NUMERICAL SIMULATIONS OF HEAT TRANSFER OF HOT-WIRE ANEMOMETER LI WENZHONG

Transcription

1 NUMERICAL SIMULATIONS OF HEAT TRANSFER OF HOT-WIRE ANEMOMETER LI WENZHONG (M.Eng. NUAA) A THESIS SUBMITTED FOR THE DEGREE OF PHILOSOPHY OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003

2 ACKNOWLEDGEMENTS I would like to express my deepest appreciation and gratitude to my supervisors, Associate Professor Khoo Boon Cheong and Dr. Xu Diao for their invaluable guidance, advice and support throughout the entire courses of my research study. I also wish to acknowledge Prof. P. Freymuth from University of Colorado, Boulder for the discussions we had, the interest he has shown in this work and the advice he has given. I would also wish to acknowledge the staff in the Supercomputing Visualization Unit of National University of Singapore for their excellent service and great help. My entire family deserves a special gratitude for their unlimited support, encouragement and love throughout my stay in NUS. Specially thanks to my wife Ma Nan for her understanding and assistance. Last, but not the least, I would also like to my acknowledgement to the National University of Singapore for its Research Scholarship. Li Wenzhong ii

3 CONTENTS ACKNOWLEDGEMENTS...ii CONTENTS...iii SUMMARY... vi NOMENCLATURE...viii LIST OF FIGURES... xi LIST OF TABLES... xiv CHAPTER 1 INTRODUCTION Introduction to hot-wire anemometry Electronic principles of constant temperature hot wire Advantage of hot-wire anemometry The thermal response of a hot wire in a fluctuating flow Hot-wire correction under influence of wall proximity Motivation and objective of the study Structure of thesis CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE Introduction Basic equations and non-dimensional parameters Boundary conditions Discretization Second-order upwind scheme Linearized form of the discrete equation Under-relaxation Discretization of the momentum equation Discretization of the continuity equation Temporal discretization iii

4 2.11 Pressure-velocity coupling Solution method Convergence criteria CHAPTER 3 MOMENTUM AND HEAT TRANSFER FROM CYLINDER IN FREESTREAM STEADY LAMINAR FLOW Introduction Governing equations and boundary conditions Numerical calculations Numerical results and discussions Conclusions CHAPTER 4 THE THERMAL RESPONSE OF A HOT WIRE IN NEAR WALL REGION MEASUREMENTS Introduction Literature survey on hot-wire heat transfer near a wall Governing equations and boundary conditions Numerical calculations Grid distribution, domain size independent check and numerical accuracy Plausible causes for the discrepancy between the near-wall hot-wire correction curves of Chew et al. and Lange et al. for adiabatic wall correction Main parameters affecting the near-wall hot-wire correction factor: Comparison with the experiment based on Y + and h Near-wall hot-wire correction curves based on Re and h Simulation results based on Y + and U On the critical Y + c and h On the velocity correction factor, C ui, based on temperature loading of iv

5 4.13 Conclusions CHAPTER 5 THE THERMAL RESPONSE OF A HOT WIRE IN A FLUCTUATING FREESTREAM FLOW Introduction Governing equations and boundary conditions Numerical calculations Heat transfer characteristics of the hot wire in free stream fluctuating flow Conclusions CHAPTER 6 SUMMARY OF CONCLUSIONS v

6 SUMMARY A thorough numerical investigation of the laminar flow and heat transfer around a single circular cylinder was performed. In the first part of this dissertation, a summary of results of numerical investigations of the two-dimensional flow around a heated circular cylinder located in a laminar crossflow is presented. Numerical investigations were carried out for the low Reynolds number range normally encountered in near-wall hot-wire measurements, and for temperature loading of 1.8, which is typical for hot wire measurement. The computations yield information on N u with Reynolds number. The temperature dependence of the fluid properties (air) was taken into account and this resulted in a temperature dependence of the N u -Re results. The results obtained are important, as they form the reference against which the subsequent computations with wall effects are compared in order to obtain the near-wall corrections. Based on the results about domain size, natural convection, viscous dissipation and temperature influence on the fluid properties in the first part of this dissertation, in the second part of the dissertation, a numerical study was carried out to obtain the nearwall measurement correction curve for the hot wire having been calibrated under freestream condition at the two extreme cases of isothermal and adiabatic wall conditions. Unlike previous studies particularly in experiments where the correction curve is primarily based on only the distance (h) between the wall and the wire expressed in wall units (Y + hu τ υ ), it is found that a second dimensionless parameter h 0 ( h D ) accounting for the effect of the hot-wire diameter (D) is necessary to fully describe the vi

7 overall near-wall correction curve. Calculations also reveal the reason for the apparent discrepancy between the near-wall hot-wire correction curves of Chew et al. (1995) and Lange et al. (1999b) next to a thermally non-conducting wall. In the third part of the dissertation, a numerical study was carried out to obtain the thermal response of a hot wire in a fluctuating freestream flow. The ranges of the parameters considered in this study are Re 200, 0 S c 0.32 (S c is nondimensional frequency) and A 0.6 (A is dimensionless amplitude of the imposed streamwise velocity pulsation). The results showed that for any fluctuating flow, a hot wire having been calibrated under imposed mean free stream condition could be used to measure it. The very rapid thermal response of the hot wire has enable the faithful measurement of fluctuating velocity in the typical range of frequency and amplitude encountered in a turbulent flow. vii

