nriTY fMM/REC) 1 . 0 0 0 0 1 . PPPp - 0 ; " 7 7-R-0 . 9 « 6 5 0 , R 9 7 0 •" . P S 3 ! 0 . 8 1 1 0 0 , 7 6 0 5 H > v 7 i - 8 5 -0 . 6 7 1 6 0 . 6 0 2 0 0 . 5 6 7 0 4 . 750P '1.6400 -4-.-5^1-0 P-4.3RP0 4 . 1 3 0 0 3.8R0p-0 . 5 6 7 0 0 . 5 2 4 5 0 . ' J 6 0 5 -0 . 4 0 3 0 P . 3 4 7 R rV-3 . 6 7 0 0 3 . '•• 3 o 0 -3-T21-00-2 . 9 6 0 0 2 . 6 R P 0 2.44 0 0 -; 2 9 8 5 0 . 2 7 2 0 0 . 2 4 3 0 O . - 2 3 3 0 7 . R 5 0 0 7.2000 —6 .?6"" 5 .050c 4 . P 4 0 0 3 . 4 5 P 0 2.54PO 2. ) ? r o t-,-R+-Pf\-0 . P R 7 4 0 . 0 fl 7 4 — 0 - . - A R 7 4 -0 . O R 74 0 . 0 R 7 S - 0 . P 8 7 6 -0 . 0 8 7 6 0.0677 -O-.-O-R 77-0 . O R 7 P 0 . 0 R 7 9 0 . 0 8 7 O P . 0 8 7 9 0 . ORRO P , . 0 . 0 P. R 1 0 . O R R 2 0 . P is R 3 -0 . 0 B R 3 0 . P P R 3 -f>vOPR4— 23.7500 23.2PPP - ? 2 y 7 o p f t -21 . 9 P 0 P 2 0 . 6 5 0 P 1 " . 4 0 0 P -1 R . 3 5 0 0 1 7 . 1 S P 0 — 1 - 6 - . - 0 5 0 0 - -1 4 . R 0 0 0 1 3 . 4 0 0 0 — t r h r 2 ^ - f r O — 1 1 . 7 7 5 0 1 0 . ROOO 9y3«pn— 7 . 5 7 5 0 6 . P 6 0 P 5 . 1 7 5 P — 0 . 0 0 4 8 0,P 0 4 a 0 - . 7 4 5 5 -1 , 7 « 9 3 3 . 4 3 « 1 4-.-P«fH-6 . 2 9 6 8 7 . 9 7 2 3 9 - . - 3 6 3 6 -1 0 . 9 1 5 2 1 3 . 2 1 3 9 — 1 - 4 - ^ - 6 - 8 - 3 -\t. 1 5 —1. 7 19 21 - 2 - 3 -3 . 8 1 0 0 3 . 1 POO -2-,-7-tso-. 3 6 8 3 , 7 6 78 i 8 7 2 7 -. 7 6 0 3 , 5 6 0 4 2 4 , 0 4 8 7 2 4 . 9 9 5 7 6 4 . 7 4 7 6 6 3 . 2 4 8 1 — 6 1". 8 6 4 9 -5 9 . 6 S 7 6 5 6 . 2 1 1 9 r ' ? . 7 7 g q -4 9 . R R B 6 4 6 . 5 9 2 0 - 4 - 3 v 5 7 6 9 -4 0 . 1 5 5 8 3 6 . 3 2 0 7 - i H - . 051 3 3 1 . 8 9 9 9 ? 9 . 2 U 0 5 - ? 5 ; 3 9 9 5 -2 0 . 4 7 3 0 1 6 . 3 6 5 4 -1-3-T9-6-5-5— 1 0 . 2 7 7 9 8 . 5 7 4 8 — 7 - ; 3 1 9 9 -" = 2 5 0 0 HIITFR C V I . I N D F R VFl. OCTTY = 9 . 5 6 C M / S F C POR 1 f TV 1 1 . P O O P 0-;f>(,H^-0 . 9 4 P 0 0 , 9 P 7 p P - . - 8 J 7 ^ -0 . P /m 0 0 . R C P ••P . R P o . 7 3 1 r 0 . 6 9 3 0 -fl-r6^-R-r-0 . 6 A ? P vni. Ts 7. o1 op —ArOfdf.-6 . 7 7 0 P ^ . 5 6 P 0 — 6 - r 4 P - P f l -6 . ? r, r r, 5.9300 • 5 . 7 S 1 1 -5 . 4 7 0 0 5 , 2 5 0 0 4 . 6 4 " 0 T H F T A P , P R 7 4 0 - P R 7 4 0 . 0 8 7/1 0.P875 0—O-P-7-5-0 . P P 7 6 0 . 0 8 7 6 ' 7 " , 0 8 7 7 0 . 0 8 7 8 P - ^ W T - B 0.0879 F R F fM I r' I r Y 35,0500 -3 4~; 5 0O-0-3 <• . 8 5 P P 3 2 . 8 P P P 3 - 2 - r l ^ - f -o -3 1 . 0 0 0 0 2 9 . 6 5 0 p — ; • ' " . » . « . A •-. 2 7 . ^ 5 o 2 6 . 2 5 0 0 2-4-j-P 5 oi— 2 3 . 2 0 P 0 P 0s R ' f i( M ^ P , P 0 4 R 7 2 6 -2 . o o 6;-, 3 . 1 0 5 2 ^ r l - 7 ^ - 1 -5 . 1 6 7 8 6 . 4 9 6 ) - - 7 . 7 2 ^ 6 -8 . 9 u o 5 1 O . 2 0 7 5 1 - 1 - r 6 ° 5 2 -1 3 . 2 ! ' 4 9 VFI O C T T Y ( M M / S E C ) ^ ^ . 5 5 3 8 —«rf±--0«6-a —• ° 2 . 2 0 ! 