A STUDY OF CIRCULAR COUETTE FLOW BY LASER DOPPLER MEASUREMENT TECHNIQUES by NIGEL D.S. GEACH B.A. S c . , University of B r i t i s h 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 presenting an advanced degree at the U n i v e r s i t y the I Library further for this shall agree scholarly thesis make i t freely that permission h i s representatives. of this thesis of British f o r financial gain Columbia, copying o f t h i s shall that /hrif lÂ£ { /<? 74 thesis copying o r p u b l i c a t i o n n o t be a l l o w e d w i t h o u t A-^*^t-Ly- Columbia that by t h e H e a d o f my D e p a r t m e n t o r permission. The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a I agree f o r r e f e r e n c e a n d study,. f o rextensive I t i s understood Department o f / ^ ^ A ^ L A / Date f u l f i l m e n t o f the requirements f o r available p u r p o s e s may be g r a n t e d by written in partial my ii ABSTRACT A laser Doppler velocimeter is constructed and used to make flow measurements in c i r c u l a r Couette flow. The flow is created between concentric cylinders with a small gap-to-radius r a t i o , and measurements of the velocity profiles 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 i o n 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. iii TABLE OF CONTENTS Chapter Page 1. INTRODUCTION 1 2. THEORY 3 2.1 3. 4. Laminar Couette Flow . 3 2.2 Turbulent Couette Flow 4 2.3 6 Reynolds Stresses INSTRUMENTATION 8 3.1 Background 8 3.2 Components 3.3 Calibration . . . . 3.4 Signal Broadening . . . . . 8 9 10 MEASUREMENTS AND RESULTS 13 4.1 The Flow Apparatus 13 4.2 Procedures 13 4.3 Analysis 15 4.4 Laminar Profiles 4.5 Turbulent Profiles 16 4.6 Spectral Broadening 17 4.7 Measurements of/ u'2 . . . . . . . 17 4.8 Measurement of Reynolds Stresses . . . . . . . 18 . . 16 4.9 .Measurements of ^ w ' ^ 19 5. DISCUSSION . . . . . . 20 6. CONCLUSIONS 24 REFERENCES . . . . . . . . . . . . . . . . . . . . . . 26 . iv Chapter Page APPENDIX APPENDIX I - Exact Solution of the Navier-Stokes Equations for Laminar Circular Couette Flow II - A Three Region Model for Turbulent Couette Flow APPENDIX III APPENDIX APPENDIX APPENDIX - Measurements of Reynolds Stresses by Laser Doppler Velocimetry â€¢â€¢ . 32 . 37 IV - Theoretical Description of the Laser Doppler Velocimeter . . . . . 40 V - Tracker Calibration VI - Data 2 8 43 , . 45 V LIST OF FIGURES Figure 1.1 2.1 3.1 Page Typical LDV signals from particles of approximately uniform size . . . . . . . . . . . . 53 Theoretical velocity profiles for plane Couette flow 54 Reference beam operation 55 3.2 Formation of LDV fringe pattern through interference of intersecting laser beams . . . 3.3(a) Schematic i l l u s t r a t i o n of the laser-Doppler system 56 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 4.1 Schematic of Couette flow apparatus 4.2 Laminar flow profiles . . .. 62 4.3 Laminar flow profiles . . 63 4.4 Turbulent flow profiles 4.5 Turbulent flow profiles 4.6 4.7 Core region slope as.a function of Reynolds number . Typical laminar flow spectrum 4.8 4.9 u turbulence intensities vs normalized position Measurement of Reynolds Stresses . 60 . 61 64 . . . . . . . 65 66 67 2 . . 68 69 . . 70 2 4.10 w 5.1 Determination of C- from the Clauser curves 1 turbulence intensities vs normalized position 71 vi Figure 5.2 â€¢5.3 Page Skin f r i c t i o n coefficient vs Reynolds number by various workers 72 Representative semi-Tog plot showing the logarithmic wall region 73 vii 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 i c t i o n E mean voltage output of LDV tracker e fluctuating voltage output of LDV tracker 1 f f Â»f-j frequency components of LDV signal H curvilinear 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 5 0 I > lQ radial and circumferential velocity components R Reynolds number based on cylinder velocity and gap width, c r c v R, Reynolds number based on midstream velocity and half gap, U c b - ~ r^,r^ S c â€¢ (R/4) radii of inner and outer cylinders, respectively . , b 9U core region slope, -g- ^TT c y=b circumferential velocity of Couette flow y U U Â£ center!ine flow velocity U o outer cylinder velocity, 2U c U ,u' components of velocity normal to the clockwise rotated fringe J pattern J fluctuating velocity components of flow in Cartesian coordinates f r i c t i o n 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 light beams von Karman constant, 0.4 wavelength of light viscosity (absolute) viscosity (kinematic) density shear stress, shear stress at wall laminar and turbulent contributions to shearing stress curvilinear coordinate in Hamel spiral motion frequency of turbulent fluctuations angular velocity of inner, outer cylinder respectively components of v o r t i c i t y stream function of flow ACKNOWLEDGEMENTS The author would l i k e 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 B r i t i s h Columbia were used for the reduction of data contained herein. This research was supported by the University of British Columbia and the National Research Council of Canada. â€¢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 c u l t i e s 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 simplicity. The shear flow between rotating concentric cylinders is also interesting in i t s 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 circular Couette flow since i t was i n i t i a l l y studied by Couette 1 [1870] is small. Some of the work includes the studies of Sir G.I. T a y l o r ' 2 S.I. Pai 4 [1939], and D.C. McPhail [1941]. 3 [1923 and 1936] 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 7 fi 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 circular 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 Cummins ^ with their 1 "laser-Doppler" velocimeter. This type of measuring technique lends i t s e l f to velocity measurement in circular Couette flow because the probe is simply an e l l i p s o i d of l i g h t , with no potentially disturbing intrusions into the flow. For this reason, and because there are few known published measurements of turbulent circular 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 light which cross. The small volume where they cross is the point of measurement with small particles which move with the f l u i d generating a frequency proportional to velocity. Figure 1.1. Typical laser Doppler signals are shown in The measurement of any mean velocity merely requires the a b i l i t y to measure the mean frequency; while to measure a fluctuating velocity requires an a b 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 velocities have been measured, as well as some representative measurements of turbulence intensities and core region profile slopes. 