8 NOMENCLATURE A Dimensionless amplitude of the imposed axial velocity pulsation C d C p F d Drag coefficient = 1 ρu 2 Specific heat of fluid at a constant pressure 2 D C u Correction factor = U 0 U meas C ua C ui D Correction factor for adiabatic wall case Correction factor for isothermal wall case Diameter of hot wire du + Correction factor = U meas U 0 U τ d + Reynolds number = UD τ υ Ec Eckert number = c P U ( T ) W 2 0 T f Pulsating frequency of the incoming flow g Gravitational acceleration ( ) gβ T T D 3 Gr Grashof number = w υ 2 h The distance from the center of the hot wire to the wall H Heat flux through the closed circulation which surrounds the cylindrical hot wire h 0 The non-dimensional distance from the center of the hot wire to the wall = D h viii

9 k L Thermal conductivity of fluid Hot wire length N u Nusselt number = ' 1 qd H da = A k T ( T T ) πk ( T ) w w N u0 N um N uf Pe Nusselt number at free stream Measured Nusselt number Nusselt number based on film temperature Peclet number = RePr Pr Prandtl number = µ Cp k Re R ef Reynolds number = U D υ Reynolds number based on film fluid property Strouhal number which is used as a non-dimensional quantity to describe the vortex S D shedding frequency S = τu 0 S c Non-dimensional frequency S c = f f 0 2 fd = Reν t Time T Temperature U 0 U meas The true upstream incoming flow velocity at the location of hot wire center Measured velocity value U τ Friction velocity Y + Non-dimensional wall distance = hu τ U = τ D h 0 υ υ Y c + Critical Y +, beyond which wall influence can be neglected ix

10 β ν ω Volume coefficient of expansion Kinematic viscosity (µ/ρ) Frequency of the pulsation T τ Temperature loading = w T τ s i,j Period of the vortex shedding Denotes Cartesian coordinate directions At the inlet of computational domain x

11 LIST OF FIGURES Figure 1-1 Basic circuit for constant temperature thermal anemometer Figure 2-1 Control volume used to illustrate discretization of a scalar transport equation Figure 2-2 Overview of the solution method Figure 3-1 The schematic drawing of computing region Figure 3-2 Grid distribution Figure 3-3 Nusselt number relative change with the Eckert number Figure 3-4 Nusselt number relative change with the temperature loading Figure 3-5 Heat transfer from a circular cylinder in uniform flow Figure 3-6 N u variation with Re for temperature loading Figure 4-1 The schematic drawing of computing region Figure 4-2 Grid distribution, region A surrounding the hot wire is further depicted in Figure Figure 4-3 The grid distribution around the hot wire Figure 4-4 C u variation with h 0 for Y + = Figure 4-5 C u variation with Y + and h Figure 4-6 C u variation with Y + and h Figure 4-7 C u variation with Y + and h 0 for isothermal wall Figure 4-8 C u variation with Y + and h Figure 4-9 Contour of static temperature distribution for Y + =2.0, h 0 =15 (C ua =1.21) and adiabatic wall Figure 4-10 Contour of static temperature distribution for Y + =4.0, h 0 =15 (C ua =0.94) and adiabatic wall xi

12 Figure 4-11 Contour of static temperature distribution for Y + =2.0, h 0 =15 (C ui =0.52) and isothermal wall Figure 4-12 Contour of static temperature distribution for Y + =4.0, h 0 =15 (C ui =0.91) and isothermal wall Figure 4-13 C u variation with Re and h Figure 4-14 C ua variation with Re and h Figure 4-15 C ui variation with Re and h Figure 4-16 C u variation with Re and h Figure 4-17 C u variation with Re and h 0 (0.1m/s U τ 1.0m/s) Figure 4-18 C u variation with Re and h Figure 4-19 U + m variation with Y + and h Figure 4-20 U + m variation with Y + and h 0 (=5, 90 and 150) Figure 4-21 U + m variation with Y + and h 0 for isothermal wall Figure 4-22 C u variation with Y + and h Figure 4-23 Critical Y + versus h 0 for adiabatic wall conditions Figure 4-24 Critical Y + versus h 0 for isothermal wall conditions Figure 4-25 Critical Y + versus h Figure 4-26 Variation of C ui with Re and h 0 for τ = 1.1 and Figure 4-27 Variation of C u with Re and h 0 for τ=1.1 in the low Re range Figure 4-28 Variation of C u with Re between temperature loading 1.1 and Figure 5-1 A schematic drawing of the computed region Figure 5-2 Traces of drag coefficient (C d ) on cylinder for Re=150 and S c = Figure 5-3 Traces of surface Nusselt number on cylinder for Re=150 and S c = Figure 5-4 Traces of drag coefficient on cylinder for Re=150 and S c = Figure 5-5 Traces of surface Nusselt number on cylinder for Re=150 and S c = xii

13 Figure 5-6 Variation of Nusselt number ratio with Re at various S c and A= Figure 5-7 Variation of Nusselt number ratio with Re in the low Re range at various S c and A= Figure 5-8 Variation of Nusselt number ratio with S c at various amplitude A and Re= Figure 5-9 Variation of Nusselt number ratio with S c at various amplitude A and Re= Figure 5-10 Variation of Nusselt number ratio with S c at various amplitude A and Re= Figure 5-11 Variation of N u with frequency Figure 5-12 Velocity variation with frequency for freestream flow Figure 5-13 Statistic quantities versus frequency for mean flow Re = xiii

