Master’s Thesis Academic Year 2016

Revisiting Hot-Wire AnemometryClose to Solid Walls

Yuta IKEYA(Student ID: 81418734)

Supervisors: Koji FUKAGATA, Ramis ORLU,P. Henrik ALFREDSSON

March 2017

School of Science for Open and Environmental SystemsGraduate school of Science and Technology

Keio University

Fluid Physics Laboratory, Dept. MechanicsKTH Royal Institute of Technology

論文要旨

乱流計測に用いられる熱線流速計は，壁面近傍での計測の際，熱線から壁面への熱損失

が原因で速度を過大評価することが知られている．この現象の壁面材質や過熱比，熱線部

分の寸法といったパラメータへの依存性について，これまで数多くの研究がなされてきた．

これらの先行研究のあいだで，パラメータ依存性に関する見解は概ね一致しているものの，

パラメータの変化が計測値の変動に及ぼす影響は研究されておらず，また壁面への熱損失

のメカニズムそのものについても未だ十分な説明がなされていない．そこで本研究では，

この分野における研究に新たな知見をもたらすことを目的とし，計測される速度の平均値

および変動のパラメータ依存性を調査する．加えて，熱線・壁面間の熱交換を説明する理

論モデルを提示する．

始めに，強制対流と自然対流の影響を分離して考察するため，無風条件下での測定およ

び風洞を用いた測定を行う．本実験から得られたパラメータ依存性は多くの先行研究の示

す傾向と一致しており，熱伝導係数の大きい壁面，高過熱比，およびセンサー表面積の増

加により，熱線流速計からの出力値は大きくなることが観測された．なお，異なる過熱比

の下での熱伝導・自然対流の出力値は無次元化により，一意に定まることが示唆された．

また，出力値の変動は壁面熱伝導率の上昇および過熱比の上昇に伴い，実際より低く見積

もられることがわかった．

次に OpenFOAM を用いた二次元定常計算を通して，壁面材質・過熱比に関する熱損失

のパラメータ依存性を調査する．計算結果は実験結果と定性的に一致することが確認され

たが，乱流を議論する上で用いられる壁指標は，異なる熱線高さの下での出力を一意にス

ケーリングできないことがわかった．

最後に，本論文は熱対流と熱伝導の重ね合わせにより，熱線・固体壁面間の熱交換を再

現する理論モデルを提示する．実用的な使用に関しては課題が残るものの，モデルに含ま

れる係数を経験的に設定することにより，実験ならびに数値計算で見られた傾向を定性的

に再現することができることが確認された．

Thesis Abstract

A well-known problem of hot-wire anemometry (HWA), is the “wall effect”, namely theoverestimation of the measured velocity near a wall. The overestimation occurs due toadditional heat loss from the heated wire-sensor to the wall. The extra heat loss dependson parameters such as the heat conductivity of the wall material, the overheat ratio ofthe wire, and the sensor geometry. This problem has been studied for quite some timeand there are several suggestions with regard to the effect of these parameters for meanflow corrections, however the effect on measurements of turbulent fluctuation has notbeen investigated. The present work aims at providing further insight on this topic, byelucidating how these parameters affect measurements of both the mean and fluctuatingvelocity. Furthermore, the present study proposes a theoretical model on the total heattransfer from hot-wire sensor to explain the phenomenon.

In the experimental part of the study, the measurements under both no flow and flowconditions are carried out to consider natural convection and forced convection separately.The results showed that the effect of the parameters is consistent with what is agreedwidely: higher wall conductivity, higher overheat ratio, and larger wire exposed area leadto higher output from an anemometer. On the other hand, it is observed that the conduc-tion under natural convection can be scaled with the overheat ratio. Velocity fluctuationsare found to decrease by employing higher overheat ratio and for walls with higher heatconductivity.

In the numerical part of the study, a two-dimensional steady calculation using Open-FOAM is performed and the parameter dependency with respect to the overheat ratio andwall heat conductivity is investigated. The results qualitatively agree with the experi-mental results. Moreover, the inner scaling commonly employed in wall-turbulence isfound to be inadequate to resolve the wall effect of HWA when various sensor heights areconcerned.

Lastly, a theoretical model on the total heat transfer from the wire close to solid wallsis established based on a superposition of the convection and the conduction contributions.The proposed model with the empirically determined coefficients is found to be capableof capturing the qualitative behaviours found in the experiment and numerical analysis,however for more practical use it leaves several issues to be further analysed.

Acknowledgments

This master’s thesis project is a collaborative work between Fukagata Lab. in Keio Uni-versity, Japan and the Linne Flow Centre in KTH Royal Institute of Technology, Sweden.The experimental part of the research was performed at KTH, where the facilities of theFluid Physics Laboratory were used. Following numerical analysis and further discussionwas done at Keio University.

I greatly thank my supervisors, Koji Fukagata, Ramis Orlu and P. Henrik Alfredssonfor enlightening me on the knowledge in fluid dynamics and other physics, and for lead-ing the present thesis to completion. The present work could not have been done withouttheir help. Frequent discussions with them have got me interested more in the field offluid mechanics and motivated me for this thesis work.

The experience that I learned from both universities through my double degree programis priceless, thereupon, I also appreciate the support and advice from members of Fuka-gata Lab. and Linne Flow Centre. Sharing the knowledge with them working in the samefield has inspired me a lot to make the present work richer in terms of its background.Especially, discussion with Yusuke Kondo, Kaoruko Eto and Rintaro Kaneko who stud-ied together with me in the same group in Fukagata Lab has taught me ideas related withthe present topic, thereby I would like to thank them. A weekly seminar with the peoplefrom Obi Lab. and Ando Lab. in Keio University was also very helpful for me to updatemy work. I appreciate the comments and advice from the professors and students there.For Professor Shinosuke Obi, particularly, I show my gratitude for supporting me both inacademic respect and in my double degree program.

Above all, I sincerely appreciate the steadfast support and love from my parents andbrother. I am very happy to have such a family standing by me all the time.

Contents

List of Figures iii

List of Tables vi

Nomenclature vii

1 Introduction 11.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Objective of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theoretical Background and Preliminaries 82.1 Hot-Wire Anemometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Physical principle . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Mode of operation . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Calibration relation . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Introduction of turbulence quantities . . . . . . . . . . . . . . . . . . . . 122.2.1 Statical analysis of velocity fluctuations . . . . . . . . . . . . . . 122.2.2 Inner-scaling in wall-bounded turbulence . . . . . . . . . . . . . 14

3 Experimental Part 153.1 Measurement apparatus and procedures . . . . . . . . . . . . . . . . . . 15

3.1.1 Probe manufacturing . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 Natural convection measurement . . . . . . . . . . . . . . . . . . 163.1.3 Wind-tunnel measurements . . . . . . . . . . . . . . . . . . . . . 18

3.2 Experimental results and discussion . . . . . . . . . . . . . . . . . . . . 213.2.1 HWA output in quiescent air . . . . . . . . . . . . . . . . . . . . 213.2.2 HWA output in laminar and turbulent flow . . . . . . . . . . . . . 22

4 Numerical Analysis 334.1 Computational setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Calculation procedure . . . . . . . . . . . . . . . . . . . . . . . 344.1.3 Computational domain and boundary conditions . . . . . . . . . 354.1.4 Post processing of the simulation results . . . . . . . . . . . . . . 37

4.2 Numerical calibration of HWA . . . . . . . . . . . . . . . . . . . . . . . 384.3 Numerical errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

i

4.3.1 Truncation error . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.2 Convergence error . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Validation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4.1 Validity of flow field around a circular cylinder . . . . . . . . . . 454.4.2 Simulation of natural convection in a closed cavity . . . . . . . . 45

4.5 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . 49

5 Theoretical Model on Wire-Wall Heat Transfer in a Fluid Flow 595.1 Components of the overall heat transfer . . . . . . . . . . . . . . . . . . 595.2 Revisiting the calibration curve . . . . . . . . . . . . . . . . . . . . . . . 605.3 The effect of heat conduction . . . . . . . . . . . . . . . . . . . . . . . . 615.4 Final form of the model on the wire-wall heat transfer and its possibility

for generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.5 Simulation of the fluctuating output . . . . . . . . . . . . . . . . . . . . 685.6 Possible issues of the proposed model . . . . . . . . . . . . . . . . . . . 70

6 Conclusions 72

Reference 74

Appendices 79

ii

List of Figures

2.1 A hot-wire heated by a current. . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 Components of the in-house built hot-wire probes. . . . . . . . . . . . . . 153.2 Examples of the in-house built hot-wire probes. . . . . . . . . . . . . . . 163.3 The experimental apparatus for the natural convection measurement. . . . 183.4 A close-up view of the prongs and the wall. . . . . . . . . . . . . . . . . 193.5 Wall-sensor arrangement variations for natural convection measurement. . 193.6 A schematic of the MTL windtunnel. . . . . . . . . . . . . . . . . . . . . 213.7 The experimental setup for the wind-tunnel measurement. . . . . . . . . . 213.8 HWA output in quiescent air on different wall materials. . . . . . . . . . . 243.9 HWA output in quiescent air at different resistant overheat ratios. . . . . . 253.10 HWA output from probes with different wire lengths in quiescent air. . . . 253.11 HWA output from probes at different wall-sensor arrangement. . . . . . . 253.12 Inner-scaled velocity profile in a laminar boundary layer (Reθ ≈ 400). . . 263.13 Inner-scaled streamwise velocity profiles in a turbulent boundary layer

(Reθ ≈ 950) on different material surfaces. . . . . . . . . . . . . . . . . . 263.14 Inner-scaled streamwise velocity profiles in a turbulent boundary layer

(Reθ ≈ 950) at different overheat ratios. . . . . . . . . . . . . . . . . . . 273.15 Inner-scaled streamwise rms profiles in a turbulent boundary layer (Reθ ≈

950) on different material surfaces. . . . . . . . . . . . . . . . . . . . . . 273.16 Inner-scaled streamwise rms profiles in a turbulent boundary layer (Reθ ≈

950) at different overheat ratios. . . . . . . . . . . . . . . . . . . . . . . 283.17 Diagnostic plot of the HWA output in a turbulent boundary layer (Reθ ≈

950) on different material surfaces. . . . . . . . . . . . . . . . . . . . . . 283.18 Diagnostic plot of the HWA output in a turbulent boundary layer (Reθ ≈

950) at different overheat ratios. . . . . . . . . . . . . . . . . . . . . . . 283.19 Comparison of inner-scaled velocity PDF in a turbulent boundary layer

(Reθ ≈ 950) on different material surfaces. . . . . . . . . . . . . . . . . 293.20 Comparison of inner-scaled velocity PDF in a turbulent boundary layer

(Reθ ≈ 950) at different overheat ratios. . . . . . . . . . . . . . . . . . . 293.21 Third-order moment of the measured turbulent boundary layer (Reθ ≈

950) on different material surfaces. . . . . . . . . . . . . . . . . . . . . . 303.22 Third-order moment of the measured turbulent boundary layer (Reθ ≈

950) at different overheat ratios. . . . . . . . . . . . . . . . . . . . . . . 303.23 Fourth-order moment of the measured turbulent boundary layer (Reθ ≈

950) on different material surfaces. . . . . . . . . . . . . . . . . . . . . . 303.24 Fourth-order moment of the measured turbulent boundary layer (Reθ ≈

950) at different overheat ratios. . . . . . . . . . . . . . . . . . . . . . . 31

iii

3.25 Comparison of power spectra of the measured turbulent boundary layer(Reθ ≈ 950) on different material surfaces. . . . . . . . . . . . . . . . . . 31

3.26 Comparison of power spectra of the measured turbulent boundary layer(Reθ ≈ 950) at different overheat ratios. . . . . . . . . . . . . . . . . . . 32

4.1 Computational domain and boundary conditions. . . . . . . . . . . . . . 364.2 Detail of the mesh around the cylinder. . . . . . . . . . . . . . . . . . . . 374.3 Computational domain and boundary conditions for the simulations in

freestream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Heat transfer from the heated cylinder at aT = 0.27 in freestream and

compared with results from the literature. . . . . . . . . . . . . . . . . . 404.5 Heat transfer from the heated cylinder at aT = 0.96 in freestream and

compared with results from the literature. . . . . . . . . . . . . . . . . . 404.6 Local Nusselt number distribution along the wire surface at different tem-

perature overheat ratios in freestream. . . . . . . . . . . . . . . . . . . . 414.7 Local temperature distribution around the wire in freestream. . . . . . . . 424.8 The grid resolution dependency of the calculated result investigated at

aT = 0.27 on aluminum with the inlet velocity gradient of S = 10. Thesensor is located at yw = 100/d. . . . . . . . . . . . . . . . . . . . . . . 44

4.9 The iteration dependency of the calculated result investigated at aT = 0.27on aluminum with the inlet velocity gradient of S = 10. The sensor islocated at yw = 100/d. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.10 The calculated drag coefficient of a circular cylinder placed in freestreamcompared with the literature. . . . . . . . . . . . . . . . . . . . . . . . . 45

4.11 Computational domain for the simulation of the natural convection in asquare cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.12 Heat transfer from the heated cylinder at aT = 0.27 on an aluminum wall. 504.13 Heat transfer from the heated cylinder at aT = 0.27 on a Plexiglas wall. . . 514.14 Heat transfer from the heated cylinder at different overheat ratios and walls. 514.15 The measured velocity by the hot-wire at aT = 0.27 on an aluminum wall. 524.16 The measured velocity by the hot-wire at aT = 0.27 on a Plexiglas wall. . 524.17 Comparison of the measured velocity of the hot-wire at different overheat

ratios and walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.18 Local Nusselt number distribution along the wire surface at aT = 0.27 for

various wire hights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.19 Local Nusselt number distribution along the wire surface at aT = 0.96 for

the wire hight y∗w = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.20 Local temperature distribution around the wire at aT = 0.27 at the location

of y∗w = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.21 Local temperature distribution around the wire at aT = 0.27 at the location

of y∗w = 300. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.22 Local temperature distribution around the wire at aT = 0.96 at the location

of y∗w = 100 on an aluminum wall. . . . . . . . . . . . . . . . . . . . . . 58

5.1 Modeled heat transfer due to heat convection at aT = 0.27 in the freestream. 625.2 Modeled heat transfer due to heat convection at aT = 0.96 in the freestream. 62

iv

5.3 A long cylinder with the diameter d, the length l and the surface tem-perature T1 located yw away from an unbounded isothermal wall with thetemperature T2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 Heat transfer due to heat conduction at aT = 0.27 as a function of thesensor location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.5 Local temperature distribution around the wire at aT = 0.27 at the locationof y∗w = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.6 The modeled heat transfer from the wire at aT = 0.27 on an aluminumwall at y∗w = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.7 The modeled heat transfer from the wire at aT = 0.96 on an aluminumwall at y∗w = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.8 The hypothetical one-dimensional temperature distribution in the regions. 685.9 The modeled heat transfer from the wire at different overheat ratios on

various walls at y∗w = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . 685.10 The modeled fluctuating output from HWA at different overheat ratios on

aluminum at y∗w = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.11 The modeled turbulence intensity output from HWA at different overheat

ratios on aluminum at y∗w = 100. . . . . . . . . . . . . . . . . . . . . . . 70

B.1 A long cylinder with the diameter d, the length l and the surface tem-perature T1 located yw away from a symmetrically placed infinitely longrectangular flat plate with the temperature T2. . . . . . . . . . . . . . . . 82

v

List of Tables

3.1 Properties of wires used in the experiment. . . . . . . . . . . . . . . . . . 163.2 Properties of the in-house built probes . . . . . . . . . . . . . . . . . . . 17