3 2. 2.1 THEORY Laminar Couette Flow The study of the laminar regime in c i r c u l a r 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 i n f i n i t e cylinders. Laminar flow is also free of turbulent velocity fluctuations, so measurements can be made of the spectral or ambiguous broadening of the s i g n a l , an effect which w i l l be discussed later in the text. The ideal plane Couette flow profile is shown in Figure 2.1(a). This is created by an i n f i n i t e upper plate moving with a velocity U with respect to an i n f i n i t e stationary lower plate. 0 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 r e l a t i o n : U = 2.1 h where h is the distance between plates. of a Newtonian f l u i d the shearing stress Furthermore, for laminar flow is proportional to the slope of the velocity profile i . e . : T = y dU dy 2.2 4 This shearing stress increases rapidly upon transition from laminar to turbulent flow. The exact profile of Couette flow between i n f i n i t e 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 l U = 2 ~2 2 " rl 2 2 C ( 2 2 " r u r w 2 l l^ r ^ 2 ~ l ^ r w w 2 - 3 r r where r-j and 2 1^2 are the radii of the inner and outer cylinders respectively, which rotate with angular velocities co-j and u ^ . A l l measurements reported in this study have been made with the inner cylinder f i x e d , i . e . , = 0, so that Equation 2 . 3 reduces to: U = â€”p r 1 2 p [roj r ? ? r 2 2 l 2 r u)p ] 2.4 2 " 1 r 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 line 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 prof i l e s 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 w a l l , a log-law region further from the w a l 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 13 more, experimental evidence from pipe flow [Hinze, 1959] suggests the profile 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. U and slope At the matching point, the core region velocity 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 restriction 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 w a l 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 p r o f i l e s , estimates of shear stresses are reported in Chapter 4. These shear stresses, or Reynolds stresses, arise from the interaction between the u' and v components of the turbulence, as long as a shear layer 1 exists, and can be demonstrated as the mechanism by which the wall stress is imparted to the opposite w a l l . wal1, r t Â» For turbulent flow far from the T ^ , hence . T = 9U xay A = PÂ£ W 3U 2.7 where p is density, and e is eddy viscosity. From mixing length con- siderations, the following equalities are v a l i d : â€¢p u v = pe 8U â€ž2, dU = pA ( ^ ) (see reference 12) 2 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 l d of flow. The mixing length Â£ can be chosen as the characteristic linear dimension for the fluctuation, 7 A f r i c t i o n velocity, u*, which i s characteristic of the turbulent motion, can be defined in terms of the shear stress as follows: Thus T also s a t i s f i e s the following: x = pu; = -p u'v' As seen from Appendix II, 2.10 the value of u* can be arrived at through the measurement of the velocity p r o f i l e s . From these measurements, the Reynolds stress is estimated for turbulent circular Couette flow, and reported in Appendix VI. 8 3. 3.1 INSTRUMENTATION Background The fact that the Doppler s h i f t of laser light could be used to measure flow velocities was f i r s t demonstrated by Yeh and Cummins^ [1964], and subsequent investigations by Goldstein and K r e i d ^ [1967], Rudd 15 [1969], Durst and Whitelaw have a l l served to extend the technique. 16 [1970], and Greated 17 [1971] 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 setup. The beam source was a 15 milliwatt Spectra Physics Helium-Neon laser operating in the TEM-00 mode. The l i g h t wavelength was 6328 Angstroms and the beam diameter at point of s p l i t t i n g was 1.2 millimeters. The s p l i t t i n g was accomplished using a f i f t y percent beam s p l i t t e r 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 m i l l i w a t t s , indicating a certain amount of loss from the reflecting surfaces. The gap between the beams was measured as 11.68 millimeters. An off-the-shelf 100 millimeter focal length lens was used to focus the light 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 r c u 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 i g h t from particles passing through the bright fringes at varying speeds. It was then band pass f i l t e r e d 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. Figure 3.4(a). Calibration curves appear in The AC capabilities 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 i s of the form f = f sinwt + f, o 1 where f-| is the DC component, f frequency of fluctuation. 3.1 the range of fluctuation, and w the Ideally i f the probe volume were i n f i n i t e 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 i g h t , and 9 the half angle of intersection. The frequency f^ is directly proportional to the DC voltage from the frequency tracker. S i m i l a r l y , 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 signal, which adds an uncertainty 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 3.3 9 In practice, however, i t is often more advisable to measure the broadening 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: - U where Af^ ) [<f T l 2 - turb ] T l lam is the measured broadening in laminar flow, It must be noted that 3.4 1 / 2 (see Reference 19) the use of Equation 3.4 as shown above represents a simplified approach to the problem of broadening. in turbulent flow there exist the following effects: Generally broadening due to variations in velocity across the scattering (probe) volume, Af^; and broadening due to the fluctuations of volume averaged velocity, A f u . 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 p a r t i c l e s , Afg; and the non-monochromaticity of the laser light source, Af<.. Assuming these effects to be Gaussian, the bandwidth observed would be given as follows: 2 Af* = ? 