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Validation of Computational Fluid Dynamics Simulations of
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Validation of Computational Fluid Dynamics Simulations of
Turbulent Heat Transfer Associated with the Hot Soaking
Phenomenon
By

Takesha Sattiewhite

A THESIS

Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of

MASTER’S OF SCIENCE
MECHANICAL ENGINEERING

2007

ABSTRACT

Validation of Computational Fluid Dynamics Simulations of
Turbulent Heat Transfer Associated with the Hot Soaking
Phenomenon
By

Takesha Sattiewhite

Hot soaking is a process that is used during the testing of a vehicle to study
the effects of exposure to heat on vehicle components. The occurrence of a
rapid increase of temperature during this process is important since it can

affect the durability and performance of vehicle components.

In the present work, computational fluid dynamics (CFD) simulations were
performed to study the performance of various turbulence models, needed in
computations associated with convective heat transfer with the help

of invariant or realizability diagrams. Three cases exhibiting phenomena
associated with hot soaking were evaluated to understand the performance of
turbulence models and CFD for simulating turbulent heat transfer: single
phase turbulent flow through a pipe with a 90 degree bend, natural convection
in a ZD square enclosure, and a two dimensional simulation of a cylinder
above a heated plate. These cases aided in the development of a numerical
simulation of the hot soaking process. Numerical results of the simulation
illustrate the high temperature regions developed on the cylinder surface and

the production of realizable turbulent heat transfer from the heated plate.

Copyright by
Takesha Sattiewhite
2007

ACKNOWLEDGEMENTS

I would like to thank my advisor, Dr. Andre Benard, for his mentoring and
support throughout my enrollment in the graduate program. His guidance
was essential to the development of this thesis and has expanded my

knowledge of turbulent heat transfer.

I would also like to thank Dr. Tom Shih for his guidance and support during
my first year of the graduate program. My thanks also go to the members of
my thesis committee, Dr. Charles A. Petty and Dr. Farhad Jaberi, for
providing valuable insights during my thesis defense which was used to

improve this thesis.

Last, but not least, I would like to thank my family and close friends for their

encouragement and support throughout my graduate studies.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................................. vii
LIST OF FIGURES .................................................................. x
NOMENCLATURE ................................................................. xv
I. INTRODUCTION .................................................................. 1
a. Hot Soaking and Its Importance ....................................... 1
b. Focus of Thesis ......................................................... 6
c. Objective .................................................................. 8

II. SINGLE PHASE FLUID FLOW THROUGH A PIPE WITH 90 DEGREE

BEND ........................................................................... 9
a. Problem Description ................................................ 9
b. Problem Formulation .............................................. 12

i. Initial Conditions .............................................. 12

ii. Boundary Conditions ..................................... 12

c. Governing Equations .............................................. 15
i. Turbulence Modeling ..................................... 15

ii. Heat Transfer .............................................. 16

iii. Velocity Profile Computation ............................ 17

d. Anisotropic Invariant Map .............................................. 17
e. Numerical Method of Solution ..................................... 21
f. Numerical Results ....................................................... 22
Ill. NATURAL CONVECTION IN A SQUARE ENCLOSURE .......... 49
a. Problem Description .............................................. 49
b. Problem Formulation .............................................. 52
i. Operating Conditions ..................................... 52

ii. Boundary Conditions ..................................... 53

c. Governing Equations .............................................. 54
i. Turbulence Modeling ..................................... 54

ii. Heat Transfer .............................................. 55

iii. Nusselt Number Computation ............................ 55

d. Numerical Method of Solution ..................................... 56
e. Numerical Results ....................................................... 56

IV. TWO DIMENSIONAL MODEL OF CYLINDER ABOVE A

HEATED PLATE ................................................................ 70

a. Problem Description .............................................. 70

b. Problem Formulation .............................................. 77

i. Initial Conditions .............................................. 77

ii. Boundary Conditions for Transient Analysis . ...78

c. Governing Equations .............................................. 82

i. Turbulence Modeling ..................................... 83

ii. Heat Transfer .............................................. 83

iii. Pressure Distribution Computation ................... 84

d. Anisotropic Invariant Map .............................................. 84

e. Numerical Method of Solution ..................................... 89

i. Grid Sensitivity Study ..................................... 90

f. Numerical Results ....................................................... 91

i. Case I: Grid Sensitivity Study ............................ 92

ii. Case II: Turbulence Model Study ........................... 112

V. CONCLUSIONS AND RECOMMENDATIONS ........................... 123
APPENDIX ................................................................................. 127
APPENDIX A ........................................................................ 128
APPENDIX B ....................................................................... 133
APPENDIX C ........................................................................ 135
BIBLIOGRAPHY ........................................................................ 150

vi

LIST OF TABLES

Table 2.1 lists the material properties for Air and Aluminum . . ....11

Table 2.2 lists the vorticity magnitude generated in a cross section of the pipe
for the different turbulence models and Reynolds number values .......... 31

Table 3.1 lists the material properties for Air and Aluminum ................... 52

Table 3.2 lists the average Nusselt number published by
different sources ......................................................................... 69

Table 4.1 lists the materials used for the components
of the 2D model ......................................................................... 76

Table 4.2 lists the properties of Air, Aluminum, and Steel ................... 76

Table 4.3 lists the second (II) and third (Ill) scalar invariants of anisotropy
over a range of time from zero seconds to 1,173 seconds generated from
calculating the maximum anisotropy value of model for the Grid

Type 4 K-Epsilon Model ............................................................. 102

Table A-1: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map for single phase fluid flow through a pipe using the
K-Epsilon Turbulence Model at Re N0. = 1000 .................................... 132

Table A-2: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map generated for single phase fluid flow through a pipe
using the K-Epsilon Turbulence Model at Re No. = 9500 ................. 132

Table A-3: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map generated for single phase fluid flow through a pipe
using the K-Epsilon Turbulence Model at Re N0. = 95,000 ................. 133

Table A-4: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map for single phase fluid flow through a pipe using the
Reynolds Stress Turbulence Model at Re N0. = 1000 .......................... 133

Table A-5: List of the anisotropic values calculated to generate the

Anisotropic Invan'ant Map for single phase fluid flow through a pipe using the
K-Epsilon Turbulence Model at Re No. 9500 .................................... 134

vii

Table A-6: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map for single phase fluid flow through a pipe using the
K-Epsilon Turbulence Model at Re No. 95,000 .................................... 134

Table A-7: List of the dimensionless values of radius and velocity used to
create the plot of the Velocity Profile of Fully Developed Flow for single
phase fluid flow through a pipe using the K-Epsilon and Reynolds Stress
Turbulence Model at different Reynolds numbers ......................... 135

Table B-1: List of the dimensionless parameters used to create the plot of
dimensionless height (in the y-direction) versus dimensionless temperature
generated inside the 2D Enclosure for the Realizable K-Epsilon turbulence
model without radiation ............................................................... 137

Table C-1: List of the temperature profile specified for the inner surface of the
heated plate .................................................................................. 139

Table C-2: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map of the steady state calculation of the Reynolds
Stress Model for Grid Types 1 through 4 140

Table C-3: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map of the transient calculation of the K-Epsilon Model
for Grid Type 4 (Maximum Anisotropy Value of Mode and value measured
near bottom cylinder wall) ............................................................... 141

Table C-3 (cont'd): List of the anisotropic values calculated to generate the
Anisotropic Invariant Map of the transient calculation of the K-Epsilon Model
for Grid Type 4 (Maximum Anisotropy Value of Mode and value measured
near bottom cylinder wall) ............................................................... 142

Table C-4: List of the Y+ values measured to create the plot of the Grid
Refinement versus the Turbulence Y+ values generated at the bottom
cylinder wall ................................................................................. 142

Table 0-5: List of the temperature values measured to create the Hot Soaking
temperature profile of the bottom wall from the cylinder for the transient
calculation of the K-Epsilon Model for Grid Type 4 ............................... 143

Table C-6: List of the temperature values calculated to create the temperature

profiles of the bottom wall of the cylinder from the transient calculation of the
K—Omega Turbulence Model for Grid Type 4 .................................... 144

viii

Table C-7: List of the temperature values calculated to create the temperature
profiles of the bottom wall of the cylinder from the transient calculation of the
RSM Turbulence Model for Grid Type 4 ............................................. 145

Table C-8: List of the heat flux values calculated to generate the plot of the
dominant mode of heat transfer generated as a result of the hot soaking
phenomenon on the top wall of the plate for K—Epsilon Model . ..1 46

Table C-9: List of the heat flux values calculated to generate the plot of the
modes of heat transfer generated as a result of the hot soaking phenomenon
on the bottom wall of the cylinder for K-Epsilon Model ................. 146

Table C-10: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map from the use of different turbulence models for Grid
Type 4, steady state (Anisotropy Value Near Wall of Cylinder) ........ 147

Table 0-1 1: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map from the K-Omega turbulence models for Grid Type
4, unsteady (Anisotropy Value Near Wall of Cylinder) .......................... 148

Table C-11 (cont’d): List of the anisotropic values calculated to generate the
Anisotropic Invariant Map from the K-Omega turbulence models for Grid Type
4, unsteady (Anisotropy Value Near Wall of Cylinder) .......................... 148

Table C-12: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map from the Reynolds Stress turbulence models for
Grid Type 4, unsteady (Anisotropy Value Near Wall of Cylinder) ........149

Table 0-13: List of the pressure coefficient values calculated to generate the
plot of the surface pressure distribution generated on the bottom
wall of the cylinder ....................................................................... 150

Table C-14: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map generated from the combination of the Reynolds
stresses and the mean temperature evaluated using the steady state K-
Epsilon turbulence model for Grid Type 4 (Measured between Cylinder &
Heated Plate) ....................................................................... 151

Table C-15: List of the anisotropic values calculated to generate the
Anisotropic Invariant Map generated from the combination of the Reynolds
stresses and the mean temperature evaluated using the unsteady K-Epsilon
turbulence model for Grid Type 4 at t = 60 seconds (Measured between
Cylinder & Heated Plate) ............................................................... 152

ix

LIST OF FIGURES

Figure 2.1 illustrates the triangular grid mesh used for the 3D Pipe

Model for Single Phase Flow. ....................................................... 10
Figure 2.2 illustrates a closer view of the triangular grid mesh used

for the 3D Pipe Model for Single Phase Flow. ..................................... 11
Figure 2.3 illustrates the Anisotropic Invariant Map

(Krogstag and Simonsen 2005). ....................................................... 19
Figure 2.4 illustrates the ZD cross section cut from the pipe. ................... 24

Figure 2.5 illustrates the vector plot of the vorticity magnitude generated in a
cross section of the pipe for the K-Epsilon Turbulence Model with Re N0. =
1000 ........................................................................................... 25

Figure 2.6 illustrates the vector plot of the vorticity magnitude generated in a
cross section of the pipe for the RSM Turbulence Model with Re
No. = 1000 .................................................................................. 26

Figure 2.7 illustrates the vector plot of the vorticity magnitude generated in a
cross section of the pipe for the K-Epsilon Turbulence Model with Re No. =
9500 ........................................................................................... 27

Figure 2.8 illustrates the vector plot of the vorticity magnitude generated in a
cross section of the pipe for the RSM Turbulence Model with Re
N0. = 9500 .................................................................................. 28

Figure 2.9 illustrates the vector plot of the vorticity magnitude generated in a
cross section of the pipe for the K-Epsilon Turbulence Model with Re No. =
95,000 .................................................................................. 29

Figure 2.10 illustrates the vector plot of the vorticity magnitude generated in a
cross section of the pipe for the RSM Turbulence Model with Re
N0. = 95,000 ......................................................................... 30

Figure 2.11 illustrates the static temperature contours generated in a cross
section of the pipe for the K-Epsilon Turbulence Model with
Re No. = 1000 ............................................................................... 33

Figure 2.12 illustrates the static temperature contours generated in a cross
section of the pipe for the RSM Turbulence Model with
Re No. = 1000 ......................................................................... 34

Figure 2.13 illustrates the static temperature contours generated in a cross
section of the pipe for the K-Epsilon Turbulence Model with
Re N0. = 9500 ............................................................................... 35

Figure 2.14 illustrates the static temperature contours generated in a cross
section of the pipe for the RSM Turbulence Model with
Re N0. = 9500 ......................................................................... 36

Figure 2.15 illustrates the static temperature contours generated in a cross
section of the pipe for the K-Epsilon Turbulence Model with
Re N0. = 95,000 ............................................................................ 37

Figure 2.16 illustrates the static temperature contours generated in a cross
section of the pipe for the RSM Turbulence Model with
Re N0. = 95,000 ......................................................................... 38

Figure 2.17 illustrates the Anisotropic Invariant Map generated for single
phase fluid flow through a pipe using the K-Epsilon Turbulence
Model at Re N0. = 1000 ................................................................ 41

Figure 2.18 illustrates the Anisotropic Invariant Map generated for single
phase fluid flow through a pipe using the K-Epsilon Turbulence
Model at Re No. = 9500 ................................................................ 42

Figure 2.19 illustrates the Anisotropic Invariant Map generated for single
phase fluid flow through a pipe using the K-Epsilon Turbulence
Model at Re N0. = 95,000 ................................................................ 43

Figure 2.20 illustrates the Anisotropic Invariant Map generated for single
phase fluid flow through a pipe using the Reynolds Stress Turbulence Model
at Re N0. = 1000 ......................................................................... 44

Figure 2.21 illustrates the Anisotropic Invariant Map generated for single
phase fluid flow through a pipe using the K-Epsilon Turbulence
Model at Re No. 9500 ................................................................ 45

