Design of a miniature optical-based velocity probe

Anemometry is the measurement of wind force and velocity. Though anemom-etry technology is fairly well developed, many velocity measurement devices are expensive, large, and/or fragile. Due to these limitations, deploying large numbers of anemometers is difficult, and often unrealistic. The miniature optical-based velocity probe is a new instrument that o ers advantages over current state-of-the-art anemometers in terms of lower expense, lower power consumption and lighter weight. The probe consists of a high performance plastic optical ber, a vertical-cavity surface-emitting laser (VCSEL), and a position sensitive detector (PSD). Light transmitted by the VCSEL shines through the free end of the optical ber illuminating the surface of the PSD. A drag force, induced by an approach ow, causes the optical ber to de ect. The PSD measures the de ection of the optical ber and outputs an analog voltage, which can be directly related to velocity through a calibration curve. Equations for the de ection and natural frequency of the optical ber, along with numerical simulations in FLUENT were used to make key design decisions in order to optimize the probe to meet the target speci cations for atmospheric research. Preliminary calibration experiments show that the velocity probe has the potential to be a viable replacement for other research-quality anemometers.

DESIGN OF A MINIATURE OPTICAL-BASED VELOCITY PROBE by Jacob Butterworth A thesis submitted to the faculty of The University of Utah in partial ful llment of the requirements for the degree of Master of Science Department of Mechanical Engineering The University of Utah May 2011 Copyright c Jacob Butterworth 2011 All Rights Reserved Th e Un i v e r s i t y o f Ut a h Gr a d u a t e S c h o o l STATEMENT OF THESIS APPROVAL The thesis of has been approved by the following supervisory committee members: , Chair Date Approved , Member Date Approved , Member Date Approved and by , Chair of the Department of and by Charles A. Wight, Dean of The Graduate School. Mechanical Engineering Jacob Butterworth Meredith Metzger 12/2/2010 Robert Stoll 12/2/2010 Patrick McMurtry 12/2/2010 Timothy Ameel ABSTRACT Anemometry is the measurement of wind force and velocity. Though anemom- etry technology is fairly well developed, many velocity measurement devices are expensive, large, and/or fragile. Due to these limitations, deploying large numbers of anemometers is di cult, and often unrealistic. The miniature optical-based velocity probe is a new instrument that o ers advantages over current state-of- the-art anemometers in terms of lower expense, lower power consumption and lighter weight. The probe consists of a high performance plastic optical ber, a vertical-cavity surface-emitting laser (VCSEL), and a position sensitive detector (PSD). Light transmitted by the VCSEL shines through the free end of the optical ber illuminating the surface of the PSD. A drag force, induced by an approach ow, causes the optical ber to de ect. The PSD measures the de ection of the optical ber and outputs an analog voltage, which can be directly related to velocity through a calibration curve. Equations for the de ection and natural frequency of the optical ber, along with numerical simulations in FLUENT were used to make key design decisions in order to optimize the probe to meet the target speci cations for atmospheric research. Preliminary calibration experiments show that the velocity probe has the potential to be a viable replacement for other research-quality anemometers. CONTENTS ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii LIST OF FIGURES: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vi NOMENCLATURE: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii ACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ix CHAPTERS 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 US Patent Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. VELOCITY PROBE DESIGN : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1 Target Design Speci cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Design Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 PSD Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Light Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.3 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.5 Fiber Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Shell Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1.1 Flow Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1.2 Fluid Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . 29 3. PROBE CHARACTERIZATION AND TESTING: : : : : : : : : : : 32 3.1 Natural Frequency Veri cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Velocity Probe Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Turbulence Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4. SUMMARY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41 4.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 APPENDICES A. DERIVATION OF DESIGN EQUATIONS: : : : : : : : : : : : : : : : : : 44 B. SENSITIVITY AND UNCERTAINTY ANALYSIS : : : : : : : : : : 48 C. DESIGN TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 53 D. FLUENT AND GAMBIT TUTORIALS : : : : : : : : : : : : : : : : : : : : 55 REFERENCES: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93 v LIST OF FIGURES 2.1 Schematic displaying the basic principle operation of the velocity probe. The dashed lines represent the deformed position created by the incoming uid ow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Experimental data provided by Swope (2009) (top) and the drag coe cient of an optical ber compared to the drag coe cient of a circular cylinder as functions of Reynolds number (bottom). The solid line is data taken from Schlichting (1979). . . . . . . . . . . . . . . . . . . 15 2.3 Expected vortex shedding frequency as a function of ow speed over the current optical ber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Outer (left) and inner (right) dimensioned schematics of the sensor shell. The units are in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Isometric view (top) and inside view (bottom) of nal CAD drawings. The nominal ow direction is labeled in the isometric view. . . . . . . . . 22 2.6 Scaled schematic of the compuational mesh used in the numerical simulations. All numerical values have units of mm. . . . . . . . . . . . . . . 24 2.7 Velocity contour plot of the ow through the medial plane of the velocity probe. The colorbar represents the velocity magnitude in m/s. The expanded view shows the region near the optical ber, and the dashed line indicates the location of the ber. . . . . . . . . . . . . . . . . 25 2.8 (Top )Velocity pro le between the two posts of the aerodynamic shell at the location of the ber. The dashed line indicates the edge of the 2 mm boundary layer.(Bottom) Numerical simulation data displaying the boundary layer thickness as the inlet velocity increases . . . . . . . . . 26 2.9 Velocity pro les for 3 di erent distances between the posts: 17.5mm, 19.7mm, 4 21.13mm deg (corresponds to data in Figure 2.8). The distance between the posts was normalized for easier comparison. . . . 27 2.10 Schematic of the con guration including the hypodermic tubing at the clamped end of the ber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.11 Results from the FSI numerical simulations:(top) inlet velocity of 2 m/s (bottom) inlet of 8 m/s. The dashed lines indicate the corre- sponding time averaged de ection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.12 Cd versus Re data from the FSI numerical simulations compared to the Cd versus Re curve for ow over a cylinder and for the data obtained from Swope (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Free vibration response of the optical ber. The actual data (top) provide information for the logarithmic decrement analysis. The inset plot shows an enlarged view of the oscillations with labels correspond- ing to veri cation equations. The variables y1 and y2 represent two successive amplitudes and T represents the damped period. The Fourier transform data (bottom) display the dominant frequency. The dotted line indicates the dominant frequency of 212.8 Hz. . . . . . . . . . . 34 3.2 Calibration data displayed with theoretical predictions from data for ow over an in nite cylinder and the corrected prediction from sec- tion 2.3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Calibration data along with the regression curve in equation 3.5 The data correspond to the data in Figure 3.2 . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Expected uncertainty in velocity measurements due to the uncertainty of the PSD chip. The data points correspond to the calibration data in Figures 3.2 and 3.3. The errorbars represent the expected uncertainty in the velocity measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Spectra plots of turbulence data. | velocity probe, 􀀀􀀀 hot-wire probe. Inside jet core: (a) Uj = 5:6 m/s, (b) Uj = 7:7 m/s, (c) Uj = 8:7 m/s. Near edge of jet: (d) Uj = 4:4 m/s, (e) Uj = 8:6 m/s. The vertical lines denote the natural frequency of the ber. . . . . . . . . 40 A.1 Schematic illustrating the di erent de ection measurements ym, yg and yc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 vii NOMENCLATURE ` distance along longitudinal axis of ber y de ection of ber from static equilibrium yc de ection as output from the PSD t time !n natural frequency in rad s􀀀1 fn natural frequency in Hz d diameter of ber db diameter of the hypodermic tubing L length of ber Lb length of the hypodermic tubing g gap size between free-end of ber and face of PSD chip m mass of exposed ber k sti ness characteristic of ber E modulus of elasticity of ber I area moment of inertia of ber, I = 1 4 (d=2)4 A cross-sectional area of ber, A = (d=2)2 density of ber a density of air absolute viscosity of air U1 approach ow speed Uj velocity at slot jet calibration facility exit f force per unit length acting on ber fs vortex shedding frequency Cd drag coe cient, Cd = Fd=(1=2 a U21 d `) Re Reynolds number for cylinder, Re = a U1 d= Rex Reynolds number for at plate, Re = a U1 x= St Strouhal Number, St = fs d=U1 logarithmic decrement BL Blasius boundary layer thickness, BL = 0:382 xRe􀀀1=5 x ACKNOWLEDGEMENTS Financial support for this work was provided by the University of Utah Research Foundation through the Technology Commericialization Project program. The author would like to thank Dr. Andras Pungor who designed the electronics circuit board for the sensor. A special thanks to Prof. Metzger for being my advisor for the project and providing invaluable insight and direction. Also thanks to my wife Stefeni for her patience and support as I worked on this project. CHAPTER 1 INTRODUCTION Anemometry is the measurement of wind force and velocity. An anemometer is an instrument used to measure one or more components of wind velocity at a single point in the ow. Though anemometry technology is fairly well developed, many velocity measurement devices are expensive, large and fragile. Due to these limitations, deploying large numbers of anemometers in eld campaigns to study atmospheric turbulence is di cult, expensive and often unrealistic. The velocity probe presented in this paper is designed to overcome these limitations, while maintaining relatively high spatiotemporal resolution. Table 1.1 shows three readily available anemometers in the market today. The miniature optical-based velocity probe described herein is a new instrument speci cally designed to bridge the gap between the sonic and hot-wire anemometers in terms of measurement volume (spatial resolution) and frequency response. Importantly, the velocity probe will also be inexpensive, lightweight, energy e cient and easily deployable in larger numbers. Being especially designed for atmospheric turbulence measurements, the probe can be used in many di erent environmental scenarios. For example, it can be suspended from pilot balloons, mounted on unmanned aerial vehicles, placed on Table 1.1. Comparison between a few di erent anemometers Type Measurement Volume Frequency Response (cm3) (Hz) Pitot-static1 0.64 0.64 2.54 10 Sonic2 10.3 6.0 6.0 10{50 Hot-wire3 0.1 0.1 0.0005 10000 1 Dwyer 167{6 tube with MKS Baratron 698 Pressure Transducer 2 Campbell Scienti c, CSAT3 3 Dantec Dynamics, CTA system with 55P01 probe 2 towers, or any other stable platform. Note, the ideas underlying the principle of operation of the velocity probe and its intended utilities have already been patented (Metzger & King, 2010). A patent search was a necessary part of that patent application process, the results of which are discussed in the next section. 