4.1 Properties of meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Calculated result of natural convective flow in a square cavity. . . . . . . 48

vi

Nomenclature

Symbol Description

Roman alphabetsaR Resistance overheat ratioaT Temperature overheat ratioAkl, Bkl, nkl Calibration constants in King’s lawAmk, Bmk, nmk Calibration constants in modified King’s lawB Probability density function (PDF)c The speed of soundcp Specific heat at constant temperaturecw Specific heat of a wireC0, C1, C2, C3 Coefficients in polynomial relationsCD Drag coefficientd Diameter of a wire or a circular cylinderE Electrical potential, voltage output from an anemometerE0 Voltage output from an anemometer at zero velocityEc Eckert numberf Frequencyf1, f2 Functions of the convection part and the conduction part in the modelFrad Radiation shape factorFd Drag force per unit length of a wireg, g Gravitational accelerationGr Grashof numberGr∞ Grashof number evaluated at the wall-remote temperatureh Convective heat transfer coefficientHcond Conduction shape factorI Electrical currentk Thermal conductivityKn Knudsen numberl Length of a wire or a circular cylinderL Thickness of the solid wallM Moment of the velocity fluctuationmw Mass of a wireMa Mach numberN The number of samplesncal1, ncal2, ncal3 Numerical calibration coefficientsnconv Coefficient for modeled heat convection

vii

Symbol Description

ncond1, ncond2, ncond3 Coefficients for the modeled heat conductionNu Nusselt numberNutotal Modeled total Nusselt numberP Heating power applied to a wirePuu Power spectral density of streamwise velocity fluctuationPr Prandtl numberQ Internal energy of a sensorq Heat fluxr Polar coordinate in radial directionR0 Cold resistance of a wireRw Electrical resistance of a wireRe Reynolds numberReθ Reynolds number based on the momentum thicknessRe∞ Reynolds number evaluated at the wall-remote temperatureS Inlet velocity gradientT TemperatureT0 Target temperature of measurementsTw Wire surface temperatureu Instantaneous streamwise velocityurms RMS velocityuτ Friction velocityU, V Mean velocity in the streamwise and the wall-normal directionW Total heat energy loss from a wirex, y Cartesian coordinates in streamwise, wall-normal directionsyw Height of a wire or a circular cylinder

Greek symbolsα Temperature coefficient of electrical resistance of wires∆ Difference of valuesθ Momentum thicknessλ Molecular pathµ Dynamic viscosityν Kinematic viscosityρ Densityϕ Polar coordinate in angular direction

Subscriptscond Value originated from forced conductionconv Value originated from forced convectionf Value evaluated at the film temperaturefc Value originated from forced convectionmax Maximum of the valuemin Minimum of the valuenc Value originated from natural convectionrad Value originated from radiation

viii

Symbol Description

∞ Value at the wall-remote region in air

Superscripts′ Fluctuation of the value+ Turbulence inner-scaled value∗ Scaled value according to numerical setup

Other symbols˙ Value per unit time

Time-average of the value

ix

Chapter 1

Introduction

1.1 Background and motivation

Hot-wire anemometry (HWA) has been the most widely used laboratory method to mea-

sure local fluid velocities in experimental fluid mechanics and it was the first technique

which enabled the study of turbulent fluctuations quantitatively. Furthermore, it was the

only method capable of measuring high frequency and amplitude velocity fluctuations

with a high spatial resolution and has been dominant in the experimental area until the

relatively new techniques such as laser Doppler velocimetry (LDV) and particle image

velocimetry (PIV) were developed.

Especially, for measurements of wall-bounded turbulent flows, HWA is prominently in

use. The accurate measurement of turbulent flows in the near-wall region is very important

for several reasons, e.g. the velocity gradient in the wall proximity is used for calculating

shear stress, and a large part of the turbulent energy is produced in this region. The seeding

particles used in LDV and PIV often tend to rotate and lift off due to large velocity gradient

in this region, which results in data with poor frequency resolution (Chew et al., 1998).

The reflection of the laser from the wall produces background noise in the acquired data,

in addition to the difficulty of seeding the area close to the wall.

However, one well-known major drawback in HWA is that a probe calibrated in the

wall-remote region registers a seemingly higher velocity in the near-wall region, which

is known as the wall effect. This is thought to be due to additional heat losses because

of heat transfer between the hot-wire itself and the wall, even inside the wall, and/or

1

distortion of the flow field through the approaching probe. The wall effect becomes a

problem especially when one wants to determine the friction velocity from the velocity

gradient at the wall, which is important for the scaling of wall-bounded turbulent flows.

Although the erroneous velocity reading has been a matter of debate for more than

several decades and there is a vast amount of literature on it, their conclusions are not

necessarily consistent and there are still many points left unclear. Differences in their

experimental conditions and other known problems of HWA such as spatial resolution

issues (see e.g. Hutchins et al., 2009) might have biased those results, thus it is difficult

to compare them in a fair manner. In addition, the previous studies are mostly concerned

about mean velocity measurements and there is little information about the wall effect on

turbulence statistics.

1.2 Literature review

Reviewing available literature in this topic gives better understanding of what is widely

accepted and what is not clear regarding the wall effect of HWA. This section summarizes

the conclusions of some representative previous studies in order to motivate the goal of

the present work.

Wills (1962) studied the reading of a hot-wire anemometer in a known velocity distri-

bution in a well-defined laminar channel flow and proposed a correction for the velocity

reading, in which he introduced an empirically determined heat-loss term and derived a

correction based on the study by Collis & Williams (1959). The heat-loss term is ex-

pressed as a function of the ratio of the distance between the wire and the wall to the

radius of the wire yw/d as a new parameter, besides the Reynolds number Re and tem-

perature loading Tw/T∞ as parameters being considered already by Collis & Williams,

i.e.

Nu = f(

Tw

T∞,

yw

d, Re

). (1.1)

Although his correction was devised for laminar flow, applying half the value of the lam-

inar correction to turbulent flows is proposed without rigorous physical explanation.

2

The experimental findings of Oka & Kostic (1972) and Hebbar (1980) in the viscous

sublayer of turbulent boundary layers, on the other hand, indicated that the corrections for

different friction velocities uτ could fall on one curve of ∆u/uτ = f (y+) when parameters

are normalized in wall coordinates y+ (to be introduced in Chapter 2) with ∆u denoting

the difference between the measured velocity and the theoretical velocity. Janke (1987)

found that the corrections for turbulent and laminar flows are the same under the same

wall-shear stress, which implies that the findings by Oka and Kostic could be extended to

laminar boundary layers, in contrast to the findings of Wills (1962).

The relatively new study by Shi et al. (2002) investigated the heat exchange process

between the hot-wire and the wall by numerical analysis and suggested the need for a

negative correction, i.e. the lower apparent velocity for the poorly conducting wall. The

authors concluded that this is due to thermal feedback from the remaining heat in the

poorly conducting walls, which occurs when the sensor height is in a certain range.

There are many studies about the effect of wall properties on the velocity misreading.

Singh & Shaw (1972) and Alcaraz & Mathieu (1975) expressed the idea that the char-

acteristic of the wall would not have significant influence on the corrections; however,

later studies show the opposite. An experimental study by Polyakov & Shindin (1978) in

which steel, copper and textolite were used shows a dependence of the velocity deviation

on the wall material. It was shown in this study that highly-conducting wall materials,

i.e. steel and copper, register higher apparent velocities than poorly conducting material,

e.g. textolite. This wall material dependency has been widely validated by later studies.

Bhatia et al. (1982) conducted a numerical study for two kinds of walls, which repre-

sent ideally conducting and nonconducting materials and concluded that the correction is

required for conducting walls but not for non-conducting walls. Although their experi-

mental results on a plastic material (PVC) showed deviations from the theoretical values

in the proximity of the wall, they concluded that it was due to distortions of the velocity

field. Later, Durst & Zanoun (2002) expressed in their experimental study that there is no

3

wall material available which does not require any corrections for the heat loss.

More recently, Shi et al. (2003) took the wall thickness and the flow condition below

the wall as new parameters into account and conducted numerical analysis and concluded

that these parameters have influence on hot-wire reading especially when the wall mate-

rial is poorly conducting or heat insulating. This conclusion was experimentally validated

by a more recent study of Zanoun et al. (2009).

Apart from the aforementioned influences of the flow field and wall material, many stud-

ies have pointed out that the geometry and properties of the probe itself can be important

parameters for the velocity deviation when approaching a wall. Zemskaya et al. (1979)

found that the correction by Wills fails when different diameters for the hot-wire were

employed. In their study, another empirical term taking the wire diameter into account

was introduced. The effect of the wire diameter was validated through several later works

(Chew et al., 1998; Durst et al., 2001) and it is observed that the error in the velocity

increases as the diameter increases. The effect of the overheat ratio was also studied

by many researchers. Krishnamoorthy et al. (1985) concluded that the error in velocity

reading increases with increasing overheat ratio. Later, an experimental results by Za-

noun et al. (2009) showed the same qualitative behavior. Additionally, Chew et al. (1995)

implied the effect of the wire geometry l/d, i.e. the velocity variance due to different over-

heat ratios decreases as l/d becomes larger, which was repeated by several later studies

such as Durst et al. (2001) and Durst & Zanoun (2002).

Researchers have attempted to clarify the principle of the additional heat loss. The effects

of the flow distortion between the wire and the wall, the heat conduction and convection

are thought to be combined together; furthermore the existence of prongs can also affect

the results. To understand the principle of the phenomenon, the contribution from each

factor to the total heat loss and relation among them have to be investigated.

The flow contraction between the wire and the wall is thought to be one of the main

4

reasons (Chew & Shi, 1993; Bhatia et al., 1982); however the numerical study by Lange

et al. (1999) concluded that the velocity interference between the wire and the wall does

not contribute to the heat loss as much as the heat conduction does. In the study by

Durst & Zanoun (2002), measurements in the wall proximity without flow are presented,

by which the effect of the forced convection and the flow interference can be neglected.

They concluded that the heat conduction is the major effect of the erroneous velocity read-

ing.

As described above, many factors possibly affect the error in velocity reading in the prox-

imity of the wall, and the results and the views of the different researchers and studies

are not necessarily consistent. The discrepancy of the wire distance to the wall where

the additional heat loss starts to be observed, namely“critical distance”, also takes part in

the variance of the data. The critical distance is thought to be dependent on the afore-

mentioned parameters of the erroneous velocity (Chew et al., 1998) and it is difficult to

compare the results among different researches without bias since the information about

the experimental setup is often not complete in early works. Nevertheless, it is consistent

that the error occurs in the viscous sublayer, i.e. y+ . 5 in most of the previous study.

In experimental studies, factors such as the way to determine the distance between the

probe and the wall, the determination of the friction velocity, and how the velocity-voltage

calibration is analytically expressed are also considered to be the reason for the discrep-

ancies among various studies. In numerical studies, on the other hand, the modeling, i.e.

what assumptions or simplifications are made, can affect the results.

As a summary, generally accepted views about the near-wall velocity reading of HWA

nowadays are listed below:

• The wall conductivity has an influence. Highly conductive materials register larger

apparent velocity readings than poorly conductive materials (Polyakov & Shindin,

1978; Bhatia et al., 1982; Durst & Zanoun, 2002).

5

• The wire diameter effects the output. The larger diameter results in larger apparent

velocity reading (Krishnamoorthy et al., 1985; Chew et al., 1995).

• The over-heat ratio is also generally considered to be a parameter for the velocity

discrepancy. The larger overheat ratio, the larger the velocity reading becomes

(Krishnamoorthy et al., 1985; Zanoun et al., 2009).

• All of the aforementioned effects are observed and limited to the viscous sublayer

(Chew et al., 1998; Tay et al., 2012).

At the same time, the following features are still under dispute or relatively new:

• The detailed principle causing the additional heat loss is still not clearly understood

(Chew & Shi, 1993; Lange et al., 1999).

• The negative correction for the poorly conducting wall has been suggested for a

certain wall distance range (Shi et al., 2002; Zanoun et al., 2009).

• All of the aforementioned statements are in regards with the mean velocity reading.

There is no information when it comes to the turbulence intensity or higher-order

moments discussed on the wall effect.

1.3 Objective of the study

In light of the recent focus for increased accuracies in the determined friction velocity

and/or absolute wall-position, the interest in higher-order moments in the near-wall region

(Meneveau & Marusic, 2013) as well as its wall-limiting quantities, e.g. the fluctuating

wall-shear stress (Orlu & Schlatter, 2011), there is a need to revisit the effect of hot-wire

measurements close to solid walls.

The present thesis carries out a systematic parameter study on the misreading of HWA

in the near-wall region. Specifically, measurements under no-flow and flow conditions in

laminar and turbulent boundary layer flows have been performed by employing different

6

sets of parameters such as wall materials and overheat ratios. Furthermore, a numeri-

cal simulation using OpenFOAM has been conducted to complement the experimental

results.

The present study aims to provide further insight into this field, which will eventually

help researchers to use HWA in the proximity of the wall and to further investigate this

topic in the future.

1.4 Outline of the thesis

The present thesis is organized as follows: the working principle and preliminary knowl-

edge of HWA are outlined in Chapter 2. An introduction of turbulent properties employed

for later discussion is also stated in the same chapter. Then, the experimental part from

the setup to the result and discussion is summarized in Chapter 3. Chapter 4 explains the

numerical part of the study: governing equations, other computational setup and results.