Figure 2.22 illustrates the Anisotropic Invariant Map generated for single
phase fluid flow through a pipe using the K-Epsilon Turbulence

Model at Re No. 95,000 ................................................................ 46
Figure 2.23 illustrates the Velocity Profile of Fully Developed Flow generated
for single phase fluid flow through a pipe using the K-Epsilon and Reynolds
Stress Turbulence Model at different Reynolds numbers ................... 48

Figure 3.1 illustrates the quad grid mesh used for the 2D Enclosure 50

xi

Figure 3.2 illustrates a closer view of the quad grid mesh used for the 2D
Enclosure. ................................................................................... 51

Figure 3.3 illustrates the contours of the stream function generated inside the
ZD Enclosure for the model (with laminar flow but without radiation) published
in Fluent 6.1 tutorial guide ....................................................... 58

Figure 3.4 illustrates the contours of the stream function generated inside the
2D Enclosure for the model generated with Realizable K-Epsilon Turbulence
model but without radiation ....................................................... 59

Figure 3.5 illustrates the vectors of the velocity magnitude generated inside
the 20 Enclosure for the Realizable K-Epsilon Turbulence
model without radiation ................................................................ 61

Figure 3.6 illustrates the contours of the static temperature generated inside
the 2D Enclosure for the model (with laminar flow but without radiation)
published in Fluent 6.1 tutorial guide .............................................. 62

Figure 3.7 illustrates the contours of the static temperature generated inside
the 20 Enclosure for the model generated with Realizable K-Epsilon
Turbulence model but without radiation ..................................... 63

Figure 3.8 illustrates the of the velocity in the Y direction generated inside the
2D Enclosure for the Model (with laminar flow but without radiation) published
in Fluent 6.1 tutorial guide ....................................................... 65

Figure 3.9 illustrates the of the velocity in the Y direction generated inside the
2D Enclosure for the model generated with Realizable K-Epsilon Turbulence
model but without radiation ....................................................... 66

Figure 3.10 illustrates the plot of dimensionless height (in the y-direction)
versus dimensionless temperature at mid-width generated inside the 2D
Enclosure for: (a) the Realizable K-Epsilon turbulence model without radiation
and (b) the reference solution measure with an aspect ratio of 6 and Rayleigh
number of 7.12x1010 (Velusamy et al. (2001)) ..................................... 68

Figure 4.1 illustrates the geometry of the ZD model of a cylinder above
a heated plate ................................................................ 71

Figure 4.2 illustrates a closer view of the geometry of the ZD model
of a cylinder above a heated plate .............................................. 72

Figure 4.3 illustrates the meshed domain of the 2D model of a Cylinder
above a heated plate ................................................................ 73

xii

Figure 4.4 illustrates a closer view of the meshed domain of the 2D
model of 3 Cylinder above a heated plate ..................................... 74

Figure 4.5 illustrates the triangular and quad shaped elements used to mesh
the domain of the 2D model of 3 Cylinder above a heated plate .......... 75

Figure 4.6 illustrates the temperature profile specified for the inner
surface of the heated plate ....................................................... 81

Figure 4.7 illustrates the region (line between bottom wall of cylinder and top
of heated plate centered at x = 0) where the turbulence anisotropy was
calculated to evaluate the anisotropic invariance of the turbulent heat transfer
generated by the model in equation (4.19) .................................... 88

Figure 4.8 Illustrates the different types of grid adaption refinements that were
generated between the cylinder and heated plate: (a) represents Grid Type 1
(the course grid), (b) represents Grid Type 2 (c) represents Grid Type 3 and
(d) represents Grid Type 4 (the finest grid) ............................ 93

Figure 4.9 illustrates the Anisotropic Invariant Map generated from the steady
state calculation of the Reynolds Stress Model for Grid Type 1 .95

Figure 4.10 illustrates the Anisotropic Invariant Map generated from the
steady state and transient calculation of the Reynolds Stress
Model for Grid Type 2 ................................................................ 96

Figure 4.11 illustrates the Anisotropic Invariant Map generated from the
steady state calculation of the Reynolds Stress Model for
Grid Type 3 .................................................................................. 97

Figure 4.12 illustrates the Anisotropic Invariant Map generated from the
steady state calculation of the Reynolds Stress Model for
Grid Type 4 ................................................................. ' ................. 98

Figure 4.13 illustrates the Anisotropic Invariant Map generated from the
transient calculation of the K-Epsilon Model for Grid Type 4 (Maximum
Anisotropy Value of Model) .................................................... 100

xiii

Figure 4.14 illustrates a closer view of the Anisotropic Invariant Map
generated from the transient calculation of the K-Epsilon Model for Grid Type
4 (Maximum Anisotropy Value of Model) .................................. 101

Figure 4.15 illustrates the Anisotropic Invariant Map generated from the
transient calculation of the K-Epsilon Model for Grid Type 4 (Anisotropy Value
Near Wall of Cylinder) ............................................................ 103

Figure 4.16 illustrates a plot of the Grid Refinement versus the Turbulence Y+
values generated at the bottom cylinder wall .................................... 105

Figure 4.17 illustrates the Hot Soaking temperature profile of the bottom wall
of the cylinder generated from the transient calculation of the K-Epsilon Model
for Grid Type 4 ...................................................................... 106

Figure 4.18 illustrates the temperature profiles of the bottom wall of the
cylinder generated from the transient calculation of the K-Omega and
Reynolds Stress Turbulence Models for Grid Type 4 ......................... 108

Figure 4.19 illustrates the dominant mode of heat transfer generated as a
result of the hot soaking phenomenon on the top wall of the plate
for K-Epsilon Model .............................................................. 110

Figure 4.20 illustrates the modes of heat transfer generated as a result of the
hot soaking phenomenon on the bottom wall of the cylinder
for K-Epsilon Model .............................................................. 111

Figure 4.21 illustrates the Anisotropic Invariant Map generated from the use
of different turbulence models for Grid Type 4, steady state
(Anisotropy Value Near Wall of Cylinder) ................................... 114

Figure 4. 22 illustrates the Anisotropic Invariant Map generated from the
K-Epsilon turbulence models for Grid Type 4, unsteady (Anisotropy Value
Near Wall of Cylinder) . ...115

Figure 4.23 illustrates the Anisotropic Invariant Map generated from the K-
Omega turbulence models for Grid Type 4, unsteady (Anisotropy Value Near
Wall of Cylinder) ....................................................................... 116

Figure 4.24 illustrates the Anisotropic Invariant Map generated from the
Reynolds Stress turbulence models for Grid Type 4, unsteady (Anisotropy
Value Near Wall of Cylinder) ..................................................... 1 17

Figure 4.25 illustrates the surface pressure distribution generated on the
bottom wall of the cylinder .............................................................. 119

xiv

Figure 4.26 illustrates the Anisotropic Invariant Map generated from the
combination of the Reynolds stresses and the mean temperature evaluated
using the steady state K-Epsilon turbulence models for Grid Type 4 ........ 121

Figure 4.27 illustrates the Anisotropic Invariant Map generated from the
combination of the Reynolds stresses and the mean temperature

evaluated using the unsteady K-Epsilon turbulence models for
Grid Type 4 at T = 60 seconds ................................................... 122

“Ima es in this thesis are resented in color”.

XV

Gb

Gk

hr

1105)

Nu

Pr

NOMENCLATURE

absorption coefficient
Area (m2)

Anisotropy tensor
Constant

Heat capacity at constant pressure (J/Kg-K)

Coefficient of Pressure

Total energy (J)

Force vector (N)

Gravity (kg/m2)

Generation of turbulence kinetic energy due to buoyancy

Generation of turbulence kinetic energy due to mean velocity
gradients

Generation of specific dissipation rate
Fluid-side local heat transfer coefficient (W/mz-K)

Spectral intensity

Kinetic energy per unit mass (J/kg)
Mass Flow rate (kg/s)

Wavelength interval

Nusselt number

Pressure (Pascal)

Prandtl number

xvi

CI rad

R,r

Total Heat Flux (W/mz)

Radiation Heat Flux (W/mz)

Radius (m)

Modulus of mean rate-of-strain tensor
Non-dimensional temperature in Chapter 4 (K)
Fluctuating component of temperature (K)
Local fluid temperature (K)

Initial temperature (K)

Mean temperature (K)

Wall surface temperature (K)

Fluid temperature (K)

Velocity magnitude in the j-th direction (m/s)
Velocity magnitude in x - direction (m/s)
Average axial velocity (m/s)

Reynolds Stress (also u’u', u'v', u'w' , etc.)
Turbulent heat flux (W/mz)

Centerline velocity (m/s)

Overall velocity vector (m/s)

Dissipation of k due to turbulence

Dissipation of to due to turbulence

Second and third invariants (respectively)

xvii

Greek

Coefficient of thermal expansion (1/K)
Delta function

Kronecker delta function

Turbulent dissipation rate (m2/sz)

Density (kg/m3)

Turbulent Prandtl number for kinetic energy

Turbulent Prandtl number for turbulent dissipation rate
Stefan-Boltzmann constant (5.672 x1OW/m2-K4)

Shear Stress (Pascal)

Effective stress tensor (Pascals)

Deviatoric Stress Tensor

Dynamic viscosity (Pascal-seconds)
Turbulent Dynamic viscosity (Pascal-seconds)

Kinematic viscosity (mzls)
Specific dissipation rate
Effective diffusivity
Azimuthal angle

Time of the averaging process

xviii

I. INTRODUCTION

l.a. Hot Soaking

Hot soaking is a phenomenon often referred to as temperature soaking or heat
soaking. It is a process during which a vehicle’s engine is shut-off suddenly after
being ran through numerous test cycles. All sources of forced convection such
as the condenser, radiator, and fans are disabled. The vehicle is parked in a
chamber or near a wall, thus eliminating any source of airflow. As a result, the
temperatures of the components of the vehicle located near sources of heat
(exhaust system, engine, etc.) experience a rapid increase of temperature and
eventually decrease to ambient conditions over a period of time. “Soaking a
vehicle can also mean letting it sit (ignition off) and reach equilibrium with the
environment, whether it is hot or cold. So, hot soaking in this sense would mean
leaving the vehicle outside or in a hot chamber until the whole vehicle reached

the ambient temperature.” (Karlson 2003)

The occurrence of the rapid increase of temperature during the hot soaking
process is very important, since it can diminish the durability and performance of
vehicle components. The material of a component may degrade or melt, thus
exposing a flammable fluid that can lead to a thermal incident. The hot soaking
phenomenon is used during the testing of a vehicle in order to study the effects
of heat transfer and the thermal characteristics of its components. Some

applications of this phenomenon to vehicle testing include the study of the

exhaust system and nearby components of a vehicle, the study of catalytic
converter cool down process, the study of brake fluid temperature rise during

braking, and the study of evaporative emission testing.

One application of hot soaking is used to study the high temperatures generated
on components surrounding the exhaust system of a vehicle. During vehicle
testing, a temperature rise is generated on the exhaust system which affects the
components near the exhaust. According to David Turner, Thermal Integration
Engineer at GM, “The temperature effect of the soak is that the temperatures of
components near the exhaust system tend to have a spike in the temperatures
during the first few minutes of the soak. Then, the temperatures eventually
decrease back to ambient” (Turner 2003). He goes on to say, “Immediately
preceding the soak, the component temperatures are a result of the energy
balance on the component. Usually most components are being heated from
radiation from the exhaust and being cooled by conduction and convection.
Immediately, after soaking starts, the radiation input is similar to the pre-soak
level radiation, but convection is measurably decreased. So, a spike in the
component temperature occurs. Eventually, natural convection currents become
established and tend to cause air to rise up in the engine compartment and the
underbody of vehicles. This air is heated as it flows over the exhaust, and the
temperature of components directly over the exhaust system is bathed in this

heat, adding to the temperature increase in the components”. (Turner 2003)

The catalytic converter is one of the hottest components of the exhaust system.
Catalytic converters are typically located near the engine exhaust manifold.
Chung et al. (2003) performed a study of the use a numerical method to model
the temperature soaking phenomenon that occurs during the cool down process
of a 3D catalytic converter model. In this problem of temperature soaking, a
temperature rise is generated on the converter skin after the engine is shut-off
reaching a maximum. "T he maximum temperature, which the converter
assembly will see, is crucial in determining shell materials, types of mounting
mats (for ceramic substrate), and mat thickness in order to avoid converter
durability issues. This maximum temperature also needs to be considered in the
identification of converter configurations” (Chung et al. 2003). This study by
Chung et al. was used to develop a numerical simulation to predict the

temperature effects of the catalytic converter.

Lee (1999) studied a typical brake system model in order to predict the maximum
brake fluid temperature during the heat soaking period of a vehicle that is parked
after repetitive braking. Lee states "Long repetitive braking, such as one which
occurs during mountain descent, will result in a brake fluid temperature rise and
may cause brake fluid vaporization. This may be concern particularly for
passenger cars equipped with aluminum calipers and with a limited airflow to the
wheel brake systems”. If the performance is diminished then the functionality of
the brake system and safety of the driver becomes an issue. The brake system

components will become worn and fail, which can lead to an automobile

accident. Lee used the findings from the study of brake fluid temperature rise to
develop a numerical simulation to predict the temperature effect during the heat
soaking period. The numerical simulation was created to aid in the design of

more efficient brake systems.