14 LIST OF TABLES Table 3-1: C d & N u change with the grid number around the cylinder Table 3-2: C d & N u change with increase of the domain size for Reynolds number Table 3-3: C d & N u of cylinder in free stream for Re = and τ = Table 3-4: Values of constants for onset criterion of natural convection effects from Lange (1997) Table 4-1: C u for Y + = 1.0, τ =1.8 for adiabatic wall Table 4-2: Nusselt number for Re = 5.0, τ = Table 4-3: Nusselt number for Re = 0.007, τ = Table 4-4: Nusselt number for Re =5.0, τ =1.8, Ec= Table 4-5: Nusselt number change with increase the domain size for Re = Table 4-6: N u for Y + = 0.5, U τ = 1.0 m/s, Re = 0.15, τ =1.8, N u0 = for adiabatic wall Table 4-7: N u for Y + = 0.5, U τ = 1.0 m/s, Re = 0.15, τ =1.8, N u0 = for adiabatic wall Table 4-8: N u for Y + = 0.5, U τ = 1.0 m/s, Re = 0.15, τ =1.8, N u0 = for isothermal wall Table 4-9: N u for Y + = 0.5, U τ = 1.0 m/s, Re = 0.15, τ =1.8, N u0 = for isothermal wall Table 4-10: The ratio of the thermal conductivity, k w, of wall materials to the thermal conductivity of air, k (= Wm K ) from Turan et al. (1987) Table 4-11: Critical Y c + values for less than 5% error in near-wall cases based on h xiv

15 CHAPTER 1 INTRODUCTION CHAPTER 1 INTRODUCTION With the advancements in the numerical method and computer power, computational fluid dynamics (CFD) has become a strong valuable tool for the investigation of fluid and heat transfer problems. It provides a much great flexibility for the specification of problem conditions. The boundary and the fluid properties can be easily varied, more importantly the physical effects could be isolated or suppressed to investigate the associated physical mechanisms in the fluid dynamics. The present work is an attempt to employ CFD for the investigation of the convective heat transfer from a cylinder in the near wall laminar flow region. The results obtained could open a new window for better understanding the underlying physics related to the hot-wire anemometry used in near-wall measurements. 1.1 Introduction to hot-wire anemometry Thermal anemometry may be the most common method employed to measure instantaneous fluid velocity. The basic operating principle of the method is relatively straightforward. It mainly depends on that any fluid velocity change would cause a corresponding change of the convective heat loss to the surrounding fluid from an electrically heated sensing probe. The variation of heat loss from the thermal element can be interpreted as a measure of the fluid velocity changes. There are two fundamentally different types of thermal anemometry, cylindrical sensors and flush sensors. Cylindrical sensors (hot wires and hot films) are normally employed to measure the fluid velocity while flush sensors (hot films) are usually used to measure the wall shear stress. Hot-wire sensors are made from a short lengths of resistance circular wire. Hot-film sensors consist of a thin layer of conducting material 1

16 CHAPTER 1 INTRODUCTION deposited on a non-conducting substrate. Hot-film sensors may also be cylindrical or other forms, such as those that are flush-mounted. Hot-wire anemometers have been used for many years in the study of laminar, transitional and turbulent boundary layer flows benefited from its technique involving the use of very small probes that offer very high spatial resolution and excellent frequency response characteristics. 1.2 Electronic principles of constant temperature hot wire Hot-wire anemometers are normally operated in the constant temperature (CTA) mode, which electronic circuit is shown in Figure 1-1. It is known that the resistance of a wire is proportional to its temperature. Assuming the bridge is balanced at a certain condition, an increase in heat transfer due to the variation of flow velocity or other flow parameters will cause a fall in probe temperature, thus the resistance of the probe will decrease. The decease of the probe resistance will cause the bridge to become unbalance; hence a (positive) error voltage is produced at the input of the servoamplifer. The signal from the amplifier will increase the bridge voltage and hence also the current through the sensor. In this way, the sensor is heated and the bridge balance will be restored. Due to the very high gain of the amplifier and the very small size of the probe, the anemometer is supposed to be able to respond to very rapid velocity fluctuations while the probe temperature is kept essentially constant (Strictly, the CTA keeps the average wire resistance constant). By monitoring the bridge voltage variation, the flow parameter variation can be measured. 1.3 Advantage of hot-wire anemometry As Bruun (1995) summarized, hot-wire anemometry is likely to remain the principal research tool for most turbulent air flow studies due to its commercial availability, high 2

17 CHAPTER 1 INTRODUCTION spatial resolution, fast response to high frequency fluctuations expected in a turbulent flow, ease of operation for the related calibration, data acquisition, and analysis. Further more, the continuous analogue signal output from a hot-wire anemometry system can be analyzed based on both time-domain and frequency-domain analysis; spatially separated probes enables the measurement of spatial/temporal correlations of turbulent fluctuations; special hot-wire anemometry probe and the related signal analysis can be used to evaluate turbulent quantities such as intermittency, dissipation rate, vorticity, etc. Compared with hot wire, Khoo et al. (2001) pointed out, both the laser-doppler velocimetry (LDV) and particle image velocimetry (PIV) need seeding in the flow for measurement, hence yielding a non-continuous output. Leighton and Acrivos (1987) found that in the very near-wall region, the particle count diminishes considerably due to the shear-induced particle migration from higher shear rate to lower shear rate regions, causing the drop in data rate. Moreover, the relatively large volume of LDA may limit its application in the near-wall region in which the velocity gradient is large. In contrast, the small size of the hot wire and its corresponding low thermal inertia permit much finer spatial resolution as well as fast response to accurately follow high frequency fluctuation expected in a turbulent flow. Due to the above reason, despite the recent progress in the technique of PIV (particle image velocimetry) and LDA (laser Doppler anemometry) for use in velocity measurement, hot-wire anemometry still remains the preferred choice for turbulence measurements for many researchers. 1.4 The thermal response of a hot wire in a fluctuating flow 3