6 89.P9RR —87-j-o R-fVj? _ « 'J 8 n - 7 7, 7 'J , /1 , 3 2 2 2 6 0 3 2 8 4 2 9 2 7 0 8 24 4 4 - 6 - 7 — 4 1 - 0 8 -6 2 . 8 8 3 5 1 3 ! 1 3 2 1 33 134 -+VJ-1 3 6 1 37 + 3 8 -1 3 9 1 4 0 0.55/10 -fv-4 0f>-<\ — 0 . 4 5 7 0 0 . 4 t ? 0 4 . ?0 0 0 ' . 0 . 0 8 7 9 - 3v«H-'V*> — A .-A > 8 P 3 . 5 7 0 0 0 . 0 8 8 1 3 . 1 1 0 0 0.OR81 i<*0 2 . 5 1 0 •-'> "O 0 . 3 5 O 0 8 . 1 0 0 ' ' 0 . 0 8 8 2 0 . 3 2 0 0 6 . 8 4 0 0 0 . 0 8 8 ? —ft-;-2+-A0—• 5 - ; A + 1 1 — ; 8 8 3 -0 . 2 6 9 0 4 . 9 1 0 0 0 . 0 8 8 3 0 . 2 3 4 0 3 . 8 0 0 0 0 . 0 8 8 4 4 9 2 1 . tl 5 0 0 -•(-<>. A S I f ) -1 7 . 8 5 0 0 1 5 . 5 5 0 0 1 2 . 1 5 00 1 0 , ? A O O ~ 8- ; -4 '1-Sf l -7 . 3 6 5 0 5 . 7 0 0 0 1 4 . 7 9 6 6 - * A . <• <•-><< 1 7 . 9 8 7 7 1 9 . 4 3 2 1 21 . 2 0 2 7 2 2 . 4 70? 2< ; - 5 9 0 9 -2 4 . 1 4 6 7 2 5 , ? 8 9 4 '•.8 . 0 9 9 7 — N3.-t'8-l7>~ 48.2810 4 2 . 0 3 3 2 3 3 . 807s... 3 2 . 8 1 7 1 2 7 . 6 9 6 6 —2-?T-7 -O« j -0~ 1 9 . 8 6 7 0 1 5 . 3 6 8 0 — H H -1 4 2 1 4 3 1 44 1 4 5 1 4 6 —W-1 4 8 1 4 9 -+5$-151 1 5 2 1 5 3 • 1 5 4 1 5 5 — 1 5 6 -0 . 2 0 7 o -1 5 7 1 5 8 1 5 0 1 6 0 1 6 t - 1 6 2 -1 6 3 1 6 4 1 6 6 1 6 7 -1 6 8-1 6 9 1 7 0 - m -1 7 2 1 7 3 -+7-4-1 7 5 1 76 +7^7-1 78 1 7 9 - 1 8 0 -181 1 8 2 1 84 1 8 5 1 8 h -1 8 7 ] 88 1 * 9 — 1 9 0 191 I O? -! ° 3 ! 9 a 4 « 5 — 1 96 1 9 7 1 9 8 • 1 9Q 200 + 9 0 0 2 8 0 0 3 3 2 0 3 6 4 0 3 6 4 0 4 3 4 0 0 0 0 ( 0, 0, - A - i 0 . 5 2 1 0 0 . 5 9 5 0 0 . 6 6 7-0-0 . 8 1 1 0 0 . 8 9 0 0 1 . 0 0 0 0 2 . 650 0-2 . 1 3 0 0 5 . 1 2 0 0 6 . 9 8 0 0 7.9700 2 . 5 5 0 0 •3.1300 3 . 8 4 0 0 4 . 3 4 0 0 - 4 . 9 0 1 1 ft-5 . 8 0 0 0 6 . 2 3 0 0 (• . 8 8 0 0 — 0 . 0 8 8 4 0 , 0 8 8 4 0 . O R R T ( 0 . 0 8 8 2 0 . 0 8 8 2 0 . 0 8 8 2 0 . 0 8 8 1 •  0 . 0 8 8 0 0 . 0 8 7 9 0 . 0 8 7 6 0 . 0 8 7 5 • 0 . 0 8 74 -3 . 9 7 5 0 3. 1 9 5 0 7 . 6 8 0 0 1 0 . 4 7 0 0 1 1 . 9 5 5 0 1 2,7500 1 5 . f J 5 0 0 I' 6 . 1 7 0 4-1 9 . 2 0 0 0 2 1 . 7 0 0 0 -2*-.-5-A*A-2 6 . 2 3 . 2 2 . 2 1 . 2 1 . -+S-T 2 9 . 0 0 00 3 1 , 1 5 0 0 - ^ 4 . 4 0 0 0 15. 1 3 . 7 2 4 3 7 8 7 1 OR 6 6 0 3 8 8 0 3 8 8 7 4 3 1 • 1 0 . 7 1 3 0 8 . 6 0 8 7 2 0 . 7 2 0 0 2 8 , 2 6 8 4 3 2 . 2 9 2 7 3 4 . 4 4 0 2 - 4 i ? . 3 1 6 4 6 . 3. - * v 8 « 3 1 5 1 . 9 8 0 5 4(149 5 8 . 8 1 18 OA7-3 hh-rti-k^-T-2 9 6 8 7 8 . 8 4 3 0 6 7 1 2 8 4 . 7 8 5 7 0 0 4 8 9 3 . 7 0 1 0 -P=5186 O U T E R C Y L I N D E R V E L O C I T Y r t fl'. 1 C M / R E C POStMflM) OC VOLTS THFTA FREQUENCY PPS-^.M) V E L O C I T Y f^M / S E C ) l . o o o o 0 . 9 R 1 2 0 . 9 4 9 1 — 0 - ; - 9 + L ^ -0 . 8 6 5 5 0 . R 2 3 0 - 0 . 7 7 6 0 0 . 7 3 4 4 0.6964 -0-.A.6 3 2 -0 . 6 6 3 2 0 . 6 2 8 1 "••''2O0 0 . 0 8 7 4 6 6 . 3 0 O - O -4.1700 3 . 