In addition, a theoretical model on the wire-wall heat transfer is proposed in Chapter 5,

where the results obtained from the experiment and the simulation are integrated to build

the model. The achievements in the entire thesis are summarized as a concluding remark

in Chapter 6.

7

Chapter 2

Theoretical Background andPreliminaries

2.1 Hot-Wire Anemometry

2.1.1 Physical principle

The basic principle of hot-wire anemometry is that the amount of heat loss from a heated

wire caused by convection is correlated with the local flow velocity. The electrical re-

sistance of the wire depends on its temperature and it enables converting the heat loss to

voltage. Assume that a current I flows through a hot-wire, which is supported by two

needles called prongs as shown in Figure 2.1. In this case, the heating power P due to the

current is given by

P = IE = I2Rw =E2

Rw, (2.1)

where E and Rw represent the potential difference between the two prongs and the resis-

tance of the wire, respectively.

Heat loss from the heated wire W consists of forced convection Wfc, natural convection

Wnc, heat conduction to the prongs Wcond, and heat radiation Wrad. In most case, however,

the probe is used under flow conditions for velocities U & 0.2 m/s, the forced convection

is thought to be dominant and

W ≈ Wfc = hπdl(Tw − T∞) (2.2)

is satisfied for the wire with the diameter d and the length l, where h, Tw and T∞ denote

the convective heat transfer coefficient, the temperature of the heated wire and the sur-

8

Prongs

Wire

Fig. 2.1: A hot-wire heated by a current.

rounding fluid, respectively. Whenever the subscript w is used, a uniform distribution of

the property along the wire length is assumed, which means that its value corresponds to

the average over its length.

To express the heat transfer in non-dimensional quantities, the Nusselt number is in-

troduced:

Nu = hd/kfluid, (2.3)

where k f is the thermal conductivity of the fluid. The Nusselt number depends on every

possible property of the fluid, material and flow, but in most cases where HWA is utilized

the following assumptions are often made to simplify the problem:

• incompressible flow, i.e. Mach number Ma = U/c < 0.3 with c denoting the speed

of sound,

• ignore free convection, i.e. U & 0.2 m/s,

• wire diameters much larger than the mean free path, viz., Knudsen number Kn =

λ/d ≪ 1 with λ the molecular path, and

• large length-to-diameter ratios, i.e. l/d ≫ 1, this is to minimize the conduction

from the wire to the prongs to make the problem less three-dimensional.

The Nusselt number in the reduced problem becomes

Nu = f (Re, aR), (2.4)

where aR is called resistance overheat ratio and expressed as

aR =Rw − R0

R0. (2.5)

9

Here, the subscript 0 denotes the cold state, i.e. reference state, which is usually when the

sensor is not operated. The term “overheat ratio” often means different ratios depending

on literature. Temperature overheat ratio is described as

aT =Tw − T∞

T∞(2.6)

and it is referred to in the numerical analysis in the present study where electrical resis-

tance of the wire is not accounted for.

When the heating power and the convection are in equilibrium and constant tempera-

ture anemometry (to be discussed in the next subsection) is used, i.e. the probe resistance

is kept constant, the forced convection and the electrical heating can be coupled by com-

bining the relations (2.2) and (2.3) as

E2 ∝ E2

Rw≈ W f = πlkfluid(Tw − T )Nu. (2.7)

Since Nu is a function of Ren in general, the voltage output from the probe reduces to a

function of Un, and taking temperature effects into the calibration constants yields:

E2 = Akl + BklUnkl , (2.8)

which is widely known as King’s law in honer of King (1914).

2.1.2 Mode of operation

There are commonly two types of operation of HWA in general: constant-temperature

anemometry (CTA) and constant-current anemometry (CCA). If the heating power P and

the energy loss W are not in equilibrium, the energy balance in the wire is

dQdt= cwmw

dTdt= P(I,T ) −W(U,T ), (2.9)

where the internal energy of the sensor is denoted as Q, and cw and mw are its specific heat

and mass, respectively.

When HWA is operated in CTA mode, the temperature of the wire is kept constant via

a feedback amplifier and the left term of equation (2.9) becomes zero and the heating sup-

ply and the loss balance each others, which is the assumption made for deriving relation

(2.7).

10

On the other hand, when HWA is run in CCA mode, the wire is supplied with a

constant current and changes in velocity and thereby temperature as well as in the wire

resistance and voltage are outputted.

In the present study, all the measurements were carried out in the CTA mode, which

is the preferred mode of operation for high-frequency turbulence measurement.

2.1.3 Calibration relation

To derive the functional relation between the voltage signal from the hot-wire and the

cooling velocity out of available calibration data, all the calibration points have to be

connected through a continuous function. Although the aforementioned King’s law is one

of the curves that can be used for fitting, it is observed that the value from Eq. (2.8) at zero

velocity, namely the square root of Akl does not coincide with a measured output. This is

because the derivation of King’s law is based on a simplification that free convection can

be neglected, which is not applicable for low velocities. A modification of King’s law to

take free convection effect into account was proposed by Johansson & Alfredsson (1982)

as

U = Amk1(E2 − E20)1/nmk + Bmk2(E − E0)1/2, (2.10)

where Aml1, Bmk2 and nml are calibration constants, and E0 denotes the voltage at zero

velocity.

However, when a wider range of velocities needs to be estimated accurately, a 3rd or

4th order polynomial relation is commonly in use nowadays (see e.g. George et al., 1989):

U = C0 +C1E +C2E2 +C3E3 + .... (2.11)

The output voltage from hot-wire probes depends on the surrounding field temperature;

thus it is ideal to keep it constant and identical during calibration and measurements.

However, it can be practically difficult to achieve it due to the experimental environment

or setup. When assuming that a CTA hot-wire probe is exposed to a fluid and the fluid

temperature increases, the temperature difference between the wire and the fluid becomes

11

smaller and the feedback system of the CTA would apply a lower electrical power to

maintain the wire temperature.

The voltage output when the temperature is changed from T0 to T can be related as

E(T0) = E(T )

√Tw − T0

Tw − T= E(T )

(1 − T − T0

aR/α

)−1/2

, (2.12)

where α denotes the temperature coefficient of electrical resistance of the wire. In the

present study, the fluid temperature was always monitored and recorded during calibra-

tions and measurements in order to compensate them through Eq. (2.12).

2.2 Introduction of turbulence quantities

2.2.1 Statical analysis of velocity fluctuations

The present study focuses not only on mean value of the outputs from an anemometer,

but also on its fluctuating values. This section introduces the definitions of quantities to

indicate the characteristics of the fluctuations.

The probability density function (PDF) of the fluctuating velocity describes the like-

lihood of an instantaneous velocity existing in a certain velocity range. When the total

duration of the velocity is in a range of u < u(t) < u + ∆u is Tu during the total sampling

time T , the PDF B(u) is defined as

prob[u < u(t) < u + ∆u] = B(u)∆u = limT→∞

Tu

T, (2.13)

where B(u) satisfies

B(u) ≥ 0 and (2.14)∫ ∞−∞ B(u)du = 1. (2.15)

The n-th moment of the instantaneous velocity u is defined as

M(n)(u) =∫ ∞

−∞unB(u)du. (2.16)

The first moment of the instantaneous velocity represents the mean velocity U over the

total sampling time T :

U = M(1)(u) =∫ ∞

−∞uB(u)du = lim

T→∞

1T

∫ T

0udt, (2.17)

12

which is a general expression of the ensemble average of N samples:

U = ⟨u⟩ = 1N

N∑i=1

ui. (2.18)

The instantaneous velocity u can be decomposed into its mean value U and deviation from

the mean, namely fluctuation u′:

u = U + u′. (2.19)

The n-th central moment of velocity u corresponds to the moment of this fluctuating part

u′:

M(n)(u′) =∫ ∞

−∞u′nB(u′)du′ = lim

T→∞

1T

∫ T

0(u − U)dt. (2.20)

Hereby, the second central moment (variance) of the velocity is determined as follows:

⟨u′2

⟩= M(2)(u′) =

∫ ∞

−∞u′2B(u′)du′, (2.21)

where the square-root of the variance is called standard deviation (std) or root-mean-

square of the fluctuation:

urms =

√⟨u′2

⟩(2.22)

Likewise, the third moment and the fourth moment can be defined in a similar way and are

often expressed by normalizing them with the variance to yield the skewness and kurtosis

factors:

Skewness :

⟨u′3

⟩⟨u′2

⟩3/2 =M(3)(u′)(

M(2)(u′))3/2 , (2.23)

Kurtosis :

⟨u′4

⟩⟨u′2

⟩2 =M(4)(u′)(M(2)(u′)

)2 . (2.24)

These moments are indicators of how the fluctuating data is distributed.

Every perturbing signal can be transformed to Fourier series, viz. it can be expanded

to superpositions of multiple trigonometric functions. Thus, spectral analysis of sampled

data helps to comprehend the signal in terms of frequency components. When a narrow

13

band-pass filter of f – f + ∆ f is applied to a sampled signal u(t), the power spectra that

the signal u(t) has is calculated from the second central moment of velocity, i.e.

M(2)(u′) = limT→∞

1T

∫ ∞

−∞u′(t)2dt = lim

∆ f→0∆ f

∫ ∞

−∞u′( f )2d f , (2.25)

where u( f ) is the Fourier transform of the velocity as a function of frequency:

u′( f ) =∫ ∞

−∞e−2πi f tu′(t)dt. (2.26)

The power spectral density function Puu( f ) is defined as

Puu( f ) = u′( f )2. (2.27)

2.2.2 Inner-scaling in wall-bounded turbulence

To discuss turbulent flows in the near-wall region, inner-scaling is commonly employed

shown below. Hereby the following scales are introduced:

velocity scale, i.e. friction velocity: uτ =√τw

ρ, (2.28)

length scale, i.e. viscous length scale:ν

uτ, and (2.29)

time scale :ν

u2τ

. (2.30)

The wall shear stress τw is determined with velocity gradient at the wall:

τw = µ∂U∂y

∣∣∣∣∣wall. (2.31)

By introducing these scaling, the variables are non-dimensionalized as

u+ =u

uτ,(2.32)

y+ =uτyν, (2.33)

t+ =u2τtν. (2.34)

14

Chapter 3

Experimental Part

3.1 Measurement apparatus and procedures

3.1.1 Probe manufacturing

To investigate the influence of various parameters of anemometers themselves, several

hot-wire probes with different sensor dimensions were built by the author. The in-house

manufacturing and repair of probes have advantages since it spares time for sending

probes to their manufacturer and waiting for the repair. Furthermore, the dimensions

and the properties of the commercial probes are often limited, which makes it necessary

to manufacture probes in-house in order to study a wide parameter range.

Each probe consists of two metal prongs, a ceramic tube, a metal tube covering the

ceramic tube, electrical cables and the wire itself as shown in Figure 3.1. The wire is

coated with silver sheath when it is stored (which is a remnant of the Wollaston process

by which the small diameters can be produced), which is removed by etching. The etched

wire is then soldered on the tips of the prongs with the aid of a microscope, where the wire

faces the bottom side of the probe so that it faces the wall. The prongs are made of piano

Wire (sensor)

Prongs

Ceramic tube

Metal tube

(smoothly cemented with epoxy putty)

Electrical cables

Fig. 3.1: Components of the in-house built hot-wire probes.

15

Table 3.1: Properties of wires used in the experiment.

Hotwire Material d [µm] Resistivity [Ω/mm] Temperature coefficient α [1/K]W1 Platinum 2.54 19.37 0.0039W2 Platinum 5.08 4.86 0.0039

Fig. 3.2: Examples of the in-house built hot-wire probes.

wires by tapering their end with the aid of a grinder and by bending to obtain a boundary-

layer type probe. The properties of the wires used for the present study are summarized

in Table 3.1. After the soldering, the electrical connection is made sure and the wire

length is calculated from its measured cold resistance. Then each probe is pre-heated until

its resistance becomes stable. Figure 3.2 shows examples of the in-house manufactured

probes, some of which were built and used for the present study. Table 3.2 shows the

specifications of the probes used for the measurements, where the cold resistance R0 is

the value after the pre-heating.

3.1.2 Natural convection measurement

Firstly, measurements in an enclosure were carried out to investigate the effect of the

parameters in the absence of flow. Hereby a probe was isolated from major disturbance

of the surrounding air in order to diminish the effect of forced convection.

A setup shown in Figure 3.3 was built which mainly consists of a hot-wire probe, a

micrometer and a laser distance meter. The setup allows the placement of various wall

16

Table 3.2: Properties of the in-house built probes

Probe wire l [mm] l/d Cold resistance R0 [Ω]P1 W1 0.7 280 14.0P2 W1 0.5 200 9.4P3 W1 1.1 430 20.8P4 W1 0.5 220 11.2P5 W1 0.6 240 11.5P6 W2 1.5 300 10.2

materials that can be easily exchanged. A metallic arm holding the probe is mounted on

the micrometer and can be traversed vertically. The close-up view of the tip of the probe

and the wall is shown in Figure 3.4 obtained via a digital camera (Nikon D7100) with a

macro lens (Nikon 200mm f/4 ED-IF AF Micro-NIKKOR) mounted with an extension

tube. The optical arrangement was always used to monitor the setup so that the wire does

not touch the wall, which was also helpful to align the two prongs horizontally. The probe

was carefully driven manually by means of the micrometer, and when it reached the point

closest to the wall, the micrometer was set to zero, i.e. yw = 0 mm and the setup was

covered with a plastic box so that any major disturbance from the surroundings would

not affect the output. Then, the output voltage was acquired at thirty-four heights in total

from yw = 0 mm up to yw = 2 mm, where the sampling frequency and the sampling time

at each point were 1000 Hz and five seconds, respectively. After these thirty-five points,

the data at a point yw ≈ 5 mm was also acquired as the voltage in which the wall effect is

likely to be negligible.

The first attempt of this measurement was carried out by traversing the probe contin-

uously while acquiring the output both from the anemometer and from the laser distance

meter pointing at one of the prongs, from which the red dot in Figure 3.4 originates.

The distance between the probe can be determined later and the relation between the

anemometer height and voltage is obtained promptly. This attempt, however, failed be-

cause of large scatter of the output from the distance meter: this is thought to be because

of the oscillation of the probe itself, caused by the movement induced by the micrometer

traversing.

17

Probe

Plastic box

CTASystem Computer

Wall material

Micrometer

Temperature probe

Fig. 3.3: The experimental apparatus for the natural convection measurement.

In this no-flow measurement, the effect of the following parameters were investigated:

• wall conductivity, i.e. material of the wall,

• overheat ratio,

• length-to-diameter ratio where only the length was changed while the diameter was

left constant, and

• wall-sensor arrangement.

