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In memory of my late father whose wisdom and guidance
have been an inspiration to me.
ABSTRACT
Numerical invgtigations of fullydeveloped, turbulent, singlephase and gasliquid,
bubbly two-phase flows in straight ducts of arbitrary, but uniform, cross-sections
are prrsented in this thesis. The main focus of this thesis is on the formulation,
implementation, and demomtration of numerical procedures for computer simulations
of the aforementioned flows in ducts of nondrcular cross-sections
The Nlydeveloped, turbulent, singlephase flow investigations were camd out in
this work in order to m l v e some shortcomings in the numerical solution procedures,
and also to lay a h n foundation for the two-phase flow investigations. In particular, a
new procedure was developed for appropriate implementations of the socalled wall-
function treatment for bridging the n e a r - d regions in fullydeveloped, turbulent
flows in ducts of complex cross-sections, in the context of Control-Volume Finite
Element Methods (CVFEMs). Four turbulence models, a linear-eddy viscosity two-
equation k - c model, a nonlinear quadratic eddy-viscosity model, a nonlinear cubic
eddy-viscosity model, and an explicit algebraic stress model were considered.
The proposed CVFEM, with the four turbulence models and the proposed wall-
function treatment, was used for computer simulations of fullydeveloped, turbulent,
single-phase flows in ducts with four different duct c-sections a square, a tilted
square, a sector, and a triangle. The numerical predictions were compared with
available experimental data and other numerical predictions: Very good agreements
were observed. The performance of the explicit algebraic stress model was found to
be the best among the four turbulence models investigated.
In two-phase flow investigations, only fullydeveloped, turbulent, dilute, bubbly,
gas-liquid flows in vertical ducts, in which both phases flow upward, in a direction
opposite to the gravitational acceleration vector, are considered. The mathemati-
cal model of the above-mentioned two-phase flows is based on ideas bornwed from
available two-fluid models, correlations for interfacial forces, and a two-timescale
turbulence model. The proposed CVFEM. an adaption of the aforementioned two-
t i m e - d e turbulence model, and a wall-function treatment were amalgamated to
demonstrate a method for computer simulatiors of the two-phase flows of interest.
The shortcomings of the available Bcperimentd data for proper testing of the propased
method are discused.
The numerical predictions of the two-phase fows agree qualitatively with the avail-
able experimental data More research is, however, needed for a better understanding
of the two-phase flows considered in this thgis In particular, better modeling of in-
terfacial forces in the vicinity of the wall, formulations that account for bubble-ske
distribation, and a wall-function treatment that is specially designed for such flows,
are r q i
Nous prhntons dans cette these, des .F-;mnlxtions nnm6riques d'fcoulements pleine-
ment d+lopp& turbulents monophasiques ainsi que d 'hulements diphasiques tur-
bulents a bu l l s dans des conduites h i t s de d o n s transversaes uniformes mais
im&dZerer Le point central de cette thbe &ide dans la formulation, I ' i p l h e n t a -
tion et la dimonstration de p d u r e s num6riques pour la simulation sur ordinatear
des koulements mentionnk adessus dam des conduites de sections non drrulaires
Dans ce travail nous awns 6tudi6 des hulements d&eIopp& turbulents monopha-
siques a6n d e pallier certaines lacunes dans les pnddures num6riques e t d'etablir une
base solide pour I'Ctude des hulements diphadques. En particulier, nous awns de-
veloppc? une nouvelle pro&ure pour des impl6mentations appropri6es de ce qui est
appel6 le traitement des fonctions de paroi pour la modelisation des r6gior.s prZs des
p a d s dans les hu lements turbulents pleiuement d M o p p & dans des conduits de
sections complexes dans le contexte de la rncthode de9 Voluma de ConfrcjIes ~ ~ h m t s
Fin& (CVFEMs). En outre, quatre models de turbulence out 6t6 consid& B sawir
un modele B B i t 6 turbulente lin&re B deux 6quations k-c, un modMe non-lin*
B viscosite turbulente quadratique, un modde non-liniaire Q viscosit6 turbulente cu-
bique et un modele algbbrique explicite.
La technique CVFEhl couplb aux quatre moddes de turbulence et B la nou-
velle loi de paroi a dtd appliqub B la simulation num6rique dhulements turbu-
lents pleinement d6velop& et monophasiques dans des conduits avec quatre sections
diffkentes: un d, un d incliner, un secteur et un triangle. Nous avons com-
pard nos pr6dictions num6riques aux d o n n h exp6rimentales diionibles et B d'autres
prdictions num6riques. Nous awns observC une tris bonne concordance et la per-
formance du modele algdbrique explicite s'est avdrb la meilleure parmi les quatres
modeles de turbulence dtudik.
Pour les Coulements diphasiques, notre Ctude a i t6 limit& i des Coulements
pleinement d6velopp5s turbulents dans des conduites verticales pour des fluides gaz-
liquids, diluC et B bulles. Pour les deux phases, 1 'Coulement est ding6 vers le haut
dans la direction opposk a celle la gravit6. Le modkle mathematique u t i l i pour
cet bulement diphasique a Ptd empruntP de modkles disponibles pour bulements H
deux fluid- de cordations pour les forces interfadales et d'un modde de turbulence
B deux &elk de temps. Ce modde de turbulence ainsi que la loi de la paroi out
aloa Ptd adapt& aux hulements diphasiques qui nous inttkesent puis coupl& au
& h a CVFEM. Nous discutons aus i du manque de d o n n b exp&hentales pour
valider les pr6dictions num6riques de ces bulements
Les predictions numCriques des hulements diphasiques se comparent qualita-
tivement aux d o n n k exp6rhentales disponibles. XI est neesake, cependant, de
continuer la recherche dam ce domaine pour mieux m e r les hulcments tlubu-
lents diphasiques consider& dans cette these. En particulier, il est nkesaire de bien
mod* les form interfades au voisinage des parois, de murir H des formula-
tions qui tiennent compte de la distribution de la taille des bulles et d'utiliser des lois
de p;uois qui sont d&isign&s pour ce type d'hulements.
I muld like to take this opportunity to thank my supervisor, Professor B.RBaliga,
for his guidance during the course of this work H i valuable suggestions, friendly
discusions and the time spent in reviewing the manuscript are well acknowledged.