8 6 0 0 - 3 - . - A 4 0 - 0 — 3 . 3 8 0 0 3 . 2 1 0 0 3 . 0 4 0 0 — 0 . 5 9 4 6 0 . 5 4 4 8 0 . 5 0 4 5 - 0 - r 4 + r 2 5 -0 . 4 1 1 6 0 . 3 5 2 0 2 . 9 5 0 0 2 . 8 4 0 0 -2-.-7-4 OO-7 . 9 5 0 0 7.650Q 0 . 087<t 0 . P R 7 4 -O^ -AM-7.5-0 . 0 R 7 5 0 . P R 7 6 0.O877 6 2 . 5 5 0 0 5 7 . 9 P 0 P -54~i -6 0 l O -5 0.7000 4 8 . 1 4 9 9 - 4 5 . 6 0 0 O -, 0 0 4 8 18 0 . 7 4 0 0 -7 . 3 2 0 0 7 , 1 0 0 0 6 . B 0 O 0 6 . 4 5 0 1 -0 . 0 8 7 7 4 4 . 2 5 0 0 R 0 . 0 8 7 8 4 2 . 6 0 0 0 j o - ^ - . - 0 8 78 4 +.-1-0 0 - 0 }-l-0 . 0 8 7 8 3 0 . 7 S a n j , 0 . O 8 7 8 3 8 . 2 5 0 0 1 2 " . 0 0 7 9 3 6 . 60 0 0 M -6 3 2 1 . 7 0 2 9 .055 3 . « " M , 8 9 8 5 4 W -n.2055 0 . 2 2 6 0 n . 1 8 6 2 -tx- a ^ o -5 . 8 5 P 0 5 , 0 3 0 0 4 . 3 6 0 0 -O . O P f l O 3 5 . 5 0 0 0 15 0 . 0 8 8 0 3 4 . 0 0 0 0 16 Hhr^WH-—3-2T2S-OP 5-7. . 8 < 6 9 .0950 v 1-9 2 7 -0 1 9 2 7 . 3 5 2 4 T-4V7-R-' 7 0 . 4 7 8 1 157.7308 - M 8 . - 6 5 - 9 5 -17 7 . 9 4 8 5 1 3 0 . 9 2 9 4 12 3 . 9 1 0 9 3 . 1 8 0 0 2 . 5 0 0 0 -?- j -<tf j i1-0 . 0 8 R 1 0 . 0 8 8 ? —O . P 8 0 3 0 .O8P4 0 . 0 8 8 4 — P T O R 8 + -2 9 . 2 5 0 0 2 5 . 1 5 0 0 -^P-0-1 9 21 - 2 * 1 5 . 9 P P 0 1 2 . 5 P P 0 - 3 6 - , - 7 5 0 0 -2 5 2 6 -+8-. P99R . 4 2 6 0 7*07-0-, 4780 . 4 3 1 8 -r2 8-0-7-.5503 .8483 i-5-7-«9-1 2 0 , 1 6 9 q 1 1 5 . 6 2 5 1 - H - 1 - . -5 0 0 2 -1 0 7 . 8 3 7 8 1 0 3 . 7 1 5 7 - 9 9 . 1 935-9 6 . 1 4 2 9 9 2 . 0 2 6 9 - 8 T - - . - 2 - 3 - 7 - 4 -7 9 . 0 6 4 0 6 7 . 9 2 3 2 — 5 8 . 8 2 7 7 -4 2 . 8 6 3 6 3 3 . 6 7 8 5 - 9 9 - ^ 5 7 ^ 2 -P = 5 3 3 8 O U T E R F.Yt I'iOF'R V E L O C I T Y = 1 8 . 6 C ' V S F C 9 0 S 1 fi f T V ) i . PIPP--0 . 9 R 0 0 0 . 9 5 6 8 0 - ^ * 4 - 5 -K VO|..TS THETA E P F. QI) F f • C Y POS'NO-M) VFI. nr ITY (MM/SFC J " . B 9 1 5 0 . 8 5 9 4 - n . = ->~5 ' . 7""5 0.7385 -^ •.-6^ -2+-0 . 6 •'! 1 P r. 5>'7fi - 0.5*20 -0 . 4 4 3 3 0,4045 -4 ;-32 0 0-4 . 2 3 P P 4 . 0 i 0 0 . 3 . 5 3 0 0 3 . 3 1 0 0 3 . < 3.02 0 8 . 3 3 " 0 0--A R 7 «-0 . 0 8 74 0 . o ft 7 4 — P - ^ 8 7 - 5 -0 . 0 F• 7 5 O . 0 6 7 6 - " .-PR7-A A , .-. K 7 0,0877 — 6 4 j8 P 0 P 6 3 . 4 5 0 0 6 0 „ '.' 5 0 0 —5-6nr+6-0* — O.-00 48 ^ . 6 7 2 1 1 .4458 7 . 7 ] 00 7 . 2 5 0O 6 , 7f , n n -6 . ?', p 0 5 . " 8 0 0 o , A H 7 p, P . O f - 7 9 - P f. K d -A 0 U 8 1 0 , 0 8 R 1 50 - ^ H 0.3635 5.6000 n.PHng 2 8 . o o o " 21 .0552 7 5 . 6 3 2 8 2 0 2 0 . 3 0 2 0 5.0000 0 . O R R 3 25,0000 2.3.o6fli 6 7 . 4 6 9 4 2 0 5 0 . 2 9 4 5 4 . 6 0 0 0 . 0 . 0 * 8 3 23.00nr> 2 3 . 3 1 3 1 6 2 . 0 6 5 1 2 0 " 0 . 2 7-0 4 . 3 0 0 0 0 . 0 8 R 3 2 1 . , S O O O 2 3 . Q 8 3 3 5 8 . 0 0 0 3 2 0 S 0 . 2 - 6 0 3.6500 0 . 0 8 8 3 1 8 . 2 5 0 0 2 4 . 