2 2 2 2 ? A f V + Aff. + Afâ€ž + Af* + Af* + Af* 0 T Jc G B S 3.5 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 l e f t with: Af* - Af: J6 = Af* u o + Af* â€¢ 3.6 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 ( -fâ€” ). The j u s t i f i c a t i o n for not attempting to compensate for this l T factor i s 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 t s 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 slight variation of the percentage with the output voltage of the tracker. Correction of turbulence measurements was carried out u t i l i z i n g this curve, i . e . , the value of ambiguous broadening was chosen depending on the mean D.C. voltage at the measuring point. 13 4. 4.1 MEASUREMENTS AND RESULTS 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 radii 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 electric 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 d r 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 p r o f i l e s , 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 l a t 0.5 inch thick base plate with rubber mat supports as vibration i s o l a t i o n . Profiles were taken at varying heights above the base of the cylinders in an attempt to find a region of the flow which was relatively 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 i g h t to the detector. Drop- out (loss of signal) due to insufficient 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 w i 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 wall. Closer proximity resulted in a D.C. flow frequency below the lower l i m i 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 p a r t i a l l y turbulent (transition). was attempted using dye. Flow visualization It was noted in the case of transition that the streaks exhibited laminar s t a b 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 slight 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 u n i t ) ; 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 circumference. 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 position, this effect has been minimized. 16 4.4 Laminar Profiles 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 slight 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 4.5 Appendix VI. 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 profile is highly dependent on the value of the f r i c t i o n 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 Since the analyzing equipment was readily available in the 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. 4.7 Measurements of Measurements of - ~ â€” 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 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 slight Reynolds number dependency evident from Figure 4.8, and this is contrary to the observations of Robertson and Johnson, which indicated that turbulence intensities were independent of flow Reynolds number. 4.8 Measurement of Reynolds Stresses As described in Appendix III, stress u'v 1 the values of the Reynolds 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 45Â°. Sample measurements of u'w' were made with the LDV probe volume directed normal to the cylinder w a l 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 light intensity caused by an increased angle of incidence, combined with d i f f i c u l t i e s involved in the location of the light receiving optics. 4.9 Measurements of As described in Appendix III, permits the measurement of thev w the slant fringe technique 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 u values. 1 The large scatter encountered in measuring the rms voltages in the slant configurations make the accuracy of the w measurements open to question, however i t is 1 probable that the indicated trend is accurate. 20 5. DISCUSSION From the values obtained for mean flow v e l o c i t i e s , i t is seen that circular Couette flow in water is consistent and predictable. Using the mean p r o f i l e s , and law-of-the-wall assumptions, i t is possible to arrive at estimates of the f r i c t i o n coefficient C^, the f r i c t i o n velocity u*, and the shear stress T . These values can then be compared with previous results and appropriate conclusions drawn. 1o 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 for the turbulent profiles reported, with U/U yU plotted against log-jQ â€” ~ . As can be seen, allowing for scatter f c yields a f r i c t i o n 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 i o n velocity u* is related to the shear stress T by 5.1 while the coefficient of f r i c t i o n is 5.2 Thus u* can be estimated directly from the C value by f Alternate values of u* are arrived at by solving the equation for the velocity profile 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 Ub c S 5.6 e These values also appear in Appendix VI, accompanied by some representative results from previous work. As justification of log law relationships in the wall region, 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 RMS values of the turbulent velocity fluctuations in the circumferential (x) direction (i.e. / u ' / u ) displayed a consis 2 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. 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 f a i r l y 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. 24 6. CONCLUSIONS Laser Doppler velocimetry has been successfully used to make measurements of velocity profiles in both laminar and turbulent circular Couette flow. Physical limitations inherent in the apparatus have introduced an uncertainty to measured values close to the moving w a l l , while the natural limitations of laser Doppler systems have prevented measurements from being taken in the low velocity region close to the stationary w a l l . These limitations are the f i n i t e dimensions of the focal volume (0.64 mm in length) and the lower l i m i 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 i o n coefficient for the plexiglas cylinder, and subsequent estimates of shear stress and Reynolds stress. Furthermore, the complete turbulent profile 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 profile d i s t o r t i o n , probably due to end effects, as observed by Coles. Laser Doppler methods as applied to the measurement of turbulence intensities produced results which had somewhat greater scatter than those observed by conventional techniques in a i r . Also, turbulence intensities showed a slight 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. u'v , 1 The technique was also applied in an effort to measure 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 wall. 