1.1 US Patent Search Many sensors and devices have been developed to measure uid ow parame- ters and conditions. The methods used are varied and have di erent advantages depending on the particular uid ows. Table 1.2 lists the most relevant devices as found during a recent patent search. A brief description of each device is given below. The clear air turbulence detector (Hara, 1980) detects clear air turbulence through the collection and analyses of back-scattered laser light from a region where clear air turbulence may exist. The interference pattern is determined through an ultra high-resolution spectroscope and analyzed with an image dissector. The spectrum of back-scattered light is correlated with a spectrum representing the absence of clear air turbulence. This instrument detects distance, direction and intensity of clear air turbulence. The gas velocity meter (Hartmann & Siersch, 1980) measures the mean ow velocity inside a duct. The meter generates two light beams in the form of shallow wide bands, which traverse the duct across two transverse planes arranged at right angles to the mean ow direction. The two light beams emerge from the duct and are received on respective photoelectric detectors. The output signals of the photoelectric detectors have a small di erence due to the time required for the gas to ow between the two planes. The di erence in time is correlated through a circuit and an output proportional to the mean ow velocity of the gas is generated. In a laser-doppler-anemometer (John & Olldag, 1986) two frequency-displaced partial beams are transmitted to a measuring probe. The probe is xed at a distance away from the laser source. The two beams are focused on each other and the probe detects the light. As particles pass through the light beams they re ect scattered 3 Table 1.2. List of reviewed patents Patent Issue Date Number U.S. Patent Title Inventor Name 04/01/1980 4195931 Clear air turbulence detector Small 05/06/1980 4201467 Gas velocity meter Hartmann, et al. 05/11/1986 4575238 Laser-doppler anemometer Knuhsten, et al. Fiber optic thermal 11/11/1986 4621929 anemometer Phillips Van Cauwenberghe, 12/30/1986 4631958 Force-balance drag anemometer et, al. 01/20/1987 4637716 Fiber-optical Doppler anemometer Auweter, et al. Fiber optic probe and system for 05/05/1987 4662749 particle size and velocity Hatton, et al. measurement 04/04/1989 4818071 Fiber optic doppler anemometer Dyott Remote measurement of physical 04/02/1991 500493 variables with ber optic systems - Kleinerman methods, materials and devices Embedded ber optic beam 06/11/1991 5023845 displacemnt sensor Crane, et al. Optical ber sensor for measuring 05/19/1992 5115127 physical properties of uids Bobb, et al. Optoelectronic motion and uid 06/09/1992 5120951 sensor with resilient member Small de ected by uid ow System for characterizeing ow 01/30/1996 5488224 pattern, pressure and movement Fagan, et al. of a uid Fluid sensing apparatus with a 06/10/1997 5638174 rotatable member utilizing di erent length light pipes for alternately Henderson transmitting a light beam Laser-based forward scatter 02/02/1999 5865871 liquid ow meter Simundich Fiber optic catheter for accurate 12/26/2000 6166806 ow measurements Tjin 01/28/2003 6510842 04/05/2005 6874480 Flow meter Ismailov 01/25/2005 6847437 Laser anemometer Bruel, et al. 4 light into the photo detector. The Doppler frequency shift of the scattered light is then used to calculate the velocity of the particles in the uid, which corresponds to the velocity of the uid. A ber optic thermal anemometer (Phillips, 1986) can be used to measure the heat transfer coe cient of a uid sample. An element with temperature sensitive optical properties is placed in contact with a uid sample. The element is heated or cooled to an equilibrium temperature. The rate of heating or cooling and the temperature di erence between the element and the uid sample indicate the heat transfer coe cient. The heat transfer coe cient of the uid sample is also a product of its composition and other physical properties. This optical method can detect the composition of gasses, liquids, uid levels, and the presence of bubbles in a liquid. It can also detect the pressure and ow rates of the uid. The force-balance drag anemometer (Cauwenberghe & Motycka, 1986) can be used to measure two orthogonal velocity components of a uid. The anemometer consists of a sphere attached to a shaft. The drag force acting on the sphere causes the shaft to de ect from its neutral position. The de ection can be measured using optical or electrical sensors. The sensors control an electromagnetic actuator, which generates a force opposing the drag force. The actuator returns the shaft to its neutral position. The force is measured and related to the velocity of the uid using a coe cient of drag versus Reynolds number curve. The ber-optical Doppler anemometer (Auweter et al., 1987) uses moving parti- cles to re ect incident light into a con guration of two optical bers. Re ected light, with a frequency di erent from the incident beam, is coupled with incident light and received by a photoelectric transducer. The photoelectric transducer records the scatter and the Doppler shift is used to determine the ow velocity. The ber optic probe and system for a particle size and velocity measurement (Hatton & Plawsky, 1987) is a system that measures the size and velocity of bubbles or drops in a multiphase process environment. The probe consists of a transmitting coherent ber bundle, a lens for projecting the transferred fringe image into a measurement zone within the uid, and receiving ber optic bundles that have a 5 distal end lens for collecting the light re ected or refracted from the bubbles or drops. The probe then transfers received light to a signal processing apparatus and converts the light into an electrical signal corresponding to the phase and amplitude components of the received light The velocity or size of the bubbles or drops are then determined from the data received. The ber optic Doppler anemometer (Dyott, 1989) uses coherent light, a direc- tional coupler formed by the combination of a pair of single-mode optical bers and a photoelectric transducer. A light source provides the rst optical ber with an incident beam. The second end of the optical ber is located adjacent to a body of moving particles. The moving particles re ect a portion of the incident light back into the rst ber with a di erent frequency than the incident light. The light is directed back through the directional coupler, which directs the light toward the second optical ber and converts it into electrical signals. Using the Doppler e ect the system can measure the velocity of moving particles. Remote measurement of physical variable with ber optic systems-methods ma- terials and devices (Kleinerman, 1991) uses a light source to generate a signal beam and a reference beam. The transmission of both beams through a single multi-mode optical ber to a single photo detector produces photoelectric signals. These signals can sense variations in the magnitude of a physical parameter. Properties such as forces applied to the optical ber, measurement changes in light intensity and velocity can be measured with these methods. The embedded ber optic beam displacement sensor (Crane & Fischer, 1991) measures two-dimensional displacements of a sample material. Embedding an opti- cal ber into an object allows for detection of its displacement with a photo-detector grid array. Light is transmitted through the optical ber and emerges from an open end. The photo-detector will detect object's movement. Forces acting on the object can be determined by how much it moves or de ects. The optical ber sensor for measuring physical properties of uids (Bobb et al., 1992) uses a ber optic cable coated in a conductive material, usually gold, which is placed into a uid. Electrical energy is applied to the conductive material to heat 6 a region of the ber. Heating the optical ber changes the path of the light inside the ber. A physical property is determined from the changes in the optical path length or phase of the light received at the end of the optical ber. Measuring the phase change and applied electrical energy from heating a ber to an equilibrium temperature provides a means of nding the ow rate of a uid. The optoelectronic motion and uid ow sensor (Small, 1992) uses a light source and a photo-sensor placed on opposite sides of a uid ow passageway. A resilient or elastic member is placed in the ow, which varies the amount of light incident on the photo-sensor from the light source. The member can be any number of things, including an optical ber. As the uid ows past the member, viscous and pressure forces act on it, causing it to de ect or move. This change causes the light incident on the photo-sensor to change. The change can then be related to the velocity or ow rate of the uid. The system for characterizing ow pattern, pressure and movement of a uid (Fagan et al., 1996) characterizes properties of the uid under high pressure in a test cell. Adjustable rock facings line the interior of the test cell. Pressure is measured using a device with pressure-distortable optical bers. The uids velocity including direction is measured with laser Doppler velocimetry. The ow pattern is also viewed using arrays of optical bers. The uid sensing apparatus with a rotatable member (Henderson, 1997) is capa- ble of determining uid ow rates, density, particulate content, light transmittance, spectral attributes and other uid characteristics. The apparatus alternates two di erent light beams emerging from light pipes of two di erent lengths. The uid ows into the rotating member. As the member rotates, due to the moving uid, the light pipes will alternately align with the emitter and detector. The frequency at which the lights change can be related to the ow rate. Because the two light pipes are di erent lengths, other uid properties can be determined as well. The change in the gap between the end of the light pipe and the detector allow for the detection of other properties such as density or particulate content. The laser-based forward scatter liquid ow meter (Simundich, 1999) measures 7 the e ect of a ow on a laser beam traveling through the ow to a detector. The ow interferes with the laser causing the laser beam to refract. These refractions cause variations in the lasers energy strength at audio and super audio frequencies. The ow rate is proportional to the audio frequency correlated with known ow rates for the particular uid. The ber optic catheter for accurate ow measurements (Tjin, 2000) can perform accurate measurements of uids owing within pipes, veins, or arteries. The catheter uses two optical bers. A re ective surface intercepts light transmitted by the rst ber and re ects the light through an optically transparent window and into the uid. Some of the light returns as backscatter and is re ected into the terminal ends of the second ber. The scattered light is collected and transmitted to an anemometer. The velocity of the uid is determined by analyzing the Doppler shift between the transmitted light and the scattered light. The ow meters (Ismailov, 2003, 2005) use a laser Doppler anemometer to measure the instantaneous centerline velocity of uid ow in a pipe. It also uses the velocity to compute the volumetric ow rate, mass rate and other ow char- acteristics. The electronic processing method is used to obtain an exact solution to the Navier-Stokes equations for periodically oscillating ow. These ow meters are speci cally designed to measure ow characteristics of high-pressure automotive fuel injection systems. A laser light source is split in to two beams, which intersect the uid ow. A photo detector is used to detect forward scatter caused by the light's interaction with the uid. The Doppler frequency shift is converted into instantaneous velocity. A laser anemometer (Bruel & Combe, 2005) measures the relative velocity between the anemometer and a uid medium. A laser beam is focused in a mea- surement zone containing particles. The particles create backscattered radiation from the laser, which is collected by an optical mixer that measures the Doppler shift between the emitted laser and the backscatter from the particles. The shift is related to the relative velocity between the laser anemometer and the particles in the uid. 8 1.2 Outline of Thesis The miniature optical-based velocity