I am very gratefd for the very useful inputs, clari6cations, and valuable advice
provided by several leading researchers in the fields of turbulence and bm-phase
flows. LQ particular, I would like to acknowledge the following persons in this regard:
Professor B.E. Launder, Professor DA. Drew, Professor A. Serizawa, Dr. M. Lopez
de Bertodano, Dr. J.L. MariC, Dr. T.B. Gatski, Dr. TJ. Craft, and Dr. FS. Lien.
Special thanks to Dr. J.L. Marib, Dr. TJ. Liu, and Dr. S.P. Antal for providing
experimental data related to turbulent, dilute, bubbly *phase flows.
The Iinanfd support provided by the National Iranian Oil Company through
study leave program is highly appreciated. The finandal support of Hydro Quebec
in the form of McGill university major f e l l d p is greatly acknowledged. Partial
financial support in the form of research anistantships from Professor B.R Baliga
is gratefully acknowledged. Also, the computer facilities which were provided by the
CEntre de Recherche en Calcul Applique' (CERC.4) are greatly appreciated. The
valuable services of the department staff, in particular Ms. C i d , are appriciated.
My long stay at McGill university gave me the opportunity to find many new
friends. It is a long list, sc, without recalling their names, I thank all of them for their
friendship, particularly the 550 Sherbrooke club, and my Iranian friends
This work could not have been attempted without the love, patience, and moral
support of my family, in particular my wife, Sousan, and my daughters, Sareh and
Sahar. Their sacrifices were great: my gratitude is profound.
In my life, I had the privilege of knowing certain people with big souls and brave
hearts: They sam$ced their lives to salvage humanity; their memories continue to be
an inspiration to me.
Finally, I would l i e to dedicate this work to the soul of my late father, who passed
away during my graduate studies abroad. Hi loss was great to me, but hi love stays
in my heart forever.
Contents
... Abstract ...............................*.... m
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
... ListofFigures ................................ XIII
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 Introduction 1
1.1 Aims.................................... 1
1.2 Overview of the Thesis and Mot:vation . . . . . . . . . . . . . . . . . 2
1.2.1 SinglePhase Fully-Developed Wbulent Flows . . . . . . . . . 3
1.2.2 Two-Phase Fully-Developed Turbulent Flows . . . . . . . . . . 10
1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Synopsis of Relevant Publications 19
2.1 Gas-Liquid Tw*Phase Flow . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Turbulence Modeling for SinglePhase Flows . . . . . . . . . . . . . . 22
2.3 Fully-Developed Turbulent Flows in Straight Ducts . . . . . . . . . . 28
2.3.1 Singlephase Flow. . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Bubbly TwPhase Flow . . . . . . . . . . . . . . . . . . . . . 32
2.4 Two-Phase Numerical Methods . . . . . . . . . . . . . . . . . . . . . 35
CONTENTS
2-5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Mathematical Models
............ 3.1 Fully.developed. lhrbulent. Single-Phase l h v s
3.1.1 Scope ............................... ....................... 3.15 Governing Equation
. . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Thbulence Models
. . . . . . . . . . . . . . . . . . . . . 3.1.4 Wd-Function Treatment
. . . . . . . . . . . . . 3.2 Fully.Developed, lhrbulent, T-Phase Flows
3.2.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Governing Equations
. . . . . . . . . 3.2.3 Expressions for interfacial momentum transfer
. . . . . . . . 3.2.4 'hbulence Models for Bubbly Two-Phase Flow
. . . . . . . . . . . 3.2.5 Specialization of the Momentum Equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Boundary Conditions
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Numerical Model
4.1 General Form of Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Domain Discretization
. . . . . . . . . . . . . . . . . . 4.3 Integral Consemtion Equation
. . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Interpolation hnctions
. . . . . . . . . . . 4.4.1 Diffusion Cbeffiaents. Density. and Sou-
. . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Mass Flow Rates
4.4.3 6 in Diffusion Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 4 in Convection Tenns
. . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Reduced Pressures
. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Void Fraction
. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Discretized Equations
4.5.1 Discretization Equations for 6 . . . . . . . . . . . . . . . . . . 4.5.2 Discretized Liquid-Phase Momentum Equations . . . . . . . .
CONTENTS X
4.5.3 D i t i z e d Equations for C~ossSectional Pressure and Void
Raa ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Complementary Algebraic Equations . . . . . . . . . . . . . . . . . .
4.6.1 Gashfomenturn Equations . . . . . . . . . . . . . . . . . . . . 4.6.2 Algebraic Equations for the Reynolds S t r e s s . . . . . . . . . 4.6.3 Bubble-induced Kinetic Energy of Turbulence and M o l d s
StrSses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Boundary Conditims . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.1 Special treatments . . . . . . . . . . . . . . . . . . . . . . -- 4.8 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 S- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Fully-Developed Turbulent Flows in Ducts- Implementat ion of Wall-
h c t i o n s in CVEEMs 9 1
5.1 Near-Wall h t m e n t for Single-Phase Fluid Flous . . . . . . . . . . . 92
. . . . . . . . . . . . . . . 5.1.1 D i t i z a t i o n of Near-Wall Regions 92
5.1.2 Specialization of Wall Functions . . . . . . . . . . . . . . . . . 95
. . . . . . . . . . . . . . . . . . . . 5.1.3 Implementation Procedure 96
. . . . . . . . . . . . . 5.1.4 Comments on an Alternative Approach 105
5.2 Near-Wall Treatment for Dilute, Bubbly , Two-Phase Flows . . . . . 106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Summary 109
6 Applications t o Single-Phase Fully-Developed Turbulent Flows 114
6.1 Fully.Developed. Turbulent Flow in a Straight Duct of Square Cross-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 115
