Numerical simulations of heat transfer of hot wire anemometer

57 CHAPTER 4 THE THERMAL RESPONSE OF A HOT WIRE IN NEAR WALL REGION MEASUREMENTS.... Based on the results about domain size, natural convection, viscous dissipation and temperature influ

2003

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

CONTENTS

ACKNOWLEDGEMENTS ii

CONTENTS iii

SUMMARY vi

NOMENCLATURE viii

LIST OF FIGURES xi

LIST OF TABLES xiv

CHAPTER 1 INTRODUCTION 1

1.1 Introduction to hot-wire anemometry 1

1.2 Electronic principles of constant temperature hot wire 2

1.3 Advantage of hot-wire anemometry 2

1.4 The thermal response of a hot wire in a fluctuating flow 3

1.5 Hot-wire correction under influence of wall proximity 5

1.6 Motivation and objective of the study 9

1.7 Structure of thesis 10

CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE 13

2.1 Introduction 13

2.2 Basic equations and non-dimensional parameters 13

2.3 Boundary conditions 19

2.4 Discretization 20

2.5 Second-order upwind scheme 23

2.6 Linearized form of the discrete equation 24

2.7 Under-relaxation 25

2.8 Discretization of the momentum equation 25

2.9 Discretization of the continuity equation 26

2.10 Temporal discretization 27

2.11 Pressure-velocity coupling 29

2.12 Solution method 32

2.13 Convergence criteria 33

CHAPTER 3 MOMENTUM AND HEAT TRANSFER FROM CYLINDER IN FREESTREAM STEADY LAMINAR FLOW 37

3.1 Introduction 37

3.2 Governing equations and boundary conditions 40

3.3 Numerical calculations 42

3.4 Numerical results and discussions 43

3.5 Conclusions 57

CHAPTER 4 THE THERMAL RESPONSE OF A HOT WIRE IN NEAR WALL REGION MEASUREMENTS 70

4.1 Introduction 70

4.2 Literature survey on hot-wire heat transfer near a wall 71

4.3 Governing equations and boundary conditions 76

4.4 Numerical calculations 77

4.5 Grid distribution, domain size independent check and numerical accuracy 77 4.6 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 78

4.7 Main parameters affecting the near-wall hot-wire correction factor: 84

4.8 Comparison with the experiment based on Y + and h 0 86

4.9 Near-wall hot-wire correction curves based on Re and h 0 92

4.10 Simulation results based on Y + and U + 94

4.11 On the critical Y c + and h 0 96

4.12On the velocity correction factor, C ui, based on temperature loading of 1.1 ……… 99

4.13 Conclusions 101

CHAPTER 5 THE THERMAL RESPONSE OF A HOT WIRE IN A FLUCTUATING FREESTREAM FLOW 142

5.1 Introduction 142

5.2 Governing equations and boundary conditions 145

5.3 Numerical calculations 146

5.4 Heat transfer characteristics of the hot wire in free stream fluctuating flow ……….147

5.5 Conclusions 155

CHAPTER 6 SUMMARY OF CONCLUSIONS 170

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 near-wall measurement correction curve for the hot wire having been calibrated under free-stream 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 + hUτ

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

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 0.025≤ Re≤ 200, 0≤ S c ≤ 0.32 (S c is non-

dimensional 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

C p Specific heat of fluid at a constant pressure

C u Correction factor =

meas U

U0

C ua Correction factor for adiabatic wall case

C ui Correction factor for isothermal wall case

D Diameter of hot wire

2 0

f Pulsating frequency of the incoming flow

g Gravitational acceleration

Gr Grashof number = ( )

2 3

∞

∞

− υ

w T g

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

k Thermal conductivity of fluid

L Hot wire length

qD

'1

N u0 Nusselt number at free stream

N um Measured Nusselt number

N uf Nusselt number based on film temperature

Pe Peclet number = RePr

0

fD f

f

S c

t Time

T Temperature

U 0 The true upstream incoming flow velocity at the location of hot wire center

h0

Y c + Critical Y +, beyond which wall influence can be neglected

Volume coefficient of expansion

τs Period of the vortex shedding

i,j Denotes Cartesian coordinate directions

∞ At the inlet of computational domain

LIST OF FIGURES

Figure 1-1 Basic circuit for constant temperature thermal anemometer 12

Figure 2-1 Control volume used to illustrate discretization of a scalar transport equation 35