probe (MOBV) is a small, inexpensive sensor for measuring wind speed and small-scale turbulence. It operates on the principle of drag-force (DeLucia & Manfrida, 1989; Fralick, 1980; Krause & Fralick, 1982). A small cantilever beam protrudes into the ow. The protruding beam de ects due to the drag force induced by the approach ow. The drag force is determined through the measurement of the de ection of the beam. The approach ow velocity can then be determined through a calibration curve. Much like the embedded ber optic beam displacement sensor described above, the velocity probe uses optical ber technology to measure the beam de ection. In this case, the cantilever beam is the ber optic cable. The ber de ects due to drag forces created by an approach ow and transmits its position to a position sensitive detector (PSD). The current design has several limitations. Because a one-dimensional PSD is used to measure the de ection of the optical ber, the velocity probe can only measure the velocity and its uctuations in one direction. In addition, the frequency response of the probe largely depends on the choice of optical ber characteristics. The current design has a frequency response of about 210 Hz, meaning that this particular sensor will not detect turbulent motions with frequencies higher than 210 Hz. The previous thesis of Swope (2009) described the initial concept underlying of the present velocity probe and characterized several early-generation prototypes. However, Swope utilized an ad hoc design process. The contribution of this thesis lies in its focus on a theory-based design approach. In addition, a battery-powered working prototype was successfully built. The outline of this thesis is as follows. First, the target design speci cations and principle of operation are stated. The de- sign methodology, including design equations, sensitivity, and uncertainty analysis, optical ber selection is presented. Results from numerical simulations of the ow around the aerodynamic shell, velocity pro les in the region of the optical ber, and basic uid-structure interaction behavior are discussed. All simulations were 9 performed in FLUENT. Calibration results are presented, indicating consistency and repeatability of the velocity measurements. Finally, the spectral response of the velocity probe and its ability to accurately capture turbulence is compared to that of a hot-wire anemometer through bench top experiments using a slotted jet. CHAPTER 2 VELOCITY PROBE DESIGN 2.1 Target Design Speci cations The current-generation velocity sensor has been designed according to the target speci cations listed in Table 2.1. Note, successfully achieving these target speci- cations would yield a velocity sensor that bridges the gap between commercially available sonic and hot-wire anemometers in terms of spatiotemporal resolution, thus ful lling a need in the atmospheric sciences community allowing for more cost-a ordable, yet sophisticated, large-scale eld campaigns. Note, the speci ed velocity range is typical of rear-surface atmospheric wind conditions and is also within the operating range of the test facility used to calibrate the velocity probe. Limitations in the state-of-the-art PSD technology precluded the current-generation velocity sensor from attaining all of the desired characteristics, as listed in Table 2.1. Nevertheless, very good progress was made in this direction in all categories. Spe- ci c design strategies, constraints and outcomes are discussed in detail below. Table 2.1. Target speci cations for the velocity probe Characteristic Desired Value Frequency Response 500 { 1000 (Hz) Velocity Range 0.5 { 12 (m/s) Uncertainty 5 cm/s Spatial Resolution 10 mm Power Consumption 5 mW Weight (with battery) 100 g Package Size Hand-held 11 2.2 Principle of Operation The velocity probe operates upon the principles of simple cantilever beam de ection and aerodynamic drag over a cylinder. Figure 2.1 shows a representative schematic, including relevant geometric parameters, of the application of these principles. Aerodynamic drag from the incoming uid ow causes the free-end of a cantilevered optical ber (diameter d, length L) to de ect. A light source illuminates the anchored end of the ber and shines through to its free-end. A one-dimensional position sensitive detector (PSD), mounted on a post opposite the optical ber, detects the position (yc) of a spotlight created by the light shining out of the free-end of the ber. A small gap (length g) separates the PSD from the free-end of the ber. As the optical ber de ects due to changes in the uid ow velocity, the PSD outputs the spotlight's change in position. Applying a characteristic mathematical model to the output of the PSD allows for a fast and accurate calculation of the velocity of the incoming uid ow. 2.3 Design Methodology The design strategy described below outlines the rationale used in selecting the ber (type, diameter, length), PSD chip (model S4583{04), and light source. !"#$%&!"'( "$)*+&,-./& 0'1$+$'2& %-+-3+'4 "$)*+& 1'#43- 5$,-4& '0+$3 L g d 1#00'4+& 54./- yc Figure 2.1. Schematic displaying the basic principle operation of the velocity probe. The dashed lines represent the deformed position created by the incoming uid ow. 12 The most critical of these is the ber, since it governs the overall response of the sensor to changes in the approach ow. Selection of the PSD chip depends on the ber used, because the ber determines the expected maximum de ection, which sets the required full-scale range of the PSD. In addition, the PSD dictates the overall uncertainty in the subsequent velocity measurements acquired from the probe. Finally, the selection of the light source depends on the type of ber selected as well as the radiant energy requirements of the incident spotlight on the PSD chip. 2.3.1 PSD Chip One of the primary constraints in the sensor design is the PSD chip itself. A low pro le, compact PSD chip was necessary in order to reduce the overall size of the velocity sensor, thus guaranteeing the desired spatial resolution. The two most important characteristics of the PSD are the range (speci ed by the active area of the chip) and the uncertainty. The current-generation velocity sensor utilizes a one-dimensional PSD (Hamamatsue S4583{04) with a range of 3 mm (in the ow direction) and uncertainty of 10 m. This e ectively yields an uncertainty of 0.67% at full-scale, which arguably for research purposes is only marginally acceptable. The rise time of the PSD, i.e., the time required for the output to change from 10% to 90% of the steady output value for a step-input measures 10 s, which is high enough to allow the sensor to respond to even the fastest turbulent uctuations expected in the atmosphere. The PSD outputs an analog voltage signal proportional to the centroid of the spotlight. There are several other viable options of one-dimensional PSDs, besides the model S4583-04 selected here, with ranges between 2 mm and 6 mm. Importantly though, the 3mm PSD selected has the best compromise between range and absolute uncertainty. In fact, at present, there are no other PSDs available on the market to the author's knowledge, with an uncertainty less than 10 m. The PSDs with a range lower than 3 mm have an equivalent uncertainty; while those with higher ranges have larger uncertainty. Alternatively, a two-dimensional PSD would The leading manufacturer of PSD technology in terms of quality and variety, is Hamamatsu 13 provide measurements of two velocity components with out any additional change in the aerodynamic shell of the velocity probe. However, the absolute error of two- dimensional PSDs remains twice that of one-dimensional PSDs. Because a twofold decrease in accuracy was deemed unacceptable for the target research application, the present velocity probe was designed for a single, one-dimensional PSD. The spectral response range of the PSD lies between 760 nm to 1100 nm with a peak sensitivity wavelength of 960 nm. 2.3.2 Light Source As it turns out, the PSD chip used in the present application produces accurate position data independent of incident light intensity. Swope (2009) performed a light intensity test revealing that changes in intensity had little a ect on the PSD output. The only exception that yielded a large change in the PSD output occurred when the intensity became too high causing the PSD to saturate. One goal of the present velocity probe, compared to earlier design of Swope (2009), was to avoid the need for a costly optical coupler between the ber and the light source. After investigating several options including various LEDs and laser diodes, a vertical cavity surface emitting laser (VCSEL) was chosen (Optek model OPV322). The VCSEL is speci cally designed for noncontact position sensing, making it ideal for this application. The VCSEL also has a wavelength (850 nm) well within the spectral range of the PSD. The three main attractive features, through, are: (i) low power consumption (ii) small size, 3.5 mm long and 1.6 mm diameter, and (iii) narrow light beam, 6 divergence angle. The latter is achieved by a dome lens incorporated into the packaging of the VCSEL. The narrow beam angle combined with the small diameter of the emitted light beam, eliminate the necessity of an optical coupler. Simply butting the clamped end of the ber against the dome lens of the VCSEL was su cient enough for purposes of this application. 2.3.3 Mathematical Models A mathematic model of the system shown in Figure 2.1, along with previous experimental data provided by Swope (2009), was used to design the ber for the 14 current-generation velocity probe using a theory-based approach. In essence, the model provides predictions of the beam position, yc and the natural frequency, !n as a function of both the geometric and material properties of the ber. Derivations based on simple cantilever beam de ection model are outlined in the Appendix. The resultant equations are, yc = Cd a d U21 L4 16E I 1 + 4 g 3 L ; (2.1) and !n = 1:8752 L2 s E I A ; (2.2) where Cd is the coe cient of drag, a is the density of the air, U1 is the approach ow velocity, E is Youngs Modulus of the ber, I is the area moment of Inertia, and A is the cross sectional area of the ber. In order to evaluate yc over a range of input parameters, an empirical relationship between Cd and Re is necessary. Prior experimental data in the form of yc versus U1, used for this purpose, is displayed in Figure 2.2. Equation 2.1 may be rearranged to give Cd = 16E I yc a d U21 L4 1 + 4 g 3 L 􀀀1 : (2.3) Substituting the experimental data of Figure 2.2 into (2.3) along with the appro- priate parameter values for Swope's sensor, allowed Cd to be plotted versus Re, as displayed in Figure 2.2. Also shown for comparison are data for ow over an in nite cylinder. Data from the sensor are generally two to three times higher than that for an in nite cylinder. MATLAB was used to determine equations for a best t line through the data points. Cd versus Re obeys the empirical relationship Cd = 10:495(Re􀀀0:3371). This regression line was then used to calculate Cd over a range of 1 m/s to 8 m/s for di erent parameter values in the present design. Values for yc and !n were found for combinations of the three design variables: L, d, and E. This method allowed fast, direct comparison between di erent optical ber types, and sizes. 15 0 2 4 6 8 0 0.05 0.1 0.15 0.2 0.25 0.3 U∞ (m/s) y (mm) 10/25/07 11/09/07 101000 101 102 101 Re Cd Infinte Cylinder 11/09/07 10/25/07 Data Curve Fit Figure 2.2. Experimental data provided by Swope (2009) (top) and the drag coe cient of an optical ber compared to the drag coe cient of a circular cylinder as functions of Reynolds number (bottom). The solid line is data taken from Schlichting (1979). 