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 i s low. It is f e l t that LDV methods can be significant 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 flowdisturbing probes combined with a linear response, make i t the most practical tool available for this type of measurement. The v e r s a t i l - i t y 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 measurements 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 p o s s i b i l i t y has been removed by the use of the LDV technique. 26 REFERENCES 1. Couette, M. "Etudes sur l e frottement des l i q u i d s , " Ann. de Chemie et de Physique, Ser. 6, V o l . 21, 1890, pp. 433-510. 2. T a y l o r , G.I. " S t a b i l i t y of a viscous l i q u i d contained between two r o t a t i n g c y l i n d e r s , " P h i l . Trans. 1923, A 223, 289. 3. T a y l o r , G.I. " F l u i d F r i c t i o n Between Rotating C y l i n d e r s , " Proc. Roy. S o c , A 157, pp. 546-564. 4. Pai, S.I. "Turbulent Flow Between Rotating C y l i n d e r s , " NACA TN 892, March 1943. 5. Reichardt, H. "Uber d i e geschwindigkeitsverteilung i n einer geradlinigen turbulenten Couette stromung," ZAMM Sonderheft, V o l . 36, 1956, pp. S26-29. 6. Robertson, J . M . "On Turbulent Plane Couette Flow," Proceedings of the S i x t h Midwest Conference on F l u i d Mechanics, 1959, pp. 169-182. 7. Van A t t a , C , "Exploratory measurements in s p i r a l turbulence," J . F l u i d Mech. (1966), V o 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 l u i d Mech. (1966), V o 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, V o l . 96, No. EM6, Proc. Paper 7754, Dec. 1970, pp. 1171-1182. 10. Yeh, H., and Cummins, H.Z., "Localized f l u i d flow measurements with He-Ne l a s e r spectrometer," A p p l . Phys. L e t t . 4 , 176, 1964. 11. Reynolds, A . J . "Analysis of Turbulent Bearing F i l m s . " Journal Mechanical Engineering Science, V o l . 5 , No. 3 , 1963, pp. 258-272. 12. Tennekes, H . , and Lumley, J . L . The MIT P r e s s , 1972. 13. Hinze, J.O. "Turbulence, An Introduction to Its Mechanism and Theory." McGraw-Hill, 1959. "A F i r s t Course i n Turbulence." 27 Goldstein, R . J . , and Kreid, D.K. "Measurement of Laminar Flow Development in a Square Duct using a Laser-Doppler Flowmeter." Journal of Applied Mechanics, E, Vol. 34, pp. 813818, 1967. Rudd, M.J. "A New Theoretical Model for the Laser Dopplermeter." J . S c 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: S c i e n t i f i c 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. â€” - + u. â€” - 3x 3t = 2 + v 3x. p k 3 u. 3p 1 3u. = 3x 1 3xÂ£ 0 k Since the flow i s p a r a l l e l , u 1 = u (x , t) u 2 = u^ = 1 2 0 Defining v o r t i c i t y , i . e . : 3u. - a 3u. -a.-A 3x 3x, k leads to the v o r t i c i t y equation: Deo. Dt 3u. * 3x k 3o). 3t + u 3io.J 3u. 2 3d). 3x 3x 3x k k \ 29 The f i r s t term on the l e f t represents total variation of v o r t i c i t y with time, while the second term represents deformation of a vortex tube. The right represents diffusion of v o r t i c i t y due to viscosity. In two dimensional flow, deformation terms vanish, and 4 becomes: 2 3 a) 9o) 3u â€” + u, 3t 3x k = 1,2 = v - y 3x k k Introducing the stream function ty, where dty 3x dty U l ' 77 = 3x 2 â€¢ 2 U 1 and 2 2 3 ty 3 ty = - ( â€”Â£ + 3x^ â€”2 3xÂ£ ) ~ hp = gives dAty 3t + 3i|> 3A^ 3^ 3x 3x-j 3x 2 3ATJJ 1 3x = vAAty 2 This is the v o r t i c i t y 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 A f 0 = f(*) , }\i Introducing an analytic function W(Z) = W(x-j + i x ) such that 2 W(z) = <> j + iH 10 Hamel found that i f the analytic function W satisfied the following condition, 2 d W l = a + i b = c o n s t > n (dW/dz) 2 the function f(<j>) s a t i s f i e s the following: f" f b = v[f 1 v + f " (a +b )+ 2f'"a] 2 2 12 (primes refer to differentiation w . r . t . <J)) Integration of 11 gives 2 * = ~?â€”9 a^+tT (a log r + b 9) + (j) . 0 where <j> i s a constant of integration, and the polar coordinates r Q and 6 are defined by 13 31 z - z The streamline i.e. = re Q i9 with <j> = constant a log z Q = const. i s a logarithmic s p i r a l , r + b 9 = constant. are concentric c i r c l e s 14 When b = 0, the streamlines r = constant. The velocity components are: a = -ab f 2a a +b f r For b = 0, Equation 12 gives f = C + r (A + B log r) 2 17 with q r = 0 ' % = \ <7 + A r + B r 1Â°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, q 9 = IÂ£^ > \ = +A 0 19 32 Applying appropriate boundary conditions leads to the velocity components as stated in the text. Since the gap is small with respect to the radius, we may let r - r tr Â« l - + A , where r ] ] is the inner cylinder radius, A is variable, and 1 everywhere. erywh< r Then, l C % = T + C 2 r C r l ( 1 + 20 ^ + C 2 h + A ) Expanding by the binomial theorem gives q e * C r 2 ] + Â£i + C A 2 + C, ( - ^ ) 1 A' + A' = C r ? 21 1 C + -1 , 1 Since q = 0 when A = 0, Q So to a f i r s t approximation, q the plane case. l B'A C where r Q T B' = C + - L 1 9 r A' = 0, and q = B ' A = B'Cr-r-j) Q is a linear d i s t r i b u t i o n , as in This result indicates that the small gap-to-radius ratio j u s t i f i e d 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 p r i o r i assumption of 10 the viscous sublayer thickness is - 7 ^ ) V t n e the log law in the wall layer. Region 1. Viscous sublayer: T = 0 2 PU* 9 = y U â€” sy Integrating gives: 2 u = u* y v Region 2. Log law region 9U u* 9y icy where K is von Kantian's constant. Integrating gives: U 1 = log y + C <_ 10 ( i . e . that velocity can be matched to Matching velocities at II y =â€” u* i K uT u 1 0 gives C,, and 4 becomes: 1 i * y ( â€” ) - ^ o g l 0 9 + 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 â€” 3y where a-j is a coefficient of order 1 and Â£ is a length scale of the turbulence. Assuming Â£ Â« b in the core, 6 above integrates to c^b u* L 2 We know U = U at y = b, thus: c Matching derivatives between log and linear regions at y = y : m 3U ay = y=y b^T ^ ' = m and s 171 1 y = ba â€” - o b a i = Â« y 35 Matching the velocities at y , we obtain a relationship between m â€” and a , , as follows: I ll* _y_ U 1 K yy r 1 1 u+ a, b a i U * 1 = or K U U -â€” + log â€” u* u* 3 bU bU = l o g (vâ€” )' + log(â€”~) + v ' 3 V 3 V a, + (10ic-l) + â€” ' ^oq(â€¢~-) y v 10K' x bU For any given Reynolds number (â€”-) this equation relates a, to U If a, is chosen by means of a best f i t to a measured p r o f i l e , â€” v U (â€”). * can u be estimated from this equation; alternatively, i f â€” 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 0.05 to 0.1). in the range of In either case, the equation above assumes a known viscous sublayer thickness, which is built 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 ) i s constant in the core region and varies linearly in the log region, since T i s 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 Â£ 0.07 bu*, then the value of a-j is equal to 0.07, i . e . : 36 2 from the definition of e, and the fact that T = pu* = constant. 