Figure 2-2 Overview of the solution method 36

Figure 3-1 The schematic drawing of computing region 60

Figure 3-2 Grid distribution 61

Figure 3-3 Nusselt number relative change with the Eckert number 62

Figure 3-4 Nusselt number relative change with the temperature loading 63

Figure 3-5 Heat transfer from a circular cylinder in uniform flow 64

Figure 3-6 N u variation with Re for temperature loading 1.8 65

Figure 4-1 The schematic drawing of computing region 103

Figure 4-2 Grid distribution, region A surrounding the hot wire is further depicted in Figure 4-3 104

Figure 4-3 The grid distribution around the hot wire 105

Figure 4-4 C u variation with h 0 for Y +=5.0 106

Figure 4-5 C u variation with Y + and h 0 107

Figure 4-6 C u variation with Y + and h 0 108

Figure 4-7 C u variation with Y + and h 0 for isothermal wall 109

Figure 4-8 C u variation with Y + and h 0 110

Figure 4-9 Contour of static temperature distribution for Y +=2.0, h 0=15 (C ua=1.21) and adiabatic wall 111

Figure 4-10 Contour of static temperature distribution for Y +=4.0, h 0=15 (C ua=0.94) and adiabatic wall 112

Figure 4-11 Contour of static temperature distribution for Y +=2.0, h 0=15 (C ui=0.52)

and isothermal wall 113

Figure 4-12 Contour of static temperature distribution for Y +=4.0, h 0=15 (C ui=0.91) and isothermal wall 114

Figure 4-13 C u variation with Re and h 0 115

Figure 4-14 C ua variation with Re and h 0 116

Figure 4-15 C ui variation with Re and h 0 117

Figure 4-16 C u variation with Re and h 0 118

Figure 4-17 C u variation with Re and h 0 (0.1m/s U 1.0m/s) 119

Figure 4-18 C u variation with Re and h 0 120

Figure 4-19 U m+ variation with Y + and h 0 121

Figure 4-20 U m+ variation with Y + and h 0 (=5, 90 and 150) 122

Figure 4-21 U m+ variation with Y + and h 0 for isothermal wall 123

Figure 4-22 C u variation with Y + and h 0 124

Figure 4-23 Critical Y + versus h 0 for adiabatic wall conditions 125

Figure 4-24 Critical Y + versus h 0 for isothermal wall conditions 126

Figure 4-25 Critical Y + versus h 0 127

Figure 4-26 Variation of C ui with Re and h 0 for = 1.1 and 1.8 128

Figure 4-27 Variation of C u with Re and h 0 for =1.1 in the low Re range 129

Figure 4-28 Variation of ∆C u with Re between temperature loading 1.1 and 1.8 130

Figure 5-1 A schematic drawing of the computed region 157

Figure 5-2 Traces of drag coefficient (C d) on cylinder for Re=150 and S c=0.0 158

Figure 5-3 Traces of surface Nusselt number on cylinder for Re=150 and S c=0.0 159

Figure 5-4 Traces of drag coefficient on cylinder for Re=150 and S c=0.16 160

Figure 5-5 Traces of surface Nusselt number on cylinder for Re=150 and S c=0.16 161

Figure 5-6 Variation of Nusselt number ratio with Re at various S c and A=0.6 162

Figure 5-7 Variation of Nusselt number ratio with Re in the low Re range at various S c and A=0.6 163

Figure 5-8 Variation of Nusselt number ratio with S c at various amplitude A and Re=0.1 164

Figure 5-9 Variation of Nusselt number ratio with S c at various amplitude A and Re=0.9 165

Figure 5-10 Variation of Nusselt number ratio with S c at various amplitude A and Re=2.5 166

Figure 5-11 Variation of N u with frequency 167

Figure 5-12 Velocity variation with frequency for freestream flow 168

Figure 5-13 Statistic quantities versus frequency for mean flow Re = 0.9 169

LIST OF TABLES

Table 3-1: C d & N u change with the grid number around the cylinder 66

Table 3-2: C d & N u change with increase of the domain size for Reynolds number 1.5×10-2 67

Table 3-3: C d & N u of cylinder in free stream for Re =4.0×10-3 and =1.003 68

Table 3-4: Values of constants for onset criterion of natural convection effects from Lange (1997) 69