Hsu et al. (2003) studied the speciation of organic gas hot soak emissions of
California light duty vehicles. Hot soaking emission test are used to estimate the
chemical composition of emissions from organic gas. According to Hsu, “Hot
soak emissions are comprised of fuel vapors emitted from a vehicle after the
engine is turned off. The elevated engine temperature causes fuel vaporization
from different sources such as fuel delivery lines, purge line to the canister, and
gas cap”. Hsu also states that “Speciation profiles are used in emission
Inventories, health risk assessments and photochemical modeling. The
Benzene mass fraction (3.43% of TOG) in the current gasoline vehicular hot
soak profile may over estimate benzene emissions from motor vehicles. In
response to Benzene over estimation concerns, recent hot soak speciation data
were requested from MSOD to update the hot soak profile.” As a result of
inaccurate estimations of the chemical composition of emissions, enhanced hot
soaking profiles have to be developed in order to correct this inaccuracy. The
accuracy of the hot soaking emissions is essential to appropriately assess

emission inventory, health risk, and ozone photochemical modeling.

Hot soaking allows the study and determination of the high temperatures that are

generated on the vehicle components to aid in the design of theses components

and to prevent the degradation of the components. Determining these high
temperatures is important because it will allow the prediction of thermal issues

concerning heat management.

The literature review on the applications of the hot soaking process to the
automotive industry demonstrated the usefulness of the hot soaking process in
studying temperature rises generated by sources of heat. Due to time
constraints and the constant demand for fast problem resolution, many auto
manufacturers use Computational Fluid Dynamics (CFD) programs that can
produce a numerical prediction of performance of a component in a short
amount of time. They cannot afford the time to develop an in depth study of the
use of CFD to accurately predict the Hot Soaking phenomenon with transient

analysis.

Most of the applications of the hot soaking process to study vehicle components
mentioned previously consisted of the study of the hot soaking process to create
a numerical simulation, except one: The study of the exhaust system mentioned
by Turner was an application of the hot soaking process that was based on he

thermal test procedure and not an numerical simulation. This study will be used
in the development of a numerical simulation of the Hot Soaking process through

an in depth evaluation of turbulence modeling and the heat transfer process.

I.b. Focus of Thesis

The focus of this thesis is to investigate the hot soaking phenomenon by
quantifying the effect of the temperature generated on a hollow cylindrical
component above a heated hollow horizontal plate. The investigation will begin
with Case I consisting of a simple study of a single phase fluid flow through a
pipe with a 90 degree angled bend. The simply study will be used to gain an
understanding of how to effectively use CFD software to predict turbulence and
heat transfer. The results will be computed using different turbulence models,
evaluated at different Reynolds number values, and using the anisotropic
invariant Map as a tool. The knowledge obtained from this study will then be
used to conduct Case III which is the numerical simulation of the Hot Soaking

process.

The Case II study will be conducted using a two-dimensional model of natural
convection within a four-sided enclosure. This study will be used to gain an
understanding of how to effectively use CFD software to predict natural
convection with turbulence modeling and heat transfer. This study will be
evaluated using the K-Epsilon turbulence models. The knowledge obtained from
this study will then be used in addition to Case I to aid in the numerical

simulation of the Hot Soaking process (Case III).

The Case III study will be conducted using a two-dimensional simplified model of

a hollow cylindrical component above a heated hollow horizontal plate contained

within a four-sided enclosure. As stated previously, convection is measurably
decreased immediately after soaking starts and thus a spike in component
temperature occurs. This temperature spike is a result of conduction and
radiation heat transfer. Conjugate heat transfer will be used to determine the
conduction through the wall of the exhaust system as well as the radiation from
the surface of the exhaust system. Since the hot soaking process does not
contain forced convection, natural convection will be of interest. Earlier it was
mentioned that during the duration of hot soaking the temperatures eventually
decrease to ambient. Natural convection will become established thus creating
currents of air that will be heated as it flows over the heated component. The
hollow cylinder located above the heated plate will be soaked in the hot air thus
contributing to the temperature increase of the hollow cylinder. The magnitude
of the modes of heat transfer generated is needed to ascertain its contribution to

the high temperatures that will develop on the hollow cylinder.

This investigation will be used to create a simple numerical simulation of the hot
soaking process to aid in further study of the thermal characteristics of the
exhaust system and components contained within the underhood and underbody
of a vehicle. The numerical simulations will be performed using Computational
Fluid Dynamics (CFD). CFD is an analytical tool that can be used to simulate
thermal phenomenon of vehicle underhood and underbody components to
“identify and resolve vehicle thermal issues early in the design phase. It also is

an efficient tool for generating parametric studies with the potential of

significantly reducing the amount of experimentation required to optimize

performance of a design”. (Damodaran 2000)

I.c. Objective

The following thesis describes the study of two simple models created to
understand how to use CFD to simulate turbulent flow and heat transfer, then
describes the creation of a more progressive model that was used to study the
hot soaking phenomenon. The body of this thesis is developed through
Chapters ll through V. Chapter II describes the numerical simulation of a Single
Phase Fluid Flow through 3 Pipe with a 90 Degree Bend. Chapter III describes
the numerical simulation of Natural Convection in a Two Dimensional Square
Enclosure. Chapter IV describes the numerical simulation of Two Dimensional
Model of a Cylinder above a Heated Plate. Chapter V summarizes the

Conclusions and Recommendations of the thesis.

II. SINGLE PHASE FLOW THROUGH A PIPE WITH 90 DEGREE BEND

This chapter focuses on a three dimensional model of a single phase flow
through a pipe with a 90 degree angled bend. The three-dimensional model was
computed with Fluent (CFD Software) to predict turbulence and heat transfer
throughout the pipe. The 3D computations were performed with the Realizable
K-Epsilon and Reynolds Stress turbulence models. Computations were also

done at three different values of the Reynolds number.

II.a. Problem Description

A three dimensional model of a pipe with a 90 degree angled bend was created
in order to learn how CFD could be used to numerically simulate turbulence and
heat transfer generated during single phase flow through a pipe. This
understanding of the use of CFD will be used in addition to the knowledge gained
from the next chapter to aid in the numerical simulation of the hot soaking

phenomenon.

The pipe flow model is three-dimensional consisting of a pipe bent at a 90 degree
angle which contains an inlet and an outlet. Figure 2-1 illustrates the geometry
and triangular mesh used to create the 3D pipe of the model. Figure 22
illustrates a closer view of the triangular mesh used to create the 30 pipe of the

model.

 

 

 

ZD cross sectlon
out here

 

 

 

 

 

 

 

 

Outlet

 

 

 

 

 

Grid

 

Feb 12, 2007
FLUENT 6.2 (3d, dp. segregated, rko)

 

Figure 2.1: Plot of the triangular grid mesh used for the 3D Pipe

Model for Single Phase Flow.

10

 

 

 

L.

 

 

 

Figure 2.2: Plot of a closer view of the triangular grid mesh used

for the 3D Pipe Model for Single Phase Flow.

The 30 pipe model with a 90 degree bend has a height and width of 2.0 meters.
The diameter of the pipe is 0.3 meters. The fluid flowing through the pipe was
modeled as air. The pipe wall was modeled with an aluminum material. The
properties of the air and aluminum materials used in this model are listed in

Table 2-1.

Table 2.1: List of the material properties for Air and Aluminum.

 

II.b. Problem Formulation
The initial conditions and boundary conditions used for CFD computations of the
3D pipe flow model are provided below. The model was analyzed using a steady

state calculation.

Qperating Conditions
The operating conditions used in this model were specified for the flow field. The
temperature of the flow field was specified as 300 degrees Kelvin. The operating

pressure was specified as 101,325 Pascals for the entire flow field.

Mndary Conditions

The model for this study contains surfaces also known as zones that were used
for the application of boundary conditions on the surfaces of the model. Four
different boundary types were used to define the boundary conditions for the
different zones in the model. These boundary types are fluid, fixed temperature

wall, velocity inlet, and pressure outlet.

The fluid zone was used to define the boundary condition of the fluid that is used
to represent air. A fluid zone represents a group of cells that is used to calculate
the governing equations of fluid flow and heat transfer. The fluid zone also

participates in radiation.

Wall zones were used to define the thermal boundary conditions of the pipe wall.

The pipe wall represents the source of heat in this study. A fixed temperature

12

wall boundary condition was applied to simulate the temperature of the hot
exhaust gas inside the pipe. The temperature of the pipe wall was specified as
373.15 degrees Kelvin. The fixed temperature wall boundary condition can be
referred to as the Dirichlet condition or a boundary condition of the first kind. The
fixed temperature was used to calculate the heat flux generated to the wall from

a fluid cell. The heat flux from a fluid cell is defined as (Fluent 2005):
q=hf(Tw—Tf)+qrad (2.1)

When using the FLUENT program, the heat transfer coefficient of a fluid is
computed based on the local flow-field conditions (e.g., turbulence level,

temperature, and velocity profiles).

The pipe entrance was defined as a velocity inlet boundary. A velocity inlet
boundary is used to define the characteristics and properties of the velocity flow
at the inlet of the boundary. The flow direction of the velocity was specified
normal to the boundary. A turbulence intensity of 1% and a hydraulic diameter of
0.3 meters were defined for the turbulence specification method. These inputs
were then used to calculate the mass flow rate, momentum fluxes, and energy
fluxes at the inlet. The equation used to calculate the mass flow rate at the

entrance of the pipe is as follows (Fluent 2005):

m =jpv-d2 (2.2)

13

As mentioned previously, the 3D model was used to compute predictions at four
Reynolds numbers, therefore different velocity values were specified for each of
the iterations. A velocity of 4.63 m/s was specified for the Reynolds number
value of 95000, a velocity of 0.46 m/s was specified for the Reynolds number
value of 9500, and a velocity of 0.049 m/s was specified for the Reynolds number
value of 1000. Under normal flow conditions, the transition from laminar flow to
turbulent flow occurs at Reynolds numbers of 2000 to 3000. A Reynolds number
below 2000 represents a flow that is completely stable and that will always be
laminar (Sabersky et al. 1999). This range of Reynolds numbers will allow the

evaluation of the prediction of turbulent flow through a 30 pipe.

The pipe exit was defined as a pressure outlet boundary. A pressure outlet
boundary is used to define and calculate the characteristics of the flow at the exit
of the boundary. A gauge pressure was specified as 0 Pascals for the exit
pressure. The gauge pressure is used as a static pressure along with other
conditions which are extrapolated from the interior of the pipe to calculate the
flow conditions at the exit. The backflow direction of the flow was specified
normal to the exit boundary. A turbulence intensity of 1% and a hydraulic

diameter of 0.3 meters were defined for the turbulence specification method.

The Realizable K—Epsilon and Reynolds Stress turbulence models were used in

the evaluation of the predictions of turbulent flow for a single phase fluid using

14

the Fluent CFD Software. The no-slip shear boundary condition was used for
wall of the pipe in this model. Since the no-slip condition has to be satisfied at
the wall, the mean velocity is affected thus in turn affecting the turbulent flow.
The Enhanced Wall Treatment method in Fluent was used to model the turbulent
flow in the near-wall region of each pipe wall zone for the k-epsilon and Reynolds

Stress Model turbulence models.

Il.c. Governing Equations
The fluid flow is solved by the calculation of the following conservation of mass

and momentum equations (Fluent 2005):

6,0 _ _
-a—t+V-(pU)—Sm (2-3)

63(p5)+v.(p5r3)=—Vp+v-(:)+pg+fi (2.4)
t

Turbulence Modeling
The Realizable K-Epsilon model was selected first. The following are transport
equations that FLUENT uses to calculate turbulence with the realizable k-epsilon

model (Fluent 2005):

i.

,u
6x [p+—i]flc— +Gk+Gb-pa (2.5)
1'

a a
5W)+E‘T(pk"j)‘ 0' ax.
J k J

15

a a _ a r! 2 a (2.6)

The Reynolds Stress Model was also used to evaluate the predictions of
turbulent flow in this study. The following are transport equations that FLUENT

uses to calculate turbulence with the Reynolds stress model (Fluent 2005):

 

a l l a l l a l l l l l
5(puiuj) +——(puk uiuj) — ——a—x;|:puiujuk + p(6kjui +6ikuj):]

 

 

axk
a —-'—'— all]. aui — —
+——— — + — 6+ 6 2.7
0x #6304 uj) puuka “1’“ka pfl(giuj gjui) ( )
k k xk xk
Heat Transfer

 

The energy equation was used for the calculation of heat transfer. All modes of
heat transfer were calculated for this study. The modes of heat transfer are
conduction, convection, and radiation. The following form of the energy equation
was specified for the calculation of conduction and convection heat transfer

(Fluent 2005):

a_a 31+
—(pE) + —a-(u (pE + p))= kefl a +“'(’g‘)efl (28)
6x1

16

Velocity Profile Computation
The following analytical formula was calculated to determine the velocity profile

for turbulent flow through a smooth pipe:

_ l/n
[1‘]. z [1 _%J , (Fox and McDonald, 1998) (2.9)

where

n = -1.7 + 1.8 log Rev (2.10)

Il.d. Anisotropic Invariant Map

In order to properly evaluate the numerical predictions of turbulence, a method

must be used to validate these predictions. One such method has been

developed by Lumley (1978, 2001 ). This method states that the state of

turbulent flow can be analyzed and distinguished by its anisotropy. The

anisotropy of the turbulent flow can be determined from the Reynolds stresses
ti]. = — p??? (2.11)

u—'t7 - X component Reynolds stress

77 - Y component Reynolds stress

w'w' - Z component Reynolds stress
W - X and Y component Reynolds stress

 

The anisotropy is determined by subtracting the isotropic part of the Reynolds

stress from itself and normalizing byW. This mathematical manipulation forms

the Reynolds anisotropy tensor (Lumley 1978):

17

 

 

b.. = Li —_L (2.12)
’1 14.14. 3

l I
u.u. = 2k (2.13)

Where k represents the turbulent kinetic energy and 61.]. represents the Kronecker
delta function. The second (II) and third (Ill) invariants of the tensor by. can be

calculated and

—II =b323 —b”b22 +1)?2 (2.14)
111 = 1),, (bub22 -b,22) (2.15)

then plotted to create what is known as the Anisotropic Invariant Map (Lumley
1978). Figure 2.3 illustrates an Anisotropic Invariant Map created by Krogstag

and Simonsen (2005).