3U * 1 However, â€” = -g- â€” in the core, so that u 2 0.07 bu* = , b or a, = 0.07 . 11 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 c for any {â€”^-) within the assumptions of the three region model with the 1 u assumed viscous sublayer thickness. The f r i c t i o n coefficient u* Ub (Cf (TJ~) ) found as a function of - ^ - w i t h i n Hinze's assumption 2 = 2 c a n D e 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 simplistic 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 = [) = Â± Ke' - â€” u; For the two configurations shown, we have the following equations: u x + i u â€” = (U + u' + w') JT V, + v' 1 = â€” (U + u' - w') Taking root mean squares of Equation 2 yields the following: (U + u;) A 2 = ul + 2U u ' + u ' x = x x 2 K (E + 2E ^ + e] ) 2 2 2 ]e e' = KE-,0 + \ - g - + higher order terms) 2 J (U + UJL' A E l 4 Operating now on the right hand side of Equation 3 yields: ^(G+u'+w ) = l(U +u +w +2U(u +w')+ 2 u V ) 1 2 2 9 ,2 ,2 - , /l U /, u . A ,2 1 u"" ^ 1 ' U . I W , w " , u'w' + higher order terms) Similarly Equation 3 yields Q (V + v ^ ' ) A 2 ,2 = KE (1 + \^r+ higher order terms) 2 and / (U + u'-wT = ^ U T 2 D 2 T 2 r 2 â€” D 2 + higher order terms) Subtracting Equations 6 and 7 from Equations 4 and 5 results in 4 E where Eâ€¢ = E = E . ] 2 * Recalling Equation 1, this simplifies to 39 uV = \- (ej - 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 , -p IT I"l -2=2 -2 U* If a value of u' D* 2 of turbulence can be found. is known, then estimates of the w' component APPENDIX IV THEORETICAL DESCRIPTION OF THE LASER DOPPLER VELOCIMETER (a) Reference Beam Operation Figure 3.1 shows geometrically the Doppler s h i f t of laser light incident on a particle moving in a f l u i d . The number of wave- fronts incident on the particle per unit time i s : c-vk. After scattering, an observer in the direction k s c would observe an apparent wavelength of: c-vk SC c-vk V v v -> . ' ' p c-vk i The frequency of this scattered radiation is given by: -Â»â€¢ r c ~ * i = c ( ) i c-vk sc c v S C v k A vk. â€¢ c = A c [ i 1 vk ] sc v*k sc and the frequency s h i f t is given by: 1 - vk v o = v sc - i v = XT C I TJâ€” 1 _ v k sc ] - v 1 4 This frequency difference can be measured when the scattered light 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 i g h t 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, light is emitted with a frequency which corresponds to the rate at which the bright fringes are cut. This frequency i s given by: v = v d -x = 2v T- x. 1 sin 6 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 slight 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 slight non-linearity of the tracker. 45 APPENDIX IV CALCULATED FLOW PARAMETERS R U c sec c m S u * sec c m bU c e Â£ 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 18.85 .425 .79 - .80 .112 - .115 242,236 0.0035 24.04 .360 1.1 27.35 .393 13,800 ] 15,700 .19 181 1.14 - 1.13 1.74 226 0.0035 18,400 2 - .240 - - - 0.0054 23,200 2 - .195 - - - 0.0052 2 - .217 - - 0.0048 27,OOO ! Murguly (1971, unpublished). 2 Robertson (1959). 47 â€¢ ". 1ST PRHFT-ts PuTpp R=IO2O 10 11 -tf13 14 -+516 17 -f810 ?0 1 . 0000 8.7000 0.9426 0.8971 0 .8578 0 .8030 0.7646 â€¢O.72P0 0 .6878 8.3000 8.0400 7.790(1 7,a000 7.0700 6.9000 6.6000 6.3600 0 .62U? 0 .5938 0.5455 -OvWW)â€” 0.461 1 0.U158 5.fl4 n o 5.3700 -a-j-ft-fr'HVa. a o o o 3.8300 CYLINDER \'FL rJTY r i 0, O048 35.5771 13,0500 -i-?;-905-fV- â€”fhr6-l fl-7â€”35.2813 O.0R7O 1 .9196 33.9130 12.4500 O.0R75 3. '1318 12,060 o 32.8289 0 . 0876 â€¢1 1 .6850 a.7422 31.7899 0.OR76 6.5625 11 .1000 30.1743 O.0R77 7.P361 10.6050 28.8127 -O.0R77 -H>T3irfra- â€”9-.~frttft9- -?R-rl-fra-90, 0P78 10.3796 9.9000 26.8673 0.0878 ! i . a 4 a 325.87R2 9,5400 0 . 0P7" 12.481? 9.0900 ?a.6a63 0.0879 13.4R42 fl.7600 23.7411 0 . ORRO P.0550 15.0767 21 .R152 -fi-j-fVRR-f)- â€”7-rVi?ft-ft- -i-<>T3i-ftPâ€” -t^i-fH-3-90.0881 17.R533 6.6000 17.8528 0.OPR1 5.7450 19,34 0 1 15.5299 0 . Aflft ] ft.3950 3.a7"0 5ft 3,1300 0 .3605 0 . 0PR2 4 ,6950 2.6200 0 .3225 0 . 0HR2 3.9300 2.2RO0 0.2965 0 . 0Â«83 3.4200 1.R50O 0 ,26'IP 0 . ftPR3 2.7750 1.500A .23P5 0 , 0 R P 3 2. ?500 ? . 2 a 0 ft 0.299 1â€¢ 3. 36 "0 0.3a0fl 2.7600 0 . 0 Pfl2 4.1400 0.Opp t 3.4600 0.3962 5,1900 rtâ€”pJtfio... -ft-j-ftH-R-ft- â€”6-,a?ftft-o-iWWfra . R ! 0 0 0.OPRO 0 .52R0 7.2150 5.4100 0.0R79 0 .5R00 8.1150 ft . Of 78 Q . 0 I, 0 "i h . 0 aft0 â€¢ o. hacn 0 , 6R65 6.3700 <Â».5550 0 , 0P7R 6 . O 0 o o 0.0877 0.75^5 10.3500 -ft-i-7 5-/4-5- -â€”6-.-Â«2ftftâ€” -1 4-i-3flft-ftâ€” 7.4400 0.821 0 0 . 0P76 11,1600 7.9500 A . 0 P 7 5 1 1 .9250 0.P930 8.3700 0 .05 3+0.OP74 â€¢ 1 2.5550 0 .9932 12,9300 0.0P7a 1 . 0 0 0 o P.6200 13.3500 8,9000 0.0P7U â€¢1 .OOOft- -R-j-5-ft-fl-fl -+-?-,-7 jftflâ€” 46 47 -ftft- 49 50 -5452 53 - 5455 56 58 5Â° 6061 6? n J P=1ISO POS'NdKO HC Vm.TR M 0,0874 >ft 22 23 2a 25 26 -?72P 29 -3 ft31 32 -433a 35 -3637 38 -^9ao ai -42a3 aa = 3,56c / s r c 20.022'J 1 a .0660 21.1535 12.6814 22.3071 1 0.6093 23.2477 0.2291 24.2837 7.4P51 25.1424 6.0667 23.1630 â€”0, 0f,7'j 21 . 11.1 792 1 9 . 9 8 % 4 14.0257 -1 7.5A72 -H-7-T-36-8 215.6526 19.5353 1 3. "395 21.9RR7 1 1 . 6-9^â€”-2-4-^5^3-51 0.4226 25.9306 P. 1 71 0 28. 1157 â€”R-H-7~!-ft- -2R-I-! 9-7-230.3454 5.9647 32â€ža595 3.5713 â€”Hr5-6-*2- â€¢34.2 0-4-20.2315 35.2a64 0.004P 36.3950 - o - y o o a f t - - -34-J-7-5-92- nilTFR CYl.INnFR Vfc'inriTY = 4.02 CM/SEC THFTA FPFOIIFMCY POP ' M ( N'M ) VELOCITY (""/SEC) O - i - f t f t V I R â€¢~2vÂ°5fift- -~ft-,-ft-P-7-Â«- -1-4--.-75ftft-Â«-0-.-2H72.7300 0 . 0 R 7 4 1 3.6500 1.6393 37.1862 2.4400 3.53P0 0.np75 1 2. 2000 33.20P5 2.2400 0 . 0 8 7 6 -Hâ€”5-i-4ftft"iâ€” 6.3100 o . 0A77 7.6904 .4650 ?5.7171 9 0.0P7R 5.6200 O.P765 22.RR30 .4300 p il-.-HAA-O-. _n.--fift.7-su â€”7 -j 2 0ftftâ€¢ -H-.620 4- -1 9.-529 1) a , n 3 R 51 6. 1 352 3 . 9 7 0 ft 0.0079 5 .9550 12.750 2 3 . 1 a 0 0 ft , 0 P p ,1 /1,710ft 16,113* 0 . r (i.Â«1 â€” 3 T^fftft1 0 . 1 a 3-72.500,-1 o.aOPO 2.26^0 3. 3 o 0 0 -H7-I-P5-621 9 , 5 -' * 2 Â«. 1629 0.oaa j 0 . 4 P a ft <, . oft0 0 . 0 - <5 0 1 6, Q <M 10. 5S37 -n.A.fcO/y. -4-.