Table 4-1: C u for Y + = 1.0, =1.8 for adiabatic wall 131

Table 4-2: Nusselt number for Re = 5.0, =1.3 132

Table 4-3: Nusselt number for Re = 0.007, =1.8 133

Table 4-4: Nusselt number for Re =5.0, =1.8, Ec=0.01 134

Table 4-5: Nusselt number change with increase the domain size for Re =1.5×10-2 135

Table 4-6: N uforY += 0.5, U = 1.0 m/s, Re = 0.15, =1.8, N u0 = 0.495 for adiabatic wall 136

Table 4-7: N u for Y + = 0.5, U = 1.0 m/s, Re = 0.15, =1.8, N u0 = 0.495 for adiabatic wall 137

Table 4-8: N uforY += 0.5, U = 1.0 m/s, Re = 0.15, =1.8, N u0 = 0.495 for isothermal wall 138

Table 4-9: N u for Y + = 0.5, U = 1.0 m/s, Re = 0.15, =1.8, N u0 = 0.495 for isothermal wall 139

Table 4-10: The ratio of the thermal conductivity, k w, of wall materials to the thermal conductivity of air, k (=0.02559 Wm−1K−1) from Turan et al (1987) 140

Table 4-11: Critical Y c + values for less than 5% error in near-wall cases based on h 0 .141

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

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

servo-amplifer 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

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

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

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

1.5 Hot-wire correction under influence of wall proximity

1.5.1 The need for hot-wire correction in near-wall configuration

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

1.5.2 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

velocity and distance from the wall are normalized by the wall parameters of U and /

U , 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

1.5.3 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

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

non-conducting 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

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 εf, εw (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

1.6.1 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

CHAPTER 1 INTRODUCTION

1.6.2 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

1.6.3 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

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.

CHAPTER 1 INTRODUCTION

Probe

Servo Amplifier

Bridge Voltage

Error Voltage

Figure 1-1 Basic circuit for constant temperature thermal anemometer

CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE

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

CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE

i

ij i

j i i

x x

p u

u x

u

∂

∂ +

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

( ) ( )

p i p i i

T c

k x T u x

T

t

Φ+

i

x

u x

u x

∂

∂

=

Here k is the molecular conductivity, Φ is the viscous dissipation function

2.2.1 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,

T T T

the non-dimensionalised governing equations of continuity, momentum and energy can

be expressed respectively as follows:

Gr x

u T x

Re x

p u

x

u

i j i

j j

1

i

(2-8)

CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE

T T k x Pr Re T

Here u i and x i are the Cartesian velocity components and coordinates, respectively; the

subscript ∞ refers to the upstream condition

j j

i

x

u x

u x

−

+ +

T S T

µ µ

τ

τµ

0 1 1

1 0

1 1

* 2

−

++

T

S T

k

k k

0.11

10

.11

* 2

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 (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)) Sµand S k are effective temperatures, called the Sutherland constants, which are characteristic of the gas For air, Sµand S k

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

qD A

N

w w

'1

(2-17)

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

2.2.2 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,

T T T

Gr x

u T x

Re x

p u

x u

u

j i

j j

i i

∂

2 ,

T T k x Pr Re T

u x

T

j j

i

x

u x

u x

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 (

D

tU 0

)

Equations (2-18), (2-19) and (2-20) reveal the same dimensionless parameters as that

given in the previous section 2.2.1 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

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.)

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 and T, together with fluid properties are prescribed In time- i

dependent 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,

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

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

time-dependent flows, an implicit finite difference scheme is employed for the discretization

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:

V S dV A

d A

CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE

v velocity vector

Ar

surface area vector

φ 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

A

N f

n f

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

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 φf 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 φf is derived from quantities in the cell upstream,

or ‘upwind’, relative to the direction of the normal velocity v in Equation (2-24) We n

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 cell-

centered solution about the cell centroid Thus as second-order upwinding is employed, the face value φf is computed using the following expression:

CHAPTER 2 NUMERICAL METHOD AND SOLUTION PROCEDURE

φ are computed by averaging φ from the two cells adjacent to

the face Finally, the gradient ∇ is limited so that no new maximum or minima are φ

introduced

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

=

nb

nb nb

where the subscript nb refers to neighbor cells, and a P and a are the linearized nb

coefficients for φ and φnb

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)

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, φold, the computed change in φ, ∆ , and the φ

under-relaxation factor, α , as follows:

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:

=

nb

f nb

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

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 co-located 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

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