16 2.3.4 Sensitivity Analysis Using equations (2.1) and (2.2) to both make design decisions and analyze data requires an understanding of the how sensitive yc and !n are to the various pa- rameters, and how uncertainty in the values of the parameters propagates through the equations. Sensitivity analysis quanti es the e ect of small perturbations in the design parameters on the subsequent value of the output. A Taylor series approximation of equation (2.1) (details of which are provided in the Appendix) was used to calculate the sensitivity of yc, denoted by yc . The resultant expression is yc = yc Cd Cd + yc a a 􀀀 3 yc d d + 2 yc U1 U1 + 12 L + g 3L + 4g yc L L + 4 g 3L + 4g yc g g 􀀀 yc E E: (2.4) The coe cient in front of each variable is proportional to the sensitivity of the output to that particular variable, i.e., the higher the coe cient, the more sensitive yc is to that variable. A close examination of equation (2.4) reveals that the variables with the highest coe cients are: length L, diameter d, and velocity U1. The values of the coe cients are 16, 9, and 4, respectively (assuming g 0). Being sensitive to small changes in velocity makes it possible to detect turbulence in the ow, and thus is a desirable feature. On the other hand, high sensitivity of yc to L and d means that care must be taken in the design, manufacture and assembly processes, because small variations in L could dramatically impact the expected performance of the probe. In addition the negative signs in (2.4) indicate that an increase in d or E translates into a corresponding decrease in yc; whereas an increase in all the res of the parameters leasds to an increase in yc. A similar analysis was performed on equation (2.2) (details given in the Ap- pendix), to obtain the sensitively of !n, denoted by !n, and given as !n = 􀀀2 !n L L + 1 2 !n E E + !n d d + 􀀀 1 2 !n : (2.5) As before, the geometric properties of the ber (i.e., L and d) have the largest coe cients, while the material properties of the ber (E and ) have the smallest 17 coe cients. Thus, !n is highly sensitive to changes in L, and moderately sensitive to changes in d. Again the negative sign reveals that an increase in L leads to a decrease in !n and an increase in d leads to an increase in !n. 2.3.5 Fiber Selection The general parameters characterizing optical bers that can be varied in the present application include: diameter size, material type, and modes. As indicated by the analysis in section 2.3.4, the performance of the velocity probe depends heavily on the geometric properties of the optical ber, and to a lesser extent the material properties. As stated earlier, the primary constraints in the sensor design are the range and uncertainty of the PSD chip. The PSD chip possesses a range of 3 mm. However, since the ber is initially placed in the middle of the chip, under conditions of static equilibrium (no ow), an overall de ection limit of 1.5 mm exists. Therefore, the ber needs to be designed such that yc 1:5 mm at the highest expected ow speed. In addition, because the PSD possess an absolute uncertainty of 0:01 mm, the ber must also be designed so that yc 0:01 mm at the lowest expected ow speed. These criteria set an important design constraint, namely that the ber should be selected to guarantee 0.01 mm yc 1.5 mm over the target velocity range, which is speci ed as 0:5 􀀀 12 m/s. The challenge, however, lies in trying to simultaneously achieve the target natural frequency of 1000 rad s􀀀1 The present design approach was to select several di erent types of commercially available bers, and then calculate their response, i.e., yc and !n using equations (2.1) and (2.2) for a range of L and U1. Table 2.2 displays a subset of the results generated; a more complete set of design tables can be found in the Appendix. Note, the listed material properties of the bers such as density and modulus of elasticity E are based on published values from the manufacturer; the listed ber diameter is that of the core plus the cladding with the insulation stripped. Optimizing both de ection and natural frequency proved to be very di cult. Ideally, the ber would have a relatively large de ection (yc) and a large natural frequency (!n). However, as mentioned 18 Table 2.2. Optical ber options Fiber Modulus Length Diameter yc (mm) yc (mm) !n Type (GPa) (mm) (mm) U1= 1 m/s U1= 8 m/s (rad/s) Silicone 70 15 0.125 3.11e-3 9.87e-2 2.53e3 Silicone 70 17.5 0.125 5.63e-2 1.69e-1 1.86e3 Silicone 70 20 0.125 9.45e-3 3.00e-1 1.43e3 Silicone 70 17.5 0.245 5.96e-4 1.89e-2 1.43e3 Silicone 70 20 0.245 1.00e-3 3.18e-2 2.79e3 SuperEska 4 15 0.25 5.38e-3 1.71e-1 1.81e3 SuperEska 4 17.5 0.25 9.76e-3 3.10e-1 1.33e3 SuperEska 4 20 0.25 1.64e-2 5.20e-1 1.02e3 SuperEska 4 15 0.5 5.33e-4 1.69e-2 3.61e3 SuperEska 4 17.5 0.5 9.66e-4 3.07e-2 2.56e3 SuperEska 4 20 0.5 1.62e-3 5.14e-2 2.03e3 in Section 2.3.4, both of the design equations are especially sensitive to changes in optical ber length (L). In fact, an increase in L tends to increase yc but simultaneously lowers !n. Similarly a decrease in diameter (d) will result in an increase in !n but a decrease in yc. Table 2.2 shows that the SuperEska ber with a diameter of 0.25mm and a length of L = 17:5 mm o ers the best compromise. For this ber, the expected natural frequency is !n = 1333 Hz, while the expected de ection is yc = 0:00097 mm at U1 = 1 m/s and yc = 0:31 mm at U1 = 8 m/s. Because the uncertainty of the PSD is 0.01mm, additional calculations were performed to re ne the ber length so that yc at U1 = 1 m/s at least matched the uncertainty of the PSD. The results from theses calculations are shown in Table 2.3. The nal selected ber length Table 2.3. Design table used to dertmine optimal length of the SuperEska ber. yc (mm) Length (mm) at U1 = 1 m/s !n (rad/s) 17.4 9.54e-3 1.34e3 17.5 9.76e-3 1.33e3 17.6 9.97e-3 1.31e3 17.7 1.02e-2 1.30e3 17.8 1.04e-2 1.28e3 19 is L = 17:7 mm which yields a natural frequency of !n = 1300 rad s􀀀1. Based on the yc calculations relative to the speci ed uncertainty of the PSD the selected ber will only be able to measure velocities greater than 1 m/s. Furthermore, because !n = 1300 Hz, the selected ber will only be capable of responding to turbulent uctuations with frequencies less than 207 Hz. Clearly the selected ber falls slightly short of meeting the target frequency response and velocity range speci cations given in Table 2.1; however, for this work, the selected ber is deemed suitable as far as demonstration purposes. From Table 2.2, one can see that for a given length and diameter the silicone and SuperEska bers perform similarly. From a practical standpoint, the SuperEska ber is also much easier to use than the silicone ber. The diameter of the plastic optical ber is twice the diameter of the silicone ber making it easier to line up with the VSCEL laser. The plastic ber is also easier to handle than the silicone ber because the silicone ber is very brittle. There is little to no risk of breaking or damaging the plastic optical ber while assembling the probe. In summary, The SuperEska high-performance plastic optical ber with a cladding diameter of 0.25 mm, a core diameter of 0.24 mm and a length of 17.7 mm was chosen for this velocity probe. It is not brittle, de ects su ciently over a reasonable velocity range, and e ectively transmits the light from the VCSEL. The geometric and physical properties of the optical ber are displayed in Table 2.4. Vortex shedding caused by ow over the cylinder could lead to resonance if it coincides with the natural frequency. Resonance causes the ber to vibrate Table 2.4. Geometric and material properties of the optical ber Quantity Value Units E 4 0:2 GPa 1.182 0:05 g/cm3 d 250 20 m L 15 1 mm g 2 1 mm a 1 :01 kg/m3 based on the material properties of Polymethyl - Methacrylate Resin 20 erratically, potentially leading to erroneous data. Experimental data derived from Roshko (1954) shows that the Strouhal number, St versus Re obeys the empirical relationship St = 0:212(1􀀀21:2=Re) between 46 < Re < 180. Figure 2.3 shows the equivalent dimensional relationship between the actual shedding frequency fs, and the approach ow speed, U1, obtained by assuming a ber diameter of 0.25 mm along with standard properties of air. Note below U1 = 3 m/s, vortex shedding ceases to exist. At the onset of vortex shedding, fs 1250 Hz. This value increases proportionally with increasing U1. Since the range of calculated fs lies well above the natural frequency of the current ber, vortex shedding is not expected to cause resonance in the present design. 2.4 Shell Geometry Table 2.5 lists the three basic components described in the previous section. These components along with the required circuitry are packaged into a single plas- tic aerodynamic shell. Figure 2.4 and Figure 2.5 show the dimensional schematics and isometric views of the nal shell. The shell is constructed of plastic and formed using a rapid prototyping method. 02 4 6 8 10 1 2 3 4 5 6 7 8 U! (m/s) fs (kHz) Figure 2.3. Expected vortex shedding frequency as a function of ow speed over the current optical ber. 21 Table 2.5. Hardware components of velocity probe Position Sensitive Diode Hamamatsu Active Area: 3.0 x 1.0 mm2 S4583-04 Spectral Range: 760{1100 nm Detection Error: 10 m Rise Time: 10 s Vertical Cavity Surface TT electronics Output Wavelength: 850 nm Emmitting Laser Optek OPV322 Minimum Output Power: 1.5 mW Package Size: ; 1.57 mm Operating Voltage: 2.2 V High-Perfomance Plastic Mitsubishi Single Mode Type Optical Fiber Rayon Co. Design Wavelength: 600{1000 nm SuperESKA Youngs Modulus: 4 GPa SK-10 Cladding Diameter: 250 m Core Diameter: 240 m Figure 2.4. Outer (left) and inner (right) dimensioned schematics of the sensor shell. The units are in mm. 22 Flow Secondary Circuit Board Main Circuit Board Power Switch Output Connector PSD Retaining Groove for Optical Fiber LED Compartment Batteries Power Connection Ciruit Board Figure 2.5. Isometric view (top) and inside view (bottom) of nal CAD drawings. The nominal ow direction is labeled in the isometric view. 23 The length of the sensor body from the tip to the trailing edge is 110.5 mm and the width and height of the probe are 38 mm and 12.24 mm, respectively. A hollow cavity inside the aerodynamic shell provides housing for the electronic circuit boards, batteries, VSCEL, optical ber and PSD. Each inner component ts securely into a hollow compartment designed speci cally to hold it. The rest of the shell is made solid to iprove durability and reduce exure. Two posts protruding out of the main shell cavity house the PSD, a circuit board and the optical ber. The tips of the posts have been desinged in attempt to minimize the ow disturbance close to the optical ber. The cylindrical posts have a diameter of 12.24 mm. The distance between the two faces at faces on the posts is 21.1 mm. 2.4.1 Numerical Simulations Numerical simulations performed in FLUENT were used to examine possible ow disturbance of the aerodynamic shell on the ber, and observe the uid- structure interaction between the optical ber and the air. Two di erent sets of simulations were performed to investigate these e ects. The rst set focuses on the ow over the aerodynamic shell and thus does not speci cally include the ber. In contrast, the second set of simulations focuses only on the ow over the ber and therefore does not speci cally include the aerodynamic shell. 2.4.1.1 Flow Disturbance The ow eld around the entire velocity probe body was simulated using the im- plicit, second order, steady state solver with the Spalart-Allmaras one-dimensional turbulence model (Spalart & Allmaras, 1992) option in FLUENT. The computa- tional domain was a rectangular box measuring 600 x 200 x 1000 mm3 containing 725,477nodes. The leading edge of the velocity probe was placed in the domain 300 mm from the inlet face as shown in Figure 2.6. A triangular mesh was applied to all of the velocity probe surfaces and a tetrahedral mesh was applied to the inner domain. The meshing around the tips of the velocity probe had ne spacing of 0.1 mm. The inlet boundary condition was speci ed with a constant uniform velocity of 8 m/s in the z direction. The outlet boundary condition was de ned 24 Figure 2.6. Scaled schematic of the compuational mesh used in the numerical simulations. All numerical values have units of mm. as a pressure outlet boundary with zero gauge pressure. Because the size of the computational domain was much larger than the velocity probe, the four remaining sides were assigned symmetry boundary conditions. A constant air density of 1.225 kg/m3 and a constant viscosity of 1.7894e-5 kg/(m s) were used in the analysis. Figure 2.7 shows the velocity contours of the medial plane of the velocity probe for the case of U1 = 8 m/s. The velocity contours reveal the ow disturbance, caused mainly by the tips, lies mainly in front of the probe. Boundary layers forming along the posts are also visible in the contour plot. These boundary layers vary with ow speed. Of particular interest in terms of the present design is the ow eld in the region around the optical ber. Figure 2.8 shows the velocity pro le between the posts 25 10 9 8 7 6 5 4 2 3 1 0 Figure 2.7. Velocity contour plot of the ow through the medial plane of the velocity probe. The colorbar represents the velocity magnitude in m/s. The expanded view shows the region near the optical ber, and the dashed line indicates the location of the ber. at the location of the optical ber for the case of U1 = 8 m/s. Boundary layers form equally along both posts. As shown in the gure the boundary layers extend about 2mm away from the edge of each post. Outside the boundary layers, the velocity pro le remains uniform across the span between the posts with an average to within 1% of the inlet ow speed. This provides critical information in terms of selecting the gap size. If the free end of the optical ber were to extend into the boundary layer, the distributed force along the optical ber would not be uniform. The boundary layer thickness increases at the ow speed increases. Figure 2.8 also displays numerical simulation results observing the change in the boundary layer thickness as the inlet velocity increases. To ensure that the free end of the optical ber does not penetrate into the boundary layeralong the opposite post, at least 26 0 0 5 10 15 20 1 2 3 4 5 6 7 8 9 Distance (mm) U (m/s) 1 4 6 8 10 1.5 2 2.5 3 3.5 Inlet Velocity (m/s) Boundary Layer Thickness (mm) Figure 2.8. (Top )Velocity pro le between the two posts of the aerodynamic shell at the location of the ber. The dashed line indicates the edge of the 2 mm boundary layer.