. . . . . . . . . . . . . . . . . . . . . . 6.1.1 Domain Discretization 117
. . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Numerical Details 118
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Results 119
6.2 Fully.Developed. Turbulent Flow in a Straight Tilted Duct of Square
Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2.1 Numerical Details . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2.2 Rgults ............................... 6.3 My.Developed, Turbulent Flow in a Sector of a S e t Duct of
C i Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . .. 6.3.1 Domain Discretization . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Numerical Details ......................... 6.3.3 Mts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 My.Developed, Turbulent Flow in astraight duct of l b n g u h Cross-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section
6.4.1 Domain D i t i z a t i o n . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Numerical Details . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Results
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Applications t o Fully.Developed. Turbulent. Bubbly Two-Phase Flows156
7.1 Background Notes on the Test Problem . . . . . . . . . . . . . . . . . 156
7.2 Domain D i t i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3 Numerical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.3.1 Grid-Refinement Ch& . . . . . . . . . . . . . . . . . . . . . 160
7.3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 160
7.3.3 Boundary Conditions for the Diretized Void Raction Equations161
7.3.4 Solution Strategy and Convergence Criteria . . . . . . . . . . 163
7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.4.1 Numerical Predictions of Single-Phase Flows . . . . . . . . . . 165
7.4.2 Numerical Predictions of Bubbly TwwPhase Flows . . . . . . 166
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8 Conclusion 179
8.1 Review of the Thesis and Its Main Contributions . . . . . . . . . . . 179
8.2 Recommendations for Extensions of This Work . . . . . . . . . . . . 183
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
coCONTENTSmVTS
A Calculations of Relative-Velocity Components
List of Figures
Gas-Liquid Two-Phase Flow Patterns In a Vertical C i Pipe. . . 18
Gas-Liquid TuwPhase Flow Patterns In a Horizontal Circular Pipe. . 18
hUy-Developed. Single-Phase Flow in a Square-Duct: Secondary Flows
in the Cros-Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Fully-Developed, Single-Phase Flow in a Square-Dua: Bulging of the
&&I-velocity Contours . . . . . . . . . . . . . . . . . . . . . . . . . 39
Plane-Channel Turbulent Flow: Comparison of LRR and SSG models. 40
Distribution of Wall Static Pressure -dent. . . . . . . . . . . . . 40
Void Coring in Tu~Phase , Bubbly, Downward Flow in a Pipe. . . . . 41
Void Paddng in TuwPhase, Bubbly, Upuwd Flow in a Mangular Duct. 41
Example of a Straight Duct of Uniform Cross-Section. . . . . . . . . . 65
Example of a Straight Duct of Uniform Cross-Section, and the Global
Coordinate System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Discretization of a Calculation Domain into Three-Node Triangular
Elements and Polygonal Control Volumes. . . . . . . . . . . . . . . . 89
Typical Control Volumes. . . . . . . . . . . . . . . . . . . . . . . . . 90
Typical Triangular Element and Associated Nomenclature. . . . . . . 90
T-ypical Elements and Control Volumes .Associated With a Near-Wall
Node Adjacent to a Smooth Wall. . . . . . . . . . . . . . . . . . . . . 110
Typical Elements and Control Volumes Associated With a Near-Wall
Node Adjacent to a Re-entrant Comer. . . . . . . . . . . . . . . . . . 111
LIST OF FIGURES xiv
5.3 Typical Elements and Control Volume .l\sodated With a Near-Wall
Node Adjacent to an Outmud Corner. . . . . . . . . . . . . . . . . . 111 5.4 Typical Elements and Control Volumes Assodated With a Near-Wall
Node Adjacent to a Corner Betweea a Wall and a Symmetry Lime. . . 112 5.5 Velocity Components in the Cartesian Coordinate D i o n s and in
the Tangential and Xed D i i o n s to a Smooth Wall. . . . . . . . 112 5.6 Wall Shear Force Components in the Cartesian Coordinate D i o n s
and in the Tangential and Normal D i i o n s to a Smooth Wall. . . . 113
6.1 Fully-Developed Flow in a Square Duct Schematic Configuration and
Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Fully-Developed Flow in a Square Duct: Typical Uniform F i t e Ele-
ment Mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.3 Fully-Dweloped Flow in a Full Square Cross-Section: Re = 250000. . 138 6.4 Wy-Developed Flow in a Quarter Square CrossSeaion: Re = 250000.138
6.5 Fully-Developed Flow in a Square Duct: Axial Velocity Contours. . . 139 6.6 Fully-Developed Flow in a Square Duct: Variation of Average Friction
Factor.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.7 Fully-Developed Flow in a Square Duct: Wall Shear Stress Distribution. 140
6.8 Fully-Developed Flow in a Square Duct: hid-Velocity Variations
Along the Wall B i i o r . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.9 Fully-Developed Flow in a Square Duct: Axial-Velocity Variations
Along the Corner Bisector. . . . . . . . . . . . . . . . . . . . . . . . . 141 6.10 Fully-Developed Flow in a Square Duct: Turbulent Kinetic Energy
Distribution Along the Wall B i i t o r . . . . . . . . . . . . . . . . . . . 142 6.11 Fully-Developed Flow in a Square Duct: Turbulent Kinetic Energy
Distribution Along the Comer Bisector. . . . . . . . . . . . . . . . . . 142 6.12 Fully-Developed Flow in a Square Duct: v-Velocity Distribution Along
the Wall Bisector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
m OF FIGURES XV
6.13 Fully-Developed Flow in a Square Duct: v-Velocity Diribution Along
the Corner B i i r . . . . . . . . . . . . . . . . . . . . . ---- - . - . 143
6.14 My-Developed Flow in a Square D u 6 -u'b/uz D i b u t i o n Along
the Wall Bisector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.15 My-Dadoped Flow in a Square D u 6 -dv'/uz Diibut ion Along
theCorner B i i o r . . . . . . . . . . . . . . . . . . . . . . . - . . - . 144
6.16 Fully-Developed Flow in a Tilted Square D u 6 Axial Velocity Contours.145
6.17 Fully-Developed Flow in a Tilted Square Duct: Secondary Veloaty
Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.18 Nly-Developed Flow in a Tilted Square Duct: Axial-Velocity D i r i -
bution Along the Wall Bisector. . . . . . . . . . . . . . . . . . . . . . 146
6.19 Fully-Developed Flow in a Tilted Square Duct: Axial-Velocity D i r i -
bution Along the Corner B i i o r . . . . . . . . . . . . . . . . . . . . 146
6.20 Fully-Developed Flow in a Sector: Geometric finfiguration and N e
tation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.21 My-Developed Flow in a Sector: Typical Uniform F i t e Element Mesh.147
6.22 Nly-Developed Flow in a Sector: kdal Velocity Contours. . . . . . . 148
6.23 Nly-Developed Flow in a Sector: Axial Veloaty Contour. . . . . . . 148
6.24 Fully-Developed Flow in a Sector: Axial Velocity Profiles Along the
Nondimensional Wall-Normal Distance. . . . . . . . . . . . . . . . . . 149
6.25 Flow in a Sector: Turbulent Kinetic Energy Profiles Along the Radius. 149
6.26 Flow in a Sector: Rate of Turbulent Kinetic Energy Diipation Profiles
Along the Radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.27 Flow in a Sector: Shear Stres Profiles Along the Radius. . . . . . . . 150
6.28 Fully-Developed Flow in a Triangular Duct: Geometrical Configuration
and Piotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.29 Fully-Developed Flow in a Triangular Duct: Typical Uniform Finite
Element Mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.30 Fully-Developed Flow in a Triangular Duct: Axial Velocity Contours. 152
6.31 Fully-Developed Flow in a Triangular Duct: Secondary Velocity Vectors152
LIST OF FIGURES xvi
6.32 Flow in a h g u l a r Duct: Axial velocity Contours, Compii-hn of
Predicted and Experimental Results . . . . . . . . . . . . . . . . . . 153 . . . . . . . . . . 6.33 Flow in a h g u l a r Duct: Average f ict ion Faaor. 153
6.34 Flow in a Mangular Duct: *bulent Kinetic Energy Distribution. . 154 . . 6.35 Flow in a Mangular Duct: Local Wall Shear Stress Distribution. 154
6.36 Flow in a Triangular Duct: Reynolds Normal Stresses D i b u t i o n . . 155
7.1 Fully-Developed, Tu-phase Bubbly Flows in a Triangular Duct: Schematic
Configuration and Notation. . . . . . . . . . . . . . . . . . . . . . . . 172 7.2 My-Developed, Two-phase Bubbly Flows in a Triangular Duct: T y p
ical Uniform ~ h t e Element Mgh. . . . . . . . . . . . . . . . . . . . 172 7.3 Rtlly-Developed, Two-phase Bubbly Flows in a Mangular Duct: Liquid-
Phase Axial Velocity Profiles. . . . . . . . . . . . . . . . . . . . . . . 173 7.4 Fully-Developed, Two-phase Bubbly Flows in aTriangular Duct: Liquid-
Phase Axial Veloaty Profiles. . . . . . . . . . . . . . . . . . . . . . . 173 7.5 Fully-Developed, Two-phase Bubbly Flows in aTriangular Duct: Liquid-
Phase Axial Velocity Profiles, JL = 1.0. . . . . . . . . . . . . . . . . . 174 7.6 filly-Developed, Two-phase Bubbly Flows in a Mangular Duct: Void
Fraction Profiles, JL = 1.0. . . . . . . . . . . . . . . . . . . . . . . . . 174 7.7 Fully-Developed, Two-phase Bubbly Flom in a Triangular Duct: Liquid-
Phase Axial Velocity Profiles, JL = 0.5. . . . . . . . . . . . . . . . . . 175 7.8 Fully-Developed, Twwphase Bubbly Flows in a Triangular Duct: Void
Fraction Profiles, JL = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . 175 7.9 Fully-Developed, Two-phase Bubbly Flows in a Triangular Duct: Liquid-
Phase Axial Velocity Profiles, JL = 1.0, < (I >= 0.08: Effect of the
. . . . Assumed Void Fraction Distribution in the Near-Wall Regions. 176
7.10 Fully-Developed, Two-phase Bubbly Flows in a Triangular Duct: Void
Fraction Profiles, JL = 1.0, < ct >= 0.08: Effect of the Asumed Void
Fraction Distribution in the Xear-Wall Regions. . . . . . . . . . . . . 176
LIST OF FIGURES xvii
7.11 My-Dewloped, Tmphase Bubbly Flows in a %angular Duct Liquid-
Phase Axial Velocity Profiles, JL = 1.0, < a >= 0.08: Effect of the
. . . . . . . . . . . . . . . . . . . . . . . . . . . Lift Force Coeffiaent. 177
7.12 My-Developed, Tapphase Bubbly Flows in a Triangular Duct: Void
Raaion Profiles, JL = 1.0, < a >= 0.08: Effect of the Lift Force
Coeffiaent. ................................. 177
7.13 My-Developed, Tapphase Bubbly Flows in aTriangular Duct: Liquid-
Phase Axial Veloaty Profiles, JL = 1.0, < a >= 0.08: Effect of the
Bubble Diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.14 Fully-Developed, Tapphase Bubbly Flows in a Triangular Duct: Void
M i o n Profiles, JL = 1.0, < a >= 0.08: Effect of the Bubble Die ter .178
List of Tables
3.1 Coefficient Values in the Non-Linear Eddy-Viscosity Models and the
Explicit Algebraic Stress Model . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Specific Forms of the General Differential Equation . . . . . . . . . . . 69
6.1 Flow in a Quarter Square Duct: Results for Re = 250000 . . . . . . . 120 6.2 Flow in a Quarter Square Cros-Section: Average Friction Factor xl@
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for Re = 250000 121
6.3 Flow in a Sector: Results for Re = 420000 . . . . . . . . . . . . . . . . 128 6.4 Flow in a Sector, Average Friction Factor x102. Re = 420000 . . . . . 130 6.5 Flow in a Mangular Duct: Results for Re = 53000 . . . . . . . . . . 133 6.6 Flow in a Triangular Duct: Average Friction Factor xlOZ for Re =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53000 134
7.1 Liquid- m d Gas-Phase Superficial Velocity Values in the Experiments
of Lopa de Bertodano [lll] . . . . . . . . . . . . . . . . . . . . . . . . 158 7.2 Test Cases Investigated: Liquid-Phase Superficial Velocity and Aver-
age Void Fraction Values . . . . . . . . . . . . . . . . . . . . . . . . . 165
xviii
LIST OF ZABLES
0 NOMENCLATURE
Description
coe5aents in the discretized
pressure equation
coe5cients in the discretized
z-momentum equation
coefficients in the discretized
y-momentum equation
coe5aents in the diret ized
z-momentum equation
element surface area
coe5aents in the linear interpolation of q5
integration points
constant in the logarithmic law of the wall
drag force coe5dent
lift force coe5aent
turbulent dispersion term coefficient
virtual mass coe5aent
hydraulic diameter
bubble diameter
pressure coe5cients derived from y- and z-momentum equations
constant in the logarithmic law of the wall
friction factor
gravity vector
width of duct
refers to node or element (i j)
LIST OF TABLES
J
k
L
M
m
P
P
P
P'
& Re
Reb
Re..