18 CHAPTER 1 INTRODUCTION Strictly, the response of hot wire can and should be checked and verified independently by subjecting it to an accurately known fluctuating velocity as the input and then observing the measured output. In this way, the users can obtain the information on the dynamics response of the instrument which may be employed to validate and track an unsteady flow. This can also be used to verify the hot-wire manufacturer s stated specifications on the instrument response. However, there is no easy way of experimentally generating an accurately known fluctuating velocity field where hot-wire sensor is subjected to in order to obtain the true dynamic response. To some extent, the user has to just rely on the manufacture s specifications of conducting the necessary standard electronic perturbation tests to determine the cut-off frequency, which usually taken synonymously as the dynamic response frequency of the hot-wire sensor. Specifically, a hot-wire anemometer should first be calibrated in a known flow (usually a steady flow) and then subsequently subjected to imposed fluctuating flow with known amplitude and frequency. In this way, the output of the hot wire via the calibration curve can be compared directly to the imposed fluctuating flow to determine the overall response of the hot-wire system. One may also consider a dynamic calibration as opposed to the commonly employed static calibration where the imposed flow is steady. Bruun (1995) summarized that dynamic calibration procedure were previously applied by Dryden and Kuethe (1929) and Schubauer and Klebanoff (1946), but Perry and Morrison (1971) developed this method into a somewhat fairly standard calibration procedure which involved shaking the hot-wire probe at low frequencies in a uniform flow of known velocity, thereby subjecting the wire to a known velocity fluctuation. 4

19 CHAPTER 1 INTRODUCTION However, due to the complex and low frequency limitation of the dynamic calibration involved (Perry and Morrison s (1971) dynamic calibrator produced known small sinusoidal motion only from 0 to 10 Hz, which may seem too low for hot-wire anemometry measurement application; Khoo et al. recently (1995) established a feasible means of imposing a fluctuating flow with known frequency and amplitude very near the wall, by which the frequency of the fluctuation imposed could be obtained from the angular velocity and the number of recesses on the top rotating disc.), the majority of researchers still use the static calibration method. The underlying assumption is that since the hot wire is very small, it is reckoned the thermal inertia should be very small and hence is able to follow a fluctuating flow very well. Comparison between the static and the dynamic calibration were carried out by Bruun (1976), Bremhorst and Gilmore (1976) and Soria and Norton (1990); they found good agreement between the static and dynamic calibration in the lower range of the frequency. So it becomes important to investigate (numerically) the thermal response of a hot wire subjected to an imposed fluctuating flow. Because the response of the electrical system in the anemometer is invariably much better than the physical thermal response, the findings of the thermal response of the hot wire will reveal the overall response and sensitivities of the hot-wire system. 1.5 Hot-wire correction under influence of wall proximity The need for hot-wire correction in near-wall configuration 5

20 CHAPTER 1 INTRODUCTION The hot wire, when operating under wall-remote conditions, is incumbent on the heat transfer characteristics of hot wire as exposed in the measured flow to be the same as that during the calibration. Such assumption, however, is no longer valid when the same hot wire is used in near-wall measurements. The wall may change the heat transfer characteristics of the hot wire with its calibration curve obtained under free stream condition, since more or even less heat is released from the hot wire due to the influence of wall effects. Further, the presence of the hot-wire prongs may alter the flow field thereby resulting in larger convective heat loss. The subject of increased aerodynamic interference effects in near-wall hot-wire operations has been discussed by Comte Bellot et al. (1971), Azad (1983) and in the recent work of Chew et al. (1998). Chew et al. (1998) also systematically investigated the other effects of wire diameters and over-heat ratio imposed. Therefore, some corrections on the measured velocity are needed for the near-wall measurement for the hot wire having calibrated under free stream flow condition Experimental investigations on hot-wire near-wall correction A number of experimental investigations on the near-wall effects on hot-wire measurements have been conducted but with diverse results. Wills (1962) suggested the incorporation of an additional empirically determined heat loss term, which is a function of the distance from the wall and Reynolds number, to the conventional heat loss from hot-wire equation to account for the wall effect in laminar flow. For turbulent flow, half of the laminar flow correction is suggested without physical explanation. Oka and Kostic (1972) together with Hebber (1980) confirmed that the correction could fall to a single curve of Y + for different wire diameters when the 6