(Bottom) Numerical simulation data displaying the boundary layer thickness as the inlet velocity increases 27 for approach ow velocities less than U1 = 8 m/s, the gap length was set at g = 2 mm. In order to investigate how changes in post spacing alters the observed ow in the vicinity of the ber, additional simulations were done for the three cases listed in Table 2.6. Note, the third case with a post spacing of 21.23 mm matches that previously shown in Figures 2.7 and 2.8. The three di erent velocity pro les versus normalized distance across the posts are compared in Figure 2.9. Although, little di erence appears to exist between the three cases, the ow does accelerate more in Table 2.6. Flow disturbance comparison for di erent distances between the velocity robe posts. The percent di erence is for an 8 m/s inlet ow velocity. Post Spacing U1 Percent (mm) (m=s) Di erence 17.5 8.12 1.48 19.7 8.09 1.10 21.23 8.08 0.99 0 0.2 0.4 0.6 0.8 1 5 5.5 6 6.5 7 7.5 8 l/L U (m/s) Figure 2.9. Velocity pro les for 3 di erent distances between the posts: 17.5mm, 19.7mm, 4 21.13mm deg (corresponds to data in Figure 2.8). The distance between the posts was normalized for easier comparison. 28 the vicinity of the ber as the post spacing decreases. Therefore, in order to achieve less than 1% di erence due ow disturbance e ects, the analysis indicates that the post spacing should remain at 20 mm or larger. Therefore, the aerodynamic shell based on the previous generation designs of Swope (2009) which had a post spacing of 21.3 mm, was continued to be used for the current velocity probe. This posed a minor problem because the length of the selected ber, L = 17:7 mm plus the desired gap, g = 2 mm, measured less than the recommended post spacing. To overcome this the clamped-end of the optical ber was extended out of the post using hypodermic tubing, as shown in Figure 2.10. The hypodermic tubing is made of stainless steel and has a modulus of elasticity of 190 GPa. Because it is almost fty times more rigid than the optical ber, it does not signi cantly de ect due to the aerodynamic drag caused by the approach ow. As discussed previously, the boundary layers along the two posts are about 2 mm thick. By using the hypodermic tubing to extend the clamped-end of the optical ber away from the posts, the optical ber itself will be almost completely out of the boundary layer, increasing the validity of the present model and the assumption of a uniform force distribution along the entire length of the optical ber. Three-dimensional numerical simulations were performed in FLUENT to !"#$%&!"'( "$)*+&,-./& 0'1$+$'2& %-+-3+'4 "$)*+& 1'#43- 5$,-4& '0+$3 L g d 1#00'4+& 54./- db Lb yc Figure 2.10. Schematic of the con guration including the hypodermic tubing at the clamped end of the ber. 29 examine the aerodynamic drag force along the ber both with and without the tubing. Simulations of ow over a cylinder with a step increase in diameter were compared to simulations of ow over a uniform diameter cylinder. The aerodynamic shell was not included in these simulations. The computational grid was re ned twice to ensure that grid convergence had been obtained. The initial grid spacing in the vicinity of the optical ber was 0.1mm. Subsequent re nements yielded grid spacing in the vicinity of the optical ber of 0.05 mm and 0.025 mm. The drag force acting on the optical ber was computed for all cases; the results are in Table 2.7. The drag force with the hypodermic tubing inserted is about 0.24 percent higher than the drag force without the tubing. Therefore at least for ratios d=db < 0:5, use of the tubing is not expected to adversely a ect the performance of the velocity probe. 2.4.1.2 Fluid Structure Interaction Fluid structure interaction (FSI) between the ber and the air is the principle upon which the velocity probe is based. Two-dimensional numerical simulations were performed to examine the unsteady interaction between the air and the optical ber, without the presence of the aerodynamic shell. Dynamic mesh capabilities in FLUENT along with user de ned functions (UDF) programming were used to model the FSI problem. To simplify the numerics, a two-dimensional solver was utilized. Although this greatly reduced computational time and coding, the cost trade o was that realistic boundary conditions were di cult to reproduce. The velocity probe operates in a three-dimensional environment and uid motion in the Table 2.7. Di erence between the force along a bare optical ber and an optical ber with tubing attached to it. Two grid re nements were performed to ensure grid independence Grid Spacing Bare Fiber With Tubing Percent (mm) (N) (N) Di erence 0.1 4.26e-4 4.26e-4 0.01 0.05 4.60e-4 4.62e-4 0.34 0.025 4.49e-4 4.50e-4 0.23 30 third direction clearly cannot be captured in a two dimensional simulation. Another problem with the FSI simulation relates to the fact that the two-dimensional solver assumes the shape to be rectangular, whereas the actual optical ber is circular. Because the ow does not move in the third direction the ow does not match the ow over the cylinder. The custom written UDF was used to calculate beam de ection, and control the motion of the beam using the dynamic mesh feature in FLUENT (The appendix contains a custom tutorial that was developed to document this procedure). The procedure for performing the FSI simulations was as follows. First, the steady-state laminar ow over the unde ected beam (oriented with its longitudinal axis perpen- dicular to the ow) was obtained. From this solution, the equivalent distributed load, f on the beam was calculated by integrating the normal stresses acting on the beam. The subsequent de ection as a function of distance along the beam is then calculated from the static beam equation. y(`) = f L4 24E I (`=L)4 􀀀 4 (`=L)3 + 6 (`=L)2 : (2.6) The solid boundaries de ning the beam are moved accordingly; and, the mesh in a neighborhood surrounding the de ected beam is redrawn dynamically. This process is repeated for each time step. Note, the actual value of the time step used in these simulations was arbitrary because the de ection equation being used in the UDF represents that for a steady load. More complicated simulations including additional drag force from the movement of the beam itself were not attempted. Figure 2.11 illustrates the temporal behavior of yc for two separate cases using an inlet of velocity of 2 m/s and 8 m/s. The data reveal that the end of the optical ber oscillates around an average of 0.0165 mm and 0.221 mm, respectively. These values are 49% and 31% di erent than those from equation (2.1) using the empirical curve in Figure 2.2, which yielded de ections of 0.324 mm and 0.0323 mm respectively for U1 = 2 m/s and 8 m/s. Figure 2.12 displays the data compared to the empirical curve t. 31 00 100 200 0.005 0.01 0.015 0.02 0.025 Time Step yc (mm) 00 50 100 150 200 0.1 0.2 0.3 0.4 Time Step yc (mm) Figure 2.11. Results from the FSI numerical simulations:(top) inlet velocity of 2 m/s (bottom) inlet of 8 m/s. The dashed lines indicate the corresponding time averaged de ection. 101000 101 102 101 Re Cd Infinte Cylinder Emperical Curve Fit FSI data Figure 2.12. Cd versus Re data from the FSI numerical simulations compared to the Cd versus Re curve for ow over a cylinder and for the data obtained from Swope (2009). CHAPTER 3 PROBE CHARACTERIZATION AND TESTING 3.1 Natural Frequency Veri cation Verifying the natural frequency of the optical ber serves two purposes. First it veri es the design equation used to optimize the velocity probe; and, second, it veri es the physical properties of the optical ber. To accomplish this a free vibra- tion experiment was performed. The optical ber tested was the same SuperEska ber as that listed in row 8 of Table 2.2, except with a length L = 17:7mm. In the experiments the free end of the optical ber was pulled back, held at a xed position, and then released. During this procedure the output signal of the PSD was recorded at a frequency of 1.5 kHz. The natural frequency of the optical ber was subsequently determined using a logarithmic decrement analysis. The logarithmic decrement is de ned as (Thomson, 1972) = ln y1 y2 ; (3.1) where y1 and y2 are the amplitudes of two successive peaks in the free vibration response. The decrement is used to determine the damping factor, , and the natural frequency, !n. The damping factor is related to the logarithmic decrement as follows = 1 q 1 + 􀀀2 2 : (3.2) The time period between the two successive peaks is called the damped period, T. The natural frequency is related to the damping factor ( ) and damped period (T) by 33 !n = 2 T p 1 􀀀 2 : (3.3) The logarithmic decrement analysis was performed using 8 successive amplitude peaks. The experimental data, displayed in Figure 3.1, yielded measured values of T = 4:7 ms and = 0:203. Substituting these values in (3.2) and (3.3) results in values of = 0:03 and !n = 1340 rad/s. Another method for determining the natural frequency of the optical ber is to perform a discrete Fourier transform of the experimental data. Figure 3.1 shows the results of the Fourier analysis. The dominant spike occurs at a frequency of 213 Hz, which translates into a natural frequency of 1337 rad/s, identical to the result obtained from the logarithmic decrement analysis. Note, the predicted natural frequency from Table 2.3 is 1300 rad/s. The dif- ference between the predicted and actual is about 2.9%, an acceptable agreement given the uncertainties in the length of the ber, and its physical properties. The experimental results are important because they verify the mathematical model along with the material/physical properties of the ber. 3.2 Velocity Probe Calibration The velocity probe was calibrated by placing it near the exit plane of a slot jet calibration facility and comparing the output of signal to the average velocity measured using a pitot-static tube connected to a high precision di erential pressure transducer. The slot jet facility is described in Metzger (2002). The approach ow velocity was controlled by adjusting the voltage of the power supply used to drive the axial fans mounted at one end of the jet facility. The output of the PSD and pressure transducer was sampled with an analog-to-digital converter at 1.5 kHz for 15 seconds for each velocity setting. The raw data were then averaged to obtain a mean value and standard deviation at each ow velocity. It was observed that as the ow speed increased the optical ber oscillated with a higher amplitude, resulting in a larger standard deviation with velocity. Figure 3.2 displays the static calibration data in Table 3.1 compared to the prediction for yc obtained from equation 2.1 using the material and geometric properties in Table 2.4 along with the Cd versus 34 −20.2 0.25 0.3 0.35 0.4 0.45 −1 0 1 2 3 4 time (s) yc(mm) −00.5.265 0.27 0.275 0.28 0 0.5 y1 y2 T 0 100 150 200 250 300 0.002 0.004 0.006 0.008 0.01 0.012 Magnitude Frequency (Hz) Figure 3.1. Free vibration response of the optical ber. The actual data (top) provide information for the logarithmic decrement analysis. The inset plot shows an enlarged view of the oscillations with labels corresponding to veri cation equations. The variables y1 and y2 represent two successive amplitudes and T represents the damped period. The Fourier transform data (bottom) display the dominant frequency. The dotted line indicates the dominant frequency of 212.8 Hz. ( )' I 35 00 2 4 6 8 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 U!