-/(-2.1ft.A...nA7R â€” â€” 6-.6 3 0 0 - -4 1 .2"HP â€”! 7 ,-QR5 /6.1 2 o 0.7660 9. 1 8 0 0 7.7MO-J 0 . 0P77 24. , Â« 1 6 0.9390 R . 3 a 0 1 2. 5 1 0 0 3/!, 0 7 4 5 ?. 0 39.< r . OP75 -4-j-O-AAO- J 0 .951 0 0.R940 0.838ft0.7690 0.7030 ,6Â«-.ftft 0.5770 *.5140 0.461ft- 0 A 48 6 a 8 3 CÂ»VSFC 65 67 68 â€¢â€¢Pfl^-f^.) l.oooo n . o y t r, 60 70 71 -?27? 7a -7-576 77 â€¢â€¢7879 80 3.5400 0.7615 3.30O0 -0.7126 3.0600 0.6471 2.7000 0 .5905 - l . ^ a - H â€¢ -2-^3210- . 4 69P 030 P"S'M(TV , 1 . PPPp 95 9697 98 - 0 ; " 7 o<j- â€¢" . P S 3 ! 7-R- 0.9Â«65 0 , R9 70 100 0.8110 101 102- H>v7i-85- 103 104 0.5670 2.44 0.4030 P. -+++11 2 11 0.2720 0.2430 O.-2330 - H u - ns 1. 1 R 3 4 7 rV- ; 2 9 8 5 3 00- 00 7.2000 0 .'J605 - 0 3 .â€¢ 'â€¢ 3 o 0 -3-T21-002.9600 0.5245 POR 1 f TV 1 1 .POOP 0- f>(,H^; 0.94 P 0 0,9P7p P-.-8 7^- 24 125 1 26 1 27 1?R -}-?<*13 0 1 0 /m 0 . P 0 . R C P â€¢â€¢P . RP o . 0 73 1 5.050c 4 . P400 3 . 45P 0 2.54PO 2. ) ? r o t-,-R+-Pf\- ^ . 56P 0 . 6 A 0 â€”6-r4P-Pfl6 . ? r, r r, â€¢ 5 . 7 34.8241 5,25 00 ? P 4.64"0 0 . 0 0,P 0.0677 0 . 0R79 1 0 . ROOO . 9y3Â«pnâ€” - . PPR 3 -f>vOPR4â€” 0 64.7476 63.2481 â€”6 864 9- 56.2119 r '?.77gq- -4-3v5769- 10.9152 40.1558 13.2139 36.3207 1 5 , 7 6 78 â€”1. 7 i 8 7 2 7 19 . 7 6 0 3 21 , 5 6 0 4 - i H - . 051 3 31.8999 ?9.2U05 -?5;399520.4730 16.3654 -1-3-T9-6-5-5â€” 10.2779 8.5748 â€”7-; OCTTY VFl. 1". 59.6S76 46.5920 -2-,-7-tso- CVI.INDFR 9 49.RRB6 5 . 1 7 5 P â€” -2-33.8100 24,0487 3.1 POO 24.9957 0 . 0 B R 3 eÂ«/ EC V I nriTY fMM/REC) 7.9723 6 . P60P 3 ".101S- 6.2669 048 04 a 9-.-3636- 7.5750 0.ORR2 . P is R r 12.3851 6 . ^ 5- â€”trhr2^-frOâ€” â€”1-4-^-6-8-311.7750 \t.. 3 6 8 3 0 . 0 P. R 1 0 19.1687 l '.91S6 6.2968 13.4000 1737 22.2974 3.43Â«1 17.1SP0 -O-.-O-R 77- â€” 1 - 6 - . - 0 5 0 0 - 0.OR7P 14.R000 P, 26. 0-.7 4 551,7Â«93 1R.3500 0.087O 29.2319 28.2169 1 " . 4 0 0 P - 4-.-PÂ«fH- - .5371â€¢ 36.6157 -*t-.-4-i-a-i- POS'NfMM) -?2y7opft21 . 9 P 0 P 20.650P 1 186 13.5935 P -Vfc-|_ OC~J T V - 23.7500 23.2PPP 0 . 0876 = 9.56 3199CM/SFC F R F M I r' I r Y P 0 R ' fi ( M ^ VFI O C T T Y ( M M / S E C ) P , P R 7 4 35,0500 P , P04R ^^.5538 0-PR7 4 -3 4~; 5 0O-07 2 6 - â€”Â«rfÂ±--0Â«6-a â€”â€¢ 0 . 0 8 7/1 3 <â€¢ . 8 5 P P 2 . o o 6;-, Â° 2.20 ! 6 32.8PPP 0.P875 89.P9RR 3.1052 3 - 2 - r l ^ -f-o0â€”O-P-7-5_ ^ r l - 7 ^ - 1 - â€”87-j-o R-fVj? 0. PP 7 6 3 1 . 00 00 Â« 'J 3 2 2 2 5.1678 THFTA f 0.0876 '7 S11- 5 . 4 7 0 0 "WE â€¢^1 -17.5935 18.8019 19.7603 20.5961 22.16Â«2 23.4763 -24.7fl33 25.R604 FÂ«F<3UFMCY THFTA HIITFR vni. Ts 7. o1 op â€”ArOfdf.- . 6 9 3 0 -fl-r6^-R-r- -fHt-TFR-Cr Y11 â€”6 .?6"" 5.9300 r 0,0RR2 0.0RR3 0 . ORRO 6 . 7 7 0 P J 2,3250 0.ORR2 P .0879 -H-7â€” 11 R 119 1 20 ~ 121 12? 0.PRR4 0.ORR1 7.R5 "=2500 6 â€¢0.0003 -0.675P8 . 9 1 PP R.250P 7.0950 5.RO50 4 . 590(1 3.3750- 4 . 750P 0.PR74 '1.6400 0 . 0fl7 4 -4-.-5^1-0 P- â€” 0 - . - A R 7 4 4.3RP0 0 . O R 74 4 . 1 300 0 . 0R7 S 3.8R0p- - 0 . P 8 7 6 2.6RP0 107 J 09 VOLTS o 0 n i- ) 1 . 4-4-1-7- -+5-r2 21-7- 0.PRR1 0.6716 0.5670 R C 12.P500 6.45 0 0 5.9400 0 . 53.6439 9 . 5 'j 9 0 â€¢ 1 1 .7250 13.5000 16.442P 0.6020 H51 Of. 44.8267 16.4428 3.6700 0 , 7605 7.9392 IO.ROOO 1 .5500 n 1 . 0000 48. 10,4250 -Â»=i-B11- 94 6.4300 0.0R79 3.0600 0.2165 91 17.7000 16.5000 i-g. 3 o o p ORRO 0.2495 90 0.OR76 0.OR77 o .on77 â€¢ 55.2994 r o. 3.9300 5*.341 0 57.3629 2.6655 0 , ORflO 0 .3775 -vt t nr T Tv-fÂ«*vse-e-j- 1 .7726 6.9500 0 .3295 ^. u 2.1600 5.5000 4.7300 0 .2895 â€ž f 19,7000 -4-fl-r^O-frft- O.0R7B 2.5700 0.4322 Rfi 20.3000 0.0875 o r v g o. ooai n . Q r; q.R- f -n . Of!75- 0.5040 0.5040 R9 11 4 . 06 0 0 0.8070 . 4 21 , a -i n o . n, o o n 7a 0.0R74 0 .887-6- 0 FPf-nilf-v^.y.â€” o. o 3.9Â«on 0 92 -95- -T-Â«f-T-4 'i. 2 n o r. 4 . 2 1 n o- i.<Â»a70 82 A3 Ra RS 86 -6^7- -j n ^(i-vflfc-T-S- s 29.650p â€” ; â€¢ ' " . Â» . Â« . A â€¢-. " , 0 8 7 7 2 7 . ^ 5 0 . 2 6 . 2 5 00 0878 P-^WT-B 0.0879 o 2-4-j-P 5 o i â€” 2 3 . 20 P 0 6 . 4 9 6 ) - - 7.72^68 . 9 u o 5 1 O.2075 1 -1-r6Â°5 2- 1 3 . 2 ! '4 9 8 n 6032 - 7 7, 8 4 2 9 7 'J ,2 7 0 8 /1 , 2 4 4 4 -6-7â€”41-0 862.8835 13! 1 32 1 33 134 0.4570 0.207 o0 + 900 0 0 2800 3320 0, 3 6 4 0 0, 3 6 4 0 -A-i 4 3 4 0 0.521 0 â€”W148 1 49 ( 151 152 153 â€¢ 154 155 l.oooo 166 1 67 -1 6 81 69 170 0 . 7 3 4 4 0.6964 -0-.A.6 3 2 - -m- -2-.-7-4 OO- 0 . 6 6 3 2 7.9500 0 . 6 2 8 1 7.650Q 7.3200 7,1000 6.B0O0 6.4 5 01- 0.594 6 1 72 173 -+7-475 1 76 0.5448 0.5045 -0-r4+r25- 1 0.4116 5.85P0 5,0300 4.36003.1800 2.5000 0.3520 +7^7- n 1 78 179 -1 8 0 181 182 200 0 .2055 0.2260 n . -tx- 1862 a^o- -?-j-<tfji1- 0 . fi f T V ) 1 0.9R0 0 OUTER THFTA 0.9568 0-^*4-5" . 91 5 0.8594 - . = ->~5 ' . 7""5 0.7385 B n -^â€¢.-6^-2+0 . 6 â€¢'! 1 r . 5>'7fi .3.5300 3.3100 4 -O^-AM-7.50.0R75 0.PR76 0.O877 8. 33" 0 P7 . 7 ] 00 7 . 2 5 0O - 0.5*20 - 6 , 7 f , n n - 0 . 4433 6 . ?', p 0 0,4045 5 . " 800 7 8.1 2< ; - 5 9 0 9 - â€”2-?T-7-OÂ«j-0~ 1467 19.8670 ?894 15.3680 9 R jo - ^ - . - 0 8 78 4 +.-1-0 0 - 0 30. 3 8 . 2 5 0 0 7 S a n " . 0 0 7 9 3 6 . 60 0 0 O.OPflO 0.088 0 35.5000 34.0000 Hhr^WH-â€”3-2T2S-OP 29.2500 25. 1 500 ^-P-0- 0.O8P4 15.9PP0 12.5PP0 0.0884 -36-,-7500- â€” P T O R 8 + - F.Yt I'iOF'R 64 63.4500 0. â€¢7 6 0 â€ž '.' 5 A , .-. K 7 0,0877 o , A H 7 p, 081 0, 08R1 A U VELOCITYf^M/SEC) '70.4781 157.7308 - M 8.-65-9517 7 . 9 4 8 5 130.9294 12 3 . 9 1 0 9 91 2 0 , 1 6 9 q 15.6251 }-l-v 1-9 2 7 - H - 1 -.-5 0 0 2 j , 0 1 927 107.8378 1 2 .3524 1 03.7157 M - T-4V7-R- - 9 9 . 1 9351 5 . P99R 96.1429 16 . 4 2 6 0 92.0269 5-7. 7 * 0 7 - 0 -8T--.-2-3-7-41 9 , 4780 79.0640 1 21 . 4 3 1 8 67.9232 - 2 * -r2 8-0-7- â€” 5 8 . 8 2 7 7 2 5 .5503 42.8636 2 6 .8483 33.6785 -+8-i-5-7-Â«9- j8 P 0 P â€” 00 â€”5-6nr+6-0* 5 CM/REC 18 0 . 7 4 0 0 - -99-^57^2- = 18.6 E P F. QI) F f â€¢ C Y POS'NO-M) 0--A R 7 Â«-â€” P. Of-79 - P f. K d - .8 <6 .0950 VELOCITY 0 . 0 8 74 0 . oft7 4 â€”P-^87-50F O.0676 78.8430 84.7857 93.7010t fl'. 1 ,0048 6321 .7029 .055 3 . Â« " M ,8985 4 W - 4 4 . 2 5 0 0 â€”O . P 8 0 3 r hh-rti-k^-T- PPS-^.M) 4 2 . 6 0 0 0 0 . 0 8 7 8 51.9805 5 8 . 8 1 18 1 3 . 4(149 OA7-3 6 . 2968 3. 6 7 1 2 - * v 0 0 48 0 . 0 8 7 7 0.08R1 0.088? .1 15. 8 Â« 3 1 0 . 0 8 7 8 0 . O 8 7 8 32.8171 27.6966 24. 25, 62.5500 57.9P0P -54~i-6 0 l O - 5 0.7000 42.0332 3 3 . 80s... 21 . 2 0 2 7 2 2 . 4 70? VELOCITY 4 49 -45.600O- N3.-t'8-l7>~ 48.2810 1 7.98 77 19.4321 66.30O-O- - " .-PR7-A 3 . < 3.02 0 .f FREQUENCY 0.0874 0 . 087<t 0 . PR7 'â€¢.8 . 0 9 9 7 â€” 7 0 4- 1 0 . 7 1 3 0 724 3 8.6087 7871 20.7200 OR 6 6 28,2684 0388 32.2927 0388 34.4402 -+S-T 7 4 3 1 â€¢ - 4 i ? . 3 1 6 4 19.2000 21.7000 CYLINDER 14.7966 - * A . <â€¢ <â€¢-><< I' 6 26. 