8 9 7 7 4 9 . 2 1 2 9 0 . 2 1 4 Q 3.0000 0 . 0 8 8 4 15.0000 2 5 . 9 4 2 0 4 0 . 4 3 0 4 2 0 8 R - 5 9 4 0 OUTER CYLINDER VFLOCITY = 2 0 . 2 CM/SEC 2 0 9 2 1 0 P f l W T H F T A R € W E - H e * P f l S- ' -V+MH-} V E L-fl£-f T Y (**/5€-ei-211 21? 1.0000 3 .4400 0 . 0 8 7 4 5 1 , 6 0 0 0 0 . 0 0 4 8 1 4 0 . 6 7 2 6 —2+3 0 . 9 6 6 0 3 . 3 2 0 0 0 . 0 8 7 4 49.0000 1 . 1 3 8 9 1 3 5 . 6 9 8 8 2 1 4 0 . 9 3 0 0 3 . 1 4 0 0 0 . 0 8 7 5 47.1000 2.33"3 1 2 8 . 2 7 3 7 215 0 . 8 9 1 0 2.0000 0 . 0 R 7 5 4 3 . 5 0 0 0 3 . 6 3 7 6 1 1 8 . 4 0 2 2 — 2 Hr 0- j 891-0 R - . R 4 0 0 -0-rf>*-7-5 4 4 - r 2-fHI 3 -.- * -37* 1 2 0 ; - ? ft 7b 2 1 7 0 . 8 3 2 0 8 . 5 6 0 0 0 . 0 8 7 6 4 2 . 8 0 0 0 5.59Q5 1 1 6 . 3 9 7 0 2 1 8 0 . 7 6 1 0 7 . 4 8 0 0 0 . 0 8 7 7 3 7 . 4 0 0 0 7.9^4 1 01 . 6 0 6 7 2 1 9 0 . 6 8 0 0 6.3000 0 , 0 8 7 8 31 . 5 0 0 0 1 0 . 6 3 7 3 8 5 . 4 7 7 1 2 2 0 0 . 6 3 R O 5 , 9 3 0 0 0 . 0 8 7 8 2 9 , 6 5 0 0 1 2 . 0 2 5 6 R 0 . 4 0 8 1 221 1.0000 3 . 5 4 0 0 0 . 0 8 7 4 5 3 . 1 0 0 0 0 . 0 0 4 8 1 4 4 , 7 6 2 0 — ? ? 2 — fi-795-00— 3;3-7-ftfr— -O-rOfl-7-4- 50-^5500 t v 6 7 2 6 — i 37 -v?fl97 223 0 . 9 0 2 0 3 . 0 5 0 0 0 . O R 7 5 4 5 . 7 5 0 0 3 . 2 7 1 7 1 2 4 . 5 4 6 2 2 2 4 0 . 8 6 2 0 2 . 8 2 0 0 0.0R75 4 2 . 3 0 0 0 4 . 6 0 2 5 1 1 5 . 0 8 7 4 —2/26 0-. R 6 2 0 R.37^-0 0 . 0 8 7 5 4 1 . 8 5 0 0 4^6 -0^5 1 1 3 . 8 6 3 1 2 2 6 0 . 8 2 6 0 7 . 7 9 0 0 0 . 0 8 7 6 3 8 . 9 5 0 0 5.70R6 1 0 5 . 9 1 7 6 2 2 7 0 . 7 R"0 7 . 2 8 0 0 0 . 0 R 7 7 3 6 . 4 0 0 0 7 . 3 2 5 5 9 R . 9 1 7 0 - -228 0 . 7 2 7 0 6 , 7 9 f i f ! f>.0 "77 3 3 . O 5 0 « O . 0 R 2 4 9 2 . 1 8 8 3 . 2 2 9 0 . 6 6 9 0 6 , 3 6 0 0 0 . 0 R 7 R 3 1 . 8 0 0 0 1 1 . 0 0 1 2 8 6 . 2 7 7 4 2 3 0 0 . 6 0 P 0 5 . R 9 0 0 0 . O R 7 9 2 9 . 4 5 0 0 1 3.0 ) 5 8 7 9 . 8 3 1 0 2 3 1 1 . 0000 4 . o 3 0 n 0 . 0 8 7 4 7 3 . 9 5 0-0 0 . 0 0 4 8 2 0 1 .6035 232 0 . 9 4 3 0 4 . 5 5 0 0 0 . 0 8 7 4 6 8 . 2 5 0 0 1 . 9 0 6 0 1 8 5 . 9 0 9 5 2 3 3 0 . 8 9 2 0 4 . 0 3 0 0 0 . O R 7 5 6 0 . 4 5 0 0 3 . 6 0 4 6 1 6 4 . 5 4 0 6 — 2 3 4 : ft-8 4 3 0 3.~A-<Vfrfl ft-.-ftB-'f, 57 . f t f tOO 5 - . - 2 3 3 9 1 5 5 . 0 3 9 5 2 3 5 0 . 7 P 8 0 3 . 5 2 0 0 0 . 0 8 7 6 5 2 . 8 0 0 0 7.0&01 1 4 3 . 5 0 0 7 2 3 6 0 . 7 1 6 0 3 . 2 0 0 0 0.OR77 48.0000 9 . 4 4 6 4 1 3 0 . 3 1 9 0 —237 • • • •—• • • 2 3 8 R - 6 0 2 7 ntJTFR CYLINDER VELOCITY = 2 1'. 0 CM/SEC 2 3 9 —2 4 0 FMlS-^H-f+H-) -*e~V-TrtrT-S T~HF-T~A F - R F T W E M C Y K1S- 4 *J-<WM> Vf+n(M"T'Y"f t**t/Sf 2 4 1 2 4 2 1.0000 5 . 1 3 0 0 0 . 0 8 7 4 7 6 . 9 5 0 0 0 . 0 0 4 8 2 0 9 . 7 8 2 2 2 4 3 0 . 9 8 6 8 5. 1 000 0 . 0 8 7 4 7 6 . 5 0 0 0 0 . 4 4 4 8 2 0 8 . 