(m/s) yc(mm) 10/12/2010 10/27/2010 Prediction Figure 3.2. Calibration data displayed with theoretical predictions from data for ow over an in nite cylinder and the corrected prediction from section 2.3.3. Table 3.1. Calibration data for calibration tests on October 12, 2010 and October 27, 2010 displayed with average de ection ( yc) and standard deviation ( yc ). 10/12/2010 Test 10/27/2010 Test U1 (m/s) yc (mm) yc (mm) U1 (m/s) yc (mm) yc (mm) 1.20 0.02 2.51e-6 1.40 0.02 2.04e-6 1.75 0.04 2.45e-6 2.17 0.04 2.34e-6 3.02 0.07 2.66e-6 3.06 0.06 1.96e-6 3.13 0.07 2.99e-6 3.25 0.07 2.89e-6 4.46 0.12 3.37e-6 4.20 0.10 2.46e-6 7.67 0.32 4.34e-6 6.77 0.26 4.61e-6 8.69 0.39 5.53e-6 8.61 0.41 8.55e-6 36 Re curve provided by Swope (2009) as shown in Figure 2.2.The data displayed in Figure 3.2 were used to create a calibration curve using a least squares regression t to a second order polynomial. The regression analysis produced the relationship, yc = 􀀀 4:40 10􀀀6 U21 + 􀀀 7:51 10􀀀6 U1 + 2:49 10􀀀6: (3.4) Using the quadratic formula to solve (3.4) for U1 as a function of optical ber de ection results in U1 = (􀀀7:51 10􀀀6) + p 1:25 10􀀀11 + 1:76 10􀀀5yc 8:80 10􀀀6 : (3.5) Figure 3.3 displays the data with the calibration curve generated from equation 3.5. 3.3 Uncertainty Analysis The uncertainty of the output can be estimated through an analysis similar to the sensitivity analysis provided previously (details given in the Appendix). To 00 0.1 0.2 0.3 0.4 1 2 3 4 5 6 7 8 9 U!(m/s) yc (mm) Figure 3.3. Calibration data along with the regression curve in equation 3.5 The data correspond to the data in Figure 3.2 37 determine the uncertainty in velocity measurements recorded by the velocity probe equation (2.1) is rearranged to solve for U1, resulting in U1 = 16ycEI Cd adL4 1 + 4g 3L 1=2 : (3.6) Performing the same di erential analysis on equation (3.6)(details given in the Appendix) to obtain the uncertainty of U1 measurements based on de ection measurement only is denoted by U1 , and given as U1 = 1 2 yc yc U1: (3.7) The uncertainty of U1 based only on the uncertainty of the PSD measurements decreases slightly as U1 increases. Figure 3.4 shows the prediction curve along with expected error bars. The uncertainty corresponding to 1.05 m/s and 8.03 m/s are U1 = 0:47 m/s (45%) and U1 = 0:12 m/s (1.5%), respectively. 0.05 0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 6 7 yc (mm) U!(m/s) Figure 3.4. Expected uncertainty in velocity measurements due to the uncertainty of the PSD chip. The data points correspond to the calibration data in Figures 3.2 and 3.3. The errorbars represent the expected uncertainty in the velocity measurements. 38 3.4 Turbulence Testing The velocity probe is speci cally designed for turbulence measurements in air- ow. The same slot jet facility used for calibration was also used to evaluate the turbulence measurement capability of the velocity probe in comparison to a typical hot-wire probe, which has superior spatiotemporal resolution. The velocity probe was placed such that the optical ber was located in the central core of the jet and 1:9h downstream from the jet exit plane, where h denotes the height of the jet ori ce. Data were collected at three di erent ow speeds over a time span of 1 minute each, with a sampling frequency of 5 kHz. Data from a hot-wire probe (TSI 2073) were simultaneously recorded at the same sampling rate and from the same location, albeit slightly o set in the transverse direction. Table 3.2 displays the statistical properties of the ow as recorded by each probe. The mean ow speed recorded by each probe has a maximum di erence of 1.48%. As the ow speed increases the standard deviation of the velocity probe increases at a greater rate than the hot-wire probe. For example the hot-wire measured a turbulence intensity of 5.6% at U1 = 8:7m/s while the velocity probe measured a turbulence intensity of 9.2%. This discrepancy may be attributed to resonance of the optical ber, as described further below. An additional test was performed in which the optical ber was located in the shear layer along the edge of the jet at 1:4h down stream of the jet ori ce and 1:9h above the centerline. Data were recorded at two ow velocities; the statistics are presented in Table 3.3. Note, the mean velocities recorded by the hot-wire at this location were close to or less than 1 m/s. As previously mentioned the velocity Table 3.2. Statistical results of turbulence data taken inside the jet core at 1:9h down stream. Uj = 5:65 m/s Uj = 7:66 m/s Uj = 8:71 m/s Probe Hot-wire Probe Hot-wire Probe Hot-wire Mean (m/s) 5.83 5.92 7.69 7.75 8.54 8.66 Std (m/s) 0.34 0.38 0.60 0.48 0.79 0.49 Skewness -0.31 0.16 -0.41 0.27 -0.36 0.32 Kurtosis 3.22 3.04 3.22 3.10 3.27 3.21 39 Table 3.3. Statistical results of turbulence data near edgeof jet at 1:4h down stream and 1:9h above the jet centerline. Uj = 4:39 m/s Uj = 8:63 m/s Probe Hot-wire Probe Hot-wire Mean (m/s) 0.53 0.41 2.08 1.12 Std (m/s) 0.42 0.23 0.48 0.64 Skewness 0.56 0.91 0.22 0.74 Kurtosis 3.03 3.50 2.46 3.12 probe is designed for ow speeds above 1 m/s. Hence, only turbulent motions with characteristic velocities greater than 1 m/s are accurately captured by the velocity probe, leaving the motions with smaller velocities completely neglected. This necessarily biases the measurements toward higher mean values. This e ect is likely exacerbated by the poor spatial resolution experienced by the velocity probe in this particular ow, i.e. the length of the optical ber measures only slightly less than the width of the jet. Therefore, turbulent motions with length scales less than h will not be accurately captured by the velocity probe. At the edge of the jet, these motions also likely have smaller velocities. The corresponding spectra from the turbulence tests are displayed in Figure 3.5. Note, the area under each spectral curve represents the measured variance. In all cases, the spectra produced from the data collected with the velocity probe show a spike at a frequency near the natural frequency (207 Hz). This spike represents additional energy due to resonance. Below the natural, or resonate frequency, the velocity probe performs well in terms of capturing the same spectra energy of the turbulence compared to the hot-wire. This holds true for each of the ve cases examined. In some instances a slight o set between the hotwire and velocity probe spectra may be observed. This is due to di erences in the variance measured between the two sensors; recall that the spectra were normalized by the variance. Additional energy at the resonant frequency causes a bias on the variance of the velocity probe toward higher values. 40 10−10 100 102 104 10−5 Frequency (Hz) Spectra (a) 10−10 100 102 104 10−5 Frequency (Hz) Spectra (b) 10−10 100 102 104 10−5 Frequency (Hz) Spectra (c) 10−6 100 102 104 10−4 10−2 100 Frequency (Hz) Spectra (d) 10−6 100 102 104 10−4 10−2 100 Frequency (Hz) Spectra (e) Figure 3.5. Spectra plots of turbulence data. | velocity probe, 􀀀􀀀 hot-wire probe. Inside jet core: (a) Uj = 5:6 m/s, (b) Uj = 7:7 m/s, (c) Uj = 8:7 m/s. Near edge of jet: (d) Uj = 4:4 m/s, (e) Uj = 8:6 m/s. The vertical lines denote the natural frequency of the ber. " • ; ~", ,i i .. , '. ", , , , , , , ',. ,, ,,, " " " " l " ," ' CHAPTER 4 SUMMARY The MOBV probe lls the gap between the sonic anemometers and hot-wire anemometers by obtaining a frequency response of 207 Hz and being more robust than a hotwire. It di ers from these and other anemometers through its mode of capturing wind velocity and velocity uctuations. A single mode plastic optical ber is cantilevered perpendicular to an oncoming ow. A vertical cavity surface emitting laser (VCSEL) illuminates the clamped end of the ber, which transmits light through to a position sensitive detector (PSD). The ber de ects from the drag force created by the oncoming ow and the position of the light beam is detected by the PSD. The output of the PSD is recorded using a digital to analog converter and through minimal post processing the ow velocity and its uctuations can be determined. The probe was developed using a theory-based design approach. A sensitivity analysis of the two design equations describing optical ber de ection (yc) and natu- ral frequency (!n) showed that the probe's performance depends signi cantly on the geometric properties of the optical ber. Using this analysis it was determined that the optimal ber length was 17.7 mm. Numerical simulations provided information about the ow eld in the area of the optical ber. Boundary layers that form along the posts of the aerodynamic shell grew to a maximum of 2mm for the target velocity range; therefore the gap between the free end of the ber and the PSD was determined to be 2mm. To avoid a potentially costly redesign of the aerodynamic shell, hypodermic tubing was utilized to extend the clamped-end of the ber away from one of the posts. This addition not only shortened the length of the optical ber to 17.7 mm, but also moved the clamped end of the ber out of the boundary 42 layer. With the gap set at 2 mm and the hypodermic tubing extending the clamped end out of the boundary layer, the entire exposed portion of ber was subject to a uniform drag force, making the current mathematical model more valid. Additional numerical simulations were used to show basic uid-structure interaction between the ber and ow. The simulations demonstrated the oscillatory behavior of the ber. Because the simulations were very simple, however, they did not accurately predict the actual de ection or behavior of the ber in real time. Once the probe design was complete calibration and testing was possible. Tests using a slot jet calibration facility were used to calibrate the probe. The data collected showed that the probe behaved as the mathematic model predicted, with only slight variation. The natural frequency of the optical ber was tested and also shown to be 212 Hz, which is within 3% of the expected value of 207 predicted by the mathematical model. Turbulence testing utilizing the same slot jet calibration facility was also completed along side of a hot-wire probe. The velocity probe data was compared to the hot-wire data and shown to match when air ow is within the range speci ed for the probe. The calibration and turbulence data demonstrated the viability of using the probe for velocity and turbulence data collection. 4.1 Future Work Though substantial progress was made in terms of fabricating a working proto- type, all of the desired characteristics for the velocity probe were not met. Future work and development can be done to yet improve upon the current design. A few suggestions are given here. Improvements in the electronic circuitry of the probe are necessary to increase the usability and performance of the sensor. An op-amp failure in the custom electronics caused a variety of problems including a delay in testing. Therefore, e ort needs to be focused on a more robust circuit design. Including a wireless data acquisition (DAQ) system would greatly increase the sensor's versatility. If data could be collected wirelessly, setting up sensors in various locations would become almost trivial. Wireless DAQ would also facilitate 43 experiments utilizing very large arrays of sensors. One of the greatest motivations for designing the velocity probe was to produce an inexpensive sensor that could easily be deployed in large numbers. Another improvement that would facilitate large sensor arrays would be an external, removable battery pack that can be attached directly to the aerodynamic shell of the probe. This would allow for the use of the probe remotely. Being removable it would accommodate the use of other power sources as well. If a battery pack were to be used, it would be advantages to reduce the power consumption. One method to lower power consumption is to strobe the VCSEL laser. Strobing the laser at a low duty cycle will decrease power consumption proportionally without a ecting the output, especially if the VCSEL is strobed at a frequency higher than the natural frequency of the optical ber. The current probe only measures velocity in one direction thereby signi cantly restricting its application. Incorporating two-dimensional velocity readings would permit direct measurement momentum ux and allow for a more diverse set of experimental scenarios. Current PSD technology limits this because the two- dimensional PSDs have twice the absolute error as the one-dimensional PSDs. As PSD technology advances, the capabilities of