18

 

  
   
  

 

 

 

 

0 4 _ i i I 2 Component 1 Cofnponent
- l3— -(l273+179) I ,\k
2 Component T ‘3
0.3 ~ Podsymmetric i -. -
0-1 ‘ “ Rod-like tu ul nc
K \ \-., Axisymmet trlrbc, T3>0 é
Disk-like turbulence t I 3=2(- -| 23013)
0 - Axisymmetric, I3<0 - a _
"3='2( '2/3)” . ”ck ‘1 Isotropic .
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

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Figure 2.3: Plot of the Anisotropic Invariant Map (Krogstag and Simonsen
2005)

The Anisotropic Invariant Map is used to characterize the turbulent flow in terms
on the Reynolds anisotropy tensor. If the anisotropy calculated is plotted and
lies within the triangle domain of the anisotropic invariant map the turbulent flow
is considered physically realizable. If the plotted anisotropy lies outside of the
triangle then the turbulence is considered to be invalid. In the graph above “—l2”

and “I3” represent the second (II) and third (Ill) invariants of the tensor bij

defined previously.

In this study, the Reynolds stress tensors will be determined for the Realizable K-

Epsilon and Reynolds Stress turbulence models. In the Fluent CFD program,

19

once the Reynolds Stress turbulence model is specified, the Reynolds stresses
are obtain feasibly through post processing. The values can be either plotted or
reported numerically. However, the Reynolds stresses have to be derived for the
Realizable K—Epsilon turbulence model.

In order to derive the Reynolds stress from the Realizable K-Epsilon turbulence

model, the Boussinesq hypothesis has to be used (Fluent 2005).

a“. auj 2 Bu.
_ r r. = __l_ _ __ + __l_ 6” 2.16
puluj ’ut 6x.+6x. 3 pk [at 6x. U ( )
J l '
m=-_(2a_:]+3 k+— ”t 9i“— (2.17)
p 8 3 p ax
__ l1
,r,.=___t[2.61]2 ._ k+: (2.18)
p 6y) 3
_ .U I”
W'W' .__ __t__[2£w_)+_2: k +421) (2.19)
p 62 3 ,0 52
_ fl
u'v'=——t(@-+-a:] (2.20)
p 6y fix

The Reynolds stresses obtained from the two turbulence models can be used to
calculate the anisotropic components of the Reynolds stress tensor. The values
of the anisotropy generated from the Realizable K—Epsilon and Reynolds Stress
turbulence models will be plotted on an anisotropic invariant map to determine
the validity of the turbulence model predictions. The results and plots of the

evaluation of the turbulent flow can be found in the numerical results section

20

below.

II.e. Numerical Method of Solution
Computational Fluid Dynamics (CFD) was used to simulate the hot soaking
process with a computer by computing the solutions to differential equations.
The differential equations computed are the governing equations of fluid flow and
heat transfer. CFD works best when used in conjunction with grid generation
and post-processing analysis. Typically, the CFD process is executed as follows:
I) Pre-processors are used to create 2D and 3D geometric models
and meshes
ll) Pre-processors are then used to generate volume meshes from
the geometric models
Ill) Solvers are used to numerically simulate fluid flow and heat
transfer by computing solutions to the differential equations
IV) Post-processing analysis is used to evaluate the numerical

solutions computed

HyperMesh, TGrid, and FLUENT CFD programs were used in this study to
numerically simulate the single phase flow through the pipe containing a 90
degree angled bend as mentioned above. To begin, a 3D model of a 90 degree
angled was constructed with triangular elements using HyperMesh. The 3D

model was then imported into the TGrid program to generate the volume

21

elements. The volume elements were created from the triangular elements
forming a tetrahedral mesh. The volume in this model was used to simulate the
air flowing through the pipe. After the volume was generated, the model was
imported into the FLUENT CFD program to numerically simulate the single phase

flow.

Since the focus of this study is to predict the turbulence and heat transfer
generated by single phase flow through a pipe, two cases created. The first case
was used to examine turbulence modeling. The second case was used to

simulate different values of the Reynolds number by creating a range of models.

ll.f. Numerical Results

As mentioned previously, two different cases were examined to evaluate the use
of the CFD software to predict the turbulence and heat transfer of the single
phase fluid flow through a pipe with a 90 degree angled bend. These two cases
were used to study the single phase flow through turbulence modeling and the
use of different values of Reynolds numbers. To examine turbulence modeling,
the Realizable K-Epsilon and Reynolds Stress turbulence models were specified
so that the turbulent flow could be generated and analyzed. Finally, the 3D
model was modified by creating three different models calculated with different

velocities to simulate different values of the Reynolds number.

22

A 2D cross section was cut from the pipe as shown in Figure 2.1 and 2.4. The
cross section was used as a plane to display the plot of the vorticity vectors and
the static temperature contour plots. Figures 2.5 through 2.10 illustrate the
vector plot of the vorticity magnitude generated in a cross section of the pipe for
the two different turbulence models and the Reynolds numbers values of 1000,
9500, and 95000. This figure shows that the vorticity magnitude increase as the
Reynolds number increases. This trend is expected because as the Reynolds
number increases, the velocity increases (See Table 2.2 ). It also displays a
symmetrical vortex circulation pattern as a result of using the realizable K-Epsilon
turbulence model. The vortex patterns generated for the Reynolds Stress
turbulence models are symmetric except for at the Reynolds number value of
9500. At the Reynolds number value of 9500, the use of the Reyndlds Stress
turbulence model produces a large vortex circulation zone near the bottom wall

of the pipe cross-section.

23

 

 

 

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cross section of the pipe for the K-Epsilon Turbulence Model with Re N0. =

Figure 2.5
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Figure 2.6

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cross section of the pipe for the K-Epsilon Turbulence Model with Re N0. =

9500.

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cross section of the pipe for the K-Epsilon Turbulence Model with Re N0. =
95000.

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cross section of the pipe for the RSM Turbulence Model with Re N0. =
95000.

30

Table 2.2: List of the vorticity magnitude generated in a cross section of the
pipe for the different turbulence models and Reynolds number values.

 

 

 

 

 

 

 

 

 

 

 

 

Vorticity
Reynolds . Magnitude
Turbulence Model Number Velocrty (m/s) (od 2D cross-

? _ section)
Realizable K-Epsilon 1000 0.049 18.16
Realizable K-Epsilon 9500 0.46 97.8
Realizable K-Epsilon 95000 4.63 469.4
I'thmolds Stress Model 1000 0.049 20.44
Reynolds Stress Model 9500 0.46 167.83
Reynolds Stress Model 95000 4.63 473

 

Table 2.2 lists the vorticity magnitude generated in cross sections of the pipe
for the different turbulence models and Reynolds number values. This table also
illustrates that the vorticity magnitude increases as the Reynolds number

increases. The Reynolds stress model generates a higher vorticity magnitude.

Figures 2.11 through 2.16 illustrate the static temperature contours generated in
a cross section of the pipe for the following Reynolds numbers values of 1000,
9500, and 95000. These figures show that the temperature distribution changes
as the Reynolds number increases. At the lower Reynolds number value, the
higher temperature value is dominant. The higher temperature value tends to be
distributed more towards the bottom wall of the pipe. As the Reynolds number
increases, the higher temperature value becomes more evenly distributed around
the walls of the pipe. This trend is expected because as the Reynolds number

increases, the velocity increases thus creating more airflow. The increased

31

airflow decreases the hot spot that is generated and creates a more even

distribution of the temperature.

Also, it is shown that the Reynolds Stress Model produces a higher temperature
distribution than the K-Epsilon model. As mentioned previously, the Reynolds
Stress model produces a higher vorticity magnitude. The rotation and swirling
motion of the hot air in the pipe can increase the thermal energy thus increasing
the temperature of the air. It can be concluded that the higher vorticity

magnitude creates higher temperature zones.

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section of the pipe for the K-Epsilon Turbulence Model with Re N0. = 1000.

33

 

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Figure 2.12

section of the pipe for the RSM Turbulence Model with Re No. = 1000.

34

 

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section of the pipe for the following models: K-Epsilon Turbulence Model

with Re N0. = 9500.

Figure 2.13

35

 

 

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section of the pipe for the following models: RSM Turbulence Model with

Figure 2.14
Re N0. = 9500.

36

 

 

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with Re N0. = 95,000.

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37

 

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section of the pipe for the following models: RSM Turbulence Model wi

Re N0. = 95,000.

Figure 2.16

38

Figure 2.17 illustrates the Anisotropic Invariant Map generated for single phase
fluid flow through a pipe using the K-Epsilon Turbulence Model at a Reynolds
number of 1000. The anisotropy was calculated from the Reynolds stress values
of the 20 cross-section cut from the pipe. The turbulence flow in this model is
physically realizable. The turbulence generated is isotropic and slightly tends
toward axis-symmetric expansion (“Rod-like”) turbulence. The “Rod-like”
turbulence occurs when one component of the turbulent kinetic energy is greater
than the other two components of turbulent kinetic energy which are equal
(Krogstag and Simonsen, 2005). Figure 2.18 illustrates the Anisotropic Invariant
Map generated for single phase fluid flow through a pipe using the K-Epsilon
Turbulence Model at a Reynolds number of 9500. It is shown in this figure that
all of the anisotropy generated is realizable except for at the radius of 0.6. The
realizable turbulent flow in this model is isotropic turbulence. In isotropic
turbulence the anisotropy is zero. Figure 2.19 illustrates the Anisotropic
Invariant Map generated for single phase fluid flow through a pipe using the K—

Epsilon Turbulence Model at a Reynolds number of 95,000. The turbulence flow

in this model is physically realizable and isotropic.

Figure 2.20 illustrates the Anisotropic Invariant Map generated for single phase
fluid flow through a pipe using the Reynolds Stress Turbulence Model at a
Reynolds number of 1000. The turbulent flow in this model is physically
realizable except for at the center of the pipe (r = 0). However, the realizable

turbulence ranges from axis-symmetric expansion to two-component turbulence.

39

Two-component turbulence can be found in the viscous sub layers at the wall. At
this location the wall-normal component of velocity fluctuations tends to zero,
leaving only wall-parallel components (Jovicic, 2006). Figure 2.21 illustrates the
Anisotropic Invariant Map generated for single phase fluid flow through a pipe
using the Reynolds Stress Turbulence Model at Reynolds number of 9500. The
turbulent flow in this model is physically realizable and ranges from isotropic
turbulence to axis-symmetric expansion. Figure 2.22 illustrates the Anisotropic
Invariant Map generated for single phase fluid flow through a pipe using the
Reynolds Stress Turbulence Model at Reynolds number of 95,000. The turbulent
flow in this model is physically realizable except for at the center of the pipe and
at the radius equal to 0.86. However, the realizable turbulence produced in this

model tends toward axis-symmetric expansion.

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Figure 2.17: Plot of the Anisotropic Invariant Map generated for single
Re No. = 1000.

 

 

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Re No. = 9500.

42

 

 

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Figure 2.19

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phase fluid flow through a pipe using the Reynolds Stress Turbulence
46

Figure 2.22: Plot of the Anisotropic Invariant Map generated for single
Model at Re No. = 95,000.

Figure 2.23 illustrates the Velocity Profile of Fully Developed Flow generated for
single phase fluid flow through a pipe using the K-Epsilon and Reynolds Stress
Turbulence Model at different Reynolds numbers. The theoretical data was
calculated from equations 2.9 and 2.10 above. This figure shows that the
Reynolds number value of 95,000 produces a velocity profile closest to the
theoretical value for both turbulence models. The remaining velocity profiles over
predict the turbulent flow at the lower Reynolds number values and especially the

Realizable K-Epsilon model at a Reynolds number value of 1000.

47

 

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Plot of the Velocity Profile of Fully Developed Flow generated
48

23
for single phase fluid flow through a pipe using the K-Epsilon and

 

Reynolds Stress Turbulence Model at different Reynolds numbers.

Figure 2.

Ill. NATURAL CONVECTION IN A 20 SQUARE ENCLOSURE

This chapter focuses on the two dimensional model of natural convection in a
square enclosure. The two-dimensional model was evaluated with the use of
Fluent CFD Software to simulate natural convection inside of a square enclosure.
The 20 model was evaluated using the Realizable K-Epsilon turbulence model

without radiation.

lll.a. Problem Description

A two dimensional model of a square enclosure was created in order to learn
how CFD could be used to numerically simulate natural convection heat transfer.
This understanding of the use of CFD will be used subsequently in the next

chapter to aid in the numerical simulation of the hot soaking phenomenon.

The square enclosure model is two-dimensional consisting of four walls. Figure

3.1 illustrates the geometry and quad mesh used to create the 20 model. Figure

3.2 illustrates a closer view of the quad mesh used to create the 20 model.