S
9, S", S", S"
combined convection and diffusion flux
turbulent kinetic energy
hight of duct
interfadal forces
mzss flow rate
unit vector normal to control volume surface
production of turbulent kinetic energy
P=
cross-sectional pressure
bubble radius
Reynolds number
Reynolds number based on the bubble diameter
Reynolds number based on average zbl velocity
symmetric part of strain-rate tensor
volumetric source terms in
continuity and momentum equations, respectively
coefficients in the linearized expression for S
velocity components in the x-, y-, and z- directions
average u-velocity a t duct inlet
bulk u-velocity
relative velocity components in the x-, y-, and z- directions
friction velocity
Reynolds stresses (divided by -p)
volume
velocity component in the y-direction
total tangential velocity
pseudo velocity defined using d i re t ized y-momentum equation
pseudo velocity defined using d i re t ized z-momentum equation
LIST OF TrlBLES
a, 4 c
av
BI
CU
ele
h
1
i, j
k
corresponding to z, y and z directions in a
Cartgiaa coordinate system
Greek Symbols
gas-phase mid fraction
void fraction
distance to the wall
dissipation of turbulent kinetic energy
difbsion coeffiaent
Von Kannan constant
assymetric part of strain-rate tensor
transported scalar
mass density
shear stress
d shear stress
second time scale in two-phase bubbly flows
reduced pressure
dynamic viscosity
kinematic 1 iscosity
Subscripts
refers to integration points
refers to average value
bubble-induced quantities
control volume
refers to a given element
pertains to hydraulic diameter
refers to interfacial terms
indices that refer to nodes and elements
phase index
L r n OF TABLES
refers to liquid- and ga~phases
refers to mixture
normal to the surface of boundary control-volumes
neighboring points in numerical molecule
index used for reference quantities
shear-induced quantities
single-phase values
hvr3phase values
pertainstowallvalues
corresponding to z, y, z coordinate directions, respectively
Superscripts
nondimensional parameters
pertains to pressure
refers to turbulent
corresponding to u, v and w velocity components, respectively
Chapter 1
Introduction
1.1 Aims
. The work presented in this thesis involves numerical investigations of fdiy-
developed, turbulent, single-phase and ga~liquid *phase flows in straight ducts
of arbitrary, but uniform, cross-sections. The investigations of the single-phase flows
were undertaken first, because they are essential for laying a firm foundation for the
t-phase flow investigations.
In two-phase flow investigations, only dilute, bubbly, gas-liquid flows (Wallis, [188];
Carey, [24]; Whalley, [191]) in vertical ducts are considered, in which both phases flow
upwards, in a direction opposite to the gravitational acceleration vector. In both the
single- and two-phase flows investigations, attention is limited to adiabatic flows with
negligible viscous diiipation. Thus, the fully-developed flows considered in this thesis
are essentially isothermal; and in the two-phase florls, there is no phase change, and
the gas and the liquid flow rates are constant.
The main goal of this thesis is the formulation, implementation, and demonstra-
tion of numerical procedures for computer simulations of the abovementioned flows
of interest. Specifically, the objectives are the following:
a) adapt control-volume finite element methods (CVFEMs) (Baliga, [Ill) for nu-
merical predictions of the above-mentioned fully-developed, single-phuse, turbu-
CH.U'TER 1. INTRODUCTION 2
lent flows, with particular emphasis on a proper implementation of the so-called
wall-function treatment (Launder and Spalding, [loo]; Wdoox, [193]) to bridge
near-wall regions in complex geometrieq
b) perform a comparative evaluation of the performance of linear and nonlinear
eddy-viscosity models (Launder and Spalding, [loo]; Rubi i e in and Barton,
[l55]; Craft et al., [31]) and an expliat algebraic stress model (Gatski and
Speziale, [49]), in conjunction with suitable wall-function treatments, for the
prediction of the aforementioned fullydeveloped, single-phase, turbulent flows;
and
c) formulate and demonstrate a method for computer simulations of the fully-
developed, dilute, bubbly, gasliquid twc-phase turbulent flows of interest, in the
context of a two-fluid model (Ishii, [69]), by an amalgamation of the proposed
CVFEM, an adaptation of a twoquation, twc-time-scale, turbulence model
(Lopez de Bertodano et al., [110]), and a udl-function treatment based on a
formulation that is akin to the logarithmic law of the wall (Launder an2 Spalding
[loo]; Marie et al., [114]).
1.2 Overview of the Thesis and Motivation
An elaboration of the scope of both the single-phase and the two-phase flow in-
vestigations undertaken in this work is presented here. Furthermore, the background,
rationale, and motivation of these investigations are presented concisely in this sec-
tion. In addition, references are made to some of the important, relevant u.orks in the
published literature. However, for a detailed literature review, the reader is requested
to wait until Chapter 2.
CHAPTER I. IN7RODUCl'ION
1.2.1 Single-Phase Fully-Developed Turbulent Flows
Scope
-4s was stated in section 1.1, an important part of this work involves fully-
developed, turbulent, single-phase flous in straight ducts of arbitrary, but uniform,
cross-sections. Attention is l i i t e d to adiabatic flows with negligible viscous dis-
sipation, thus the fullydeveloped flows remain essentially isothermal. The fluid is
Newtonian, and its properties are asumed to remain essentially constant.
The main thrust in this phase of the work is the adaptation of CVFEMs (Baliga,
[Ill) for the prediction of the flous of interest. Special emphasis is placed on the
proper incorporation of d - f u n c t i o n treatments (Launder and Spalding, [loo]; Wilcox,
11931) to bridge the near-uall regions in regular and irregular geometries. Attention is
also given to a comparative evaluation of several turbulence models that are suitable
for engineering predictions of the flous of interest.
Background, Rationale, a n d Motivation
Single-phase, fully-developed, turbulent flows akin to those investigated in thiis
thesis are commonly encountered in engineering equipment. Examples indude the
following: flous in pipes and tubes used in the electric power and process industries;
axial flows in rod bundle geometries encountered in heat exchangers and nuclear
reactor cores; flows in heating, ventilation, and air condition equipment; and flows
in the passages of compact heat exchangers. Thus, the mathematical models of such
flows and the numerical solution methods put forward in this thesis have important
practical applications.