21 CHAPTER 1 INTRODUCTION velocity and distance from the wall are normalized by the wall parameters of U τ and υ/ U τ, respectively. Such a correction curve implies implicitly that the near-wall effects are at least directly independent of Reynolds number and wire diameter. Krishnamoothy et al. (1985) demonstrated the importance of the wire diameter and overheat ratio on the near wall effects. Singh and Shaw (1972) pointed out that the correction is independent of the wall conductivity, but Bhatia et al. (1982) showed otherwise. Overall, it may be mentioned that these experimental works do not elicit a consistent behavior of the hot wire near the wall Numerical investigations on hot-wire near-wall correction In view of the inherent experimental difficulties of the near-wall measurement, it is obvious that a numerical experiment to study the near-wall effects would be an attractive alternative especially with the advent of computational fluid dynamics and improved computer hardware technology. Piercy et al. (1956) used potential flow theory to study the two-dimensional flow past a hot wire near a highly conducting wall. Their velocities were much lower than the experimental results. Bhatia et al. (1982) included the viscous effect in their formulation. They considered the wire as a point heat source and numerically solved the disturbed temperature field with an assumed linear velocity profile. The initial temperature profile was based on Lauwerier s (1954) solution of the energy equation without the viscous dissipation term and wall effects. Their velocities were lower than the near-wall correction for Y + < 2.5 and higher for Y + > 2.5 when compared with previous experimental results of Oka and Kostic (1972) and Hebber (1980). It should be noted that in their calculations, the momentum equation was not solved and the influence of the wire diameter was neglected since it was assumed as a point. The influence of wire diameters is known to be important due to 7

22 CHAPTER 1 INTRODUCTION the interference to the flow with altered heat loss from the hot wire (see Wills, 1962, and Krishnamoorthy et al., 1985). Thus it is essential to solve the momentum and energy equations as a coupled problem. With the development of computer resources, more detailed investigations on hot-wire near-wall correction have been conducted. Among these numerical studies to find the near-wall measurement correction curves, Chew et al. (1995) and Lange et al. (1999a) conducted fairly extensive works. Although both obtained similar near-wall hot-wire correction curve for conducting wall with isothermal boundary condition, their respective hot-wire correction curve exhibit completely opposite trend for nonconducting wall of adiabatic thermal boundary condition. Chew et al. showed that for the non-conducting wall, as the hot wire is positioned increasingly close to the wall, the heat loss from the wire remains higher than the corresponding case without the presence of the wall; they attributed this phenomena to the distortion of velocity field by the wall and consequent alteration of the heat transfer characteristics of the hot wire such that there is on overall larger heat loss to the flow. On the other hand, Lange et al. pointed out that insulating wall will suppress the flow and cause an accumulation of heat between the hot wire and the wall, thereby reducing the temperature gradient in this region and hence causing a reduction in the measured Nusselt number. This discrepancy in the simulations is but perhaps not unexpected since there are many different factors which may influence the results, like the size of the computational domain employed, the convergence criteria for simulations and assumptions concerning the physical properties of the fluid such as use of the mean film temperature and others. Lange et al. (1999) extended the computational domain 2000D to the front and top of the cylinder and 3000D to the rear of the cylinder and took the 8

23 CHAPTER 1 INTRODUCTION maximum sum of the normalized absolute residuals in all equations to be less than 10-6 as the convergence criteria. (Here D is the diameter of the hot wire.) On the other hand, Chew et al. (1995) used a much smaller computational domain of 150D in front and top of the cylinder, and 240D to the rear of the cylinder and judged the convergence criteria of ε f, ε w (the maximum difference between the respective values of stream function and vorticity on successive iterations) to be less than 10-4 without any reference to the temperature field. So it is necessary to use a reasonable domain size for computation and to resolve the discrepancy of trend as observed by Chew et al. and Lange et al. by solving the Navier-Stokes equation together with the energy equation. Some of the important parameters like wall conductivity, wire diameter, distance from the wall, temperature loading on the near wall measurement can also be investigated. Finally the correction curves for universal applications to near-wall hot-wire measurements based on the main dimensionless parametric groupings can be obtained. 1.6 Motivation and objective of the study Hot-wire thermal response in a near wall flow Although so many above-mentioned works including experimental investigation and numerical study have been conducted to determine an accurate hot-wire near-wall correction curve, there still remains several discrepancies among them. None of the above works gives the reasons why there exist the differences among their correction curves for the near-wall measurements. Also some of the important parameters like wall conductivity, wire diameters, distance from the wall, temperature loading on the near-wall measurement require more detailed investigation. 9

24 CHAPTER 1 INTRODUCTION Hot-wire thermal response in a fluctuating freestream flow It is important to investigate numerically the thermal response of a hot wire subjected to an imposed fluctuating flow. Because the response of the electrical system in the anemometer is invariably much better than the physical thermal response, the results of the thermal response of the hot wire should reveal the overall response and sensitivities of the hot-wire system Objective of this study The purpose of this project is to conduct an accurate simulation and provide a deeper physical insight of a near-wall hot-wire operation by solving the full Navier-Stokes equations together with the energy equation. It is to determine the reasons causing the differences among the wall correction curves, and to investigate some of the important parameters like wall conductivity, wire diameters, distance from the wall, temperature loading on the near-wall measurements. Also from imposed known fluctuating flow, one can obtain the heat transfer response of a hot wire in a free-stream fluctuating flow. The results of this study can be valuable to researchers who perform near-wall turbulence measurement using the hot wire. 1.7 Structure of thesis The description of the present work will be organized as follows. Chapter 2 presents the governing equations and relevant non-dimensional parameters of the physical problem. Limits of the current investigation imposed by model assumptions and simplifications are also considered. Chapter 3 examines the momentum and heat transfer from cylinder in freestream steady laminar flow. Chapter 4 covers the thermal 10