5 1 5 9 2 4 4 0 . 9 6 2 8 4 . 8 0 0 0 0 , 0 8 7 4 72.0000 1 . 2 4 6 0 1 9 6 . 1 8 0 8 2 4 5 0.93a? 4 . 4 0 0 0 0 . O R 7 5 66 k0000 2 , 0 6 5 7 1 7 9 . 7 6 8 1 - 2 4 6 0 . -9 ft-om 4-T2/0 O-O 0-rO-P -7-5 6 3~rM> <V0 3-.-01 * 6 1"f 1~.-5 2 55 2 4 7 0 . R R 2 P 4 . 0 5 0 0 0 . 0 8 7 5 6 0 . 7 5 0 0 3 . 9 1 0 8 1 6 5 . 3 3 5 1 2 4 8 0 . R 4 6 0 3 . 5 5 0 0 0.ftR76 5 3 , 2 5 0 0 5 . 1 3 4 2 1 4 4 . 8 4 5 6 - 2 * 9 0 . 'M 4 ft -^ - r « -S-0-fi 0 . 0 8 7 6 . 51 . o*-*-fl 6 T + J 7+z s 1 3 8 . 6 6 2 3 2 5 0 1 . 7 9 2 8 3 . 3 0 f t 0 0 . 0 8 7 6 49.510(1 6 . 9 0 1 0 1 3 4 . 5 4 1 3 2 5 1 0 . 7 7 0 2 3 . 2 0 0 0 0 . 0 R 7 7 A R . O n O O 7 . 6 5 0 h 1 3 0 . 4 2 1 4 -25-2 Ov-7^ 6-0 3-rt-O-O-fl— 0-rO 8-7-7 ttfrj-^HHVO -8 T-7 -8 3R t-26"r?> 8-3-2 2 5 3 0 , 7 0 9 5 2 . 9 0 0 0 0 . 0 R 7 7 4 3 . 5 0 0 0 9.6613 1 1 8 . 0 9 0 5 2 5 4 0 . 6 7 2 5 2 . 8 0 0 0 O . O R 7 R 4 2 . 0 0 0 f t 1 0 . 8 8 5 4 1 1 3 . 9 5 7 1 -25^ ft . 6 3 4 6 2 . 8 0 0 0 0 . 0 R 7 8 4 2 . 0 0 ft-0 12.1 1 1 3.8946 2 5 6 0 . 5 9 2 5 2 . 7 0 0 0 0 , 0 8 7 9 4 0 . 5 0 0 0 1 3 . 5 2 7 1 1 0 9 . 7 5 9 9 2 5 7 0 , 5 4 7 5 2 . 5 5 O 0 O . f t R S O 3 8 . 2 5 0 0 1 5 . 0 1 0 4 1 0 3 . 5 9 4 7 - 2 5 8- 0-5 0-7-0- 2-jpiHHV* ft--ftp R ft 36 ,-0 Oftfl 1 -6-r3-4 4 0 97 y4 4 39 2 5 9 0 , ' ! 7 2 P 2,3000 O . O H R O 34,5000 1 7 . 4 6«6 9 3 . 3 3 7 6 2 6 0 0 . 4 7 2 8 7 , 4 0 0 0 O . O R R O 37,0000 1 7 . 4 6 R 6 1 0 0 . 1 0 1 2 0 . '1398 7. 1 OO0 o.ORRi 35.50-p-fl Hh-W+ 95.997g 2 f t ? 0.3015- 6 . 5 0 0 0 0 . 0 8 8 1 32.5000 20 . M 7 3 P 7 . R 2 3 6 2 6 3 1 . 3 5 0 1 5 .4f t f t f t 0 . 0 P P 2 ? 7 . 0 f t O 0 2 1 . 4 9 3 7 7 2 ° 9 1 7 h 2 6 4 0 . 3 0 7 9 4 . 4 0 0 0 0.0883 2 2 . " 0 0 0 2 2 . 8 7 5 0 5 9 * 3 7 8 1 - 0 . 2 7 4 5 3 . 3 0 0 0 0 . O P R 3 1 6 . 5 0 0 0 p - j . o ^ s ; 4 4 . S 1 2 1 2 6 6 O . ? ' i 3 0 2 , 8 00ft 0 , 0 8 8 3 l / . O O O f t j u . Q o t , - ; 3 7 . 7 5 0 7 -?-tyr . ft__p.A.1,A H - a w o-i-ft -.--i-ft0-0 2 3 1 ; i p .^I?r , rn ? 6 P 51 ?69 2-7 ft 9= 6 1 5 5 ODTTP C Y L TNDFP VFl OTT TY = ? 1 . 1 C ' V S f C 271 ?72 2^-3 90.8 1 v CI f->) 0.9020-"it VC1I.TS 5—1 s ft n THFTA n A n 7 >i F R F O U F ^ C Y POS ' •; ('•"•'•l V F L 0 C 1 T Y C " ' V S F C ) ?74 275 - 9 7 6 0.9775 0.9545 0.9197 5.1500 4 . 7 0 1 0 4.26 0-0— \> ~ i *r 0 . 0 R 7 a 0 . D R 7 4 n fl fi 7 r: 7 7.25 ft-ft— 75.7500 70.5000 i-l A A A 0.27)9-0.7553 1.5225 — 5 7 5 3 206.4432 192.0705 277 27« : 0 ,880b 0.80 81 0.8015 3.9500 3.R200 3.3800— »' • 'J n t 3 0 . 0 8 7 S 0 . 0 K 7 6 n nn?/. I* 5 (t-S 9 . 2 S 0 0 S7.^000 f ri T A A « ? - . 6 8 ? 