the velocity probe will also advance accordingly. Finally, incorporating a fast-response temperature sensor (such as a re wire thermocouple) would allow for heat ux measurements, which is of particular interest in the atmospheric sciences especially for purposes of examining the surface energy budget. APPENDIX A DERIVATION OF DESIGN EQUATIONS This section provides the derivation of the two design equations characterizing de ection and natural frequency. A.1 De ection Equation The mathematical model used to describe the de ection (yc) of the optical ber assumes that it acts as a cantilever beam under a uniformly distributed load. The governing equation is (Thomson, 1972), @2 @`2 E I @2y @`2 + A @2y @t2 = f; (A.1) where l denotes the distance along the ber, f denotes the distributed load along the ber, A is the cross-sectional area of the ber, E is the modulus of elasticity of the ber, is the optical ber's density and I is the area moment of inertial of the optical ber. The boundary conditions are zero de ection and zero rotation at the mounted end of the optical ber and zero shear and zero moment at the free end of the ber: y(` = 0; t) = 0; @y @` (`=0;t) = 0; @2y @`2 (`=L;t) = 0; and @3y @`3 (`=L;t) = 0: (A.2) The distributed force, f is determined using data for the coe cient of drag for ow over a cylinder. The total force can be calculated based on the de nition for the coe cient of drag, FD = 1 2 Cd a U21 Af ; (A.3) where Cd denotes the coe cient of drag, Af ( dL) is the frontal area, a is the air density, and U1 is the mean approach ow speed. Dividing by the length of the ber to get an expression of force per unit length results in, 45 f = 1 2 Cd a d U21 : (A.4) The steady-state solution of (A.1), i.e., the solution assuming dy dt is y(`) = f L4 24E I (`=L)4 􀀀 4 (`=L)3 + 6 (`=L)2 : (A.5) The maximum de ection (ym) occurs at the end of the optical ber (l = L), i.e., ym = y(`=L) = f L4 8E I : (A.6) Combining equations A.4 and A.6 results in, ym = Cd a d L4 U21 16E I : (A.7) Because of the gap, however, the position of the light beam on the PSD will be di erent than ym. The coordinate yc de nes the location of the light beam on the PSD, as shown in Figure A.1, and can be calculated as follows yc = ym + dy d` `=L g | {z } yg : (A.8) The additional displacement yg due to the gap is as follows, yg = dy d` `=L g = Cd a d U21 L3 12E I g = 4 g ym 3 L : (A.9) Substituting equations A.7 and A.9 into equation A.8 produces an equation for the position of the light beam as a function of the approach ow speed as well as the geometric and material properties of the ber as follows Figure A.1. Schematic illustrating the di erent de ection measurements ym, yg and yc. 46 yc = Cd a d U21 L4 16E I 1 + 4 g 3 L : (A.10) A.2 Natural Freqency Equation The natural frequency(!n) of the optical ber is calculated from the free-vibration response of the system, obtained by setting f = 0, in equation(A.1) @2 @`2 E I @2y @`2 + A @2y @t2 = 0: (A.11) By assuming harmonic motion the solution takes the form, y = 􀀀 e{ ! t: (A.12) The second derivative with respect to time of ( A.12) is, @2y @t2 = !2 e{ ! t = 􀀀!2 y: (A.13) Substituting (A.13) into (A.11) and assuming constant material properties of the optical ber results in, @4y @`4 􀀀 A E I !2 4 y = 0: (A.14) The general solution to (A.13) is y = A cosh ` + B sinh ` + C cos ` + D sin `; (A.15) where 4 = A E I !2: (A.16) Applying the boundary condtions into the general solution results in, (y)`=0 = A + C; ) A = 􀀀C (A.17) dy dx `=0 = [A sinh ` + B cosh ` 􀀀 C sin ` + Dcos `]`=0 = 0 [B + D] = 0; ) B = 􀀀D (A.18) d2y dx2 `=1 = 2 [Acosh L + B sinh L 􀀀 C cos L 􀀀 D sin L] = 0 A(cosh L + cos L) + B (sinh L + sin L) = 0 (A.19) 47 d3y dx3 `=1 = 3 [Asinh L + B cosh L + C sin L 􀀀 Dcos L] = 0 A(sinh L 􀀀 sin L) + B (cosh L + cos L) = 0 (A.20) Dividing (A.19) by (A.20) results in, cosh L + cos L sinh L 􀀀 sin L = sinh L + sin L cosh L + cos L ; (A.21) which reduces to cosh L cos L + 1 = 0: (A.22) Equation A.22 is satis ed by a number of values of L corresponding to each normal mode of oscillation. The rst three normal modes are L = 1:875, L = 4:733, and L = 7:85 respectively. The natural frequency is determined from the rst or fundamental mode of oscillation. Substituting this relationship in to equation A.16 gives an equation for the natural frequency, based on geometric and material properties of the ber, as follows !n = 1:8752 L2 s E I A : (A.23) APPENDIX B SENSITIVITY AND UNCERTAINTY ANALYSIS This section outlines the Taylor series expansion used in the sensitivity and uncertainty analysis. Results from the analysis provide insight into how sensitive the equations are to each parameter. For the general case where r is a function of J variables Xi for i = 1; :::; j,i.e, r = r (X1;X2; :::Xj) ; (B.1) A rst order Taylor series expansion provides an estimate of r at Xi + Xi for i = 1; :::; j, r (Xi + Xi) = r (Xi) + @ r @ Xi Xi; (B.2) where r(X + X) = r(Xi) + r(Xi). Solving for r yields r = @ r @ X1 X1 + ::: + @ r @ X Xj : (B.3) The best estimate for the uncertainty in r is (Coleman & Steele, 1989) r = " @ r @ X1 X1 2 + @ r @ X2 X2 2 + ::: + @ r @ XJ Xj 2 #1=2 : (B.4) For the sensitivity analysis, the partial derivatives in equation (B.4) represent the sensitivity coe cients and quantify how sensitive r is to each variable. B.1 De ection Equation Analysis Beginning with the design equation for de ection, yc = Cd a d U21 L4 16E I 1 + 4 g 3 L ; (B.5) 49 where I = d4 64 ; (B.6) the general expression for uncertainty becomes, yc = @ yc @ Cd Cd + @ yc @ a a + @ yc @ d d + @ yc @ U1 U1 + @ yc @ L L + @ yc @ g g + @ yc @ E E (B.7) The partial derivatives are: @ yc @ Cd = a d U21 L4 16E I 1 + 4 g 3 L = yc Cd ; (B.8) @ yc @ a = Cd d U21 L4 16E I 1 + 4 g 3 L = yc a ; (B.9) @ yc @ d = 􀀀3 Cd a U21 L4 16E I 1 + 4 g 3 L = 􀀀3 yc d ; (B.10) @ yc @ U1 = 2 Cd a d U1 L4 16E I 1 + 4 g 3 L = 2 yc U1 ; (B.11) @ yc @ L = 12 L + g 3L + 4g Cd a d U21 L4 16E I 1 + 4 g 3 L = 12 L + g 3L + 4g yc L ; (B.12) @ yc @ L = 4 g 3L + 4g Cd a d U21 L4 16E I 1 + 4 g 3 L = 4 g 3L + 4g yc g ; (B.13) @ yc @ E = 􀀀 Cd d U21 L4 16E2 I 1 + 4 g 3 L = 􀀀 yc E : (B.14) Substituting the partial derivatives back into (B.7) results in, yc = yc Cd Cd + yc a a + 􀀀3 yc d d + 2 yc U1 U1 + 12 L + g 3L + 4g yc L L + 4 g 3L + 4g yc g g 􀀀 yc E E: (B.15) The best estimate for the uncertainty in yc is derived from taking the square root of the sum of the squares (Coleman & Steele, 1989). Dividing each side of equation B.15 by yc and taking the square root of the sum of the squares results in, 50 yc = " Cd Cd 2 + a a 2 + 􀀀3 d d 2 + 2 U1 U1 2 + E E 2 + 12 L + g 3L + 4g 2 L L 2 + 4 g 3L + 4g 2 g g 2 #1=2 ; (B.16) where yc = yc yc : (B.17) . B.2 Natural Frequency Analysis Beginning with the design equation for the natural frequency, !n = 1:8752 L2 s E I A ; (B.18) where I = d4 64 ; (B.19) and A = d2 4 ; (B.20) and following the same procedure in the previous section, the general expression for the uncertainty in the natural frequency is given by, !n = @ !n @ L L + @ !n @ E E + @ !n @ d d + @ !n @ : (B.21) The partial derivatives are @ !n @ L = 􀀀2 1:8752 L3 s E I A = 􀀀2 !n L ; (B.22) @ !n @ E = 1 2 1:8752 L2 s E I A = 1 2 !n E ; (B.23) @ !n @ d = 1:8752 4 L2 s E = !n d ; (B.24) @ !n @ = 􀀀 1 2 1:8752 L2 s E I 3 A = 􀀀 1 2 !n : (B.25) 51 Substituting the partial derivatives back into (B.21) results in, !n = 􀀀2 !n L L + 1 2 !n E E + !n d ; d + 􀀀 1 2 !n :4 (B.26) The best estimate for the uncertainty in !n is derived from taking the square root of the sum of the squares. Dividing each side of equation B.26 by !n and taking the square root of the sum of the squares, of the right hand side, results in, !n = " 4 L L 2 + 1 4 E E 2 + 1 4 2 + d d #1=2 ; (B.27) where !n = !n !n : (B.28) B.3 Uncertainty of U1 Due To Uncertainty in the PSD Chip Beginning with the design equation for de ection yc = Cd a d U21 L4 16E I 1 + 4 g 3 L ; (B.29) where I = d4 64 ; (B.30) and rearranging to solve for U1 results in U1 = 16ycEI Cd adL4 1 + 4g 3L 1=2 : (B.31) Following the same procedure in the previous section, the general expression for the uncertainty in measurements of U1 is given by, U1 = @ U1 @ yc yc+ @ U1 @ Cd Cd+ @ U1 @ a a+ @ U1 @ d d+ @ U1 @ L L+ @ U1 @ g g+ @ U1 @ E E: (B.32) Assuming that only the uncertainty in the PSD measurement (yc) is relevant (B.32) reduces to U1 = @ U1 @ yc yc : (B.33) 52 The partial derivative is @ U1 @ yc = 1 2 16ycEI Cd adL4 1 + 4g 3L 􀀀1=2 16EI Cd adL4 􀀀 1 1 + 4g 3L ! ; (B.34) which reduces to @ U1 @ yc = 1 2 U1 yc : (B.35) Substituting the partial derivative back into (B.33) results in, U1 = 1 2 yc yc U1; (B.36) an equation for the uncertainty of U1 measurements as a result of the uncertainty in the PSD chip. APPENDIX C DESIGN TABLES This section contains the design tables used to determine the optimal optical ber for the velocity probe. Table C.1. Complete design table for single mode optical bers. (yc is the de ction of optical ber) Fiber Modulus Length Diameter yc (mm) yc (mm) !n Type (GPa) (mm) (mm) U1= 1 m/s U1= 8 m/s (rad/s) Silicone 70 15 0.125 3.11e-3 9.87e-2 2.53e3 Silicone 70 15 0.245 3.29e-4 1.04e-2 4.97e3 SuperEska 4 15 0.25 5.38e-3 1.71e-1 1.81e3 SuperEska 4 15 0.5 5.33e-4 1.69e-2 3.61e3 Silicone 70 17.5 0.125 5.63e-3-2 1.69e-1 1.86e3 Silicone 70 17.5 0.245 5.96e-4 1.89e-2 1.43e3 SuperEska 4 17.5 0.25 9.76e-3 3.10e-1 1.33e3 SuperEska 4 17.5 0.5 9.66e-4 3.07e-2 2.56e3 Silicone 70 20 0.125 9.45e-3 3.0e-1 1.43e3 Silicone 70 20 0.245 1.0-3 3.18e-2 2.79e3 SuperEska 4 20 0.25 1.64e-2 5.20e-1 1.02e3 SuperEska 4 20 0.5 1.62e-3 5.140e-2 2.03e3 54 Table C.2. Complete design table for multimode optical bers. (yc is the de ection of optical ber) Fiber Modulus Length Diameter yc (mm) yc (mm) !n Type (GPa) (mm) (mm) U1= 1 m/s U1= 8 m/s (rad/s) Silicone 70 15 0.125 3.11e-3 9.87e-2 2.53e3 Silicone 70 15 0.22 4.71e-4 1.5e-2 4.46e3 Silicone 70 15 .23 4.06e-4 1.29e-2 4.66e3 Silicone 70 15 .26 2.7e-4 8.57e-3 5.27e3 Silicone 70 15 .33 1.22e-4 3.87e-3 6.69e3 Silicone 70 15 .425 5.24e-5 1.66e-3 8.61e3 Silicone 70 15 .43 5.03e-5 1.6e-3 8.72e3 Silicone 70 15 .63 1.41e-5 4.47e-4 1.28e4 Silicone 70 15 .73 8.61e-6 2.73e-4 1.48e4 Silicone 70 15 1.24 1.47e-6 4.67e-5 2.51e4 Silicone 70 15 1.55 6.98e-7 2.22e-5 3.14e4 Silicone 70 17.5 .125 5.63e-3 1.79e-1 1.86e3 Silicone 70 17.5 .22 8.54e-4 2.71e-2 3.28e3 Silicone 70 17.5 .23 7.36e-4 2.34e-2 3.42e3 Silicone 70 17.5 .26 4.89e-4 1.55e-2 3.87e3 Silicone 70 17.5 .33 2.21e-4 7.01e-3 4.91e3 Silicone 70 17.5 .425 9.49e-5 3.01e-3 6.33e3 Silicone 70 17.5 .43e 9.13e-5 2.9e-3 6.4e3 Silicone 70 17.5 .63 2.55e-5 8.10e-4 9.38e3 Silicone 70 17.5 .73 1.56e-5 4.95e-4 1.09e4 Silicone 70 17.5 1.24 2.66e-6 8.46e-5 1.85e4 Silicone 70 17.5 1.55 1.26e-6 4.02e-5 2.31e4 Silicone 70 20 .125 9.45e-3 3.0e-1 1.43e3 Silicone 70 20 .22 1.43e-3 4.55e-2 2.51e3 Silicone 70 20 .23 1.24e-3 3.92e-2 2.62e3 Silicone 70 20 .26 8.21e-4 2.61e-2 2.96e3 Silicone 70 20 .33 3.7e-4 1.18e-2 3.76e3 Silicone 70 20 .425 1.59e-4 5.06e-3 4.85e3 Silicone 70 20 .43 1.53e-4 4.86e-3 4.9e3 Silicone 70 20 .63 4.28e-5 1.36-3 7.18e3 Silicone 70 20 .73 2.62e-5 8.313e-4 8.32e3 Silicone 70 20 1.24 4.47e-6 1.42e-4 1.41e4 Silicone 70 20 1.55 2.12e-6 6.47e-5 1.77e4 APPENDIX D FLUENT AND GAMBIT TUTORIALS !"#$%&'()*)+#,-#%".)$'/)0#12)3,")456&/)7%"68%6"#)9'%#"$8%&,'),3) $)!$'%&5#:#");#$-)&')$)<$%#")82$''#5=) 9'%",/68%&,') ! "#$!%&'%()$!(*!+#,)!+&+(',-.!,)!+(!/$0(1)+'-+$!2$(0$+'3!-1/!2',/!2$1$'-+,(1!*('!+#$! &)$!(*!+#$!/31-0,4!0$)#!4-%-5,.,+,$)!,1!*.&$1+66!!!! ! "#,)!+&+(',-.!/$0(1)+'-+$)!+#$!*(..(7,128! ! • 9'$-+,12!),0%.$!2$(0$+'36! • 9'$-+,12!0&.+,%.$!*-4$)!*('!&)$!7,+#!/31-0,4!0$)#6! • :%$4,*3,12!;(&1/-'3!9(1/,+,(1)!-1/!9(1+,1&&0!+3%$)6! •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oj 1 ••• ..> OperaUon Geometry (j) ~~~Jru Vertex ~ m~ ~" I j1 1 J:!J ~ I:r~I -.!J~ create Real Vertex COordinate Sys. "U-c.;:.s..:y.._s·,_ ____. ....1 .1 Type Cartesian ...J I Global Local x: 1'000 ! x: l~'=oo=o====:! y: to I y: to 1 z: t~o ===:1 z: t~:o ===:1 ~el ~U __________________ ~I Apply Reset Oose 1 ! G6 9'$-+$!-..!(*!+#$!>$'+,4$)!,1!+#$!+-5.$!5$.(76! Vertex x(mm) y(mm) z(mm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create Straight Edge Vertices II 1 [!] Type: Real .., Vi rtual Label !.:-II ____________ 1 Apply Reset Oose I ! ! V6 9'$-+$!-..!FV!$/2$)!,1!+#$!+-5.$!5$.(76! ! Edge Vertex 1 Vertex 2 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 1 9 8 9 10 9 10 11 10 11 12 11 12 13 12 5 ! F6 W,$7!-..!$/2$)6! G6 9.()$!+#$!$/2$)!7,1/(7! 59 =J AII->I 0 1 2 <-Alii No filter .JI aose I ! ! ! ! ! 7%#B)HD)!"#$%#)4$8#1) F6 9'$-+$!-!*-4$!53!)$.$4+,12!+#$!4(00-1/!5&++(1)!,1!+#$!*(..(7,12!('/$'6! !"#$"%&'( (!(0,+"( !