49

 

Figure 3.1: Plot of the quad grid mesh used for the 2D Enclosure.

50

 

Figure 3.2: Plot of a closer view of the quad grid mesh used for the 20
Enclosure.

51

The 20 model has a height and width of 1.3 meters. The fluid inside the
enclosure was modeled as air but with fictitious properties for comparison
purposes. The fluid was also modeled using the Boussinesq approach to
properly simulate the buoyancy forces that are created through natural
convection. The walls were modeled with an aluminum material. The properties

of the air and aluminum materials used in this model are listed in Table 3.1.

Table 3.1: List of the material properties for Air and Aluminum.

 

 

 

 

 

 

 

 

 

 

 

Material Properties Air Aluminum
Density - Boussinesq (kg/m‘3) 1000 2719
Specific Heat gJIkg-K) 1 10340 871
Thermal Conductivity (WIm-K) 1 5.309 202.4
Viscosity (kgIm-s) .001 N/A
Absorption Coefficient (1Im) 0.2 N/A
Thermal Expansion Coefficient (1IK) 0.00001 N/A

 

Ill.b. Problem Formulation

The operating conditions and boundary conditions used for CFD computations of
the 20 natural convection model are specified below. The values of the
operating and boundary conditions were selected so that the results can be
compared to those of a tutorial model created by Fluent, lnc in order to
understand how to model heat transfer. The model was analyzed using a steady

state calculation.

Operating Concmions

The operating conditions used in this model were specified for the flow field. The

52

temperature of the flow field was specified as 1000 degrees Kelvin. The
operating pressure was specified as 101,325 Pascals. Gravity was activated in

the model and set to 6.96 x 10'5 m/sz.

ficuncfly Conditions

Boundary conditions were used to set conditions for fluid flow and heat transfer
on the boundary or surfaces of the model. The model for this study contains
zone surfaces. Three different boundary types were used to define the boundary
conditions for the different zones in the model. These boundary types are fluid,
fixed temperature wall, and adiabatic wall. The fluid properties and operating

conditions were previously described.

The fixed temperature wall zones were used to define the left and right walls of
the enclosure. The fixed temperature of the right wall was specified as 2000
degrees Kelvin and the fixed temperature of the left wall was specified as 1000
degrees Kelvin. These specifications were used to describe a hot and cold wall.
This temperature difference will induce the natural convection that will occur
inside the enclosure. The adiabatic wall zones were applied to the top and
bottom wall of the enclosure. The zones were used to specify a heat flux of O

W/m2.

The laminar flow model was used in the evaluation of the predictions of natural

convection using the Fluent CFD Software. Turbulence modeling was not used

53

in this study in order to focus the buoyant flow by natural convection.

lll.c. Governing Equations
Fluid flow and heat transfer were used to simulate natural convection in a ZD
enclosure. The fluid flow is solved by the calculation of the following

conservation of mass and momentum equations (Fluent 2005):

Conservation of Mass:

6p _ _
E+V-(pu)—Sm (3.1)

And the momentum equation in an inertial reference frame:

%(p5)+V-(p55)=—Vp+V-(:)+pg+fi (3-2)

Turbulence Modeling
The variation of the k-epsilon model chose for this study is the realizable k-
epsilon model. The following are transport equations that FLUENT uses to

calculate turbulence with the realizable k-epsilon model (Fluent 2005):
2(pk)+—Q—(pku )__6_ +f—t— 2&— +G +G — a (33)
at 6x]. f 6x . ,u 0' k 6x]. k b '0 '

a 6 6 66 £2 a
— +— eu. =— +— —— + Ss— ——+C —C G (3-4)
a((pa) 6xj(p 1) 8x]. [[fl LXI] pCl pC2k+M 16k 38 b

54

Heat Transfer

The energy equation was used for the calculation of heat transfer. All modes of
heat transfer was calculated for this study. The modes of heat transfer are
conduction, convection, and radiation. The following form of the energy equation
was specified for the calculation of conduction and convection heat transfer for

the Realizable K-Epsilon model (Fluent 2005):

a 6 6 6T

— E — . E =— k — . .. 3.5

a0) )+ 6x1. (ul(p +p)) 6x]. eff 6x]. +ul(ry)eff ( )
Nusselt Number Commtation

The following are the analytical solutions for the computations of the average

nusselt number published by different sources.

Experimental Calculation of the average Nusselt number:
Nuf = 0.062xRa1’3, (Elsherbiny et al., 1982) (3.6)

NUf = 0.082xRa1’3' (Markatos and Pericleous 1984) (3.7)

Empirical Correlation:

Pr
Nu =0.18 ——
f (

0.29
R , Catton, 1978 3.8
0.2+Pr 0L) ( ) ( )

Where Pr is the Prandtl Number and RaL is the Rayleigh Number.

55

lll.d. Numerical Method of Solution

Hyper Mesh and FLUENT CFD programs were used in this study to numerically
simulate natural convection in a 20 enclosure. To begin, the volume of the 20
model of a square enclosure was constructed with quad elements using Hyper
Mesh. The volume in this model was used to simulate the air inside the
enclosure. After the volume was generated, the model was imported into the

FLUENT CFD program to numerically simulate natural convection.

Ill.e. Numerical Results

As mentioned previously, a two-dimensional model of an enclosure containing
hot and cold walls was evaluated by using CFD software to predict natural
convection in the square enclosure. The low temperature wall and a high
temperature walls were created to produce natural convection inside an
enclosure. The case was used to study the influence of turbulence on natural
convection through the use of the realizable k-epsilon turbulence model.

Figure 3.3 and Figure 3.4 illustrate the comparison between the model generated
by Fluent, Inc. and the natural convection simulation model. The model
generated by Fluent, Inc. contain was performed with laminar flow without
radiation. As shown in Figure 3.3, the natural convection heat transfer process
produces non-symmetric contours of the velocity stream function. The contours
display a trend of warm air rising near the hot wall (then becoming cooled) and
cool air falling near the cool wall (then becoming warm). Figure 3.4 shows that

the use of the Realizable K-Epsilon produces a similar velocity stream function

56

distribution through natural convection but at a large magnitude. This is
expected due to the higher velocity created by the introduction of turbulent flow.
The percent error generated by the comparison of the model with realizable K-

Epsilon turbulence to the model without turbulence is 23.86%.

57

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Figure 3.3: Plot of the contours of the stream function generated inside the
2D Enclosure for the model (with laminar flow but without radiation)
published in Fluent 6.1 tutorial guide.

58

 

 

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Figure 3.4: Plot of the contours of the stream function generated inside the

2D Enclosure for the model generated with Realizable K-Epsilon

Turbulence model but without radiation.

59

Figure 3.5 illustrates the contours of the velocity magnitude generated inside the
ZD Enclosure for the Realizable K-Epsilon Turbulence model without radiation.
Although not shown, the vectors of the velocity magnitude are very close to the
laminar flow model mentioned above. The figure shows the momentum

boundary layer along both the hot (right) and the cold (left) walls.

Figure 3.6 and Figure 3.7 illustrate the contours of the static temperature
generated inside the 2D Enclosure for both models. This figure shows that the
use of the realizable K-Epsilon turbulence model does not influence the

distribution of temperature generated between the hot and cold wall.

60

 

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Figure 3.5

ilon Turbulence model without

ble K-Eps

the 2D Enclosure for the Real

radiation.

61

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C'
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Figure 3.6: Plot of the contours of the static temperature generated inside
the 20 Enclosure for the model (with laminar flow but without radiation)
published in Fluent 6.1 tutorial guide.

62

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Figure 3.7: Plot of the contours of the static temperature generated inside
the 2D Enclosure for the model generated with Realizable K-Epsilon
Turbulence model but without radiation.

63

Figure 3.8 and Figure 3.9 illustrate the vectors of the velocity in the Y direction
generated inside the 20 Enclosure for both models. The velocity graph
represents the rising plume of lighter hot air on the right side of the enclosure,
and the falling plume of heavier cool air on the left side of the enclosure. This
behavior is caused by the buoyancy forces generated by changes in density and
gravity. The rising of the hot air and falling of the cool air occurs faster in the
model without turbulence. Figure 3.9 also show that the there is a region of cool

air that is located near the warmer side of the enclosure.

64

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Figure 3.8: Plot of the of the velocity in the Y direction generated inside the
ZD Enclosure for the model (with laminar flow but without radiation)
published in Fluent 6.1 tutorial guide.

65

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Figure 3.9: Plot of the of the velocity in the Y direction generated inside the
2D Enclosure for the model generated with Realizable K-Epsilon
Turbulence model but without radiation.

66

The model (with Laminar flow without radiation) published in Fluent 6.1 tutorial
guide produces a total heat transfer flux of 138559.44 W/mz. The Model
generated with Realizable K-Epsilon Turbulence model but without radiation
produces a total heat transfer flux of 139036 W/mz. The percent difference
between the laminar flow model and the realizable K-Epsilon turbulent flow

model is 0.34%.

Figure 3.10 illustrates the plot of dimensionless height (in the y-direction) versus
dimensionless temperature generated inside the 2D Enclosure. The graph
displays data for the Realizable K-Epsilon turbulence model without radiation and
for reference data calculated with an aspect ratio of 6 and a Rayleigh number of
Ra=7.12x1010 (Velusamy et al., 2001). It is shown that the trend of the 20
enclosure of the Realizable K-Epsilon turbulence model similar to the theoretical
reference data. This shows that the model has generated a good predicting of

natural convection in a 20 enclosure.

67

 

Dimensionless Height, YIH

p p p 5::

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Figure 3.10: Plot of dimensionless height (in the y-direction) versus
dimensionless temperature at mid-width generated inside the 2D Enclosure
for: (a) the Realizable K-Epsilon turbulence model without radiation and (b)
the reference solution measure with an aspect ratio of 6 and Rayleigh
number of 7.12x101° (Velusamy et al. (2001)).

68

Table 3.2 lists the computations of the average nusselt number as published by
different sources. The predicted nusselt number computed by the Fluent CFD
software is closest to the experimental calculation performed by Elsherbiny et al.

(1982). The percent error calculated for the predicted measurement is 13.48%.

Table 3.2: List of the average Nusselt number published by different
sources.

 

 

 

 

 

 

 

 

Rayleigh No. 10‘"

Nusselt Number (Fluent - CFD prediction) 259.5072
Nusselt Number (Elsherbiny et al. - experiment) 228.7308
Nusselt Number (Markatos and Pericleous — experiment) 302.5149
Nusselt Number (Catton — Heat Transfer Book) 212.9974

 

69

IV. TWO-DIMENSIONAL MODEL OF CYLINDER ABOVE A HEATED
PLATE

This chapter focuses on the two dimensional model which was used to
investigate the hot soaking phenomenon using turbulence modeling and heat
transfer. The two—dimensional model consists of a hollow cylinder above a
heat hollow horizontal plate contained within a four-sided enclosure. The 20
model was evaluated using the Realizable K-Epsilon, K-Omega, and
Reynolds Stress turbulence models. The 2D model was evaluated using

different grid sizes ranging from coarse to fine.

lV.a. Problem Description

A simplified model was created in order to learn how CFD could be used to
numerically the heat transfer of the hot soaking phenomenon. This
understanding of the use of CFD was used to aid in the numerical simulation
of the hot soaking process. The simplified model is two-dimensional
consisting of a heated horizontal plate and a hollow cylinder both located
inside of a four-sided enclosure. Figures 4.1 8. 4.2 illustrate the geometry of
the model. Figures 4.3, 4.4, & 4.5 illustrate the type of mesh generated for

the model.

The outside surface of the heated plate is 0.086 meters long by 0.03 meters
wide. The inside diameter of the hollow cylinder is 0.032 meters and the

outside diameter is 0.04 meters. The enclosure is 1 meter in height by 1.5

70

meters wide. The thickness of the hollow cylinder and heated plate is 0.004

meters.

 

Figure 4.1: Plot of the geometry of the 2D model of a cylinder above a
heated plate.

71

—l
E
O
r
3
0
M
U)
o
-

I;
9’.
0
co
3
a
O

E
5
C‘.
0
4

l“ V000

 

Figure 4.2: Plot of a closer view of the geometry of the 20 model of a
cylinder above a heated plate.

72

 

 
 
   

  
  
 
       

A) A ‘gM‘Lib i595”?
tifmwfiats%%
5: SNQINREVA‘NAVA
V’gzfil‘wwmmw

      
  
   

005054455230
‘V EH? ‘géiy A'Qgfil‘fl
4MVAVAVAWAVAVIAVAWMAVAVI>
VVVVVVVVVVVVVVVVVYV

  
  

 

 

      

 

Figure 4.3: Plot of the meshed domain of the 20 model of a Cylinder
above a heated plate.

73

 

 

 

J

 

Figure 4.4: Plot of a closer view of the meshed domain of the 20 model
of a Cylinder above a heated plate.

74

 

 

 

 

 

 

Figure 4.5: Plot of the triangular and quad shaped elements used to
mesh the domain of the 2D model of a Cylinder above a heated plate.

75

The thickness of each component was modeled as a solid. The materials of

the solids are aluminum for the heated plate and steel for the hollow cylinder.

The two-dimensional model component materials are listed in Table 4.1. The

properties of air and the materials used in this model are listed in Table 4.2.

Table 4.1: List of the materials used for the components of the 2D

model.

 

 

 

 

 

 

2D Model Component Materials
Component Zone Material
Enclosure Wall Aluminum
Hollow Cylinder Wall Steel
Heated Plate Wall Aluminum

 

 

 

 

Table 4.2: List of the properties of Air, Aluminum, and Steel.