25 CHAPTER 1 INTRODUCTION characteristics of a hot wire in the near-wall region. Chapter 5 investigates the thermal response of a hot wire in a fluctuating freestream flow. Finally, Chapter 6 summaries the conclusion of each above chapter. 11

26 CHAPTER 1 INTRODUCTION Error Voltage Servo Amplifier Probe Bridge Voltage Figure 1-1 Basic circuit for constant temperature thermal anemometer 12

27 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE 2.1 Introduction Based on the motivation of the study mentioned in Chapter 1, the aim of this part of the work is to provide detailed information on the laminar flow and associated heat transfer around a circular hot wire by DNS (Direct Numerical Simulation). With the incompressible flow assumption, the equations for mass, momentum and energy conservation are normalized; the non-dimensional parameters relevant to the hot-wire measurements situation are determined. Further more, the typical dimensions of the hot wire and the range of Reynolds number covered by numerous other investigations are considered in the numerical model. 2.2 Basic equations and non-dimensional parameters The continuity equation for an incompressible flow in two dimensions can be written in tensor notation as x i = i ( ρ u ) = 0 1, 2 i (2-1) where ρ is the density of the fluid, u i is the velocity component in the direction of the x i coordinate system. Conservation of momentum (Navier-Stokes equations) in the i th direction taken in an inertial (non-accelerating) reference frame is described by 13

28 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE t i, j = xi 1, 2 p x ij ( ρ ui ) + ( ρuiu j ) = + + Fi i τ x i (2-2) Here p is the static pressure; τ ij is the stress tensor (described below) and F i is a volume force applied to the fluid. For gas in the case of Newtonian fluid, the molecular momentum transport term (stress tensor τ ij ) is given by u = µ x u j + x i τ ij (2-3) j i where µ is the molecular viscosity and the second term on the right hand side is the effect of volume dilation. In our study, the volume force is due to buoyancy, which couples the momentum and the energy equations. Based on the Boussinesq approximation, the buoyancy force can be written as F i = ρ g β ( T T ) (2-4) i where g i is the gravitational acceleration in the direction x i ; β is the coefficient of volumetric thermal expansion; T is the local temperature of the fluid and the T is the fluid temperature at the undisturbed region of the flow. The energy equation can be written in terms of static temperature T as 14

29 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE 15 ( ) ( ) p i p i i i c x T c k x u T x T t Φ + = + ρ ρ 2 1, = i (2-5) j i i j j i x u x u x u + = Φ µ (2-6) Here k is the molecular conductivity, Φ is the viscous dissipation function Non-dimensional equations and non-dimensional parameters for steady, two-dimensional case For steady flow, taking the diameter of hot wire D as the characteristic length, the upstream incoming flow velocity at the wire location (U 0 ) as the characteristic velocity and the following parameters as the reference scales, D x x =, 0 U u u =, 2 0 U p p = ρ, ( ) ( ) = T T T T T w, = µ µ µ, = K K K the non-dimensionalised governing equations of continuity, momentum and energy can be expressed respectively as follows: Continuity Equation: ( ) 0 = i i u x 2 1, = i (2-7) Momentum Equations: ( ) ( ) 2 1 T Re Gr x u T x Re x p u x u i j i j j i i + + = µ 2 1,, = j i (2-8)

30 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE 16 Energy Equation: ( ) ( ) 1 Φ + = Re Ec x T T k x Pr Re T u x i i i i 2 1, = i (2-9) Here u i and x i are the Cartesian velocity components and coordinates, respectively; the subscript refers to the upstream condition. Dissipation function is j i i j j i x u x u x u + = Φ (2-10) ( ) [ ] ( ) ( ) = T S T T S T µ µ τ τ µ (2-11) ( ) [ ] ( ) ( ) = T S T T S T k k k τ τ (2-12) One may note that µ (the dimensionless dynamic viscosity based on Sutherland Formula (White 1991)) and k (the dimensionless thermal conductivity of fluid based on Sutherland Formula (White 1991)) express how the physical properties of fluid changes with the temperature as reflected in temperature loading of hot wire τ. (Here T T w τ, where T w is the constant wire temperature and T is the temperature of the undisturbed flow, both in absolute units (K)). S µ and S k are effective temperatures, called the Sutherland constants, which are characteristic of the gas. For air, S µ and S k

31 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE are taken as constants at 111K and 194K, respectively. Due to the high temperature loading as big as 1.8 used in hot-wire measurements, equation (2-11) and (2-12) are selected to accurately express the variations of dynamic viscosity and thermal conductivity of measured fluid with the variation of temperature. Equations (2-7), (2-8) and (2-9) reveal the dimensionless parameters with the characteristic velocity as the free stream velocity at the height of hot wire (U 0 ), and the characteristic length as the diameter of hot wire (D). These give Reynolds number: U 0 D Re = (2-13) υ Prandtl number: Pr µ C p = k (2-14) Grashof number: ( T T ) gβ D 3 Gr = w υ 2 (2-15) Eckert number: U 2 Ec = C p T w 0 (2-16) ( T ) The average Nusselt number is defined as: N u = ' 1 qd H da = A k T ( T T ) πk ( T ) w w (2-17) 17