When parameters independent of the probe geometry, viz. the wall conductivity and over-

heat ratio, and the wall arrangement were investigated, the same probe was used in the

same angle to eliminate the effect of geometrical difference. For the last parameter, the

positional relation of the probe against the wall was changed among four types of ar-

rangements as shown in Figure 3.5. Comparison of these different arrangements requires

precise determination of the wall-sensor distance, for which a different method from the

other three parameters investigated was performed. This procedure is described in detail

in Appendix A.

3.1.3 Wind-tunnel measurements

Secondly, measurements under flow conditions were carried out in the Minimum Turbu-

lent Level (MTL) wind-tunnel at the Fluid Physics Laboratory at KTH Mechanics, whose

test section is 7 m long and has a cross-sectional area of 0.8× 1.2 m2. A schematic of this

tunnel is shown in Figure 3.6. A hot-wire probe is mounted on a metal arm with a traverse

18

Fig. 3.4: A close-up view of the prongs and the wall mirrored on the shiny surface. Thisview was able to be magnified 19 times in actual measurements by using a function ofthe camera. The red dots are the light from a laser distance meter to measure the probedistance, whose output was not used for the present results.

wire

wall

(a) (b) (c) (d)

Fig. 3.5: Wall-sensor arrangement variety for natural convection measurement. The di-rection of the gravitational acceleration is consistent through four small images. (a): wallbeneath the sensor. (b): wall above the sensor. (c): wall beside the sensor with the grav-itational acceleration vertical to the sensor length direction. (d): wall beside the sensorwith the gravitational acceleration parallel to the sensor length direction.

system in the test section as shown in Figure 3.7 and can be controlled by a computer. The

flat plate located beneath the probe has different material surfaces: aluminum and Plexi-

glas, by which the effect of different wall materials can be investigated by traversing the

probe in the spanwise direction.

Every probe was calibrated before the measurements. The calibration was performed

outside of the boundary layer in the freestream of the test section, next to a Prandtl tube

which is also used to obtain the freestream velocity of the tunnel. The freestream velocity

was controlled in a range of about 0.5 – 15 m/s and the corresponding voltage outputs

from the probe were recorded. The voltage in the absence of flow was also recorded and

all those points were used for deriving a fitting relation based on Eq. (2.11) to estimate

the outputs for lower velocities.

19

After the calibration, the hot-wire probe was traversed to the proximity of the wall

while an enlarged view of the tip of the probe and the wall was monitored on the digital

camera as well as in the natural convection measurement. This procedure was performed

after the wind tunnel started running and the freestream velocity derived by the output

from the Prandtl tube became stable. This procedure was required since the arm support-

ing the probe bends due to the aerodynamic forces and the probe gets even closer to the

wall when there is a flow. Therefore, the absolute heights of the measuring points are not

able to be derived from the record of the traverse system, and they need to be estimated

from the output instead. This procedure of determining the absolute wall distance is to be

explained in detail in the following section 3.2.

In this windtunnel experiments, measurements both in a laminar and a turbulent bound-

ary layer were conducted. The freestream velocity was set to 12 m/s and the Reynolds

numbers based on momentum thickness are Reθ ≈ 370 for the laminar cases and Reθ ≈

950 for the turbulent cases, where the momentum thickness θ is defined as

θ =

∫ ∞

0

U(y)U∞

(1 − U(y)

U∞

)dy. (3.1)

All the data were acquired at the plane 450 mm downstream from the leading edge of the

test section.

For the laminar case, the measurement was conducted above the aluminum section

only and overheat ratio aR was taken as a parameter while we used two different probes

with different wire dimensions: P5 and P6 (cf. Table 3.2).

Meanwhile, for the turbulent case, the following two parameters were considered:

• wall conductivity, i.e. surface material: aluminum and Plexiglas, and

• overheat ratio aR.

Probe P5 was used through out all turbulent cases. The turbulent boundary layer at the

same downstream location was obtained by a trip attached on the leading edge, which is a

strip of plastic tape with continuously embossed letters “V”. The sampling frequency and

the sampling duration in these wind-tunnel measurements were 60 kHz and 5 seconds,

respectively.

20

Test section

Heat exchanger

Freestream

Freestream

Stagnation chamber

Driving unit

Fig. 3.6: A schematic of the MTL windtunnel.

Probe

Aluminum

Plexiglas

Trip(for the turbulent cases)

Traversingsystem

Freestream

Fig. 3.7: The experimental setup for the wind-tunnel measurement.

3.2 Experimental results and discussion

3.2.1 HWA output in quiescent air

Results of the measurements on different wall materials in quiescent air are depicted in

Figure 3.8 and show, as expected, the dependency of the hot-wire reading on the wall con-

ductivity. In accordance with Durst & Zanoun (2002), large differences can be observed

between poorly conducting walls (Plexiglas and styrofoam with heat conductivities of

the order of 10−1 and 10−2 W/mK, respectively), while the results from highly conduct-

ing materials (such as aluminum, brass and steel, with heat conductivities of the order of

101 – 102 W/mK) do not vary among each others. It is observed that the rate of volt-

age increase follows the same tendency of dependency on material conductivity as the

21

absolute voltage output.

The dependency on the overheat ratio for the same probe on the aluminum wall is

shown in Figure 3.9. The output voltage becomes larger with increasing wire temperature

loading, which agrees with Durst & Zanoun (2002), however, it was found that the three

curves can be scaled into a single curve with the wall-remote output. It suggests that

the conduction under the natural convection in the near-wall region is independent of the

overheat ratio. The noticeable offset of aR = 0.30 at the wall proximity is possibly due to

the uncertainty of determining the point yw = 0.

Furthermore, the effect of the sensor length is presented in Figure 3.10. It was ob-

served that increasing l while maintaining d leads to a higher output. Scaled voltage

changing rate is amplified larger with increasing the wire length in most part of the mea-

suring heights, however, the behavior of the curves do not seem to be consistent. This

is thought to be the effect of three-dimensional convective flow due to the different sen-

sor lengths, and also the aforementioned uncertainty of determining the absolute wall

distance.

Finally, the output of different wall-sensor arrangements are shown in Figure 3.11.

Taking a look at the left figure, it can be seen that the wall-remote output E0 for the ar-

rangement (a), i.e. the case with the wall beneath the wire differs from the other three

cases. This is considered to be due to damage on the sensor during the process of deter-

mining yw = 0 (see Appendix A). Otherwise, the difference of the other three arrange-

ments is small, yet noticeable, i.e. the interference between the buoyant flow and the wall

affects HWA output.

3.2.2 HWA output in laminar and turbulent flow

The measured streamwise velocity from the measurements in a laminar boundary layer is

plotted as a function of the wall-normal position in Figure 3.12, where inner-scaled units

are employed for scaling. The friction velocity was calculated from a part of the obtained

velocity profile which has a linear distribution of velocity against the wall-normal height

y. The output deviates from the theoretical line U+ = y+ due to the wall effect and

22

the deviation becomes larger as the wire gets closer to the wall. The results show the

consistent qualitative tendency with the no-flow case regarding the parameter dependency;

high overheat ratio and larger surface area of the sensor result in higher output.

The region with linear velocity profile in a turbulent boundary layer is much thinner

than that in a laminar boundary layer, which makes it difficult to determine the friction

velocity in this way. In HWA measurements, particularly, the wall-effect alters the ve-

locity profile in such proximity of walls, hence acquiring the linear profile is practically

impossible.

Therefore, in the present study, the mean velocity profile of the boundary layer was fit

to a known profile of Chauhan et al. (2009) to determine the friction velocity, although it

should be noted that this way of estimating the friction velocity is more uncertain com-

pared to the fitting in the linear velocity region as it is done for the laminar flow case.

The streamwise velocity profile in wall-units are plotted in Figures 3.13 and 3.14. The

discrepancy of the plots in the outer layer in Figures 3.13 results from slight differences

in the Reynolds number, which indicates the measurement plane was too close to the trip,

so that the laminar-turbulent transition was not complete, non-uniformly in the spanwise

direction. The difference of the parameters are not noticeable in these plots, which is

considered to be hidden due to the aforementioned uncertainty of the friction velocity.

The velocity fluctuation rms urms is plotted in the same way in Figures 3.15 and 3.16 and

it also does not show a significant parameter effect possibly due to the same reason.

To eliminate the uncertainty, now the outputs are scaled by means of pure output

values from the anemometer and shown in form of the so called diagnostic plot in Fig-

ure 3.17 and 3.18 (see Alfredsson & Orlu, 2010; Alfredsson et al., 2011b). Deviation for

the different parameters becomes visible thanks to this scaling and it can be said that urms

is measured lower when the wall with higher wall conductivity and, a probe with higher

overheat is employed: the larger amount of additional heat loss towards the wall leads

to the lower reading of the fluctuation. The difference of the parameters is observed in

the region U/U∞ < 0.25, which corresponds to the viscous sublayer (Alfredsson & Orlu,

23

10-5 10-4 10-3

yw [m]

0.88

0.9

0.92

0.94

0.96

0.98

1E

[V]

Aluminium

Brass

Steel

Plexiglas

Styrofoam

100 101 102 103

yw/d

0

0.05

0.1

0.15

(E−

E0)/E

0

Aluminium

Brass

Steel

Plexiglas

Styrofoam

Fig. 3.8: HWA output in quiescent air on different wall materials. Probe P1 at a resistanceoverheat ratio aR = 0.80 was used. Left: voltage output as a function of wire height.Right: plot of the scaled variables.

2010).

Figures 3.19 and 3.20 show probability density function, PDF of the sampled velocity

plotted as a function of the wall-normal position. In accordance with Alfredsson et al.

(2011a), PDF contour lines should be parallel to each others, which is observed for the

lines at higher velocities. Contrarily, the probability distribution at lower velocities is

found to be altered and lines are shifted upwards the closer one comes to the wall. This

tendency is more intense when a wall with higher conductivity and higher overheat ratio

is employed: the PDF gets narrower consequently, which is why the rms value measured

is lowered.

Furthermore, higher-order moments, namely skewness and kurtosis factors of the ac-

quired data were investigated although the effect of the parameters could be hidden by the

aforementioned uncertainty of deriving the friction velocity (Figures 3.21 – 3.24). Both

moments are non-dimensionalised by means of the friction velocity uτ instead of urms as it

is explained in Orlu et al. (2010). The noticeable discrepancy of the plots in Figures 3.21

and 3.23 is likely to be due to the aforementioned inadequate laminar-turbulent transition.

Power spectra of the measurement plane are shown in Figures 3.25 and 3.26. The effect

of the parameters seems trivial here again.

24

10-5 10-4 10-3

yw [m]

0.5

1

1.5

2E[V

]

aR = 0.30

aR = 0.80

aR = 1.30

100 101 102 103

yw/d

0

0.05

0.1

0.15

(E−

E0)/E

0

aR = 0.30

aR = 0.80

aR = 1.30

Fig. 3.9: HWA output in quiescent air at different resistant overheat ratios. Probe P1 wasused on an aluminum surface. Left: voltage output as a function of wire height. Right:plot of the scaled variables.

10-5 10-4 10-3

yw [m]

0

0.5

1

1.5

2

E[V

]

l/d = 2.0× 102

l/d = 2.8× 102

l/d = 4.3× 102

100 101 102 103

yw/d

0

0.05

0.1

0.15

(E−E

0)/E

0

l/d = 2.0× 102

l/d = 2.8× 102

l/d = 4.3× 102

Fig. 3.10: HWA output from probes with different wire lengths in quiescent air. ProbeP1, P2 and P3 were used at a overheat ratio aR = 0.80 on a Plexiglas wall. Left: voltageoutput as a function of wire height. Right: plot of the scaled variables.

10-5 10-4 10-3

yw [m]

0.9

0.92

0.94

0.96

0.98

1

E[V

]

arrangement (a)

arrangement (b)

arrangement (c)

arrangement (d)

100 101 102 103

yw/d

0

0.05

0.1

0.15

(E−E

0)/E

0

arrangement (a)

arrangement (b)

arrangement (c)

arrangement (d)

Fig. 3.11: HWA output from probes at different wall-sensor arrangement. Probe P4 wasused at a overheat ratio aR = 0.80 on an aluminum surface. Left: voltage output as afunction of wire height. Right: plot of the scaled variables.

25

100 101 102

y+

100

101

U+

1 3 5

3

5

Fig. 3.12: Inner-scaled velocity profile in a laminar boundary layer at Reθ ≈ 400 on thealuminum wall at an overheat ratio of aR = 0.30 (thin line) and aR = 0.80 (thick line).Probes P5 (red) and P6 (orange) were used. The black dashed line indicates the linearprofile U+ = y+.

100 101 102 103

y+

0

5

10

15

20

25

U+

aluminum, aR = 0.30

Plexiglas, aR = 0.30

100 101 102 103

y+

0

5

10

15

20

25

U+

aluminum, aR = 0.80

Plexiglas, aR = 0.80

Fig. 3.13: Inner-scaled velocity profiles in a turbulent boundary layer (Reθ ≈ 950) ondifferent material surfaces. Left: the results at a consistent overheat ratio aR = 0.30.Right: the results at a consistent overheat ratio aR = 0.80.

26

100 101 102 103

y+

0

5

10

15

20

25

U+

aluminum, aR = 0.30

aluminum, aR = 0.80

100 101 102 103

y+

0

5

10

15

20

25

U+

Plexiglas, aR = 0.30

Plexiglas, aR = 0.80

Fig. 3.14: Inner-scaled velocity profiles in a turbulent boundary layer (Reθ ≈ 950) atdifferent overheat ratios. Left: the results on an aluminum surface. Right: the results on aPlexiglas surface.

100 101 102 103

y+

0

1

2

3

u+ rm

s

aluminum, aR = 0.30

Plexiglas, aR = 0.30

100 101 102 103

y+

0

1

2

3

u+ rm

s

aluminum, aR = 0.80

Plexiglas, aR = 0.80

Fig. 3.15: Inner-scaled streamwise rms profiles in a turbulent boundary layer (Reθ ≈ 950)on different material surfaces. Left: the results at a consistent overheat ratio aR = 0.30.Right: the results at a consistent overheat ratio aR = 0.80.

27

100 101 102 103

y+

0

1

2

3u+ rm

s

aluminum, aR = 0.30

aluminum, aR = 0.80

100 101 102 103

y+

0

1

2

3

u+ rm

s

Plexiglas, aR = 0.30

Plexiglas, aR = 0.80

Fig. 3.16: Inner-scaled streamwise rms profiles in a turbulent boundary layer (Reθ ≈ 950)at different overheat ratios. Left: the results on an aluminum surface. Right: the resultson a Plexiglas surface.