5 3,9840 5.0644 —1-7-#T*6-HI 161 .2476 155.8671 280 281 2« 2 0.7713 0.7287 e - T f r 7 - * 9 — 3.2700 3.1700 2 9P0ft— " « U M / f.) 0.0877 0.0877 0—0 8-7-8-•.j i i , / o o H--49.0500 47.5500 /l/l T A ft 6.6)2? 7.614? 9.0257 — 1 37. 8205 — 133.2766 129.1211 283 ?84 0 .6220 0.5628 0 .4965— 2.9200 2 . 7 7 0 0 2. 6 0 0 0 0.0879 0.0879 0 (innfi W 'J „ M > 0 P— 1-0-7-8-0-6-)-12,5539 14.5065 — 1 - ? 1 .2873 — 1 18.7543 112.5573 2P6 287 0.4390 2.4500 l' 4 ii " tf 0.0881 jrQ . 7 S 0 0— 36,7500 1 6.6803 18,5789 — 1 07 .5780 99,3762 -28-8— 289 290 -29-1— 2 9 2 2 9 3 -294 295 296 . 297 2 9 9 2 9 9 - 3 0 0 -3 0 1 302 -3-0^ 3-304 30s 3 06 307 308 3 ft q 3 1 0 3 1 1 - 3 1 2 -3 1 3 . 3 1 4 3 1 5 3 1 6 317 - 3 1 fl~ 319 320 POS '".cm —*=6?00 OUTL~W C V L I M P F P V E L O C I T Y - ? 1'. (, C M / f l F C HC V O L T S T H F T A FBFGIIfc'MCY P O S ' M ( M M ) V F L f l C I T Y C M M / S E C ) .0000 .9860 -r9-6*fl-.9330 .0020 .8<,?p .8230 .7830 -.-7-4-2-0-69O0 .6430 .5 730 1.5220 S. 5 . 4. 4. -«-r 2 8 0 0 1 200 5500 2600 0100 3. 3. -4-r 3. 3. 7400 58O0 -3-8-0 ft-2700 1 0 0 0 •ft-g-A-O-.0874 . 0874 . 0875 .0875 .0A75 79 76 68 63 -6-0 2.6900 -P=+-0B20-.0876 .0A77 -.-ft 8-7 7--.1878 .0878 v0»79 0.0880 56 53 -5-0-49 46 -4-? .2000 .8000 •i-4-5-0*-,2500 .9000 . 0000 .1000 . 7000 -,-7-ftftft-,0500 .5000 T^ft 0 0 0,0048 0.4719 -l-T+38-9-2.2393 3.271 7 • a.6 0 25 8985 2260 58 5-3-ft092 86 0 4 1 0 1 1 14.1701 215.9161 ?09.330? - H r^va+ f r * -1 8 5 . 8 8 2 5 173.9S65 163.2446 152.5471 1 45.9361 -*3-7-rH>4-3-1 3 3 . 1 3 6 9 1?6.1131 114.6059 40.3500 15.8502 109.2418 -^+fT-FCf----e-VL7TAf'->e<*---V'Fl:'>e-I-TY-^-37 ;-7---eM/SfrCr-P 0 S " " C T v ^ OC V O L T S T H F T A F P F G l IF MC Y p n s ' M f V M j V F L O C I T Y f i-" V 5 F C 1 . 0000 .°660 -.-»1?-P-,8820 .8440 . 74 00 . 6 H 1 0 i^ -4-20-. q ft r) n 15330 9.2200 8.3600 -6-r9*-ft-0-6,4300 5.O800 5.7100 5.4100 5.2110 -4-.-9 4AA-4,6600 a.aooo 0.0874 0 . 0874 --ft-,-0* 75-.0 . 0875 0.0876 0 . 0876 0.0877 0 . 0878 8-7-8-0 , 0879 1 . 0 8 8 0 1 38.2999 125.3999 - + < H h r 3 9 9 « » -96.4499 89.7000 85.6409 81 . 1500 78.1409 —7-4-r+ftO-O-69.9ft00 66,0000 0. 0048 1 .1389 -2-.-7-7?4-3.9374 5.2010 6 . 8-9-4-4-8 . 6 5 1 6 1 0 . 6 0 4 2 - H - T * " 3 5 -1 3 . 6 4 3 0 1 5 . 4 8 8 2 377.0354 341.6978 -?rWiv-?-7-25-262.4917 243.9867 232.7981 220.3977 212.0679 -20-0^ -96*4-1 "9.4283 178.7142 3 2 ? 3 2 3 3 2 4 3 2 5 3 ? h 3 2 K 3 2 9 3 3 0 3 3 1 3 3 ? 334 * 3 5 —1.496 ft 0 . a a o ft 0.4020 0.3470 ft.2«oo 1 . 2 5 2 0 -V?3*f-0.4110 ft . a a r, 1.5630 1,6180 1 , 6 0 AO 0.8460 ft . o 2 4 o 4 . 3 1 10 4 . 1 2 0 0 3 . 8 / 0 0 3 . 6 100 3 . 2 6 O 0 ? , H 8 ft ft 3 . ' . ' ft ft ft 3.85ftft a.2?0 0 • 4 ; 5 6 0 ft 4 . P '4 0 0 5.2210 -5-r5-6-ftfV-h. 0200 7.020ft 0 . 