The single-phase, fullydeveloped, turbulent flows considered here provide a conve-
nient, computationally affordable, and yet challenging, setting for the implementation
and comparative evaluation of turbulence models, and the formulation and demon-
stration of numerical solution procedures, as is discussed in the remainder of thiis
subsection. The flows of interest are ~Pdimensional , but they could involve com-
plex (irregular-shaped) calculation domains. Furthermore, fullydeveloped turbulent
CHAPTER 1. IM'RODUCTION 4
flows in straight ducts of nonckdar aosssection are characterized by secondary
motions in the plane n o d to the mainstream direction (Prandtl, [145] ; Launder
and Ying, [loll; Demuren and Rodi, [35]; Wikox, [193]; Hanjalic, [56]). Thus, even
though such flows are two-dimensional, they inwlve velocity components in three
orthogonal coordinate directions, one axial component in the mainstream direction
and two other orthogonal components in the duct c-sectional plane.
The magnitude of the aforementioned secondary motions in the duct cmss-sectional
plane are only of the order of 2 - 3% of the magnitude of the average (bulk) veloc-
ity in the mainstream direction (Launder and Ymg, [101]; Demuren and Rodi, 1351;
Wicox, [193]; Hanjalic, [56]). However, the secondary motions can have important
consequences on the veloaty distribution in the duct cross-section, mriations in the
pall shear st- and rate of heat transfer, and the lateral spreading rates of scalars
introduced into the flow (Launder and Ymg, [loll; Demuren and Rodi, [35j; Wicox,
[193]; Hanjalic, [56]). In order to predict these phenomena, it is necessary to employ
turbulence models that provide the capability to capture the anisotropy of turbulence
in the cross section of noncircular ducts, and properly implement such models into
numerical solution methods that can handle irregular-shaped calculation domains.
These challenging and practically relevmt problems are tackled in this part of the
thesis.
Turbulence Models
Direct numerical solutions (DNS) of the full three-dimensional Navier-Stokes equa-
tions have been used successfully in fundamental studies of turbulent flows (Hinze,
[66]; Wilcox, (1931). However, because of the wide spectrum of length and time
sa les intrinsic to the turbulent flows considered here, DNS would be impractical at
present. Another approach to the simulation of turbulent flows, in which the large
eddies are computed and the small eddies are modeled, has been developed and a p
plied by many researchers. This approach is called Large Eddy Simulations (LES). Its
rationale lies in the obsermtion that the characteristics of the large eddies in turbu-
lent flows tend to be particular to the problem being studied, while the small eddies
are more universal in nature (Ciofalo, 1261). Although LES is more economical and
practical than DNS, it st i l l exerts very large demands on computer resources, for it
requires thrre-dimensional unsteady simulations Attempts to use tapdimensional
LES computations have proved inadequate, because they cannot produce stochaG
tic three-dimensional turbulent fluctuations (Murakami et al., 11241). Furthermore,
economical and effective treatments of wall b o u n w conditions in LES of complex
turbulent flows in ducts are still not available For these reasons, DNS and LES - not used in this work.
Economically f e e b l e wajs to simulate turbulent flows in ducts are still based
on some form of averaging of the governing equations (Wdcox, [193]; Hanjalic, [56];
Launder. [94]). Three fonns of averaging most commonly used in turbulence mod-
eliig are the time average, the space or volume average, and the ensemble average
(Hinze, 1661 ; LVilcox, [MI; Launder, [94]!. The simplest and most widely used
averaging procedures are based on the methodology proposed by Reynolds [I481 in
1895, in which all instantaneous quantities are expressed as the sum of time-mean
and fluctuating parts, and the governing equations are time-averaged. For the fully-
developed, statistically stationary, turbulent duct flows considered in this work, the
time- and ensemble-averaging procedures give the same result (ergodic hypothesis).
The turbulence models used in this thesis are all based on the Reynolds averaging
procedure.
Within the framework of Reynolds averaging, the full differential Reynolds Stress
Models (RSMs) are perhaps the most natural way of modeling the turbulent fluxes
that appear in the time-averaged momentum equations. This is because in RSk,
the six independent components of the Reynolds stress tensor, -px, are obtained
from the solution of full differential transport equations. In addition to these six
differential equations, RSMs require an equation for estimating the turbulent length
scale: This length scale is usually obtained by solving a differential transport equation
for either the rate of dissipation of turbulent kinetic energy, c, or the specific rate of
dissipation, w (Wilcox, [193]).
Numerous advances in RSMs have been made in recent years (Leschziner, [103];
Launder, [96]; Wilcax, [193]). Despite these advances, however, RSMs are still lim-
ited by uncertainties in the modeling of higherorder correlations that appear in the
Reynolds stress transport equations, and also in the modeling of the so-called pressure-
echo or d-reflect ion effect on the prrssure-strain correlation tensor (Wdcmt, [193];
Hanjalic, [56]). Furthermore, the six differential Re-ynolds stress transport equations
are highly nonlinear, coupled, and source dominated. Thus, computer sirnulatiom
based on RSMs are often plagued by numerical instabilities and slow convergence
rates ( Wilcox, [193]; Hanjalic, [56]; Lien and Leschziner, [102]). Furthermore, as was
mentioned in Section 1.1, one of the intentions in the single-phase flow portion of the
work pursued in this thesis was to lay a 6rm foundation for the subsequent -phase
flow investigations. There are numerous other challenges in the mathematical models
of two-phase flows considered here, as will be discussed later. Thus, after a brief
consultation with Professor Launder (Personal Communications during a Centre de
Recherche Mathematique (CRM) Workshop at the University or Montreal, 1995), it
was decided that RShk were not suitable for the work presented in this thesis.
Over the last 25 years, two-equation eddy-viscosity models (EVMs) of turbulence
have been the basis of the most widely used "fast" engineering methods for computing
complex turbulent flows (Hanjalic, 1994). In this work, the following EVMs were
considered: the linear EVM of Launder and Spalding [loo]; the nonlinear (quadratic)
EVM of Rubii tein and Barton [155]; and the nonlinear (cubic) EVM of of Craft
et al. [31]. In addition, an Explicit Algebraic Stress model (EASM) of Gatski and
Speziale [49], derived from parent RSMs by invoking mrious simplifying assumptions,
was also investigated.