23. 22. 21. 21. 1 0 .0876 2 9 . 0 0 00 0 .0875 31,1500 â€¢ 0 . 0 8 74 - - ^ 4 . 4 0 0 0 OUTER -4 ;-32 0 04. 23PP 4. 0i00 3 . 9 7 5 0 3. 950 7.6800 10.4700 11.9550 12,7500 1 5 J 5 0 0 O R R T ( K VO|..TS THETA i . PIPP-- tl -2*-.-5-A*A- P=5338 90S 0 8 8 4 0.0882 0 . 0882 0.0882 0 . 0881 â€¢ â€¢ 0 . 0880 0.0879 4.1700 0 . 8 6 5 5 1 2 . 1 5 00 10,?AOO ~ 8-;-4'1-Sfl7.3650 5.7000 A 0,0884 3.8600 -3-.-A40-0â€” 3.3800 3.2100 3.0400â€” 2.9500 2.8400 0 . R 2 3 0 "O 0.0882 0.088? 8830 .0883 0.0884 .- > 8 P â€”0. "â€¢â€¢''2O0 0.9R12 0.9491 â€”0-;-9+L^- - 0 . 7 7 6 96 2 . 650 02.1300 5.1200 6.9800 7.9700 2.5500 â€¢3.1300 OC VOLTS POStMflM) 1 21. 0.0881 P=5186 â€”156- 0.OR81 500 -â€¢(-<>. A S I f ) 17.8500 15.5500 A 3.8400 4.34 0 0 - 4 . 9 0 1 1 ft5.8000 6.2300 (â€¢ . 8 8 0 0 0.5950 0 . 6 6 7-00.8110 0.8900 1.0000 -+5$- ! Â°3 ! 9a 4 Â«5â€” 1 197 1 98 â€¢ 1 9Q .0.0879 3.1100 2 . 5 1 0 â€¢> '8.100'' 6.8400 5-;A+11â€”; 4.9100 3.8000 i<*0 0.35O0 0.3200 â€”ft-;-2+-A0â€”â€¢ 0.2690 0.2340 1 1*9â€” 190 191 I O? - ' 00 3.5700 0.4t?0 â€” H H 1 42 143 44 145 146 ] 88 0 > 136 1 37 + 38139 140 1 84 185 18h187 4 . ? -fv-4 0f>-<\ â€” - 3vÂ«H-'V* â€” -+VJ- 157 158 50 160 16t -162163 1 64 49 0.55/10 O.-00 48 ^.6721 1 .4458 C'VSFC VFI. nr ITY (MM/ SFC J 50 - ^ H 2 2 0 0.3635 0.3020 0.2945 0.27-0 0.2-60 0.214Q 2 0 5 2 0 " 2 0 S 5.6000 5.0000 4.6000 4.3000 3.6500 3.0000 208 209 R-5940 210 P f l W 21? â€”2+3 214 215 â€” 2 Hr 217 218 219 220 221 .0.0*83 0.08R3 0.0883 0.0884 Râ‚¬WE-He* 3.4400 75.6328 67.4694 62.0651 58.0003 49.2129 40.4304 21 .0552 2.3.o6fli 23.3131 23.Q833 24.8977 25.9420 OUTER CYLINDER VFLOCITY T H F T A 211 28.ooo" 25,0000 23.00nr> 21.,SOOO 18.2500 15.0000 n.PHng 0.ORR3 = 20.2 PflS-'-V+MH-} V CM/SEC E L-flÂ£-f T Y (**/5â‚¬-ei- 223 224 â€”2/26 226 227 - -228 .229 230 231 1.0000 0 .9660 0.9300 0.8910 0- j 891-0 0.8320 0.761 0 0 .6800 0.63RO 1.0000 fi-795-00â€” 0.9020 0.8620 0-. R 6 2 0 0.8260 0.7R"0 0.7270 0.6690 0.60P0 1 . 0000 232 233 â€”234 0.9430 0.8920 4 30 4.5500 4.0300 3.~A-<Vfrfl 235 0.7P80 3.5200 0.0876 52.8000 7.0&01 143.5007 236 0.7160 3.2000 0.OR77 48.0000 9.4464 130.3190 â€”??2 â€” ft-8 : â€”237 â€¢ 3.3200 3.1400 2.0000 R-.R400 8.5600 7.4800 6.3000 5,9300 3.5400 3;3-7-ftfrâ€” 3.0500 2.8200 R.37^-0 7.7900 7.2800 6,79fif! 6,3600 5.R900 4.o30 n 238 239 R-6027 â€”2 4 0 FMlS-^H-f+H-) -*e~V-TrtrT-S 0.0874 0.0874 0.0875 0.0R75 -0-rf>*-7-5 0.0876 0.0877 0 , 0878 0.0878 0.0874 -O-rOfl-7-40.OR75 0.0R75 0.0875 0.0876 0.0R77 f>.0 "77 0.0R7R 0.OR79 0.0874 0.0874 0.OR75 ft-.-ftB-'f, 51,6000 49.0000 47.1000 43.5000 4 4 - r 2-fHI 42.8000 37.4000 31 . 5 0 0 0 29,6500 53.1000 50-^5500 45.7500 42.3000 41.8500 38.9500 36.4000 33.O50 Â« 31.8000 29.4500 73.950-0 0.0048 1 . 1 389 2.33"3 3.6376 3 -.- * -37* 5.59Q5 7.9^4 1 0.6373 12.0256 0.0048 tv6726â€” 3.2717 4.6025 4^6-0^5 5.70R6 7.3255 O.0R24 11.0012 13.0)58 0 .0048 68.2500 60.4500 57 . f t f t O O 1.9060 3.6046 5-.-2339 140.6726 1 35.6988 128.2737 118.4022 1 2 0 ; - ? ft 7 b 116.3970 1 01 . 6 0 6 7 85.4771 R0.4081 144,7620 i 37-v?fl97 124.5462 115.0874 113.8631 105.9176 9R.9170 92.188 3 86.2774 79.8310 2 0 1 .6035 185.9095 164.5406 155.0395 â€¢ â€¢ â€¢â€”â€¢ â€¢ ntJTFR CYLINDER VELOCITY = 2 1'. 0 CM/SEC T~HF-T~A F-RFTWEMC Y K1S- *J-<WM> 4 Vf+n(M"T'Y"f t**t/Sf 241 242 1.0000 5.1300 0.0874 76.9500 0.0048 209.7822 243 0. 9 8 6 8 5. 1 000 0 . 0874 76.5000 0.4448 208.5159 244 245 -2 46 0.9628 0.93a? 0 . - 9 ft-om 4.8000 4.4000 4-T2/0 O-O 0,0874 0.OR75 0-rO-P -7-5 72.0000 66 0000 6 3~rM> <V0 1.2460 2,0657 3-.-01 * 6 196.1808 179.7681 1"f 1~.-5 2 55 247 248 -2*9 250 251 0.RR2P 0.R460 0 . ' M 4 ft 1.7928 0.7702 3.9108 5.1342 6 + 7+z 6.901 0 7.650h 165.3351 144.8456 138.6623 1 34.5413 130.4214 -25-2 253 254 Ov-7^6-0 0,7095 0.6725 .6346 0.5925 -25^ ft 256 257 4.0500 3.5500 -^-rÂ«-S-0-fi 3.30ft0 3.2000 3-rt-O-O-flâ€” 2.9000 2.8000 2.8000 2.7000 0,5475 2.55O0 0-5 0-7-0- 2-jpiHHV* 259 0 ,'!72P 2,3000 260 0.4728 7,4000 - 2 5 8- 2 f t ? 263 264 266 -?-tyr 0.0875 60.7500 53,2500 . 51 . o*-*-fl 49.510(1 AR.OnOO 0.ftR76 0.0876 0.0876 0.0R77 0-rO 8-7-7 0.0R77 O.OR7R 0 . 0R78 0,0879 O.ftRSO T J s ttfrj-^HHVO - 8 - 7 - 8 3R 43.5000 9.6613 42.000ft 10.8854 4 2 . 0 0 ft-0 12.1 40.5000 13.5271 t-26"r?> 8-3-2 118.0905 113.9571 1 1 3.8946 109.7599 103.5947 T 38.2500 15.0104 36 ,-0 Oftfl 1 -6-r3-4 4 0 97 y 4 4 3 9 O.OHRO 34,5000 17.46Â«6 93.3376 O.ORRO 37,0000 17.46R6 100.1012 ft--ftp R ft 0 . '1398 7. 1 OO0 o.ORRi 35.50-p-fl Hh-W+ 95.997g 0.3015- 6.5000 5.4ftftft 4.4000 3.3000 2 , 8 00ft H - a w 0 . 0881 0.0PP2 0.0883 0.OPR3 0 , 0883 o-i-ft 32.5000 ?7.0ftO0 22."000 16.5000 l/.OOOft -.--i-ft0-0 20 . M 7 3 21.4937 22.8750 p-j.o^s; ju.Qot,-; 2 3 1 ;i P7.R236 72Â°917h 59*3781 44.S121 37.7507 p.^I?r,rn 1.3501 0.3079 -0 . 2745 O.?'i30 . k ft__p.A.1,A â€¢ 51 ?6P ?69 9=6155 2-7 ft 271 ?72 2^-3 ?74 275 90.8 1 277 27Â« 280 281 2Â« 2 283 ?84 POS '".cm .0000 .9860 -r9-6*fl.9330 .0020 .8<,?p .8230 .7830 -.-7-4-2-069O0 .6430 .5 730 1.5220 299 -300301 313 .314 315 316 317 - 3 1 fl~ 319 320 3 2 ? 323 324 325 3?h 32K 3 29 3 30 331 3 3? 334 *35 2.7700 2. 6 0 0 0 2.4500 â€”*=6?00 299 - 3 1 2 - 4.26 0-0â€” 3.9500 3.R200 3.3800â€” 3.2700 3.1700 2 9P0ftâ€” 2.9200 T -294 295 296 . 297 311 4.7010 0 .6220 0.5628 0 .4965â€” 0.4390 292 293 310 VC1I.TS 5â€”1 sftn 5.1500 e- fr7-*9â€” 2P6 287 -28-8â€” 289 290 -29-1â€” 302 -3-0^3304 30s 3 06 307 308 3ftq "it 0.90200.9775 0.9545 0.9197 0 ,880b 0.80 81 0.8015 0.7713 0.7287 -976 : v CI f->) HC VOLTS S. 2 8 0 0 5 . 1 200 0 S " " C T v ^ . 0000 .Â°660 -.-Â»1?-P,8820 .8440 . 74 00 .6H1 0 i^-4-20. q ft r) n 15330 CYL TNDFP THFTA n A n 7 >i 0.087S 0.0K76 OC VOLTS 9.2200 8.3600 -6-r9*-ft-06,4300 5.O800 5.7100 5.4100 5.2110 -4-.-9 4AA4,6600 a.aooo C'VSfC 7 7.25 ft-ftâ€” 75.7500 70.5000 i-l 5 A (tI* A A 0.27)9-â€” 5 7 5 3 0.7553 206.4432 1.5225 192.0705 ? - . 6 8 ? 5 â€”1-7-#T*6-HI 3,9840 161 .2476 5.0644 155.8671 S7.^000 f ri 6.6)2? â€” 1 37. 8205 â€¢.j i i , / o o H-7.614? 133.2766 49.0500 9.0257 129.1211 47.5500 /l/l W 'J â€žT MA>ft0 Pâ€” 1-0-7-8-0-6-)-â€” 1 - ? 1 .2873 12,5539 1 18.7543 14.5065 112.5573 jrQ . 7 S 0 0â€” 1 6.6803â€” 1 07 .5780 36,7500 18,5789 99,3762 S9.2S00 "n Â« nn?/. U M / f.) 0.0877 0.0877 0â€”0 8-7-80.0879 0.0879 0 (innfi l' ii " tf 0.0881 4 OUTL~W THFTA ?1.1 POS ' â€¢â€¢; ('â€¢"â€¢'â€¢l V F L 0 C 1 T Y C " ' V S F C ) FRFOUF^CY ~ i *r 0.0R7a 0 . DR74 n â€¢fl 'Jfin7t r:3 Â»' \> = VFl OTT TY T A A CVLIMPFP Â« VELOCITY FBFGIIfc'MCY .0874 . 0874 - POS'M(MM) 79 .2000 76 .8000 â€¢i-4-5-0*4.5500 68 ,2500 . 0875 4.2600 .0875 63 .9000 .0A75 -Â«-r 0100 -6-0. 