(0#&$(0,+"! ! -6 9.,4N!(1!+#$!;.-4N!E''(7!1$=+!+(!+#$!3$..(7!)$.$4+,(1!5(=!+(!4#(()$! $/2$)6! ! ! 60 Create Face from \lAreframe Edges II I [!] Type: ... Real " Virtual ..J Initial Face II I.!J ..J Guide Edges II I.!J ..J Guide Vertices II I.!J Tolerance Auto ~I Label II I Apply Reset Oose I :$.$4+!$/2$)6F!+#'(&2#!$/2$6X!*'(0!+#$("6>9(<1=2(-1/!4.,4N!,--./6! ! ! ! ! F6 9'$-+$!-!)$4(1/!*-4$!53!)$.$4+,12!$/2$6Y!+#'(&2#!$/2$6Z!-1/!$/2$6[! +#'(&2#!$/2$6FV! G6 :->$!3(&'!7('N6! V6 9'$-+$!\$)#! 7%#B)CD)7B#8&3.)I,/#1),')%2#);#$-) F6 :%$4,*3!J(/$)!(1!-1!$/2$!53!)$.$4+,12!+#$!4(00-1/!5&++(1)!,1!+#$! *(..(7,12!('/$'6! $"?@( !("8!"( !($"?@("8!"! ! -6 9.,4N!(1!+#$!;.-4N!E''(7!1$=+!+(!+#$!3$..(7!)$.$4+,(1!5(=!+(!4#(()$! $/2$)6! 61 Edge Ust (Multiple) Available edge.9 edge.10 edge.11 edge.12 edge.13 No filter ~ I Picked ---> edge.1 edge.2 ____ I edge.3 ~ edge.4 edge.S AII->I edge.6 edge.? edqe 8 Oose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~1!e~"~'~"=====:J1 .!J Mesh Edges ' .... II ReI<. with IInI<.s Ravenel Form .... ...... , God'll ,.. Apply oefaunl T~ Succelilve Ratio .... 1 Inve~ 1 .J Dou~le , Ided ..... ...w Spacing III" App ly Default! Interv al co unt ..... I Mesn .J Remove ola mesh .J Ignore size functions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esh Faces Faces IIace.z Scheme: Apply DefaUlt1 Elements: Tri -'I Type: Pave -' Spacing: ..J Apply Defaultl Interval size ~ Options: Mesh Appty ..J Remove old mesh ...J Ignore size functions Reset 1.!J aose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pecify Boundary Types FLUENT 5/6 ,,",tion: • Add v Modify v Delete v Delete all -' J ,,==,-,,==,-' ...J Show labels ...J Show colors Name: I Beam lertside I Type: WALL Entity: Ed ges ...J I pedgel0 Remoye Reset I.!J Edit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pecify Boundary Types FLUENT 516 Action: ... Add v Modify v Delete v Delete all Name Type Beam_end WALL inlet VELOCITY _INLET Top SYMMETRY oullet PRESSURE_OUTLET stat_wall WALL ... I, ... ..J Show tabets ..J Show colors Name: [Idel wall Type: WALL ...J I Entity: Edges ...J I pedge.9 Labet edl e 1 j Edqe edge.9 Remove I.!J Type Edit I .... Apply Reset Oose ~ J I ! ! -6 !:$.$4+!H-4$!F!*'(0!+#$!H-4$!I,)+6! 56 :$.$4+!0<LC8!*'(0!+#$!%/-9!/'(%!/(71!.,)+6! 46 <1+$'!?2A2135A:/M83FA15!*('!+#$!1-0$!-1/!4.,4N!+#$!,--./!5&++(16! V6 :$.$4+!H-4$!G!*'(0!+#$!*-4$!.,)+6! -6 :$.$4+!0<LC8!*'(0!+#$!%/-9!/'(%!/(71!.,)+6! 56 <1+$'!89N3:F15>M83FA15!*('!+#$!1-0$!-1/!4.,4N!+#$!,--./!5&++(16! 68 Specify Continuum Types FLUENT 5/6 ,,",tion: • Add v Modify v Delete v Delete all Type -' J ,,:;:=====::::r:.,.. . ...J:;:=====:r:.,... I ...J Show labels ...J Show colors Name: I Stationary Domain I Type: FLUID ....I Entity: Faces ....I Il face., Remoye Reset I.!J Edit ! ! ]6 :->$!3(&'!7('N6! GOB,"%)0#12) F6 01.9(!(";-3:2(!($9=O`! G6 :$.$4+!+#$!<=%('+!GTR!@aTbC!\$)#!(%+,(16! 69 D Specify Continuum Types FLUENT 5/6 P<:tion: • Add v Modify v Delete v Delete all Stationary Domain Deforming Domain FLUID FLUID -' J =...J ==I' ..:.. ...J L==' .... ' ...J Show labels ...J Show colors "~, [cP ===:J Type: FLUID ...J Entity: Faces ...J III I~ Remoye Edit ! V6 9.,4N!,BB9-26! !,'8561&,') ! "#,)!\$)#!*,.$!,)!'$-/3!+(!,0%('+!,1+(!H.&$1+!*('!),0&.-+,(16!!:$$!5$-0+&+(',-.6/(4=! *('!/,'$4+,(1)!ARH!-1/!/31-0,4!0$)#!,1)+'&4+,(1)6! 70 Export Mesh File File Type: UNS I RAMPANT I FLUENT 5/6 File Name: Ubeamtutorial2 .msh Export 2-D(X-Y) Mesh Accept I Browse ... I aose ~---------------------------------------------- ~ :4#$0-+,4)! ! ! J369(JEFQ9:=( ! ! "6>9(JEFQ9:=( ! ! 71 2 3 7 6 10 11 1 8 9 12 5 4 2 6 11 1 7 5 3 10 12 8 9 13 4 !"#$%&'()#*(#)+&,-(+).*($/-0&1.-($"+2+)&3+.4& ,-()/%#*($/-& ! "#$!%&'%()$!(*!+#,)!+&+(',-.!,)!+(!,/+'(0&1$!-/0!0$2(/)+'-+$!+#$!&)$!(*!3)$'!4$*,/$0! 5&/1+,(/)!6345)7!*('!),2%.$!5.&,0!8+'&1+&'$!9/+$'-1+,(/:!!!! ! "#,)!+&+(',-.!0$2(/)+'-+$)!+#$!*(..(;,/<=! ! • >$-0!-/!$?,+,/<!2$)#!*,.$!,/!5@3AB":! • C$',*D!+#$!<',0!*('!0,2$/),(/)!-/0!E&-.,+D:! • F00!-!/$;!2-+$',-.!*'(2!2-+$',-.)!0-+-G-)$:! • 4$*,/$!)(.H$'!)$++,/<):! • 4$*,/$!0D/-2,1!2$)#!%'(%$'+,$):! • 4$*,/$!0D/-2,1!2$)#!I(/$):! • J',+$!-!),2%.$!345!*,.$:! • K(2%,.$!-!345:! •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w c Symmetry E ro w OJ ~ W > W -" C ro U Wall Stationary Zone "- ~ Deforming Zone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olver Fonnulation Av DPreensssiutyr eS Suuede d ~ Av IEmxpplliicciitt ~ Space Time A 'D A Steady ~ v ,,"'ymmd ", ~ v Unsteady v Axisymmetric Swirl v JD Velocity Fonnulation A Absolute v Relative ~ Gradient Option Porous Fonnulation v Green-Gauss Cell Sued A Superficial Velocity A Green-Gauss Node Sued v Physical Velocity v l eut Squares Cell Sued DK I I Cancel I Help I ! ! !!!!!!-:!! 3)$!.-2,/-'!2(0$.:! '(+:&?0&>.(+)$."7& L: K(%D!;-+$'[.,E&,0!6#M\!].^7!*'(2!+#$!5@3AB"!0-+-!G-)$! 6%9)5%!!!:'(%1)'+/V! ! ! -: K.,1Q!+#$!*+>%5(!6'('B'/%V!G&++(/!+(!(%$/!+#$!*+>%5(!6'('B'/%! :'(%1)'+/!%-/$.! 76 Model v Inviscid A Laminar v Spalart-Allmaras (1 eqn) v k-epsilon (2 eqn) v k-omega (2 eqn) v Reynolds Stress (5 eqn) OK I I Cancel I Help I - iN'~'m~',--__________ Material Type lair ~ rC~h'~m~;'='~1 F~'~'m~"=" =-______ Fluent Fluid Materials E ~~ O:~::;~::::',::"" l ~~ Fluent Database... I HixLiIT ~ ~~ User-Defined Database .. I Properties Density (kg/m3) I constant l..!l [JiL,,1 I~'~·'~'o~========~~--~~ Viscosity (kg/m-s) I constant ~..!l [JiL,,1 :========='=----== 11.7894,,-05 , Close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rFC'"c':"C':F:':":'d:"::::':":'=":':' cc,.,-____ -"",."i Material Type vinyl silylidene (h2cchsih) ~ vinyl-trichlorosilane (sicl3ch2ch) vinylidene-chloride ~h2cc~ Order Materials By -------- -- ----I jwiter-vapor(h2o"j- - - - - - - - - I Iwood-volatiles (wood_vol) A A N,m, 1 v Chemical~ '" Copy Mderiah irom c,,\\ I Delete l Properties Density (kg/m3) ~ant 1 998 .2 Cp (j/kg-k) ~ant 1 4182 Thermal Conductivity (w/m-k) ~ant Ie .0 Viscosity (kg/m-s) ~ant 1°·001003 Hew u I Edit u I Save Copy I Close I I.!I View I I .!I View I I .!I View I I .!I View I • Help ! G: K.,1Q!"%(H!G&++(/!+(!(%$/!+#$!*+>)-!%-/$.! ! ,: 8$.$1+!I'(%1=+)J>)-!*'(2!+#$!:'(%1)'+!?'&%!0'(%[0(;/!.,)+:! ,,: K.,1Q!KL!+(!1.()$!+#$!*+>)-!%-/$.:! 1: >$%$-+!%'$H,(&)!)+$%!*('!/('();5'18E-;&')5:! ! M: 8$+!+#$!G(&/0-'D!1(/0,+,(/)!*('!+#$!,/.$+:! 78 Zone Typ. beam leftside fliid beam_rightside solid det wall default-lnterlor default-interior :001 default-Interlor:o 12 <1eformln domain inlet outlet s tat_wall stationary_domain "p Set ... Copy ... I Zone Name I deforming_domain 10 [2 Close l.4aterial Name'[ -w-.C,,-,C_1C,q- 'C,Cd ----,I[!I EdiL I ..J Porous Zone ..J Source Tenns ..J Fixed Values Help 1.401ion 1 {'orOin Ivw 1 Heactkm 1 Snlree THms 1 Fixed Values 1 Rotation-Axis Origin X(m1 lo Y(m) r:c[- ---- 1.40tion TYPel Stationary [li • OK -: 8$.$1+!,/.$+!*'(2!+#$!F;5%!)$.$1+,(/!.,)+:! G: K.,1Q!+#$!"%(V!G&++(/!+(!(%$/!+#$!A%+;#)(8!45+%(!%-/$.:! ! ! ,: A/+$'!L!2N)!*('!A%+;#)(8!:'C5)(>-%:! ,,: K.,1Q!KL!+(!1.()$!+#$!A%+;#)(8!45+%(!%-/$.:! ! T: K.()$!+#$!@;>5-'18!.;5-)();5/!%-/$.:! & & & ! ! ! ! ! ! ! ! ! ! ! 79 Zone eloci inlet ID Set ... Copy ... I Close Help '(+:&B0&'(+.%A&'(.(+&'/"#($/-& L: 8$+!+#$!8(.&+,(/!1(/+'(.):! ";+<%!!!.;5(1;+/!!!";+>();5V! ! ! -: 8$.$1+!8$1(/0!_'0$'!*'(2!+#$!M1%//>1%`!-/0!:;&%5(>&!0'(%[0(;/! .,)+):! G: K.,1Q!KL!+(!1.()$!+#$!";+>();5!.;5(1;+/!%-/$.:! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 80 r~~-IC=J __ ~~~~ ____ ~ Equations • .3 Under-Relaxation Factors riiii;;i ..... iiiiiiiiiiiiiii~ ~Iow Pressure-Velocity Coupling I [SiIAPLE ~.!J I Pressure I 0.3 Density 11 Body Forces ~ Momentum I 0.7 Discretization , pressure~nd Order l..!l Momentum~nd Order UPwinQ..!l OK Default I I Cancel I Help , ! ! ! ! M: 9/,+,-.,I$!+#$!*.(;:! ";+<%!!!45)()'+)N%!!45)()'+)N%V! ! -: 8$.$1+!)5+%(!*'(2!+#$!.;&7>(%!*1;&!0'(%[0(;/!.,)+:! G: K.,1Q!45)(!-/0!#+;/%!+#$!";+>();5!45)()'+)N'();5!%-/$.:! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 81 Compute From I inlet Initial Values Reference Frame ----,[!] A R,'"'"''' c,,, Zoo. l v Absolute Gauge Pressure (pascal) I 0 X Velocity (m/s) rl ,------­YVeIOcity( m/s) l o , T: A/-G.$!%.(++,/<!(*!'$),0&-.)!0&',/<!1-.1&.-+,(/:! ";+<%!!!:;5)(;1/!!!,%/)->'+V! ! ! -: A/-G.$!M+;(!,/!+#$!K7();5/!<'(&%!G(?:! G: K.,1Q!KL!+(!1.()$!+#$!,%/)->'+!:;5)(;1/!%-/$.:! ! X: 8-H$!+#$!1-)$!*,.$!6G$-2:1-):<I7:! *)+%!!!31)(%!!!.'/%:::! 4<<#12%OCN%&(%&"-%:#+-%/#++%$*C-%#&%*$%*%H#;;-<%:#+-%&"*&%/#++%/()0%(1%7(&"% >#1<(/$%*1<%E#1'I%;+*&:(),$8% % a: 9+$'-+$!+#$!)(.&+,(/:! ";+<%!!!4(%1'(%V! -: 8$+!?>&B%1!;9!4(%1'();5/!+(!a\\:! G: K.,1Q!4(%1'(%!+(!)+-'+!+#$!1-.1&.-+,(/:! 1: K.()$!+#$!4(%1'(%!%-/$.!-*+$'!,+$'-+,(/)!-'$!1(2%.$+$:! ! O: 4,)%.-D!*,..$0!%'$))&'$!1(/+(&')! b: 6)/7+'8!!!.;5(;>1/V! 82 DI,.I==;i=-__ --::= __ Options Storage Plotting I D r Print Iterations 11000 U WindOW ro--- ~ r Plot ___ ~ Normalization I ..J Normalize r sc~ Convergence Criterion "",,;,,",,000 ~ I Axes ... I Curves .. ·1 ~ute ~.!1 Check Absolute Residual Monitor Convergence Criteria I cont i nu i t~ , , 1 0 . 001 I ~ v "l oc i t~ , , 1 0 . 001 I ~ v "l oc i t~ , , 1 0 . 001 • OK Plot Renorm I Cancel Help ! -: A/-G.$!*)++%-!,/!+#$!K7();5/!<'(&%!G(?:! G: K.,1Q!6)/7+'8! ! 1: K.()$!.;5(;>1/!%-/$.:! ! ! 83 Options Contours of Ilr Fill,d I Pressure ... ~~ r Node Values I Static Pressure I ~ r Global Range Min Max r Auto Range I I ...J Cllp to flange ...J Draw Profiles Surfaces 1 =1 beamend ...J Draw Grid beamleft levels Setup beamright channel_bottom [20;11; default-interior tl Surface Name Pattern Surface Types I@ axis I "a"h~ clip-surf exhaust-fan ran tl Display I Compute I Close I Help I - 3.i6 e·03 3.05 e·03 2.636'03 2.22 e·03 1.80 e·03 1.3g e·03 9.74a·02 ~.60 e ·02 1.-i5 e·02 -2.70e· 02 -5.8ie·02 • -1.10 e.03 -1.51 e+03 -1.93e+03 -2.3i e·03 -2.75e·03 -3.17e·03 -3.5ge+03 -4.00 e.03 - 4.4 2e·03 -4.83e+03 c: 8-H$!1-)$!-/0!4-+-!*,.$!6G$-2U)+$-0D:1-):<I7:! *)+%!!!31)(%!!!.'/%!'5-!6'(':::! '(+:&C0&'+(#:&9A-.4$*&>+7D& L: K(2%,.$!345:! 6%9)5%!!!P/%1=6%9)5%-!!!*>5#();5/!!!.;&7)+%-V! ! ! -: 3/0$'!";>1#%!*)+%/!1.,1Q!Q--O! ! 84 D Source Files "" Header Files Add ... I Delete I Add ... I Delete I library Name 11i budf load I Cancel I Help Filter I /home /butte rWO/UDF/be amtut/*. q Directories home/butterwo/UDF IbeamtuU. fhome/butterwo/UDF Ibeamtuti .. fhome/butterwo/UDF Ibeamtutilibudf ~~L:==============================r.'.,,' • "" build Files Milium! Ii@GI O'O'O"O':":oF-:":""'"'--,"'''',--_,.,,--_-,-,--,.,,--______- '"',.',' Hem oVf I fhomefbutterwofUDF lbeamtutlbeamtutori al.e ~~c:==================================~,.,,"' Filter Cancel I / ! ,: 8$.$1+!G$-2+&+(',-.:1!-/0!1.,1Q!KL:! 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G: K.,1Q!+#$!>$2$)#,/<!+-G!-/0!)$+!+#$!'$2$)#,/<!%-'-2$+$'):! 85 I I I I l.4odels Smoothing I Layering I Remeshing I > Cylinder I Six om Solver r Dynamic l.4esh Spring Constant Factor I 0. 1 ..J In-Cylinder Boundary Node Relaxation 11 ..J L5D ..J Six DOF Solver Convergence Tolerance I 0.001 l.4esh l.4ethods Number of Iterations 120 ~, r Smoothing ..J l ayering Ir Remeshingl OK I Cancel I Help I ! ! ,: K.,1Q!+#$!Y$)#!81-.$!9/*(V!+-G!+(!<$+!+#$!.$/<+#!)1-.$)!(*!+#$!<',0:! ! ! ! ! ,,: A/+$'!+#$!H-.&$)!%('H,$0!*'(2!+#$!Y$)#!81-.$!9/*(!;,/0(;:! 1: K.,1Q!KL!+(!1.()$!+#$!685'&)#!:%/$!M'1'&%(%1/!%-/$.:! ! ! ! ! ! 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G: K.,1Q!KL!+(!$?,+!P/%1=6%9)5%-!*>5#();5!V;;2/!%-/$.:! 90 Initialization none FdiLI Adjust none FdiLI Execute at End none I EdiLI Read Case none FdiLI Write Case none FdiLI Read Data none FdiLI Write Data none FdiLI Execute at Exit none FdiLI OK Cancel I Help Available Execute at End Functions. :3 Selected Execute at End Functions. :3 C"lcul"te forces fibudf Add Remove OK Cancel I Help ! 91 Initialization none FaiLI Adjust none FaiLI Execute at End Calculate Forc I EdiLI Read Case none FaiLI Write Case none FaiLI Read Data none FaiLI Write Data none FaiLI Execute at Exit none FaiLI OK Cancel I Help '(+:&E0&F-7(+.%A&>/%+"& L: 8%$1,*D!3/)+$-0D!8(.H$'! 6%9)5%!!!:;-%+/!!!";+<%1H! ! -: 8$.$1+!P5/(%'-8!";+<%1!*'(2!2$/&! 92 I I I I Solver Fonnulation Av PDreensssiutyr eS Suueed d ~ Av IEmxpplliicciitt ~ Space Time A 'D v Steady v Axisymmetric I"" UnsteadY! ~ v Axisymmetric Swirl Transient Controls v JD ..J F roUIl r:lux F onnubtkm Velocity Fonnulation Unsteady Fonnulation A Absolute ~ v Explicit v Hebtive A 1 st-Order Implicit v 2nJOrJH Implicit Gradient Option Porous Fonnulation v Green-Gauss Cell Sued A Superficial Velocity A Green-Gauss Node Sued v Physical Velocity ~ v l eut Squares Cell Sued DK I Cancel I Help I REFERENCES Auweter, H., Horn, D. & Lueddecke 1987 Fiber-optical doppler anemometer. 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