In this work, the computer programs for simulations of the fully-developed, tur-
bulent, single-phase flow of interest are designed such that the linear and non-linear
parts of the aforementioned turbulence models are treated separately. Even though
it is a known fact that the linear EVM of Launder and Spalding [loo] can not predict
the fully-developed secondary flows in the cross-sections of the straight noncircular
ducts, this model was also included in this work, to serve as baseline case in compar-
ative evaluations. Furthermore, the implementation of nonlinear EVMs hvolves this
CHriPTER 1. INTRODUCTION 7
model, as the linear component, so no additional implemartation efforts were needed
to run the program for the linear EVM.
Wall-F'unction Treatment
It should be noted that most computations of turbulent flows in industry are done
using the so-called wall-function treatment wlcax, 11931) to bridge n e a r - 4 regions
In such regions. the turbulence is usually damped out as the wall is approached,
causing wry steep gradients of the dependent variables. Thus, if a discrete numerical
solution is sought all the way to the wall, very h e grids are needed in the near-wall
regions, in order to properly m l v e the steep gradients of the dependent variables
Such full-numerical calculations, all the uay to the wall, place enormous demands on
computer resources, especially in three-dimensional flows, and are impractical ui th
current computers This di5culty is overcome by bridging the near-wall regions using
appropriate wall-functions (Wilcox, [193]).
Most of the available wall-function treatments (Launder, [96]; Wilcox, [193], Han-
jalic, [56]) are based on the d l e d logarithmic law of the ud l , derived from a
theoretical analysis of turbulent flow of an incompressible fluid over a flat plate a t
zero angle of attack (Schlichting, [162]). It has been established that though this
treatment is based on a flow uith zero pressure gradient, it applies remarkably well
to flows with favorable and adverse pressure gradients, as long as there is no flow sep
aration (Schlichting, [162]; White, [192]). Moreover, with appropriate modifications,
the wall-function treatment has been used extensively in computer simulations of two-
and three-dimensional flous uith and without recirculation (Launder, [96]; Wilcox,
11931, Hanjalic, [56]). However, the validity of this approach is highly questionable in
the vicinity of flow separation and re-attachment points.
In this thesis, attention is focused on fullydeveloped turbulent flows in straight
ducts. In these flous, there is no reversal of flow in the mainstream direction. How-
ever, in the cross-section of noncircular ducts, the fully-developed secondary flous do
recirculate. Severtheless, in the near-udl region of the fully-developed flows consid-
ered here, the resultant (total) component of the velocity vector that is tangent to the
uaU never goes to zero. Thus, such flows provide a challenging setting for the devel-
opment of procedures for the proper implementation of the wall-function treatment,
but, in a physical context where the validity of this treatment is not questionable.
Details of the wall-function treatments used in this uwk are presented in subsequent
chapters.
In threedimensional, turbulent, parabolic flows in ducts (Patankar, [137]), there
is no reversal of the mainstream velocity component, though redrculating secondary
flows in the duct cross-sections are encountered. Thus, this category of flows looks
attractive for providing test problems for the research m r k described in this thesis
Houwer, such flows haw the following characteristics which diminish their suitability
as test problems: (i) computer simulations of such flows require accurate prescription
of the dependent variable over the enfire inlet crosssection, and this type of data,
experimental or numerical, is not readily available, especially for turbulence variables
(such as k and e) and for all dependent variables in the near-wall regions; (i) the
location of the near-wall nodes must be adjusted as the flow develops in the duct, in
order to keep values of the nondiensional wall-normal distances, 6+, in the desired
range of 30 5 6+ 5 120 (Launder and Spalding, [loo]), and this adjustment requires
solution-adaptive grids, at least in the vicinity of the duct uallq and ( i ) computer
simulations of such flows are considerably more expensive than those of corresponding
fully-developed turbulent flows. For these reasons, attention in this work was limited
to the fullydewloped region, in both the single- and two-phase flow investigations.
Numerical Method
In this work, one of the goals is to propose numerical methods that would be capa-
ble of handling turbulent fluid flow in complex geometries. The candidate categories
of method were finite-difference and finite-volume methods (FDMs and FVMs) based
on boundary-fitted coordinates (BFCs) (Patanlcir, [137]; Shyy, [ lz ] ) , finite element
methods (Zienkiewin, [194]; Baker, [lo]; Reddy, [146]), and control-volume finite
element methods (CVFEMs) (Baliga, 1997). Comprehensive discussions of FVMs
based on BFCs are available in articles by Shyy and Vu [174], and Shyy [ l z ] . These
CHAPTER 1. INTRODUCTION 9
methods have been quite successful, but they are not as well suited as methods based
on finite element grids for the solution of problems with complex irregular-shaped
calculations domains. This is because finite element grids provide greater flexibility
than grids based on BFCs, specially in the context of adaptive-grid methods. Fhr-
theemore, as has been discussed by Baliga [Ill, methods based on finite element grids
can work with mathematical models written with respect to the Cartesian coordinate
system, even in the solution of problems with irregular-shaped domains. In contrast,
FVhk based on BFCs typically work with mathematical models that are written with
respect to generalized curdimear, orthogonal or nonorthogonal, coordinate systems,
and are thus more complicated than the same mathematical representations in the
Cartesian coordinate system, assuming that the dimensivnality of the problem of
interest remains the same.
Again, borrowing from the work of Baliga [Ill, CVFEMs for fluid flow may be
regarded as amalgamation of concepts that are native to FVMs and FEMs. Thus,
CVFEMs have the following desireable features: Their formulations are amenable
to easy physical interpretation; their solutions satisfy local and global conservation
requirements even on coarse grids; and they provide the geometric flexibility that is
traditionally associated ~ 4 t h FEMs. It is for these reasons that CVFEMs u-ere chosen
for the investigations in this thesis.
The CVFEhf put forward in this thesis is adapted from the works of Saabas and
Baliga [156, 1571, and hlasson et al. [118]. Modifications have been made, mainly
in the implementation of the so-called momentum interpolation technique used in
the derivation of the discretized pressure equation in these equalorder collocated
CVFEMs (Rhie and Chow, [149]; Peric et al. [139]; Baliga, [ll]), in an effort to en-
hance the rate of convergence of the method. Some modifications were also necessary
in the implementation of a mass-weighted (MAW) scheme for the discretization of the
continuity equation for the gas-phase, in the t ~ w p h a s e flow investigations. Details
will be presented in Chapter 4.