 

l'

1006.43
0.0242
1 .79E-05
28.966

76

871 502.48
202.4 16.27
N/A N/A

A A

IV.b. Problem Formulation

The initial conditions and boundary conditions that were used for the 20
model will now be specified. This model was analyzed using the steady state
condition. The steady state calculation of the 20 model was used as the

initial conditions for the transient analysis of the 2D model.

Initial Conditions

The initial conditions used in this model were specified for the flow field and
the walls of the components. The temperature of the flow field was specified
as 353.15 degrees Kelvin. The operating pressure was specified as 101,325

Pascals. The initial velocity was specified as 13.89 m/s.

The initial temperature of the interior wall of the heated plate was specified as
911.15 degrees Kelvin. The other remaining surfaces of the heated plate and
hollow cylinder were modeled as adiabatic walls with a heat flux of 0 W/mz.
The front wall of the four-sided enclosure was modeled as a velocity inlet.
The rear wall of the four-sided enclosure was modeled as a pressure outlet,
the gauge pressure was specified as 0 Pascals and the backflow total
temperature was specified as 353.15 K. The bottom wall of the four-sided
enclosure was modeled as a moving, adiabatic wall. The velocity of the
moving wall was defined as 13.89 mls. The heat flux of the wall was
specified as 0 W/m2 with an internal emissivity of 1. The top wall of the four-

sided enclosure was modeled as symmetry, due to the nature of this

77

boundary condition, the heat flux, normal velocity, and velocity gradients are

assumed to have a value of zero.

Boundary Conditions for Transient Analysis

Boundary conditions were used to set conditions for fluid flow and heat
transfer on the boundary or surfaces of the model. Five different boundary
types were used to define the boundary conditions for the different zones in
this model. These types are fluid, solid, fixed temperature wall, fixed heat flux

wall, and coupled wall boundary conditions.

The fluid zone and fixed temperature wall zones have been defined in the
previous chapter. The solid zone was used to define the boundary conditions
of the solid used to represent the thickness of the cylinder and the horizontal
plate. A solid zone represents a group of cells that is used to solve
conduction heat transfer. In this model, prisms were extruded to create
solids. Prisms were extruded on both the cylinder and plate to represent the
thickness of the components in order to calculate conjugate heat transfer.
Conjugate heat transfer was used to determine the conduction through the
wall of the horizontal plate as well as the radiation from the surface of the

horizontal plate to the cylinder. The solid zone also participates in radiation.

Wall zones were used to define the thermal boundary conditions at the wall.

The wall zones used in this model are the enclosure, the plate, and the

78

cylinder. The fixed heat flux wall boundary condition was used for the walls of
the enclosure, the outer surface of the horizontal plate, and both surfaces of
the cylinder. The heat flux was set at zero to define an adiabatic wall. The
fixed heat flux boundary condition represents the temperature gradient at the
wall of the modeled components by using Fourier’s Law and is given by

(Fluent 2005):

=—k—=O (4.1)

This boundary condition can be referred to as the Neumann condition or a
boundary condition of the second kind. The wall thickness and heat

generation rate of the walls modeled were specified as zero.

The walls of the outer plate surface and outer diameter of the cylinder were
modeled as coupled two-sided walls. When a wall zone contains a solid or a
fluid on each side, it is considered to be a two-sided wall and shadow zones
are created right next to the wall zone. Shadow zones are used to specify
boundary conditions distinctly on each side of the zone. The wall zones and
shadow zones of the plate and outer diameter of the cylinder are coupled
together. Coupled means that the both zones are grouped together and the
heat transfer is calculated directly from the solution in the adjacent cells

without specifying any additional boundary conditions.

79

The horizontal plate represents the source of heat in this study. A fixed
temperature wall boundary condition was applied to the wall of the inner
surface of the horizontal plate to simulate the temperature of the heated plate.
A temperature profile was used to represent the temperature of the inner
surface of horizontal plate as it changed with time. This temperature profile is

shown in the Figure 4.6 below.

80

 

 

Temperature (K)

no 2.3.0.0 .5000. . 1045038. p.040 2330.. mam—00V
4030040840 0403.0

 

 

 

 

 

 

 

 

.ooo
moo I III I II
ooo ., |. I ll 103N020. 0.0.0 .3304
ooo I mclmom 4030040840
08 - II N I, I 1- II-

o . . . _

 

 

0.0 No Po ob mo ..ob

4.30 3.3.3500V

 

 

Figure 4.6: Plot of the temperature profile specified for the inner surface

of the heated plate.

81

Other boundary conditions were also defined on the walls of this model. The
walls that were defined in this model (with the exception of the bottom wall of
the enclosure) were specified as stationary, non-moving walls. The shear
condition was defined at these walls as the slip wall condition that is
equivalent to zero shear stress. The roughness value of the walls was set to
zero to represent smooth walls. The radiation is defined at the walls as

diffuse and with an emissivity of one.

lV.c. Governing Equations

Fluid flow and heat transfer were used to simulate the hot soaking
phenomenon of the 20 model of a hollow cylinder above a hollow heated
plate. The fluid flow is solved by the calculation of the following conservation

of mass and momentum equations (Fluent 2005):

Conservation of Mass Equation:

6p -

— + V - u = S
at (,0 ) I.

(4.2)

Momentum Equation in an inertial reference frame:
a - - - = ~
5;(pu)+V-(puu) =-Vp+Vo(r)+pg+F

(4.3)

82

Turbulence Modeling

The governing equations used for the K-Epsilon and Reynolds Stress Model
were defined in the previous chapter. The variation of the k-omega model
chosen for this study is the standard k-omega model. The following are the
transport equations that were used to calculate turbulence with the standard

k-omega model (Fluent 2005):

a 6 6 6k
6 a 6 6w
E‘W’+a—.W“i)—§,[Fwa7,]+6~ (4'5)

Heat Transfer

The energy equation was used for the calculation of heat transfer. All modes
of

heat transfer was calculated for this study. The modes of heat transfer are

conduction, convection, and radiation. The following form of the energy
equafion

was specified for the calculation of conduction and convection heat transfer

(Fluent 2005):

(3‘ 0 6 6T
—E+—.E+ =—k—-+...
at (p ) aXi (ur(p p)) axj[ efl' ax]. “1(Ty )efl]

(4.6)

83

The following equation was used to solve radiation heat transfer for an

absorbing, emitting, and scattering medium at position 7 in the direction§ .
This

equation represents the Discrete Ordinates Model of radiation.

 

- _. 4

—61(”S) +(a+0's)I(F,§)=an2 ”T +344 1(F,§')<D(§,§')d0'
6t 7t 47Lr 0

(4.7)

Pressure Distribution Computation
The following analytical equation was used to compute the surface pressure
distributions for boundary layer flow around the cylinder above the heated

plate.

Pressure Distribution:

Cp =1- 4sin2(theta), (Munson et al., 2002)
(4.8)

lV.d. Anisotropic Invariant Map

The anisotropic invariant map was used to evaluate the numerical predictions
of

Hot soaking in this study. In this study, the Reynolds stress tensors will be
determined for the Realizable K-Epsilon, K-Omega, and Reynolds Stress
turbulence models. The derivations of anisotropy for the Realizable K-Epsilon
and Reynolds Stress model will be similar to the derivations discussed in the

previous chapter. However, since a 2D model was used for this study, the

84

Z — component of the Reynolds stress has a value of zero. The following

equations were used to determine the anisotropy of the K-Omega turbulence

model utilizing the Boussinesq Approach:

611- all} 2 all-
— 7'.= —'—+——— -— + -—‘ 6..

j I

 

(4.9)
“I I___#_I[2§y_).3 k 12.532:
p 6x 3 pax
(4.10)
7__&[2fl].3[k ii)
p 6y 3 pay
(4.11)
w'w'=0
(4.12)
W..——”-' 2.9:
p 6y 6x
(4.13)

An additional study was created in order to determine the physical realizability
of the turbulence generation influence by heat transfer. The focus of this
study was to evaluate the mean temperature in addition to the Reynolds
stresses generated by turbulence which the following:

< u'u'T >

(4.14)

85

T is the mean temperature calculated by the Fluent CFD code in degrees

Kelvin.

The instantaneous change in mean temperature can be determined by adding
the fluctuating component of temperature (T’) to the steady component of
temperature <T>:

T=T4<T>

(4.16)

Multiplying this equation by the any component of the Reynolds stress, will
produce the following:

<u075=<uhT5w+<wM><T>

(4.1 7)
< u'u’T' > was determined from the following equation of the turbulent

diffusion of the heat flux:

< u'u'T' >= —C0 [:[W au T]
5 &

 

(4.18)

where -Ce represents a coefficient with a value of 0.22. The term (a; T]1n
x

 

the expression above represents the gradient of the heat flux < u’T >. The
expression < u'u' >< T > in equation (4.18) represents the product of the

Reynolds stress and mean temperature. The addition of equation (4.17) and

86

equation (4.18) was used to evaluate the variance of turbulence and heat

transfer for this model:

< u'u'T >= —C0 £[Wa—u—Z} < u'u' >< T >
8 6x

(4.19)

This term was evaluated in a similar matter to the Reynolds stress term in
Chapter II by calculating the second and third invariants to create an
Anisotropic Invariant map of turbulent heat transfer. Figure 4.7 represents an
illustration of the region where the turbulence anisotropy was calculated to
evaluate the anisotropic invariance of the turbulent heat transfer generated in
this model. The turbulent anisotropy was calculated along the line between

bottom wall of cylinder and top of heated plate centered at x = 0.

87

 

Figure 4.7: Illustration of the region (line between bottom wall of
cylinder and top of heated plate centered at x = 0) where the turbulence
anisotropy was calculated to evaluate the anisotropic invariance of the
turbulent heat transfer generated by the model in equation (4.19).

88

lV.e. Numerical Method of Solution

HyperMesh, TGrid, and FLUENT CFD programs were also used in this study
to numerically simulate the hot soaking process as mentioned above. To
begin, a 2D model was constructed of a hollow cylinder above a hollow plate
located inside an enclosure using HyperMesh. The 20 model was then
imported into the TGrid program to generate the volume elements. Prisms
were initially grown from the surfaces of the heated plate and the cylinder for
the proper calculation of conjugate heat transfer. The prisms will be modeled
as the solid wall of the cylinder and plate. Triangular elements were then
used to generate the mesh for the remaining surfaces of the model. The
triangular volume in this model was used to simulate the air between the
cylinder and plate. After the volume was generated, the model was imported
into the FLUENT CFD program to numerically simulate the phenomenon of

the hot soaking process.

The focus of this study was to create a numerical simulation process of the
hot soaking phenomenon. This was achieved by performing an analysis
study of the heat transfer process between the cylinder and plate. To
understand the hot soaking process, the dominant mode of heat transfer had
to be determined. Conduction, convection, and radiation were studied to
establish the dominant mode of heat transfer. Two cases were studied to

determine the best process of numerical simulation of the hot soaking

89

process using CFD. These two cases were used to study the grid sensitivity
and turbulence modeling to evaluate how they both affect the hot soaking

process.

Grid Sensitim Study

Grids are elements that consist of triangular or equilateral cells and nodes
commonly known as grid points. There are many combinations of grids that
can be used to build models. Two types of grids that are most commonly
used are structured or unstructured grids. Structured grids consist of grids
that are generated in a uniform manner and that are similar in shape and size.
Unstructured grids are comprised of a grid patterns that are not uniform.
Unstructured grids are rarely the same shape or size. The geometries
primarily used in fluid flow and heat transfer problems are complex. The
construction of grids for a complex geometry using structured elements such
as quadrilateral and hexahedral is very time consuming and nearly
unfeasible. The best type of grid to use for a complex geometry is an
unstructured grid with triangular and tetrahedral elements because it is less

time consuming and allows the use of fewer elements.

Hyperrnesh and Tgrid are pre-processors that were used primarily in this
study for the creation of the 2D triangular unstructured mesh. The ZD model
was then imported into the Fluent program, which was used to create the

different topologies using solution-adaption grid refinement. Solution-

90

adaption grid refinement was used to refine the grids that are located
between the cylinder and plate. The grids in this area are important to the
evaluation the hot soaking phenomenon. To adapt grids in the FLUENT
program, the region adaption method was used. This method allows the user
to define and select the specific region of cells for grid adaption. The first 2D
volume mesh generated consisted of coarse grids that did not produce
accurate data (See Figure 4.8(a)). As a result, “‘finer” meshes were created
between the cylinder and plate to produce more accurate results. The finest

mesh which produced the most accurate results is shown in Figure 4.8(d).

Each of the grid meshes were studied in order to determine how the results
generated from the numerical simulations were affected by the size of the
grid. This is considered a Grid Sensitivity study. If the solution is dependent
upon the grid size, then an optimum grid size will need to be use when
creating a simulation of the hot soaking process. If the solution is not
dependent upon the grid size, less grids can be used which can decrease the

time it takes to create the model and increase the speed of the calculations.

lV.f. Numerical Results

As mentioned previously, two different cases were examined to determine the
optimal process to numerically simulate the hot soaking process. These two
cases were used to study the grid sensitivity and turbulence modeling. To

examine the grid sensitivity, the ZD model was modified by creating a range

91

of models containing different size elements located between the heated plate
and cylinder. The main focus of this examination is the affect the grid size
has on the accuracy of the results. Finally, to examine turbulence modeling,
the boundary conditions of the 2D model will be modified by using the

Realizable K-Epsilon, K-Omega, and Reynolds Stress turbulence models.