0 0.2 0.4 0.6 0.8 1U/U∞

0

0.1

0.2

0.3

0.4

urm

s/U

aluminum, aR = 0.30

Plexiglas, aR = 0.30

0.15 0.25

0.3

0.35

0 0.2 0.4 0.6 0.8 1U/U∞

0

0.1

0.2

0.3

0.4

urm

s/U

aluminum, aR = 0.80

Plexiglas, aR = 0.80

0.15 0.25

0.3

0.35

Fig. 3.17: Diagnostic plot of the HWA output in a turbulent boundary layer (Reθ ≈ 950)on different material surfaces. Left: the results at a consistent overheat ratio aR = 0.30.Right: the results at a consistent overheat ratio aR = 0.80.

0 0.2 0.4 0.6 0.8 1U/U∞

0

0.1

0.2

0.3

0.4

urm

s/U 0.15 0.25

0.3

0.35

aluminum, aR = 0.30

aluminum, aR = 0.80

0 0.2 0.4 0.6 0.8 1U/U∞

0

0.1

0.2

0.3

0.4

urm

s/U 0.15 0.25

0.3

0.35

Plexiglas, aR = 0.30

Plexiglas, aR = 0.80

Fig. 3.18: Diagnostic plot of the HWA output in a turbulent boundary layer (Reθ ≈ 950)read at different overheat ratios. Left: the results on an aluminum surface. Right: theresults on a Plexiglas surface.

28

101 102 103

y+

100

101

u+

5 7

1

2

3

4

101 102 103

y+

100

101

u+

5 7

2

3

4

Fig. 3.19: Inner-scaled velocity PDF in a turbulent boundary layer (Reθ ≈ 950) on dif-ferent material surfaces, i.e. aluminum (red) and Plexiglas (blue). Thin lines denote 1,5, 20, 40, 60 and 90 % of the local maximum (thick line) of the PDF. Left: the results ataR = 0.30. Right: the results at aR = 0.80.

101 102 103

y+

100

101

u+

5 7

2

3

4

101 102 103

y+

100

101

u+

5 7

1

2

3

4

Fig. 3.20: Inner-scaled velocity PDF in a turbulent boundary layer (Reθ ≈ 950) at differentoverheat ratios. Thin lines denote 1, 5, 20, 40, 60 and 90 % of the local maximum (thickline) of the PDF. Left: the results at aR = 0.30 (black) and aR = 0.80 (red) on an aluminumsurface. Right: the results at aR = 0.30 (black) and aR = 0.80 (blue) on a Plexiglas surface.

29

100 101 102 103

y+

-4

-2

0

2

4

6u′3/u3 τ

aluminum, aR = 0.30

Plexiglas, aR = 0.30

100 101 102 103

y+

-4

-2

0

2

4

6

u′3/u3 τ

aluminum, aR = 0.80

Plexiglas, aR = 0.80

Fig. 3.21: Third-order moment of the measured turbulent boundary layer (Reθ ≈ 950)on different material surfaces. Left: the results at a consistent overheat ratio aR = 0.30.Right: the results at a consistent overheat ratio aR = 0.80.

100 101 102 103

y+

-4

-2

0

2

4

6

u′3/u

3 τ

aluminum, aR = 0.30

aluminum, aR = 0.80

100 101 102 103

y+

-4

-2

0

2

4

6

u′3/u

3 τ

Plexiglas, aR = 0.30

Plexiglas, aR = 0.80

Fig. 3.22: Third-order moment of the measured turbulent boundary layer (Reθ ≈ 950) atdifferent overheat ratios. Left: the results on an aluminum surface. Right: the results on aPlexiglas surface.

100 101 102 103

y+

0

20

40

60

80

100

120

140

u′4/u

4 τ

aluminum, aR = 0.30

Plexiglas, aR = 0.30

100 101 102 103

y+

0

20

40

60

80

100

120

140

u′4/u

4 τ

aluminum, aR = 0.80

Plexiglas, aR = 0.80

Fig. 3.23: Fourth-order moment of the measured turbulent boundary layer (Reθ ≈ 950)on different material surfaces. Left: the results at a consistent overheat ratio aR = 0.30.Right: the results at a consistent overheat ratio aR = 0.80.

30

100 101 102 103

y+

0

20

40

60

80

100

120

140

u′4/u

4 τ

aluminum, aR = 0.30

aluminum, aR = 0.80

100 101 102 103

y+

0

20

40

60

80

100

120

140

u′4/u

4 τ

Plexiglas, aR = 0.30

Plexiglas, aR = 0.80

Fig. 3.24: Fourth-order moment of the measured turbulent boundary layer (Reθ ≈ 950) atdifferent overheat ratios. Left: the results on an aluminum surface. Right: the results on aPlexiglas surface.

10-2 10-1

f+

101

102

y+

10-2 10-1

f+

101

102

y+

Fig. 3.25: Comparison of power spectra of the measured turbulent boundary layer (Reθ ≈950) on different material surfaces, i.e. aluminum (red) and Plexiglas (blue). The linesdenote 10, 20, 30, 40, 50 and 60 % of the peak. Left: the results at a consistent overheatratio aR = 0.30. Right: the results at a consistent overheat ratio aR = 0.80.

31

10-2 10-1

f+

101

102

y+

10-2 10-1

f+

101

102

y+

Fig. 3.26: Comparison of power spectra of the measured turbulent boundary layer (Reθ ≈950) at different overheat ratios. The lines denote 10, 20, 30, 40, 50 and 60 % of the peak.Left: the results at aR = 0.30 (black) and aR = 0.80 (red) on an aluminum surface. Right:the results at aR = 0.30 (black) and aR = 0.80 (blue) on a Plexiglas surface.

32

Chapter 4

Numerical Analysis

4.1 Computational setup

4.1.1 Governing equations

Performing numerical analysis of the present topic brings further understanding of heat

transfer in the air surrounding the heated wire and inside walls. In the present study,

a steady, two-dimensional numerical simulation using OpenFOAM (version 2.2.2) was

conducted. A built-in solver chtMultiRegionSimpleFoam in OpenFOAM deals with con-

jugated heat transfer among different material regions by solving governing equations.

For the fluid region in the present work, the conservation of mass, momentum and

energy for steady compressible flow are solved, i.e.

∂(ρ∗U∗i

)∂x∗i

= 0, (4.1)

∂(ρ∗U∗i U∗j

)∂x∗i

= −∂P∗

∂x∗j+

1Re∂

∂x∗i

µ∗ ∂U∗j∂x∗i+∂U∗i∂x∗j− 2

3∂U∗k∂x∗kδi j

+ ρ∗g∗j, (4.2)

∂(ρ∗c∗pT ∗U∗i

)∂x∗i

+Ec2

∂(ρ∗U∗i U∗j U

∗j

)∂x∗i

=1

RePr∂

∂x∗i

(k∗

c∗p

∂(c∗pT ∗)

∂x∗i

)+ Ecρ∗U∗i g∗i . (4.3)

In the solid region, the heat-conduction equation is solved:

k∗

ρ∗c∗p

∂2T ∗

∂x∗i ∂x∗i= 0. (4.4)

The inlet velocity at the height of the wire center Uw and the wire diameter d are

employed to normalize the velocity components and coordinates, respectively, while the

temperature is scaled as T ∗ = (T − T∞)/(Tw − T∞). The thermal physical properties of

33

air ρ∗, µ∗, k∗, and c∗p (density, dynamic viscosity, thermal conductivity and specific heat at

constant pressure, respectively) in the equations are chosen as 7th polynomial functions

of temperature and normalized by the corresponding values at the inflow temperature

T∞: the equations are of compressible-type but the density is not altered by pressure, in

which sense the simulation is incompressible. The other non-dimensional parameters are

the Eckert number, the Prandtl number and the Reynolds number, which are defined as

follows:

Eckert number: Ec =Uw

2

cp∞ (Tw − T∞), (4.5)

Prandtl number: Pr =µ∞cp∞

k∞, (4.6)

Reynolds number: Re =ρ∞Uwdµ∞

. (4.7)

The radiation is completely neglected in the present simulation, since its contribution is

supposed to be trivial (see Appendix B for the detail).

4.1.2 Calculation procedure

Spatial derivatives in the governing equations were discretized with a second-order central

difference scheme, while a second-order linear upwind scheme was employed for the

advection terms.

Coupling of velocity and pressure was treated according to the SIMPLE algorithm

equipped on OpenFOAM which was first proposed by Patankar & Spalding (1972). Its

procedure is summarized as follows:

1. Set the boundary conditions.

2. Solve the discretized momentum equation and energy equation to calculate the in-

termediate velocity field.

3. Compute the mass flux at the cell faces.

4. Solve the pressure equation.

5. Correct the mass fluxes at the cell faces.

6. Correct the velocity according to the updated pressure field.

34

Table 4.1: Properties of meshes.

yw/d Fluid region cells Solid region cells100 168674 98107300 198254 98107

1000 217974 98107

7. Update the boundary conditions.

8. Repeat until convergence is achieved.

Steps 4 and 5 were repeated three times for the present case to compensate for the non-

orthogonality of the meshes (see Jasak, 1996; Ferziger & Peric, 2002). Convergence is

regarded to be achieved when averages of the residuals for all variables at all calculation

points are reduced to a value lower than 10−6.

4.1.3 Computational domain and boundary conditions

An infinitely long cylinder with a diameter d = 5 µm parallel to the wall and normal to

the flow is employed to represent the hot-wire sensor as shown in figure 4.1. The entire

computational domain is divided into fluid and solid regions representing air and a solid

wall, respectively. The wire center is located at (x/d, y/d) = (0, yw/d) where yw varies

between yw = 100d, 300d and 1000d, and the domain spreads in the streamwise direction

−3000 < x/d < 6000. The fluid region is extended from 0 < y/d < 5000 + yw/d and the

solid region is from −5000 < y/d < 0.

The unstructured meshes for the present calculation are created with ANSYS ICEM

and the detail around the cylinder is shown in Figure 4.2. The grid distribution is finer

near the wall, the interface and the cylinder, and becomes coarser as it gets farther from

these boundaries. The first layer thickness in the proximity of the cylinder is 2d × 10−3.

The number of cells for each mesh (yw = 100d, 300d and 1000d) is shown in Table 4.1.

In the fluid region, a linear velocity profile with fixed temperature T = T∞ was imple-

mented at the inlet, accordingly the velocity and the temperature at the top wall were set

to U = S y, V = 0 and T = T∞, respectively, where S is a variable to give different shear

35

,

, ,

, ,

Inflow Outflow

Interface

Wire

Fluid region

Solid region

Fig. 4.1: Computational domain and boundary conditions.

velocities to the flow field. A Neumann zero-gradient condition was applied at the outlet

for both velocity and temperature. Finally, the wire surface was set to the no-slip condi-

tion, i.e. U = V = 0 with temperature T = Tw. The temperature overheat ratio defined

by (2.6) were varied between two values; aT = 0.27 and 0.96 where T∞ = 293.15 K. A

gauge pressure value was set at the outlet and a zero-gradient pressure was implemented

at the rest of the boundaries. Also, the gravitational acceleration working downwards in

y-direction was taken into account.

The temperature T = T∞ was set at the left boundar. The zero-gradient conditions,

∂T/∂x = 0 and ∂T/∂y = 0 were set at the right and the bottom boundaries, respectively.

At the interface, the no-slip boundary condition for velocity was implemented, and

the continuity of temperature and the heat flux, i.e. Tfluid = Tsolid and kfluid ∂T/∂y|fluid =

ksolid ∂T/∂y|solid were maintained. The present work employs two different materials,

namely aluminum (k∗ = 9221) and Plexiglas (k∗ = 7.393) representing high conductive

and poor conductive materials, respectively.

36

Fig. 4.2: Detail of the mesh around the cylinder.

4.1.4 Post processing of the simulation results

The heat loss from the wire was evaluated as the mean Nusselt number Nu at the wire

surface, which is calculated by integrating the local Nusselt number Nu(ϕ). The heat flux

at a certain point on the surface q(ϕ) is calculated as

q(ϕ) = −k(Tw)∂T (r, ϕ)∂r

∣∣∣∣∣r=d/2, (4.8)

where r and ϕ are the polar coordinates originated at the wire center. Normalizing q(ϕ)

with a reference heat flux qref = k(T f )(Tw − T∞)/d returns the local Nusselt number:

Nu(ϕ) =q(ϕ)qref= −k(Tw)

k(T f )∂T ∗(r∗, ϕ∗)∂r∗

∣∣∣∣∣r∗=0.5

, (4.9)

where ϕ∗ = ϕ/(2π) and T f is the film temperature: T f = (Tw + T∞)/2. By taking the

average of Nu over the wire surface, the mean Nusselt number is derived as:

Nu =∫ 1

0Nu(ϕ∗)dϕ∗. (4.10)

37

Fluid

Solid (uncoupled)

Solid (uncoupled)

Outflow

Wire

Inflow Interfaces

Fig. 4.3: Computational domain and boundary conditions for the simulations infreestream.

4.2 Numerical calibration of HWA

In actual measurements, the output voltage from an anemometer is coupled to the velocity

through a calibration relation obtained from measurements in the freestream. Instead, for

the present numerical analysis, the heat transfer from a heated cylinder was calculated

under different freestream velocities to correlate the Nusselt number with the velocity.

A schematic of the calculation domain for the calibration cases is shown in Figure

4.3. The grid spacing of the mesh close to the wire is equal to that of the mesh shown

in Figure 4.2. To run the calculation using the same solver, solid regions were prepared

but the temperature and the heat flux are not coupled at the interfaces for the calibration

cases: a constant temperature T∞ is implemented on the walls. These solid regions are

located at a 5000d from the cylinder center and Neumann zero-gradient condition for the

velocity, namely slip condition was applied at the interfaces.