0 n « 1 0 . ft * 8 0 0 . ftp 79 0 . ft 1; 7 9 0 . ft 7 8 -ft-j-ft*»-7-7-0 . 1 P 7 6 0 . " P 7 5 1 387 8901 2335 96a 6 5031 •7-ftA9--fl , 0 ? a 0 64 . 650ft 1 6 . 7 " 5 4 1 7'I. 965c 0.OPP1 61 .8 0 0 0 18,25 0 7 1 6 7 . 1 38" ft.PPPI 58.0500 19.7"31 0.OPP? 5 4 . 1 5 1 0 ?1.5956 0.0BP3 4P.Oft 00 23,1661 0 . ft r> a. 3 4 1.2000 ? 4 . 7 0 2 0 0 . n p p ? h-l . ft ftft-ft 2 1 . 'Jft-i— 19.4977 1 7 , ! "' n 5 1 4 , 4 n o o 12.6860 ] ft . 0 4 ? ft —P ,0^" 5.13a? 2 . 5 3 9 ? 5 7.7510 • 63.3110 6 a.39oo 7?.59"9 7 P . 3 i ft ft —P3-r3«9-9-9 ft . p o o o 1 O 5 . ? o t)'.) 167, 156, 146, 131 . 116. 1 37. 156. 171. IPS. 1 0 6 . ?1 ?. -2*6-245 , 0995 , ?P 1 8 , 20^5 827P 527? -5-7-0-4-6?56 ?66,75?4 52 3 3 7 3 3 8 -559-9 = 1 5 8 7 0 O u T F ' P C V I . I M O E S V E L O C I T Y = 5 4 . 7 r M / S F C p n g ' " ( ! »• ' ) r»C V O L T f l T H E T A C R F O H E M C Y P O S ' M ( H M ) VEI O f T T Y ( M M / S F C ) 3iin 3 4 1 — 3 4 ? ~ 3 4 3 3 4 4 — 5 4 * -1 . 0 0 0 0 - f l - -959f>-o . 9 2 f , n 0 . 8 7 6 5 0 . 8 3 1 9 0 . 7 8 8 5 r . 7 " 8 5 0 . 6 4 8 8 0 . 5 6 8 2 4 . 0 1 0 0 -3 - r? -7 -Of l -3 . 0 4 0 0 2 . 7 9 0 0 2 . 6 4 0 0 2 . 5 5 ( 1 0 P . 1 o o o -f-. -R-^-O-fl-7 . 2 2 0 0 6 . 7 5 0 0 0 . 0 8 7 4 - O r O P ^ - a -0 . 0 8 7 5 0 . 0 8 7 5 6 0 . 1 4 9 9 ~4 9 - ; - 0 * 0 0 -4 5 . 6 0 0 0 41 , 8 5 0 0 0 . 0 0 4 8 - 1 - . - 3 * 6 1 -2 . 4 7 2 2 a . 1 2 0 2 5 4 6 . 6 -4 4 « i - ; 5 -4 l 3 . 9 3 7 9 . 6 0 . 0 8 7 6 0 . 0 8 7 6 0 . 0 8 7 6 -0^-0-8-7+-0 , 0 8 7 8 0 . 0 8 7 9 39.6000 3 8 , 2 5 0 0 1 2 1 . 4 9 9 9 1 1 7 . ? 9 9 « -1 0 8 . 2 9 9 9 101 , 2 5 0 0 5 . 6 0 2 8 3 5 9 . 0 34 6 3 47 - 3 4 8 -3 4 9 3 5 0 551 3 5 2 3 5 3 3 5 4 3 5 5 3 5 6 C - J P O F F I L E 7 . 0 4 3 8 7 . 0 4 3 8 —9-T211-2-1 1 . 7 3 4 9 1 4 . 3 2 8 4 3 4 6 . 5 3 3 0 . 2 1 7 0 - 5 1 * . 5 0 0 0 2 9 3 . 7 3 7 3 2 7 4 , 3 0 4 0 0 . 5 1 6 0 0 . 4 6 7 9 0 . 4 2 5 1 - o - r 3 - 6 9 * » -0 . 3 2 0 4 0 . 2 7 1 2 6 . 4 5 0 0 6 . 0 7 0 0 5 . 8 8 0 0 - 5 - . -55-0-0-5 . 3 0 0 0 5 . 0 9 0 0 0 . 0 8 8 0 0 , 0 8 8 1 0 . .1 e 8 1 9 6 . 7 5 0 0 9 1 . 0 U 9 9 8 8 . 2 0 0 0 " 8 5 - ^ - 2 5 0 - 0 -7 9 . 5 0 0 0 7 6 . 3 4 9 9 1 6 . 0 4 7 - 7 2 6 1 . 9 1 4 6 - o — i - « * 2 -0 . 0 8 8 2 0 . 0 8 8 3 1 7 . 6 2 9 7 1. 9 . 0 3 5 1 - 2 l - ; - f l 5 4 -9 -2 2 . 4 6 6 1 2 4 . 0 7 4 8 2 4 6 . 3 1 2 7 2 3 8 . 4 5 5 2 • 2 2 4 . 8 9 2 1 2 1 4 . 6 0 9 6 2 0 5 . 