Case I: Grid Sensitivity Study

Figure 3.7 Illustrates the different types of grid adaption refinements that were
generated between the cylinder and heated plate. Four different grid types
were generated for the grid sensitivity study. Figure 4.8 (3) represents the
grid adaption model Type 1. This grid is course compared to the other three
grid adaption models. Figure 4.8 (b) represents the grid adaption model Type
2 which displays one level of grid refinement. Figure 4.8 (c) represents the
grid adaption model Type 3 which displays a level of refinement greater than
the Type 2 grid refinement. Finally, Figure 4.8 (d) represents the grid

adaption model Type 4 which displays the finest grid refinement.

92

 

 

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’ - v c 1 v A . .

r :41 014.9%.S‘vflFafihgmagfifigmfl§kv $15
I.» ...

 

 

 

 

 

 

 

 

 

 

 

Figure 4.8: Plot of the different types of grid adaption refinements that
were generated between the cylinder and heated plate: (a) represents
Grid Type 1 (the course grid), (b) represents Grid Type 2 (c) represents
Grid Type 3 and (d) represents Grid Type 4 (the finest grid).

93

Figures 4.9 through 4.12 illustrate the Anisotropic invariant Maps generated
from the steady state calculation of the Reynolds Stress Model for the four
different grid types. In Figures 4.9, 4.10, and Figure 4.11, the grid types used
produced both physically unrealizable and realizable turbulent flow. The
realizable turbulence generated by these grid configurations produces
turbulent flow that is close to isotropic turbulence. Isotropic turbulence
represents turbulent flow with zero anisotropy. In Figure 4.12, the turbulence
generated by the type 4 grid refinement is physically realizable. The turbulent
flow produced by this model tends to range from axisymmetric contraction
(“Disk-Like”) turbulence to isotropic turbulence according to the diagram.
According to Krogstag and Simonsen (2005), the “Disk-Like” turbulence
occurs when one component of the turbulent kinetic energy is smaller than

the other two components of turbulent kinetic energy which are equal.

94

 

 

>3.0o...oo.0 .3<0...03. 2.00 . 930.0 2.000. 04.0 .... cam: 4:30.0300 .5000:

-ll

 

o.

c

 

 

 

ll 3.0.3.3030 06030.03

I 3.0.3.3030 003.8030:

 

 

 

I 4.20 00.300303. 4:350:00

 

 

04.0 . 00.0 00 .830 .3 .0 3.0.. 0.00 -
.0. 04004

 

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0.0m 0..

 

     

00.00.0003 . ,
03000.00 3.000. a... n ”40.3 x
000.00 .3000. 04 n 3.00 x

 

 

o...m ob ohm

 

 

tropic Invariant Map generated from the

steady state calculation of the Reynolds Stress Model for Grid Type 1.

ISO

Plot of the An'

Figure 4.9

95

 

 

0.100.308.3333. 2.00 . 2.30.0 .5000. 04.0 >000..03 0 5m: 4:350:00 .5000:

 

J90

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0 - I - 13.053.30.30 00030.0:
\ II 2.5.3.3030 003.8000:
0 m . .. , i 4.3 00.300303. 4:350:00
IT 030 M :0 00.0 30.2 .030 083.0 030 053.: 300. 20.. -
o... .0. 0.004
I .. 1.1 03.0 N :0 00.0 305. .030 033.0 030 0.333 .30.. 060
... . - .0. 04004
0 m I. 030 N 00 00.0 300.. $0: - .0. 04004
. .. \ 19.030 m 00 00.0 .30.. <50: - .0. 0400.
I. 0.0. -.----I .-...I
II /.I II
3
l I- OIIII -0 4 . .11. II IJ‘
.o 00 c 0.0... o. o..m o.» .8
0..

 

 

 

 

 

 

   

o I 0.000.. 08.0 030.5...
5 I 030.0004 032.3...

 

 

 

 

 

Figure 4.10: Plot of the Anisotropic Invariant Map generated from the

steady state and transient calculation of the Reynolds Stress Model for

Grid Type 2.

96

 

 

>3.00.30.0 55:03. :00 . 2.30.0 3000. 0:0 >000..03 a cam... 4:30.033 ...—000..

 

0......

 

  
       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.
_
.
I‘IIIA: -. 4
0.0 III . I --
_nll>x.0<.3.30.3.0 06030.03
ll 3.0533030 003.3020:
.l.. ...50 00300303. 40.00.0300
IOI 000 w 00 00.0 300. 20.. - .0. 0.00..
_+ 010 ‘0” 0.0.-.0..m.0 30x <0.:0 - .0. 00000
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.0.. . I- I I I llJ
.0.»

 

 

 

00 n 0.00% 0.0.0 03020.0
:0 s 030.003 03020.0

 

 

 

 

tropic Invariant Map generated from the

ISO

: Plot of the An'

Figure 4.11

steady state calculation of the Reynolds Stress Model for Grid Type 3.

97

 

 

>3.00...00.0 .3<0...03. 2.00 .. 0.30.0 2.000. 010 >000..03 0 ..amz. 40.60.0300 2.000..

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I0...
0.0
0.0
ll>x.0<_.3.30.30 06030.03
0 n I >x.0<3.30.30 0030.00.63
4.00 00300303. 40.00.0300
fl 0... .
+030 0 00 300. $0.. - .0. 0.00 ~30
0.004
+ 030 a 00 300. 5.0.. . ~30 0.00.
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\ +030 0 00 .30.. <0_00 - .0. 0.00
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30 01..
00 I 0.000.. 08.0 03030.0
:0 I 030.000.. 03033.0

 

 

 

 

Figure 4.12: Plot of the Anisotropic Invariant Map generated from the

steady state calculation of the Reynolds Stress Model for Grid Type 4.

98

Figure 4.13 illustrates the Anisotropic Invariant Map generated from the
model’s maximum anisotropy values of the transient calculation of the K-
Epsilon Model for Gn’d Type 4. The anisotropy was calculated from the
maximum Reynolds stress values of the entire model. It is shown in this
Figure that anisotropy decreases as time increases. The turbulent flow
becomes more realizable as the length of time to calculate the model
increases. Figure 4.14 displays the anisotropy calculations generated for the
last two intervals of time equal to 1,155.5 seconds and 1,173 seconds. The
turbulence flow becomes physically realizable after the time interval of 990.5
seconds according to Table 4.3. The turbulent flow of this model tends to
display axisymmetric expansion (“Rod-like”) turbulence. The “Rod-like”
turbulence was defined previously in Chapter II. Figure 4.15 illustrates the
Anisotropic Invariant Map generated from the transient calculation of the K-
Epsilon Model for Grid Type 4. These anisotropy values were calculated near
the bottom wall of the cylinder. The turbulence generated for this model over
time is physically realizable. The turbulence generated is near the

axisymmetric contraction and isotropic zones.

99

 

 

>3.00.400.0 .3<0q.03. .500 . 04.0 0. 030.0005 x-m00._0340...00.0300 2.000.
...—0.030.: <0.00 0. .5000. - .:<0...03.0 30000400 O<0.. 4.3.0.

 

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.- .LII..>x.0<.330.30

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I >x.0<.330.30
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Ibl 4u00 0000300

+413 0000300

+4n.mo 0000300

lel 4umao 0000300

IT 4nwmo 0000300

4&3 0000300

4u000 0000300

4&3 0000300

.0080 0000300

.. 4n. .000 0000300

.-o- 41.3 0000300

 

 

 

transient calculation of the K-Epsilon Model for Grid Type 4 (Maximum
100

Figure 4.13: Plot of the Anisotropic Invariant Map generated from the
Anisotropy Value of Model).

 

 

4|

>3.0o=.ou.0 .3<0q.03. 2.00 - 04.0 0. C300000<. x-muw..034c_&c.0300 2.000.
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‘ :‘iéa; 1‘ i 1 n ‘ 5“ i 1‘ i I w‘

 

 

 

 

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4:30.038

Il 4150 m 0000300

 

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no Invariant Map

Plot of a closer view of the Anisotrop'

generated from the transient calculation of the K-Epsilon Model for Grid

Type 4 (Maximum Anisotropy Value of Model).

Figure 4.14

101

Table 4.3: List of the second (II) and third (lll) scalar invariants of
anisotropy over a range of time from zero seconds to 1,173 seconds
generated from calculating the maximum anisotropy value of model for

the Grid Type 4 K-Epsilon Model.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time Invariant -|l Invariant III (Max
(seconds) (Max Value) Value)
0 3497.45 1165.78
60 4154.35 1384.75
120 832.78 277.56
180 544.12 181.34
240 290.30 96.73
360 4283.64 1427.84
540 210.36 70.08
[660 15.19 5.03
510 3.15 1.01
|990.5 -0.29 -0.13
1155.5 0.29 0.06
1173 0.30 0.06
Shaded area represents unrealizable

turbulence anisotropy

 

 

 

102

 

 

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Plot of the Anisotropic Invariant Map generated from the

Figure 4.15

transient calculation of the K-Epsilon Model for Grid Type 4 (Anisotropy

Value Near Wall of Cylinder).

103

Figure 4.16 illustrates a plot of the Grid Refinement versus the turbulence Y+
values generated at the bottom wall of cylinder. The Y+ value measures the
distance of the boundary layers in which laminar and turbulent flow develop.
In this model it is shown that the Y+ value measured decreases with grid
refinement of the model. This displays that out of the four models chosen for
this study, the grid type 4 model was the best at predicting the nature of the

boundaries in the near wall region of the bottom wall of the cylinder.

Figure 4.17 illustrates the Hot Soaking temperature profile of the bottom wall
of the cylinder generated from the transient calculation of the K-Epsilon Model
for Grid Type 4. As mentioned previously the objective of this thesis is to
simulate the hot soaking phenomenon through CFD with a cylinder above a
heated plate. This graph shows the temperature data measured on the
bottom of the hollow cylinder over time. The temperature of the bottom wall
of the cylinder reaches a peak at 300 seconds. The temperature of the
bottom wall of the cylinder then decreases reaching steady state. This
temperature behavior clearly displays the hot soaking phenomenon described

in Chapter I.

104

 

 

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Figure 4.16: Plot of the Grid Refinement versus the Turbulence Y+

values generated at the bottom cylinder wall.

105

 

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Figure 4.17: Plot of the Hot Soaking temperature profile of the bottom
wall of the cylinder generated from the transient calculation of the K-
Epsilon Model for Grid Type 4.

106

Figure 4.18 illustrates the temperature profiles of the bottom wall of the
cylinder generated from the transient calculation of the K-Omega and
Reynolds Stress Turbulence Models for Grid Type 4. As shown by the
temperature profiles, these models did not accurately simulate the hot
soaking process. These temperature profiles do not display an abrupt
increase in temperature on the bottom wall of the cylinder. It is also important
to determine the dominant mode of heat transfer that produces the high

temperature on the bottom wall of the cylinder.

107

 

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Figure 4.18: Plot of the temperature profiles of the bottom wall of the
cylinder generated from the transient calculation of the K-Omega and
Reynolds Stress Turbulence Models for Grid Type 4.

108

Figure 4.19 illustrates the dominant mode of heat transfer generated as a
result of the hot soaking phenomenon on the top wall of the plate. This graph
shows that that natural convection is the dominant mode of heat transfer
produced by the hot plate during the initial phase of the hot soaking process.
It also shows that the natural convection reaches equilibrium along with the
other modes of heat transfer after 400 seconds. Natural convection is the
highest contributor of temperature generation on the bottom wall of the

cylinder.

Figure 4.20 illustrates the modes of heat transfer generated as a result of the
hot soaking phenomenon on the bottom wall of the cylinder. It is shown that
radiation is the second highest mode of heat transfer during the initial phase

of the hot soaking process.

109

 

 

 

 

 

 

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Figure 4.19: Plot of the dominant mode of heat transfer generated as a
result of the hot soaking phenomenon on the top wall of the plate for K-
Epsilon Model.

110

 

 

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Figure 4.20: Plot of the modes of heat transfer generated as a result of
the hot soaking phenomenon on the bottom wall of the cylinder for K-
111

Epsilon Model.

Case ll: Turbulence Model Study

Figure 4.21 illustrates the Anisotropic Invariant Map generated from the use
of different turbulence models for Grid Type 4 measured at steady state. The
anisotropy value was measured near the bottom wall of the cylinder. It is
shown in this graph that the Reynolds Stress model, K-Epsilon and K—omega
turbulence models generate physically realizable turbulence near the bottom

wall of the cylinder.

Figure 4.22 illustrates the Anisotropic invariant Map generated from the K-
Epsilon turbulence models for Grid Type 4 measured near the bottom wall of
the cylinder during the transient state. This graph shows that the K-Epsilon
turbulence model produces physically realizable turbulence when measured
over time. The turbulence generated is centered between the axisymmetric
contraction, axisymmetric expansion, and isotropic two-component
turbulence. Figure 4.23 illustrates the Anisotropic Invariant Map generated
from the K-Omega turbulence models for Grid Type 4 measured near the
bottom wall of the cylinder during the transient state. It is shown in this graph
that the K-omega turbulence model generates physically realizable turbulence
that range from axisymmetric contraction to axisymmetric expansion over
time. Figure 4.24 illustrates the Anisotropic lnvan'ant Map generated from the
Reynolds Stress turbulence models for Grid Type 4 measured near the
bottom wall of the cylinder during the transient state. This graph shows that

the Reynolds Stress turbulence model produces unrealizable turbulence near

112

the bottom wall of the cylinder. The anisotropy becomes more unrealizable
as the length of time increases. Overall, the K-Epsilon turbulence model
produces a constant state of realizable turbulence when compared to the
varying realizable turbulence of the K-Omega model and the unrealizable

turbulence of the Reynolds Stress Model.