The gravitational acceleration is implemented so that a buoyant flow arises. Accord-

ing to the previous study by Collis & Williams (1959), the effect of buoyant flow emerges

38

on heat transfer from heated wires when Re∞ < Gr1/3∞ . In the present case, therefore,

the freestream velocity of the calibration cases is varied in a range of U∞ ≥ 0.05, ap-

proximately corresponding to Re∞ > Gr1/3∞ . The Reynolds number Re∞ and the Grashof

number Gr∞ are defined as follows:

Re∞ = Uwdν∞, (4.11)

Gr∞ =g(Tw−T∞)d3

T∞ν2∞. (4.12)

Figures 4.4 and 4.5 show the results of the calibration cases and fitting curves deter-

mined by

Nu(

T f

T∞

)−0.17

= ncal1Rencal2f + ncal3, (4.13)

where the coefficients were found to be ncal1 ≈ 0.60, ncal2 ≈ 0.40 and ncal3 ≈ 0.22 for

both temperature overheat ratios. The Reynolds number Re f is calculated based on ther-

mophysical properties at the film temperature, i.e. T f = (Tw + T∞)/2. The correction of

the Nusselt number in Eq. (4.13) concerning the overheat ratio was proposed by Collis &

Williams (1959) and it gives good agreement between two different temperature loadings

for the present case: the discrepancy of two calibration curves is within less than 1%.

Several results and correlations of Nusselt number and Reynolds number from previous

experimental and numerical studies are plotted together for comparison. The present fit-

ting curves agree well with the previous studies with 5% of maximum deviation in the

range 6 × 10−3 < Re f < 1 except for the one of Kramers (1946) which is not valid for

such low Reynolds numbers.

The local Nusselt number distribution along the wire surface and the temperature dis-

tribution are also presented in Figure 4.6 and 4.7. The heat transfer from the upper and the

lower half of the wire surface are almost equal, and temperature contour lines draw almost

symmetric ovals with respect to the horizontal line. However, for the low speed calcula-

tions, it it observed that the contour tilts from horizontal due to the natural convective

behavior in the calculation domain.

39

10-4 10-3 10-2 10-1

Ref

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Nu(T

f/T∞)−

0.17

present, calibration points, aT = 0.27

present, calibration curve, aT = 0.27

Collis & Williams (1959), Re < 0.5

Collis & Williams (1959), 0.02 < Re < 44

Shi et al. (2003), aT = 0.27

Lange et al. (1998)

Kramers (1946), 0.1 < Re < 104

Fig. 4.4: Heat transfer from the heated cylinder at aT = 0.27 in freestream and comparedwith results from the literature.

10-4 10-3 10-2 10-1

Ref

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Nu(T

f/T∞)−

0.17

present, calibration points, aT = 0.96

present, calibration curve, aT = 0.96

Collis & Williams (1959), Re < 0.5

Collis & Williams (1959), 0.02 < Re < 44

Lange et al. (1998)

Kramers (1946), 0.1 < Re < 104

Fig. 4.5: Heat transfer from the heated cylinder at aT = 0.96 in freestream and comparedwith results from the literature.

40

-0.5 -0.25 0 0.25 0.5x∗

0.3

0.35

0.4

0.45

0.5

0.55

Nu

Ref ≈

1.3× 10−1

8.0× 10−2

2.7× 10−2

2.1× 10−2

1.3× 10−2

-0.5 -0.25 0 0.25 0.5x∗

0.3

0.35

0.4

0.45

0.5

0.55Nu

Ref ≈

8.4× 10−2

5.0× 10−2

1.7× 10−2

1.3× 10−2

(i) aT = 0.27 (ii) aT = 0.96

Fig. 4.6: Local Nusselt number distribution along the wire surface at different temperatureoverheat ratios in freestream. The solid lines and the broken lines represent the values onthe upper-half and the lower-half surfaces, respectively, although they are almost identicaland hardly distinguishable. Left: the result at aT = 0.27. Right: the result at aT = 0.96.

41

x*

y*

0.01

0.050.07

0.090.10

0.15

0.02 0.03

0.20

x*

y*

0.01

0.05

0.07

0.090.10

0.15

0.20

0.02

0.03

x*

y*

0.01

0.05

0.07

0.09

0.100.1

5

0.02

0.03

x*

y*

0.01

0.05

0.07

0.090.10

0.150.2

0

0.020.03

x*

y*

0.01

0.050.07

0.02

0.03

x*

y*

0.01

0.05

0.07

0.02

0.03

(i-a) aT = 0.27, Ref ≈ 1.3 × 10−2 (i-b) aT = 0.96, Ref ≈ 1.3 × 10−2

(ii-a) aT = 0.27, Ref ≈ 2.7 × 10−2 (ii-b) aT = 0.96, Ref ≈ 1.6 × 10−2

(iii-a) aT = 0.27, Ref ≈ 1.3 × 10−1 (iii-b) aT = 0.96, Ref ≈ 8.4 × 10−2

Fig. 4.7: Local temperature distribution around the wire in freestream. Left column:aT = 0.27. Right column: aT = 0.96.

42

4.3 Numerical errors

4.3.1 Truncation error

The grid dependency of the calculation result was evaluated by varying the grid spacing

between the one with the first layer thickness of 1× 10−3, 2× 10−3 (present) and 4× 10−3.

The mesh used for the calculation was the one with yw/d = 100. Figure 4.8 shows the

calculated Nusselt number at the wire surface for the different mesh spacings, where the

overheat ratio and the inlet velocity gradient was set to aT = 0.27 and S = 10, and

the solid region is aluminum. The converging tendency of the result with refining mesh

was not significant in this range of the cell spacing; however, the mesh for the present

calculation is sufficiently fine given that the refining mesh only changes the value by less

than 1%.

4.3.2 Convergence error

The number of the iteration was varied to investigate whether the calculation is adequately

iterated. The mesh with yw/d = 100 was employed for the calculation. Figure 4.9 presents

the transition of the calculated Nusselt number at the wire surface with respect to the

convergence criterion (the residual value at which a calculation is terminated), where the

overheat ratio and the inlet velocity gradient was set to aT = 0.27 and S = 10, and the

solid region is aluminum. It was observed that the convergence criterion below 10−6 does

not influence the result significantly, whereby the present calculation is confirmed to be

sufficiently converged for the given meshes.

43

0.001 0.002 0.004First layer thickness/d

×10-3

0.33

0.335

0.34

0.345

0.35

0.355

Nu

Fig. 4.8: The grid resolution dependency of the calculated result investigated at aT = 0.27on aluminum with the inlet velocity gradient of S = 10. The sensor is located at yw =

100/d.

10-7 10-6 10-5

Convergence criteria

0.33

0.335

0.34

0.345

0.35

0.355

Nu

Fig. 4.9: The iteration dependency of the calculated result investigated at aT = 0.27 onaluminum with the inlet velocity gradient of S = 10. The sensor is located at yw = 100/d.

44

10-3 10-2 10-1 100

Ref

101

102

103

104

Cd

present

Oseen (1911)

Tomotika & Aoi (1951)

Fig. 4.10: The calculated drag coefficient of a circular cylinder placed in freestream com-pared with the literature.

4.4 Validation method

4.4.1 Validity of flow field around a circular cylinder

To check the validity of flow field around the sensor, the drag coefficients of a circular

cylinder in freestream were evaluated. The drag coefficient is defined as

CD =Fd

12ρ∞U2

wπd, (4.14)

where Fd represents the drag force per unit length of the wire. The calculation domain

for this case is identical to the one used for the calibration (see Figure 4.3), but the wire

temperature is set to the same value with the inflow air temperature T∞. Figure 4.10

displays the calculated drag force coefficient as a function of Reynolds number plotted

together with the correlations derived by Oseen (1911) and Tomotika & Aoi (1951). It

can be seen that the present calculation shows good agreement with the results of the

literature.

4.4.2 Simulation of natural convection in a closed cavity

A simulation of natural convection in a square cavity was conducted to investigate the va-

lidity of the solver simulating buoyant effect. In a closed cavity shown in Figure 4.11, the

45

Fluid

Solid (uncoupled)

Solid (uncoupled)

InterfacesLeft wall Right wall

Fig. 4.11: Computational domain for the simulation of the natural convection in a squarecavity.

convective flow arises due to the temperature difference between the walls. A frequently

cited literature de Vahl Davis (1983) presents benchmark solutions of this natural convec-

tion in various Rayleigh numbers in a range of 103 ≤ Ra ≤ 106. Calculated properties of

converged convective flow field are listed in Table 4.2 together with result of the literature,

where the variables on this table is explained as follows:

umax: the maximum velocity in the x-direction,

y|umax: the y-location of u = umax,

vmax: the maximum velocity in the y-direction,

x|vmax: the x-location of v = vmax,

Nux=0: the average Nusselt number on the left wall x = 0,

Numax: the maximum Nusselt number on the left wall x = 0,

y|Numax: the y-location of Nu = Numax,

Numin: the minimum Nusselt number on the left wall x = 0,

y|Numin: the y-location of Nu = Numin.

46

It was observed that the employed solver for the present work is capable of simulating

convective flow, however several properties do not necessarily agree with the result of

de Vahl Davis (1983). This is thought to be mainly due to the difference of the grid res-

olution: the present grid has 101 × 101 cells with finer spacing close to the walls, while

the grids employed in the literature are uniformly distributed grids of 81 × 81 cells maxi-

mum. Furthermore, the natural convection at lower Rayleigh numbers is what is actually

important for the present topic, whose calculation is validated through the comparison of

the calibration relation with the literature after all (see section 4.2).

47

Tabl

e4.

2:C

alcu

late

dre

sult

ofna

tura

lcon

vect

ive

flow

ina

squa

reca

vity

.

Ra

103

104

105

106

case

pres

ent

deV

ahlD

avis

pres

ent

deV

ahlD

avis

pres

ent

deV

ahlD

avis

pres

ent

deV

ahlD

avis

u max

3.49

33.

649

16.0

4616

.178

43.6

734

.73

125.

2764

.63

y | um

ax0.

815

0.81

30.

828

0.82

30.

885

0.85

50.

939

0.85

0

v max

3.53

03.

697

19.4

1619

.617

68.1

168

.59

215.

0021

9.36

x | vm

ax0.

172

0.17

80.

115

0.11

90.

069

0.06

60.

0363

0.03

79

Nu x=

01.

146

1.11

72.

240

2.23

84.

564

4.50

99.

150

8.81

7

Nu m

ax1.

541

1.50

53.

678

3.52

88.

057

7.71

718

.036

17.9

25y | N

u max

0.00

10.

092

0.13

70.

143

0.07

80.

081

0.04

190.

0378

Nu m

in0.

565

0.69

20.

473

0.58

60.

580

0.72

90.

779

0.98

9y | N

u min

11

11

11

11

48

4.5 Numerical results and discussion

The calculated surface-averaged Nusselt number as a function of Reynolds number Re f =

Uwd/ν f is shown in Figures 4.12 and 4.13, where Uw represents the velocity at the wire

height at the inlet. The result of the present cases on an aluminum wall is plotted together

with the numerical data of Shi et al. (2003) and it shows good agreement for the same wire

height even though the case of Shi et al. (2003) has a flow beneath the aluminum wall.

The plots deviate further from the calibration curve as the Reynolds number decreases,

and also as the wire-wall distance becomes smaller. When compared with respect to wall

materials and overheat ratio, the result indicates the dependency on the wall conductiv-

ity, yet independent of overheat ratio for low Reynolds number. This behavior of the

overheat-ratio difference implies that unscaled temperature gradient at the sensor surface

is proportional to the temperature difference Tw − T∞ given that the present derivation of

Nusselt number is based on scaling in Eq. 4.9.

Figures 4.15 – 4.17 present velocities which were converted from the Nusselt num-

bers of the present results with the obtained calibration curve. This velocity corresponds

to what is acquired as velocity in real-life HWA measurement. The inner-scale is em-

ployed in this plot, as is often the case with numerical researches on the present topic.

The“measured velocity” increases with decreasing inner-scaled height in the viscous sub-

layer: the typical wall-effect is demonstrated. Besides, the effect of the overheat ratio and

the wall conductivity is found to be consistent with the experimental result (Figure 4.17).

The present U+ on aluminum agrees well with the result of Zanoun et al. (2009) when

compared at the same sensor height yw/d = 100. However, it is observed that the curves

do not necessarily fall into a single line with respect to different wire location yw, which

means that the inner-scaling is not adequate to discuss the wire-wall heat transfer when

various sensor heights are concerned. This is thought to be simply because contribution

of the heat conduction towards the wall is not explained in the same scale and principle

with turbulence.

Taking a look at local Nusselt number at each point on the wire surface, geometrical

49

feature of the present phenomenon is illuminated. Figures 4.18 and 4.19 presents the

local Nusselt number at different sensor heights and velocities. The difference between

the upper and lower side of the wire is noticeable for relatively high velocity gradient

such as S = 300 and 1000, which is natural considering the larger velocity difference

across the wire top and bottom. Hence, the distribution becomes more identical to that of

freestream case as the velocity gradients gets smaller provided the wire is thermally far

enough from the wall so that conduction effect is negligible. It should be noted that the

cases with low velocity gradient and wire height such as S = 10 at y∗w = 100 are affected

by the heat conduction and show the distribution with intense heat transfer on the lower

side.

This tendency can be explained by referring to the temperature distribution around

the wire (Figures 4.20 – 4.21). It is apparent that, with the presence of solid wall, the

temperature drops steeply beneath the wire. Especially for the aluminum wall cases,

temperature at the interface is always approximately identical to the atmospheric value

T∞, while for the Plexiglas wall contains an evident high-temperature region which shifts

towards downstream as the flow becomes faster. In addition, the shape of the temperature

contour lines approaches to freestream state with increasing flow velocity.

10-4 10-3 10-2 10-1

Ref

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Nu

present, aluminum, yw = 100d

present, aluminum, yw = 300d

present, aluminum, yw = 1000d

Shi et al. (2003), aluminum, yw = 100d

present, calibration curve

Fig. 4.12: Heat transfer from the heated cylinder at aT = 0.27 on an aluminum wall.

50

10-4 10-3 10-2 10-1

Ref

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Nu

present, Plexiglas, yw = 100d

present, Plexiglas, yw = 300d

present, Plexiglas, yw = 1000d

present, calibration curve

Fig. 4.13: Heat transfer from the heated cylinder at aT = 0.27 on a Plexiglas wall.

10-4 10-3 10-2 10-1

Ref

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Nu

aluminum, yw = 100d, aT = 0.27

aluminum, yw = 100d, aT = 0.96

Plexiglas, yw = 100d, aT = 0.27

calibration curve, aT = 0.27

calibration curve, aT = 0.96

Fig. 4.14: Heat transfer from the heated cylinder at different overheat ratios and walls.

51

0 1 2 3 4 5 6 7 8y+w

0

2

4

6

8

10

U+

present, aluminum, yw = 100d

present, aluminum, yw = 300d

present, aluminum, yw = 1000d

Shi et al. (2003), aluminum, yw = 100d

U+ = y+

Fig. 4.15: The measured velocity by the hot-wire at aT = 0.27 on an aluminum wall.