32 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE where H is the heat flux through the closed circulation which surrounds the cylindrical hot wire and D is the hot-wire diameter Non-dimensional equations and non-dimensional parameters for unsteady case For unsteady flow, taking the diameter of hot wire D as the characteristic length, the mean upstream incoming fluctuating flow velocity at the wire location (U 0 ) as the characteristic velocity, correspondingly the characteristic time is D/U 0 and the following parameters as the reference scales, x = x D, u U p =, ρ u =, p 2 0 U 0 T = ( T T ) ( T T ) w, µ µ =, µ K = K K The non-dimensionalised governing equations of continuity, momentum and energy can be expressed respectively as follows, Continuity Equation: x i = ( u ) 0 = i i 1, 2 (2-18) Momentum Equations: t i p 1 j Gr ( u ) + u ( u ) = + µ ( T ) + T i, j = i x 1, 2 i j x j Re x i u x i Re 2 (2-19) Energy Equation: t 1 T Ec ( T ) + ( u T ) = k ( T ) + Φ x i i Re Pr x i x i Re (2-20) 18

33 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE i = 1, 2 Here u i and x i are the Cartesian velocity components and coordinates, respectively. Dissipation function is u u i j u i Φ = + x x (2-21) j i x j µ is the dimensionless dynamic viscosity defined by equation (2-11), k is the dimensionless thermal conductivity of fluid defined by equation (2-12), and t is the dimensionless time ( tu 0 ). D Equations (2-18), (2-19) and (2-20) reveal the same dimensionless parameters as that given in the previous section with the proviso that U 0 is interpreted as the mean streamwise velocity at the wire location. Accordingly, dimensionless parameters like Re, Ec and N u are taken to imply the respective mean quantities. 2.3 Boundary conditions To solve the non-linear partial differential equations (2-7), (2-8) and (2-9) for steady flow or (2-18), (2-19) and (2-20) for pulsating flow, boundary and initial conditions are needed. Any initial condition from time-dependent computations may be used. For convenience, the trivial solution (u i =0, T=0) is usually employed. (The details of the boundary conditions are given in the respective chapters.) 19

34 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE The boundary conditions for inflow and wall boundaries are specified (Dirichlet BC), and for outflow boundaries gradients are specified (Neumann BC). At inflow boundaries, U i and T, together with fluid properties are prescribed. In timedependent computations the prescribed values may also depend on time. Wall boundaries may be of pure Dirichlet type, or also of a mixed type. In the latter case, the heat flux through the wall is given instead of the temperature. The velocity components at the wall are set as zero. Outflow boundary is used to describe downstream boundary condition. It is located far downstream with respect to the region of interest of the flow and assumed that the gradients of the solution at outflow boundary are so small that it can be approximated by a zero gradient. The normal gradient of the main variables is set as, φ = 0 n (2-22) where φ stands for flow variables, such as U, V and T and n stands for the outward normal vector at the boundary. Thus, we have a closed set of equations describing the physical problem. Next, the numerical method for the equations will be outlined. 2.4 Discretization 20

35 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE The common ways for discretizing the above set of partial differential equations include finite difference, finite volumes or finite elements. In the present work, a finite volume scheme was chosen for the spatial discretization of the governing equations due to its high flexibility to handle flow problems in complex geometry. A conservative formulation of the discretization is used, thus helping to ensure the satisfaction of the global balance of conserved quantities. For the case of timedependent flows, an implicit finite difference scheme is employed for the discretization of time derivatives. After discretization, the resulting system of nonlinearly coupled equations was solved iteratively based on the SIMPLEC method. Control-volume-based technique is employed to convert the governing equations to algebraic equations. It consists of integrating the governing equations applicable to each control volume, yielding discrete equations that conserve each quantity on a control-volume basis. Discretization of the governing equations can be illustrated most easily by considering the steady-state conservation equation for transport of a scalar quantityφ. This is demonstrated by the following equation written in integral form for an arbitrary control volume V as follows: r ρφv da = Γ φ φ da + r V S φ dv (2-23) where ρ density 21

36 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE v A r Γ φ velocity vector surface area vector diffusion coefficient for φ φ gradient of φ S φ source of φ per unit volume Equation (2-23) is applied to each control volume, or cell, in the computational domain. The two-dimensional, triangular cell shown in Figure 2-1is an example of such a control volume. Discretization of Equation (2-23) on a given cell yields N N ( φ ) A S V faces faces v fφ f Af = Γφ n f + φ (2-24) f where f N faces number of faces enclosing cell φ f value of φ convected through face f v f mass flux through the face f A f area of face f ( φ ) n magnitude of φ normal to face f V cell volume The equations to be solved take the same general form as the one given above and can be applied to multi-dimensional, unstructured meshes composed of arbitrary polyhedra. 22

37 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE The computer code used stores discrete values of the scalar φ at the cell centers (see CV 0 and CV 1 in Figure 2-1). However, face values φ f are required for the convection terms in Equation (2-24) and can be interpolated from the cell center values. This is accomplished using an upwind scheme as the convection term values in a cell are mainly affected by those in the cell upstream. Upwinding means that the face value φ f is derived from quantities in the cell upstream, or upwind, relative to the direction of the normal velocity v n in Equation (2-24). We use second-order upwind to discretize convection term. The scheme is described below. Due to the diffusion term values in a cell are affected by all those values in cells surrounding it, the diffusion terms in Equation (2-24) are central-differenced and are always second-order accurate. 2.5 Second-order upwind scheme As the second-order accuracy is desired, quantities at cell faces are computed using a multi-dimensional linear reconstruction approach. In this approach, higher-order accuracy is achieved at cell faces through a Taylor series expansion of the cellcentered solution about the cell centroid. Thus as second-order upwinding is employed, the face value φ f is computed using the following expression: 23