113

 

 

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Figure 4.21: Plot of the Anisotropic Invariant Map generated from the
use of different turbulence models for Grid Type 4, steady state

(Anisotropy Value Near Wall of Cylinder).

114

 

 

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Epsilon turbulence models for Grid Type 4, unsteady (Anisotropy Value

Near Wall of Cylinder).

115

 

 

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Omega turbulence models for Grid Type 4, unsteady (Anisotropy Value
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Figure 4.24: Plot of the Anisotropic Invariant Map generated from the

Reynolds Stress turbulence models for Grid Type 4, unsteady

(Anisotropy Value Near Wall of Cylinder).

Figure 4.25 illustrates the surface pressure distribution generated on the
bottom wall of the cylinder. This graph shows a plot of the theoretical
description of the surface pressure distribution for boundary layer flow
generated around a circular cylinder. The surface pressure distribution was
predicted using the grid type 4 K-Epsilon turbulent model. The model predicts
the boundary layer similar to he theoretical data from theta = 30 to theta = to
150. The value of theta > 150 shows that the there is boundary layer

separation due to turbulence toward the rear of the cylinder.

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Figure 4.25: Plot of the surface pressure distribution generated on the
bottom wall of the cylinder.

119

Figure 4.26 illustrates the Anisotropic Invariant Map generated from the
combination of the Reynolds stresses and the mean temperature evaluated
using the steady state turbulence model. Figure 4.27 illustrates the
Anisotropic Invariant Map generated from the combination of the Reynolds
stresses and the mean temperature evaluated using the transient turbulence
model. In this figure the data was measured at t = 60 seconds. The
anisotropic invariance of the turbulent heat transfer was calculated using a
combination of the mean temperature and Reynolds stresses measured
between the bottom wall of the cylinder and the top of the heated plate (see

Figure 4.6).

As mentioned previously, the Anisotropic Invariant Map of the K-Epsilon
turbulence measured near the bottom wall of the cylinder during the transient
state shows a constant physical realizable turbulence. The evaluation of the
turbulent heat transfer anisotropy at steady state displays physically
realizable turbulent flow. The turbulence generated represents axisymmetric
contraction. The evaluation of the turbulent heat transfer anisotropy at t = 60
seconds also displays physically realizable turbulent flow. The turbulence
generated in this model ranges from axisymmetric contraction turbulence to

axisymmetric expansion turbulence.

120

 

 

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122

V. CONCLUSIONS AND RECOMMENDATIONS

The focus of this thesis is to investigate the hot soaking phenomenon by
quantifying the effect of the temperature generated on a hollow cylindrical
component above a heated hollow horizontal plate. This investigation will be
used to create a simple numerical simulation of the hot soaking process to aid
in further study of the thermal characteristics of vehicle components
contained within the underhood and underbody of a vehicle. The numerical

simulations were performed using the CFD Software Fluent.

V.a. Conclusions

Single Phase FILM Flow througfl Pipe with a 90 Degree Bend

The Reynolds Stress turbulence model and the realizable K-Epsilon
turbulence model produce both physically realizable and unrealizable
turbulence. The physically realizable turbulence produced by the Reynolds
Stress turbulence model displays turbulent flow that ranges from Two-
component turbulence to axisymetric-expansion to isotropic turbulence;
however the realizable K-Epsilon turbulence model displays isotropic
turbulent flow. The graph of the velocity profiles generated for fully developed
flow shows that the turbulent flow generated with the Reynolds number value
of 95,000 produces a velocity profile closest to the theoretical value for both

turbulence models.

123

Natural Convection in a 20 Square Enclosure

The realizable K-Epsilon model produces a numerical prediction of the natural
convection process similar to the prediction of laminar flow published in the
Fluent 6.1 Tutorial Guide. The use of the realizable K-Epsilon model with the
boussinesq approach provided an accurate simulation of the buoyancy effects
from natural convection. Figures 3-3 and 3-6 show the model’s ability to
simulate the rising plume of the lighter hot air and the falling plume of the
heavier cold air created by the changes in temperature and density of the
model. The comparison of the dimensionless height of the enclosure versus
the dimensionless fluid temperature at mid width in Figure 3-7 to the
theoretical data in Figure 3-8 further displays the model’s ability to simulate

natural convection.

Two Dimensional Mogel of a Cylinder above a Heated Plate

The Grid Type 4 refinement level produces the best prediction of realizable
turbulent flow. The K-Epsilon turbulence model produces physically
realizable turbulence when measured over time. The turbulence generated is
centered between the axis-symmetric contraction, axis-symmetric expansion,
and isotropic twp-component turbulence. Overall, the K-Epsilon turbulence
model produces a constant state of realizable turbulence when compared to
the varying realizable turbulence of the K-Omega model and the unrealizable

turbulence of the Reynolds Stress Model.

124

The K-Epsilon model temperature profile shows an accurate simulation of the
hot soaking phenomenon. It displays an abrupt increase in temperature on
the bottom wall of the hollow cylinder. The temperature spike created on the
bottom wall of the cylinder was generated from natural convection from the
top of the heated plate. The surface pressure distribution measured around
the hollow cylinder agrees with theoretical data and shows evidence of a
turbulent boundary layer near the rear of the cylinder. The evaluation of the
turbulent heat transfer anisotropy at steady state calculation and the transient
calculation oft = 60 seconds displays physically realizable turbulent flow. The
steady state calculation of anisotropy produces axisymmetric contraction
turbulence. The unsteady calculation produces turbulence that ranges from

axisymmetric contraction turbulence to axisymmetric expansion turbulence.

In conclusion, the study of the model of a hollow cylinder above a heat plate
successfully produced a numerical simulation of the hot soaking
phenomenon. Through the preliminary studies of the previous two cases of
Single Phase Fluid Flow through a Pipe with a 90 Degree Bend and Natural
Convection in a ZD Square Enclosure, the understanding of the use of CFD to
numerically predict turbulent flow and heat transfer was achieved. This
gained knowledge of the use of CFD to numerically simulate turbulence and
heat transfer aided in the creation of the numerical simulation of the hot

soaking process as well as the determination of the dominant mode of heat

125

transfer during this process. This model can now be used for studies of
vehicle component that are exposed to extreme temperature conditions in
environments without forced convection. Some recommendations are listed
below regarding the development of a more robust simulation of the hot

soaking process.

V.b. Recommendations

Based on the trend of the increased accuracy of turbulent flow predictions
through grid refinement, it is recommended that the grids modeled during a
hot soaking simulation require a high level of refinement. Although the
realizable K-Epsilon model accurately simulated the hot soaking process, a
highly refined grid model will allow the use of a variety turbulence models to
simulate hot soaking for different testing conditions. An extensive study is
required to evaluate the predictability of the turbulent heat flux through
anisotropy and invariance. This study should carefully analyze of the use of

different turbulence models as well as the different modes of heat transfer.

126

APPENDIX

127

APPENDIX A

SINGLE PHASE FLOW THROUGH A PIPE WITH 90 DEGREE BEND

128

Table A-1: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map generated for single phase fluid flow through a pipe using the K-
Epsilon Turbulence Model at Re N0. = 1000.

Flow Anisotropy Calculations

(Minder

 

Table A-2: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map for single phase fluid flow through a pipe using the K-Epsilon
Turbulence Model at Re N0. = 9500.

Flow Anisotropy Calculations

Cylinder

 

129

Table A-3: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map generated for single phase fluid flow through a pipe using the K-
Epsilon Turbulence Model at Re N0. = 95,000.

Flow Anisotropy Calculations

Cylinder

 

Table A-4: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map for single phase fluid flow through a pipe using the K-Epsilon
Turbulence Model at Re No. 1000.

Flow Anisotropy Calculations

(Minder

 

130

Table A-5: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map for single phase fluid flow through a pipe using the Reynolds
Stress Turbulence Model at Re N0. = 9500.

Flow Anisotropy Calculations

Cylinder

 

Table A-6: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map for single phase fluid flow through a pipe using the K-Epsilon
Turbulence Model at Re No. 95,000.

Flow Anisotropy Calculations

Cyllnder

 

131

Table A-7: List of the dimensionless values of radius and velocity used to create
the plot of the Velocity Profile of Fully Developed Flow for single phase fluid flow
through a pipe using the K—Epsilon and Reynolds Stress Turbulence Model at
different Reynolds numbers.

 

132

APPENDICES B

NATURAL CONVECTION IN A 2D SQUARE ENCLOSURE

133

Table B-1: List of the dimensionless parameters used to create the plot of dimensionless height
(in the y—direction) versus dimensionless temperature generated inside the 2D Enclosure for the
Realizable K-Epsilon turbulence model without radiation.

 

134

APPENDICES C

TWO-DIMENSIONAL MODEL OF CYLINDER ABOVE A HEATED PLATE

135

Table C-1: List of the temperature profile specified for the inner surface of the
heated plate.

 

136

Table C—2: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map of the steady state calculation of the Reynolds Stress Model for
Grid Types 1 through 4.

 

137

Table C-3: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map of the transient calculation of the K-Epsilon Model for Grid Type 4
(Maximum Anisotropy Value of Mode and value measured near bottom cylinder
wall).

Cylinder

 

138

Table C-3 (cont’d): List of the anisotropic values calculated to generate the
Anisotropic Invariant Map of the transient calculation of the K-Epsilon Model for
Grid Type 4 (Maximum Anisotropy Value of Mode and value measured near
bottom cylinder wall).

Cylinder

 

Table C-4: List of the Y+ values measured to create the plot of the Grid
Refinement versus the Turbulence Y+ values generated at the bottom cylinder
wall.

 

139

Table C-5: List of the temperature values measured to create the Hot Soaking
temperature profile of the bottom wall from the cylinder for the transient
calculation of the K-Epsilon Model for Grid Type 4.

 

140

Table C-6: List of the temperature values calculated to create the temperature
profiles of the bottom wall of the cylinder from the transient calculation of the K-
Omega Turbulence Model for Grid Type 4.

 

141

Table C-7: List of the temperature values calculated to create the temperature
profiles of the bottom wall of the cylinder from the transient calculation of the
RSM Turbulence Model for Grid Type 4.

 

142

Table 08: List of the heat flux values calculated to generate the plot of the
dominant mode of heat transfer generated as a result of the hot soaking
phenomenon on the top wall of the plate for K-Epsilon Model.

15920 410.4925 64881.4925

 

Table C—9: List of the heat flux values calculated to generate the plot of the
modes of heat transfer generated as a result of the hot soaking phenomenon on
the bottom wall of the cylinder for K-Epsilon Model.

 

143

Table C-10: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map from the use of different turbulence models for Grid Type 4, steady
state (Anisotropy Value Near Wall of Cylinder).

Cylinder

 

Table C-11: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map from the K—Omega turbulence models for Grid Type 4, unsteady
(Anisotropy Value Near Wall of Cylinder).

Cylinder

 

Table C—11 (cont'd): List of the anisotropic values calculated to generate the
Anisotropic Invariant Map from the K-Omega turbulence models for Grid Type 4,
unsteady (Anisotropy Value Near Wall of Cylinder).

Cylinder

 

145

Table C-12: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map from the Reynolds Stress turbulence models for Grid Type 4,
unsteady (Anisotropy Value Near Wall of Cylinder).

 

146

Table C—13: List of the pressure coefficient values calculated to generate the plot
of the surface pressure distribution generated on the bottom wall of the cylinder.

 

147

Table C-14: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map generated from the combination of the Reynolds stresses and the
mean temperature evaluated using the steady state K-Epsilon turbulence model
for Grid Type 4 (Measured between Cylinder & Heated Plate).

 

148

Table C-15: List of the anisotropic values calculated to generate the Anisotropic
Invariant Map generated from the combination of the Reynolds stresses and the
mean temperature evaluated using the unsteady K-Epsilon turbulence model for
Grid Type 4 att = 60 seconds (Measured between Cylinder & Heated Plate).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-coordinate
“1"” w invariant -11 Invariant m
cylinder and
heated plate
0.040953297 0.083329624 -0.009260496
0.039832428 0.083335496 0009258538
0.038711558 0.130022925 0.006303938
0.037590688 0.1 18422638 0.0024371 76
0.036469819 0.169985922 0.019624937
0.035348949 0.243971555 0.044286815
003422808 0.084635572 -0.00882518
0.03310721 0.178266659 0.022385183
0.031986341 0.173678426 0.020855772
0.030865471 0.181625066 0.023504652
0.029744601 0.194301167 0.027730019
0.028623732 0.201411937 0.030100275
0.027502862 0.201411937 0030100275
0.026381993 0.198632944 0.0291 73944
0.025261123 0.195123541 0.028004143
0024140254 019087553 002658814
0.023019384 0.185631227 0.024840039
0.021898514 0.188539279 0.025809389
0020777645 0.212987373 0033958754
0.019656775 0.220294374 0.036394421
0018535906 018858835 0025825746
0.017415036 0079797773 0010437779
0.016294167 0.083332445 -0.009259555
0.01522 0.083317044 0009264689
0.01517 0.083322108 -0.009263001
0.01 51 1 0083293672 000927248
0.01506 0.082032324 -0.009692929
0.01501 0.083730001 -0.009127037

 

 

 

 

149

 

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150

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