0 1 2 3 4 5 6 7 8y+w

0

2

4

6

8

10

U+

present, Plexiglas, yw = 100d

present, Plexiglas, yw = 300d

present, Plexiglas, yw = 1000d

U+ = y+

Fig. 4.16: The measured velocity by the hot-wire at aT = 0.27 on a Plexiglas wall.

52

0 1 2 3 4 5 6 7 8y+w

0

2

4

6

8

10

U+

present, aluminum, yw = 100d, aT = 0.27

present, aluminum, yw = 100d, aT = 0.96

present, Plexiglas, yw = 100d, aT = 0.27

Zanoun et al. (2009), yw = 100d, aT = 0.27

U+ = y+

Fig. 4.17: Comparison of the measured velocity of the hot-wire at different overheat ratiosand walls.

53

-0.5 -0.25 0 0.25 0.5x∗

0.3

0.35

0.4

0.45

0.5

0.55

Nu

Ref ≈

1.3× 10−1

1.3× 10−2

1.3× 10−3

4.0× 10−2

S = 300

S = 100

S = 10

S = 1000

-0.5 -0.25 0 0.25 0.5x∗

0.3

0.35

0.4

0.45

0.5

0.55

Nu

Ref ≈

1.3× 10−1

1.3× 10−2

1.3× 10−3

4.0× 10−2

S = 300

S = 100

S = 10

S = 1000

-0.5 -0.25 0 0.25 0.5x∗

0.3

0.35

0.4

0.45

0.5

0.55

Nu

Ref ≈

1.2× 10−1

4.0× 10−2

1.2× 10−2

S = 300

S = 100

S = 30

-0.5 -0.25 0 0.25 0.5x∗

0.3

0.35

0.4

0.45

0.5

0.55Nu

1.2× 10−1

Ref ≈

4.0× 10−2

1.2× 10−2

S = 300

S = 100

S = 30

-0.5 -0.25 0 0.25 0.5x∗

0.3

0.35

0.4

0.45

0.5

0.55

Nu

Ref ≈

1.3× 10−1

4.0× 10−2

1.3× 10−2

S = 100

S = 30

S = 10

-0.5 -0.25 0 0.25 0.5x∗

0.3

0.35

0.4

0.45

0.5

0.55

Nu

Ref ≈

1.3× 10−1

4.0× 10−2

1.3× 10−2

S = 100

S = 30

S = 10

(i) yw = 100

(ii) yw = 300

(iii) yw = 1000

Fig. 4.18: Local Nusselt number distribution along the wire surface at aT = 0.27 forvarious wire heights. The solid lines and the broken lines represent the values on theupper-half and the lower-half surfaces, respectively. Left column: results on an aluminumwall (red). Right column: results on a Plexiglas wall (blue).

54

-0.5 -0.25 0 0.25 0.5x∗

0.3

0.35

0.4

0.45

0.5

0.55

Nu

Ref ≈

8.4× 10−2

8.4× 10−3

8.4× 10−4S = 10

S = 100

S = 1000

2.5× 10−2

S = 300

Fig. 4.19: Local Nusselt number distribution along the wire surface at aT = 0.27 for thewire hight y∗w = 100. The solid lines and the broken lines represent the values on theupper-half and the lower-half surfaces, respectively.

55

x*

y*0.02

0.01

0.15

0.05

0.09

0.07

0.03

0.10

x*

y*

0.03

0.15

0.10

0.09

0.07

0.05

x*

y*

0.01

0.02

0.03

0.15

0.05

0.09

0.07

0.01

0.10

x*

y*

0.01 0.02

0.03

0.05

0.07

0.09

0.10

0.15

x*

y*

0.010.02

0.030.05

0.07

x*

y* 0.01

0.05

0.070.09

0.02

0.03

(i) S = 10, Ref ≈ 1.3 × 10−3

(ii) S = 100, Ref ≈ 1.3 × 10−2

(iii) S = 1000, Ref ≈ 1.3 × 10−1

Fig. 4.20: Local temperature distribution around the wire at aT = 0.27 at the locationof y∗w = 100. Left column: results on an aluminum wall. Right column: results on aPlexiglas wall.

56

x*

y*

0.07

0.02

0.01

0.10

0.15

0.20

0.03

0.090.05

x*

y*

0.07

0.02

0.10

0.15

0.20

0.03

0.09

0.05

x*

y*0.07

0.15

0.10

0.030.02

0.01

0.09

0.05

x*

y*

0.07

0.15

0.10

0.03

0.02

0.01

0.09

0.05

x*

y*0.03

0.02

0.01

0.05

x*

y*0.03

0.02

0.01

0.05

(i) S = 10, Ref ≈ 4.0 × 10−3

(ii) S = 100, Ref ≈ 4.0 × 10−2

(iii) S = 1000, Ref ≈ 4.0 × 10−1

Fig. 4.21: Local temperature distribution around the wire at aT = 0.27 at the locationof y∗w = 300. Left column: results on an aluminum wall. Right column: results on aPlexiglas wall.

57

x*

y*0.02

0.01

0.15

0.05

0.09

0.07

0.03

0.10

x*

y* 0.02

0.01

0.050.07

0.03

x*

y*

0.020.01

0.05

0.07

0.03

(i) S = 10, Ref ≈ 8.4 × 10−4

(ii) S = 100, Ref ≈ 8.4 × 10−3

(iii) S = 1000, Ref ≈ 8.4 × 10−2

Fig. 4.22: Local temperature distribution around the wire at aT = 0.96 at the location ofy∗w = 100 on an aluminum wall.

58

Chapter 5

Theoretical Model on Wire-Wall HeatTransfer in a Fluid Flow

5.1 Components of the overall heat transfer

In this chapter, a theoretical, yet occasionally empirical model on the heat transfer be-

tween a hot-wire, a solid wall and surrounding air is proposed by combining the knowl-

edge obtained thorough the experiment and the numerical analysis. On the present topic,

correlations of the contributions from convection and conduction in the near-wall region

have often been discussed in former studies but has still not been completely established.

To establish a model in a form which is as simple as possible, in the present study,

radiation is assumed to be small enough to be negligible as it was in the present numerical

analysis, and heat conduction from a wire to supporting prongs is also neglected, i.e. the

wire is assumed to be long enough compared to its diameter. Therefore, the following

two major components of heat transfer are thought to be dominant:

• convection: forced convection and natural convection,

• conduction caused by the presence of solid walls.

The total heat transfer on the wire surface is assumed to be a superposition of functions

of the two factors stated above:

Nutotal = f1 (Nuconv) + f2 (Nucond) , (5.1)

where the first term on the right hand side denotes the heat transfer due to convection, and

59

the second term on the right hand side, i.e. the conduction part is interpreted as how much

the presence of the wall distorts the temperature field from its unaffected state.

5.2 Revisiting the calibration curve

In the present numerical analysis, the calibration curve coupling heat transfer Nu to

Reynolds number Re f was derived. Although this calibration curve was found to agree

with correlations by several previous researches in the range of 6× 10−3 < Re f < 1, those

reference curves are not able to predict the heat transfer in the lower Reynolds number

range.

Simulation of natural convection is always influenced by the size of the calculation

domain, hence calculation in relatively small domain compared to real-life scale does

not bring physically meaningful results. In fact, the calibration curves in the previous

numerical works often neglect the gravitational effect (Lange et al., 1998; Shi et al., 2003)

and are not adequate for the discussion of the mixed convection.

The experimental work of Collis & Williams (1959) provides data in the region Re∞ <

Gr1/3∞ , where the natural convection starts to dominate the total heat transfer. They ob-

served that the Nusselt number deviates from their correlation in the low Reynolds num-

ber region however no correlation for this behavior was proposed. It is stated in their work

that this mixed convection registers lower Nusselt numbers than the value of pure natural

convection, i.e. without flow applied.

On the other hand, heat transfer from a heated cylinder under pure natural convection

has been studied both in experimental and numerical approaches by numerous researchers

but their estimates deviate from each other, possibly due to the difference of the experi-

mental and numerical setup. Collis & Williams (1954) employed relatively long sensors

(l/d ≥ 20000) suitable to compare with the present two-dimensional calculation, and their

measurements were conducted in the Grashof number region of 10−10 ≤ Gr f Pr f ≤ 10−3,

in which the present case is included. Morgan (1975) gave a correlation to their result:

Nunc = 0.675(Gr f Pr f )0.058, (5.2)

60

where the Grashof number Gr f is defined as

Gr f =g(Tw − T∞)d3

T f ν2f

. (5.3)

The relation below is proposed in the present study to connect the Nusselt number

Nunc at Re = 0 calculated with (5.2) and the original calibration points Nucal,orig for Re∞ >

Gr1/3∞ :

Nuconv = Nucal,new = Nunc exp(

Re f

nconv

)+ Nucal,orig

(1 − exp

(Ref

nconv

)). (5.4)

Here, the original calibration curve is simply expressed as in Eq. (4.13) in the present

numerical analysis, hence the relation (5.4) becomes

Nuconv = Nunc exp(

Re f

nconv

)+

(ncal1Rencal2

f + ncal3

) (1 − exp

(Re f

nconv

)) (T f

T∞

)0.17

. (5.5)

In accordance with the finding by Collis & Williams (1959), this relation converges

to Nunc in the limit of Ref → 0, while it asymptotes to the original calibration curve

for larger Reynolds number. Furthermore, the undershoot of the Nusselt number in the

mixed convection region 0 < Re < Gr1/3 is simulated. By choosing the coefficient so

that the improved calibration curve deviates by 1% from pure forced convection curve at

Re = Gr1/3, the heat transfer due to convection is determined with nconv = 0.29 and 0.23

for overheat ratios aT = 0.27 and 0.96, respectively, where pure forced convection was

calculated by eliminating the gravitational acceleration in the original calibration cases.

The calculated Nusselt number through Eq. (5.5) is presented in Figures 5.1 and 5.2.

5.3 The effect of heat conduction

Heat conduction is usually observed together with convection as long as measurements

or simulations are performed in a fluid, thus separating their contribution from each other

is hardly achievable. While in solid materials, the heat transfer can be always considered

as pure conduction. Thereupon, in the present work, the calculation in the same domain

as Figure 4.1 but with a solid region with the air thermal properties instead of the fluid

region was carried out to extract the effect of pure conduction.

61

10-4 10-3 10-2 10-1

Ref

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Nu

calibration points

pure natural convection

Nuconv, calibration curve (improved)

calibration curve (original)

pure forced convection

Fig. 5.1: Modeled heat transfer due to heat convection at aT = 0.27 in the freestream.

10-4 10-3 10-2 10-1

Ref

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Nu

calibration points

pure natural convection

Nuconv, calibration curve (improved)

calibration curve (original)

pure forced convection

Fig. 5.2: Modeled heat transfer due to heat convection at aT = 0.96 in the freestream.

Incidentally, heat conduction between primitive geometries is determined uniquely.

For a long cylinder with the diameter d, the length l and the temperature T1 located yw

apart from an unbounded isothermal wall with the temperature T2 as shown in Figure 5.3,

the conduction shape factor Hcond,isoth is calculated as (see Incropera et al., 2006, Ch. 4.3)

Hcond,isoth =2πl

acosh(2yw/d), (5.6)

62

Fig. 5.3: A long cylinder with the diameter d, the length l and the surface temperature T1

located yw away from an unbounded isothermal wall with the temperature T2.

and the heat transfer from this cylinder is calculated consequently as

Q = qπdl = kmedHcond,isoth(T1 − T2), (5.7)

Nucond,isoth =q

qref. (5.8)

where kmed is the thermal conductivity of the medium where the cylinder is placed, and

the reference heat flux is calculated as qref = k(T f )(T1 − T2)/d with T f = (T1 + T2)/2.

The result of pure conduction calculation and the theoretical Nusselt number for an

isothermal wall is plotted together in Figure 5.4. Heat conduction on an aluminum wall

shows good agreement with the theoretical curve of the isothermal wall. This behavior

explains the previously stated feature in the temperature contour shown in Figure 4.20,

namely that aluminum maintains its temperature identical to the atmospheric value.

Figure 5.5 shows the temperature distributions for the pure conduction cases. The

temperature gradient around the wire surface becomes more gentle as the sensor moves

away from the interface, whereby the heat transfer contribution from the conduction part

is certainly a function of the distance between wire and the wall. In other words, the

wall presence distorts the temperature distribution from the unaffected state, i.e. the state

without a wall, and the distortion becomes more significant with lowering the sensor

height.

Furthermore, it has been seen from the previously presented figures that the tempera-

ture gradient around the wire becomes gentler and the Nusselt number becomes smaller as

the flow velocity decreases if there is no wall nearby, i.e. heat diffuses farther around the

wire at slower velocities, although that influenced area of heat shrinks when a solid wall is

63

placed nearby after all. Hence, the temperature field should be altered more significantly

as the flow velocity becomes smaller when compared at the same wire height.

Here, lowering the wire height and decreasing the flow velocity have similar effects

on the alteration of the temperature field, whereby the conduction part in the total heat

transfer is proposed in the following form with the clue of Eq. (5.6):

f2(Nucond) = f2(Re f , yw) =ncond1

acosh

1 + Rencond2f

ncond3

+ 1

Nucond(yw), (5.9)

where Nucond(yw) is approximately equal to Nucond,isoth(yw) for the highly conductive walls.

Function (5.9), however, does not take into account the aforementioned peculiar under-

shooting behavior in the convection curve, but yields values increasing monotonically

with the increasing Reynolds number. Hence, the following revised function is proposed

by taking the convective heat transfer Nuconv as a variable instead of the Reynolds number,

namely as

f2(Nucond) = f2(Nuconv, yw) =ncond1

acosh(1 +

Nuconvncond2

ncond3

)+ 1

Nucond(yw). (5.10)

5.4 Final form of the model on the wire-wall heat transferand its possibility for generalization

By interpreting the total heat transfer as the sum of the pure-convection state without a

wall and the distortion from it due to the wall, relation (5.1) is written as

Nutotal = Nuconv + f2(Nucond), (5.11)

where the convection part and the conduction part are denoted as Eq. (5.4) and (5.10),

respectively. The conduction is likely to be independent of the temperature loading aT

considering the results in the natural convection measurement in the experimental part.