38 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE φ f = φ + φ S (2-25) where φ and φ are the cell-centered value and its gradient in the upstream cell, and S is the displacement vector from the upstream cell centroid to the face centroid. This formulation requires the determination of the gradient φ in each cell. This gradient is computed using the divergence theorem, which in discrete form is written as N faces 1 ~ φ = φ f A (2-26) V f ~ Here the face values φ are computed by averaging φ from the two cells adjacent to the face. Finally, the gradient introduced. f φ is limited so that no new maximum or minima are 2.6 Linearized form of the discrete equation The discretized scalar transport equation (i.e. Equation (2-24) in discretized form) contains the unknown scalar variable f at the cell center as well as the unknown values in surrounding neighboring cells. This equation will, in general, be non-linear with respect to these variables. A linearized form of Equation (2-24) can be written as apφ = anbφnb + b (2-27) nb where the subscript nb refers to neighbor cells, and a P and a nb are the linearized coefficients for φ and φ nb. The number of neighbors for each cell depends on the grid topology, but will typically equal the number of faces enclosing the cell (boundary cells being the exception). 24

39 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE Similar equations can be written for each cell in the grid. This results in a set of algebraic equations with a sparse coefficient matrix. For scalar equations, a point implicit (Gauss-Seidel) linear equation solver is used to solve this linear system. 2.7 Under-relaxation Because of the nonlinearity of the equations, it is necessary to control the change of φ. This is typically achieved by under-relaxation, which reduces the change ofφ produced during each iteration. In a simple form, the new value of the variable φ within a cell depends upon the old value, φ old, the computed change in φ, φ, and the under-relaxation factor, α, as follows: φ = φ old + α φ (2-28) In all our cases, unless otherwise stated, to ensure the stability of iteration and the iteration efficiency, an under-relaxation factor of 0.7 is used. 2.8 Discretization of the momentum equation The discretization scheme described in section 2.4 for a scalar transport equation is also used to discretize the momentum equations. For example, the x-momentum equation can be obtained by setting φ = u : anbu nb + p f A i + a Pu = S (2-29) nb i stands for the scalar in u direction. If the pressure field and face mass fluxes were known, Equation (2-23) could be solved in the manner outlined in section 2.4, and a velocity field obtained. However, the 25

40 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE pressure field and face mass fluxes are not known a priori and must be obtained as a part of the solution. There are important issues with respect to the storage of pressure and the discretization of the pressure gradient term; these are addressed next. A colocated scheme is used, whereby pressure and velocity are both stored at cell centers. However, Equation (2-23) requires the value of the pressure at the face between cells CV 0 and CV 1, shown in Figure 2-1 in Discretization. Therefore, an interpolation scheme is required to compute the face values of pressure from the cell values. 2.9 Discretization of the continuity equation Equation (2-23) may be integrated over the control volume in Figure 2-1 to yield the following discrete equation N faces f J f A f = 0 (2-30) where N J f is the mass flux through face f, i.e. ρ vn. Thus, faces ρ v n A = 0 (2-31) f f is the discrete continuity equation. In the numerical schemes, the momentum and continuity equations are solved sequentially. In this sequential procedure, the continuity equation is used as an equation for pressure. However, pressure does not appear explicitly in equation (2-24) for incompressible flows, since density is not directly related to pressure. The SIMPLEC (SIMPLE-Consistent) family of algorithms is used for introducing pressure into the continuity equation. This procedure is outlined in SIMPLEC in section 2.12 below. 26

41 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE In order to proceed further, it is necessary to relate the face values of velocity vn to the stored values of velocity at the cell centers. Linear interpolation of cell-centered velocities to the face may result in unphysical checker-boarding of pressure values. The following procedure is taken to prevent checkerboarding so that unphysical results of numerical pressure values can be avoided. The face value of velocity v n is not averaged linearly; instead, momentum-weighted averaging, using weighting factors based on the a P coefficient from equation (2-23), is performed. Using this procedure, the face flux J f may be written as J f = J f + d f ( p p ) c0 c1 (2-32) where p c0 and pc 1 are the pressures within the two cells on either side of the face, and f J is the part of mass flux containing the influence of velocities in these cells (see Figure 2-1). The term d f may be written as d f 2 ρaf = (2-33) a P The term a P is the average of the momentum equation a P coefficients for the cells on either side of face f Temporal discretization The time-dependent equations must be discretized in both space and time. The spatial discretization for the time-dependent equations is identical to the steady-state case. Temporal discretization involves the integration of every term in the differential 27

42 CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE equations over a time step t. The integration of the transient terms is straightforward, as shown below. A generic expression for the time evolution of a variable φ is given by φ t = F ( φ ) (2-34) where the function F incorporates all the spatial discretization. The time derivative is discretized using second-order, backward differences; the corresponding discretization is given by n 3φ + 1 where n n 4φ + φ 2 t 1 = F ( φ ) (2-35) φ = a scalar quantity n +1 = value at the next time level, t + t n = value at the current time level, t n 1 = value at the previous time level, t t After the time derivative has been discretized, we use implicit time integration to evaluate F ( φ ) at the future time level, = + tf φ n ( ) n +1 n φ φ +1 (2-36) i n This implicit equation can be solved iteratively by initializing φ to φ and iterating the equation i n φ = 4 3φ 1 3φ n i ( ) 3 tf φ (2-37) 28