9 6 0 2 S S I G r c Figure 1.1 Typical LDV signals from particles of approximately uniform size: (a) x = 0.2 msec/cm, y = 5.0 mv/cm; (b) x = 0.2 msec/cm, y = 0.2 mv/cm FIXED WALL (a) FIXED WALL (b) Figure 2.1 Theoretical (a) laminar velocity profiles for plane flow; (b) turbulent flow Couette flow: Reference beam operation: (a) Schematic of frequency s h i f t ; (b) Schematic of optical Figure 3.2 Formation of LDV fringe pattern through interference by intersecting laser beams cn pin photodiode in amp. circuit 1 signal out aperture c! lens 5 cm mirror FLOW laser lens beam 10 cm splitter variable filter disa pre-amp doppler signal processor frequency to voltage DC digital VM flow parameters readout Figure 3.3(a) Schematic i l lus t rat ion of the laser Doppler system used \1 Figure 3.3(b) LDV and Couette flow apparatus 30 4 0 50 100 200 500 1000 2000 4 0 0 0 (slew rote) Figure 3.4 Calibration curves for DISA tracker 10 Figure 3.5 Calibration of ambiguous broadeni 61 fo proces A electroh |sing ics photodiocje light beam lens 4.1 Schematic of Couette flow apparatus Figure 4.2 Laminar flow prof i les , concentric cylinders The theoretical flow is for in f in i te 63 Figure 4.3 Laminar How prof i les . The theoretical flow is for i n f in i concentric cylinders Figure 4.4 Turbulent flow prof i les . Theoretical profiles with and without modification proposed by Hinze 65 Figure 4.5 Turbulent flow profiles X X X xx CIRCULAR COUETTE FLOW,WATER X Present Work A Murguly (1971) (unpublished) PLANE COUETTE FLOW,AIR O Robertson (1959) 1 v b_ 78U X X  /S N , " U" v"8y/ c ( c o r e r e g i o n s l°P e> A X I.I x I 0 4 Figure 4.6 Core region slope as a function of Reynolds number 70 I I I I I I I I I ! I L 7 8 9 10 II 12 13 14 15 16 17 18 19 20 FREQUENCY (Khz) Figure 4.7 Typical laminar flow spectrum. U = 4.8 cm/sec 100 12 11 10 9 8 7 6 5 4 3 2 r~ • R = 6256 A R = 10820 o R = 15700 X R = 66000 (Johnson 1970, plane Couette flow in air) A A 0 o * A ° A ° A • A A A o a A A A * A A A A A V  N A A A • • • • • O A A A 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 fixed wall 2b 4.8 u' turbulence intensit ies vs normalized position 69 Figure 4.9 Measurement of Reynolds stresses (a) normal fringe pattern; (b) counter clockwise fringe rotation (looking from source; (c) clockwise rotation of fringes 12 II 10 8 7 xlOO 4 3 2 0 R = 10820 K o o o a o o i 0.2 0.3 04 05 0.6 07 0.8 0.9 1.0 Figure 4.10 w"1 turbulence intensities vs normalized position Figure 5.1 Determination of C f from the Clauser curves 0.016 d) Couette, air • water • Reichardt, oil • water I 0 2 I 0 3 I0 4 I 0 5 Figure 5.2 Skin f r ic t ion coefficients vs Reynolds number from various workers Figure 5.3 Representative semi-log plot showing the logarithmic wall region 

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