Choosing the coefficients empirically as (nconv1, nconv2, nconv3) ≈ (0.22, 20, 3.9 × 10−10)

and (0.23, 22, 3.9 × 10−11) for aT = 0.27 and 0.96, respectively, the modeled function of

64

0 200 400 600 800 1000y∗

w

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

Nu

theoretical, isothermal

present, conduction, aluminum

present, conduction, Plexiglas

Fig. 5.4: Heat transfer due to heat conduction at aT = 0.27 as a function of the sensorlocation. The theoretical curve for an isothermal wall are calculated based on the shapefactor in Eq. (5.6).

the total heat transfer yields the curves presented in Figures 5.6 and 5.7, and it captures

the behavior of the simulation results for the wire on an aluminum wall at y∗w = 100 for

both overheat ratios.

The aforementioned motivations for the establishment of the proposed model are

mainly oriented to highly conducting wall materials for which an approximation of an

isothermal is valid. When poorly conducting materials such as Plexiglas, glass and styro-

foam are employed, the temperature at the interface right beneath the wire varies depend-

ing on the velocity as it is seen in the temperature contours, and also on the thickness of

the wall as stated in several previous works (Shi et al., 2003; Zanoun et al., 2009).

Assume that the wall with a certain thickness L bottom has an temperature T∞, and

that the flow velocity is small enough so that the one-dimensional temperature distribution

in y-direction as shown in Figure 5.8 is approximated. Hereby, due to the continuity of

the wall-heat flux and temperature, the following is satisfied:

∂T∂y

∣∣∣∣solid=

kfluidksolid

∂T∂y

∣∣∣∣fluid, (5.12)

Tw = T∞ +(L kfluid

ksolid+ yw

)∂T∂y

∣∣∣∣fluid. (5.13)

65

x*

y*0.02

0.01

0.15

0.05

0.09

0.07

0.03

0.10

x*

y*

0.15

0.09

0.20

0.30

0.10

x*

y*

0.020.01

0.15

0.20

0.30

0.05

0.09

0.07

0.03

0.10

x*

y*

0.15

0.20

0.30

0.09

0.10

(i) S = 100

(ii) S = 300

Fig. 5.5: Local temperature distribution around the wire at aT = 0.27 at the location ofy∗w = 100. Left column: results on an aluminum wall. Right column: results on a Plexiglaswall.

Thus, the temperature gradient in the fluid region is expressed as

∂T∂y

∣∣∣∣∣fluid=

Tw − T∞L kfluid

ksolid+ yw. (5.14)

By introducing this surface temperature for the compensation, the conduction part of the

model is reformulated as

f2 (Nucond) = f2

(Nuconv, yw,

kfluid

ksolid

)=

ncond1

acosh

1 + Nuconvncond2f

ncond3

+ 1

yw

L kfluidksolid+ yw

Nucond,isoth(yw). (5.15)

The modeled Nusselt numbers for the aluminum wall and Plexiglas walls of 1000d and

66

10-4 10-3 10-2 10-1

Ref

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Nu

numeric., yw = 100d, aluminum, aT = 0.27

model, Nutotalmodel, f1(Nuconv) = Nuconvmodel, f2(Nucond)

Fig. 5.6: The modeled heat transfer from the wire at aT = 0.27 on an aluminum wall aty∗w = 100.

10-4 10-3 10-2 10-1

Ref

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Nu

numeric., yw = 100d, aluminum, aT = 0.96

model, Nutotalmodel, f1(Nuconv) = Nuconvmodel, f2(Nucond)

Fig. 5.7: The modeled heat transfer from the wire at aT = 0.96 on an aluminum wall aty∗w = 100.

5000d thickness are shown in Figure 5.9. The effect of the thickness of the poorly conduc-

tive wall is observed, i.e. a thicker wall leads to less heat transfer. The present numerical

results on the Plexiglas (5000d thick) was found to show discrepancies with the model

based on Eq. (5.15), which is thought to be due to the difference of the boundary condition

at the domain bottom: the numerical analysis set a Neumann zero-gradient temperature

67

Fluid region

Solid region

Highly conducting wall

Poorly conducting wall

Fig. 5.8: The hypothetical one-dimensional temperature distribution in the regions.

10-4 10-3 10-2 10-1

Ref

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Nu

numeric., yw = 100d, aluminum, aT = 0.27

numeric., yw = 100d, Plexiglas, aT = 0.27

model, yw = 100d, aluminum, aT = 0.27

model, yw = 100d, aluminum, aT = 0.96

model, yw = 100d, Plexiglas (1000d thick), aT = 0.27

model, yw = 100d, Plexiglas (1000d thick), aT = 0.96

model, yw = 100d, Plexiglas (5000d thick), aT = 0.27

model, yw = 100d, Plexiglas (5000d thick), aT = 0.96

Fig. 5.9: The modeled heat transfer from the wire at different overheat ratios on variouswalls at y∗w = 100.

condition at the bottom instead of the constant value.

5.5 Simulation of the fluctuating output

The fluctuating output from HWA is simulated by employing the established model. The

turbulence intensity is known to be roughly around urms/U = 0.4 according to previous

works (see Kim et al., 1987). If the velocity gradient at the inlet S is pulsated with this am-

plitude on the proposed model, the total amount of heat transfer fluctuates consequently.

The calibration curve in actual experiments should be the one in Eq. (5.5) instead of the

68

previously submitted one in the previous chapter. Hereby, applying the new calibration

function, one obtains the “measured velocity” output from HWA based on the model.

Figure 5.10 presents the maximum, minimum and mean of the measured fluctuating

velocity as a function of the inner-scaled wire height. It should be noted that the present

calculation model varies the scaled height y+ by changing the inlet velocity gradient S ,

while the actual wire height is left constant (yw = 100 for the present case). It is observed

that the difference of the overheat ratio emerged in range y+ . 5. The deviation of the

minimum value is larger than that of the maximum and the mean curves when compared

at the same y+. This tendency is qualitatively in accordance with the present experimental

results: only the lower speed region of the PDF of the velocity seems to be overestimated.

Figure 5.11 shows the output turbulence intensity, i.e. urms/U as a function of the

inner-scaled wire height. The dependency of the turbulence intensity on the overheat

ratio is consistent with the experimental result: the higher the overheat ratio, the lower

becomes the turbulence intensity. When the same velocity is applied, the Reynolds num-

ber evaluated at the film temperature Re f is always lower at higher overheat ratio due to

the difference of thermophysical properties ρ f and µ f . Furthermore, the curves of Nutotal

have gentler gradients as Re f decreases as it is seen in Figures 5.6 and 5.7. Hence smaller

fluctuation of Nu is observed at higher overheat ratio. Applying small fluctuation of the

Nusselt number to the calibration curve yields smaller velocity fluctuation consequently,

which is why the rms is registered lower at higher temperature loading of the wire. The

present model in the form of Eq. (5.11) is not able to maintain the consistent behav-

ior regarding the difference between two different overheat ratios, which appears around

y+ ≈ 1. This is a result from function (5.5) for simulating the undershooting behavior of

the convection part. Further discussion with an analytical approach should be necessary

to investigate the validity of this function.

69

0 1 2 3 4 5 6 7y+

0

1

2

3

4

5

6

7

u+

Fig. 5.10: The modeled fluctuating output from HWA at different overheat ratios on alu-minum at y∗w = 100. The solid red line and the broken red line represent the overheatratios aT = 0.27 and 0.96, respectively. The broken black line is the theoretical linear plotU+ = y+. Three curves in the same color and the line shape illustrates maximum, mean,minimum of the fluctuation.

0 1 2 3 4 5 6y+

0

0.1

0.2

0.3

0.4

urm

s/U

model, aluminum, yw = 100d, aT = 0.27

model, aluminum, yw = 100d, aT = 0.96

Fig. 5.11: The modeled rms output from HWA at different overheat ratios on aluminumat y∗w = 100.

5.6 Possible issues of the proposed model

The proposed model in the present chapter was found to be useful for the qualitative dis-

cussion on the present topic. However, for the practical use of this model, it is necessary

70

to take more considerations into account. For example, the location and the temperature

of the high temperature region inside a poorly conducting wall should be expressed as

a function of certain parameters, e.g. flow velocity and the sensor height, which has a

role in the conduction part of the model. The coefficients in the model were determined

empirically in the present study, however, it should be possible to relate them to cer-

tain parameters. Furthermore, if one applies the present model to actual measurements,

the wire length and consequently three-dimensional effects of convection and conduction

should be accounted for. Overall, it is needed to collect more numerical data in a wider

parameter range to improve the applicability of the present model.

71

Chapter 6

Conclusions

The present study investigated the parameter dependency and the principles of the so-

called “wall effect” of hot-wire anemometry, namely the overestimation of the mean ve-

locity in the near-wall region. The systematic parameter study through experiment and

numerical analysis was performed in order to revisit the general idea of the topic. In ad-

dition, a theoretical approach to the wire-wall heat transfer was carried out to propose

a modeled heat transfer correlation, which would give ideas of how the convection and

conduction are taking part in the phenomenon.

In the experimental part, the measurements without flow and under flow conditions were

carried out. The effects of the wall conductivity, the overheat ratio, and the sensor dimen-

sions were concerned, and the results agreed generally with the previous literature: higher

wall conductivity, larger overheat ratio and larger exposed wire area lead to a higher volt-

age output. However, under the no-flow measurement, it was found that the scaled voltage

output from a probe with different overheat ratios remains almost constant. This suggests

that the heat conduction in the near-wall region is not dependent on the overheat ratio.

In addition, the fluctuation of the output from wind-tunnel measurements was also dis-

cussed. It was observed that employing a higher wall conductivity and a larger overheat

ratio results in smaller rms values, i.e. smaller magnitude of the fluctuation. This effect

was found to be due to the overestimation in the low-speed region in the velocity PDF,

while the high-speed region remains constant, which narrowed down the entire probability

72

distribution. The aforesaid influences by parameters under flow condition were observed

only in the viscous sublayer, and one should note that the effect can easily be “hidden”

when the measured velocity profiles are employed to determine the absolute wall distance

and the friction velocity as it is common in the literature.

In the numerical analysis part, simulations of the flow and temperature field between the

air and the solid wall was conducted. The calculated results showed a consistent param-

eter dependency with previous studies and wind-tunnel experiments. The present study

pointed out that the inner-scaled velocity as a function of the inner-scaled wire height

does not necessarily yield a single curve. This implies that the inner scale is not adequate

to discuss the present problem of heat transfer nevertheless it has been used in numerous

publications. By investigating the temperature distribution around the wire and inside the

wall, aluminum was found to be acting approximately as isothermal, while Plexiglas ac-

cumulates heat inside, thus creating a high temperature region. The temperature gradient

beneath the wire is steepened due to the presence of the wire for the lower velocities or the

lower sensor height, where temperature drops so it meets the atmospheric value especially

for the aluminum case. It was also found that the temperature distribution resembles more

to that of the freestream state with the increasing flow velocity or the wire-wall distance.

By considering the findings from the experimental part and the numerical analysis, a

theoretical model on the wire-wall heat transfer was proposed. The model consists of the

superposition of the contributions from convection and conduction. For the convection

part, the improved calibration curve capable of simulating the natural convection effect is

considered. The proposed model shows fine agreement with the present numerical results,

and the possibility of the generalization for the various heat conductivities is suggested.

However, it is still necessary to widen the parameter range to comprehend the parameter

dependency of the model coefficients, also to improve its applicability.

73

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78

Appendices

79

Appendix A

An attempt for more accurate determination of the abso-lute wall distance

The accurate determination of the distance between a hot-wire sensor and a wall surface is

an important matter, especially when one calculates the wall-shear stress. In the previous

studies, researchers attempted to determine this distance with several types of techniques,

such as measuring the distance directly with the aid of microscopes (Chew et al., 1998),

estimating it from the no-flow data (Bhatia et al., 1982; Zanoun et al., 2009), optical

methods using lasers (Janke, 1987), etc. In the natural convection measurement in the

present work, the method using a digital camera image was used to determine the point

yw = 0 as explained in section 3.1.2, which was dependent on the observers’ practice and

subjective judgment. The difference of the output due to different parameters such as wall

material, overheat ratio, and sensor size was significant enough to be compared. However,

for the investigation of the wire-wall arrangement, such a relatively subjective estimation

of the zero-point could be too rough and possibly leads to misinterpretation of the result.

Hereby, another way of the determination is introduced, which was used only in a

measurement concerning the wall-arrangement. The procedure is explained as follows:

the probe was initially connected to a multimeter instead of the CTA system. Then the

probe was traversed closer to the wall surface as the resistance value on the multimeter

was monitored. When the aluminum wall was being employed, the value on the multi-

meter changes the moment when the probe (either the prongs or the wire to be precise)

touches the wall. At this point the micrometer was set to yw = 0 and the probe was con-

nected to the CTA system, the electrical current was applied from above yw = 0.01 mm,

80

then the measurements were carried out accordingly.

With this improved method, the determination of the absolute distance becomes little

more accurate, however, this procedure has a drawback too. The procedure requires the

probes to actually touch the surface, which can damage the wire. Considering the dis-

crepancy of the wall-remote voltage in the no-flow measurement at the arrangement (a)

in Figure 3.5 (see Figure 3.11), the probe is likely to be damaged due to this process and

changed into the state which was used for all the rest measurements.

81

Appendix B

Validation of the negligible effect of the heat radiation

The influence of heat radiation on HWA reading is usually not taken into account in

most of the previous literature. The present study also neglects this effect based on the

evaluation of its magnitude, which is explained in this appendix.

The radiation heat transfer between black bodies can be determined by their geometri-

cal arrangement. The following is the radiation shape factor for a circular cylinder of the

length l to a symmetrically placed infinitely long rectangular flat plate nearby as shown in

Figure B.1, introduced by Feingold & Gupta (1970):

Frad =1π

atanbyw. (B. 1)

The heat flux at the cylinder surface due to the radiation is

Q = qπdl = σFradπdl(T 41 − T 4

2 ), (B. 2)

where σ denotes the Stefan-Boltzman constant: σ = 5.670367 × 10−8 Wm−2K−4. For

the present case, the rectangular flat plate is infinitely large compared to the sensor height

Fig. B.1: A long cylinder with the diameter d, the length l and the surface temperatureT1 located yw away from a symmetrically placed infinitely long rectangular flat plate withthe temperature T2.

82

thus the shape factor takes the limit of Frad → 1/2. Taking the reference heat flux as

qref = kair(T1−T2)/d and the Nusselt number due to radiation yields values of the order of

10−4 – 10−3 for the concerned overheat ratios in the present numerical analysis. The calcu-

lation above is for black bodies but the hot-wire sensor and the wall are usually not black

bodies, thus the radiation heat transfer becomes even smaller. This calculated Nusselt

number is small enough compared to the contribution from conduction and convection

as it is obvious in the present numerical results, thus the effect of the heat radiation is

reasonably negligible.

83