0000 3. 7400 .0876 56 .1000 3. 58O0 .0A77 53 . 7000 -4-r-3-8-0 ft- -.-ft 8-7 7-- -5-0--,-7-ftftft3. 2700 .1878 49 ,0500 3. 1 0 0 0 .0878 46 .5000 â€¢ft-g-A-Ov0Â»79 -4-?T^ft 0 0 2.6900 0.0880 40.3500 -P=+-0B20- P ODTTP ? 1'. (, â€” â€” CM/flFC VFLflC I T Y C M M / S E C ) 0,0048 215.9161 0.4719 ?09.330? -l-T+38-9- - H r ^ v a + f r * 2.2393 1 8 5 . 8 8 2 5 3.271 7 173.9S65 â€¢ a.6 0 25 163.2446 8985 152.5471 2260 1 45.9361 58 5-3- -*3-7-rH>4-31 0 ft092 1 3 3 . 1 3 6 9 1 1 86 0 4 1?6.1131 14.1701 114.6059 15.8502 109.2418 -^+fT-FCf----e-VLTAf'->e<*---V'Fl:'>e-I-TY-^-37 ;-7---eM/SfrCr7 THFTA F P F G l IF MC Y 0.0874 1 38.2999 0 . 0874 125.3999 --ft-,-0* 75- - + < H h r 3 9 9 Â« Â» .0 . 0875 96.4499 89.7000 0.0876 85.6409 0 . 0876 81 . 1500 0.0877 78.1409 0 . 0878 8-7-8- â€”7-4-r+ftO-O0 , 0879 69.9ft00 66,0000 1 . 0880 pns'MfVMj V F L O C I T Y f i-" V 5 F C 1 0. 0048 377.0354 1 .1389 341.6978 -2-.-7-7?4- -?rWi-?-7-253.9374 262.4917 5.2010 243.9867 6 . 8-9-4-4- 232.7981 8 . 6 5 1 6 220.3977 1 0 . 6 0 4 2 212.0679 - H - T * " 3 5 - -20-0^-96*41 3 . 6 4 3 0 1 "9.4283 178.7142 1 5 . 4 8 8 2 v â€”1.496 ft 4 . 3 1 10 -fl , 0 ? a 0 64 . 650ft 1 6 . 7 " 5 4 1 7'I. 965c 0 . a a o ft 4 . 1 2 0 0 0.OPP1 61 .8 0 0 0 18,25 0 7 167, 1 6 71.387 1 38" 0.4020 3.8/00 ft.PPPI 58.0500 19.7"31 156, 8901 0.3470 3.6100 0.OPP? 54.1510 ?1.5956 146, 2335 ft.2Â«oo 3.26O0 0.0BP3 4P.Oft 00 23,1661 131 .96a 6 ? , H 8ftft 1.2520 0 .ftr> a. 3 4 1.2000 ? 4 . 7 0 2 116. 0 5031 3 . ' . 'ftftft -V?3*f0 . npp? h-l . ft ftft-ft 2 1 . 'Jft-iâ€” 1 37. â€¢7-ftA90.4110 3.85ftft 0 . 0 n Â« 1 5 7.7510 19.4977 156., 0995 ft . a a r, a.2?0 0 0 .ft* 8 0 â€¢ 63.3110 1 7 , ! " ' n 5171., ?P 1 8 1.5630 6 .39oo â€¢ 4 ; 5 6 0 ft 0 .ftp79 1 4 , 4 n o oIPS., 20^5 1,6180 4.P' 4 0 0 0 .ft1; 7 9 7?.59"9 1 0 6 827P . 12.6860 1 , 6 0 AO 5.2210 0 .ft7 8 7 P . 3 iftft]ft. 0 4 ? ft?1 ?. 527? -5-r5-6-ftfV- -ft-j-ft*Â»-7-7- â€”P3-r3Â«9-9- â€”P ,0^" -2*6- -5-7-0-40.8460 h. 0200 0 . 1 P 7 6 9ft. p o o o 5.13a? 245 6?56 ft . o 2 4 o 7.020ft 0 . " P 7 5 1 O 5 . ? o t)'.) 2 . 5 3 9 ? ?66,75?4 a 52 337 9=15870 OuTF'P CVI.IMOES VELOCITY = 54.7 rM/SFC 338 -5593iin p n g ' " ( !Â»â€¢') 341 â€” 3 4 ? ~ CRFOHEMCY POS'M(HM) VEI O f T T Y 1.0000 4.0100 0.0874 60.1499 0.0048 546.6 -fl--959f>- -3-r?-7-Ofl- -OrOP^-a- ~4 9 - ; - 0 * 0 0 - -1-.-3*612.4722 -4 4 Â« i - ; 5 4 l 3. 9 a . 1202 379.6 343 o.92f,n 3.0400 0.0875 45.6000 0.8765 2.7900 0.0875 41 0.8319 2.6400 0.0876 39.6000 5.6028 34 6 0.7885 2.55(10 0.0876 38,2500 7.0438 3 47 r.7"85 1 2 1 . 4 9 9 9 -348- ,8500 359.0 346.5 P.1ooo 0.0876 -f-.-R-^-O-fl- -0^-0-8-7+0,0878 108.2999 1 1 .7349 101 , 2 5 0 0 9 6 . 7500 14.3284 274,3040 16.047-7 261. 9 1 4 6 17.6297 246.3127 349 0.6488 7.2200 117. ?99Â«- 7.0438 â€”9-T211-2- 330.2170 -51* .5000 293.7373 350 0.5682 6.7500 0.0879 551 0.5160 6.4500 0.0880 352 353 354 0.4679 6.0700 0,0881 0.4251 5.8800 0 . .1 -o-r3-69*Â»- - 5 -.-55-0-0- -oâ€”i-Â«*2- "85-^-250-0- 355 0.3204 5.3000 0.0882 79.5000 22.4661 214.6096 356 0.2712 5.0900 0.0883 76.3499 24.0748 205.9602 C-JP S S I G c THET A 344 â€” 5 4 * - r rÂ»C V O L T f l O F F I L E e8 1 91 . 0 U 9 9 88.2000 1. 9 . 035 1 238.4552 - 2 l - ; - f l 5 4-9- â€¢ 2 2 4 . 8 9 2 1 (MM/SFC) Figure 1.1 Typical LDV signals from particles of approximately uniform s i z e : (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 velocity profiles for plane Couette flow: (a) laminar flow; (b) turbulent 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 signal out aperture c! mirror FLOW lens 5 cm laser lens 10 cm beam splitter 1 variable filter disa pre-amp doppler signal processor frequency to voltage DC digital flow Figure 3.3(a) parameters VM readout Schematic i l l u s t r a t i o n of the laser Doppler system used \1 Figure 3.3(b) LDV and Couette flow apparatus 30 40 50 100 200 500 1000 (slew rote) Figure 3.4 Calibration curves for DISA tracker 2000 4000 10 Figure 3.5 Calibration of ambiguous broadeni 61 fo proces |sing A electroh ics light beam photodiocje 4.1 lens Schematic of Couette flow apparatus Figure 4.2 Laminar flow p r o f i l e s , concentric cylinders The theoretical flow is for i n f i n i t e 63 Figure 4.3 Laminar How p r o f i l e s . concentric cylinders The theoretical flow is for i n f i n i Figure 4.4 Turbulent flow p r o f i l e s . Theoretical profiles with and without modification proposed by Hinze 65 Figure 4.5 Turbulent flow profiles CIRCULAR COUETTE FLOW,WATER X Present Work A Murguly (1971) (unpublished) PLANE COUETTE FLOW,AIR O Robertson X XX 1 xx X v b_ 78U /SUN " U" v"8y/ X A (1959) , ( c o r e c r e g i o n sl Â°P > e X I.I x I 0 Figure 4.6 Core region slope as a function of Reynolds number 4 70 7 I I 8 9 I I 10 I II I 12 I 13 I 14 I 15 FREQUENCY Figure 4.7 Typical laminar flow spectrum. 16 ! I L 17 18 19 (Khz) U = 4.8 cm/sec 20 r~ â€¢ R = 6256 A R = 10820 11 o R = 15700 12 R = 66000 (Johnson 1970, plane Couette flow in air) X 10 A A 9 0 o* A 8 O 7 100 6 Â° A A â€¢A *A 5 A Â° A A A o A A V a A A 4 A â€¢ â€¢ â€¢ A A A A A A N A â€¢ â€¢ 3 2 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 intensities 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 R = 10820 II 10 8 7 xlOO 4 o 3 2K o o 0 ao Figure 4.10 0.2 o i w" 1 0.3 04 05 0.6 07 turbulence intensities vs normalized position 0.8 0.9 1.0 Figure 5.1 Determination of C from the Clauser curves f 0.016 d) I0 2 Figure 5.2 I0 3 Couette, air â€¢ water â€¢ Reichardt, oil â€¢ water I0 4 I0 5 Skin f r i c t i o n coefficients vs Reynolds number from various workers Figure 5.3 Representative semi-log plot showing the logarithmic wall region
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A study of circular couette flow by laser doppler measurement techniques Geach, Nigel Douglas Sinclair 1974
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Title | A study of circular couette flow by laser doppler measurement techniques |
Creator |
Geach, Nigel Douglas Sinclair |
Date Issued | 1974 |
Description | A laser Doppler velocimeter is constructed and used to make flow measurements in circular Couette flow. The flow is created between concentric cylinders with a small gap-to-radius ratio, and measurements of the velocity profiles 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 friction 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. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0099878 |
URI | http://hdl.handle.net/2429/18722 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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