The main contribution of this phase of the thesis with regard to CVFEMs is in
the development of a procedure to implement udl-functions in irregular-shaped duct
CHAPTER 1. INTRODUCTION 10
crcs~sections. The work of Saabas and Baliga [156, 1571 included a procedure for the
implementation of wall-functions, but that procedure does not ensure consenation
of mass in the near-wall regions. The proposed implementation procedure overcomes
this difficulty. Details are provided in Chapter 5.
1.2.2 TwePhase Fully-Developed Turbulent Flows
Scope
As mas stated earlier in this chapter, in the two-phase flow part of this work,
attention is limited to dilute, bubbly, gasliquid, fullydeveloped, turbulent flows in
vertical ducts. with both phases flowing upwards with respect to the gravitational
acceleration vector (which points vertically domward). The flows considered are
adiabatic, and the viscous d i p a t i o n in these flows is negligible. Thus, these fully-
developed flows are essentially isothermal, there is no phase change, and the gas and
liquid flow r a t s are constant.
The main goal in this part of the work is the development and demonstration of
a numerical procedure for the solution of a twc-fluid model of the two-phase flows of
interest. Discussions of such models are available in the works of Ishii, [69], and Drew
and Wallis, [39]. The intention here is to propose a numerical tool that would be useful
in investigations aimed a t fundamental model development and/or enhancement, and
also in investigations where available two-fluid models are applied to practical prob-
lems. The proposed numerical procedure is demonstrated here by incorporating into it
an adaptation of the tw~time-scale twwquation (k -c ) turbulence model of Lopez de
Bertodano e t al. [110], and applying it to fullydeveloped, dilute, bubbly, two-phase,
upward, turbulent flows in a straight vertical duct of triangular cross-section. This
demonstration activity has helped to identify the type of experimental data that are
needed to check and fine-tune mathematical models akin to those used here. Some
comments related to these needs are given in Chapter 7. A wall-function treatment,
based on a formulation akin to the single-phase logarithmic law of the wall, and its
adaptation to the proposed extension of the two-time-scale, twosquation, turbulence
CHAPTER I. IhTRODUCTION
model of Lopez de Bertodano e t aL [110], is also put forward in this the&
Background. Rationale, and Motintion
Gas-liquid two-phase flows are commonly found in industrial processes that involve
boiling and condensation, in pipelines which nominally carry oil or gas alone, but
which actually carry a mixture of oil and gas (Wallis, [188]), in the environment,
and in biological systems. Gas-liquid flows are also encountered in gas-stirred liquid
metal ladles used in the metallurgical industries, in manufacturing plants that produce
aerated water, carbonated soft drinks, and in nudear and thermal power plants.
Gas-liquid -phase flows in ducts take on various geometric configurations,
which are referred to a s pow patterns or pow regimes in the published literature.
Examples of such flow patterns are presented and discussed in the works of IYallis
(1881, Carey [24], and Whalley (191]. The common flow patterns for upward gasliquid
flows in a vertical pipe are illustrated in Fig. 1.1, taken from the work of Whalley
[191]. As the quality is increased from zero towards one, the flow patterns obtained
are categorized by Whalley [191], as follows:
1. bubbly pow, in which the gas (or vapor) bubbles are of approximately uniform
size;
2. plug pow (sometimes called slug pow), in which the gas flows as large bullet-
shaped bubbles, along with some small gas bubbles distributed throughout the
liquid;
3. drurnJow, which is highly unstable flow of an oscillatory nature, with the liquid
near the tube wall continually pulsing up and down; and
4. annular pow, in which the liquid travels partly as an annular film on the walls
of the tube and partly as small drops distributed in the gas which flows in the
center of the tube.
The common flow patterns that are used to categorize horizontal gas-liquid flows
in a circular aoss-section tube are illustrated in Fig. 1.2, taken from Ihalley [191].
CIL.zpTER 1. INTRODUCTION 12
Again, following Whalley [191], as the quality is increased from zero tomads one, the
flow patterns obtained can be categorized as follows
1. bubbly flow, in which the gas (or ~ p o r ) bubbles tend to flow along the top of
the tube;
2. plug pow, in which the individual small gas bubbles have coalesced to produce
long plugs;
3. stmtified flow, in which the liquid-gas interface is smooth;
4. w a q flow, in which the wave amplitude increases as the gas velocity incrrases;
5. slug flow, in which the wave amplitude is so large that the wave touches the top
of the tube: and
6. annular flow, which is similar to vertical annular flow except that the liquid
film is much thicker a t the bottom of the tube than at the top.
Additional details and d i i i o n s of gas-liquid flow patterns can be found in the
works of Wallis [ l a ] , Hewitt [65], Collier and Thome [29], Carey [24], and Whalley
[191].
Mathematical modeling of the aforementioned gas-liquid &-phase fl0WS is an
enormously challenging task that has been intensively tackled by numerous researchers
over the last 60 years. D i i o n s and reviews of these efforts can be found in the
works of Nusselt [132], Jakob [?2], Roshenow [153], Zuber [195], Wallis (1881, Carey
[24], Ishii [69], Crowe [32, 331, Hewitt [65], Lahey [85], Drew and Wallis [39], and
Whalley [191], for example. These intensive research efforts have resulted in mathe-
matical models for specific gas-liquid flow patterns (or regimes), which have proved
to be quite successful, when they are fine-tuned for particular flows, using empirical
inputs. However, the physics of these flows is very complicated and not completely
understood, and major difficulties and uncertainties remain in the mathematical mod-
els of dense bubbly, plug, chum, stratified, wavy, and annular flows. Furthermore,
there is no one comprehensive model that can predict all of the aforementioned flow
regimes, and the physical mechanisms of transition from one flow pattern to another
are not well understood, so their prediction is very difficult, without empirical inputs
for speciiic situations The mathematical models of dilute bubbly gas-liquid flows,
in which the bubble motion is controlled by surface and body forces acting on the
bubble, and not by bubble-bubble interactions [33], are reasonably a.eUenablished,
relative to the models for the other aforementioned flow patterns Howew, they
still d e r from significant uncertainties and difficulties, especially with regard to
turbulence and near-wall regions
Thus, major and sustained rrsearth effort are needed to enable and/or improve
the reliability of computer simulations of gas-liquid -phase flows. The research
undertaken in this thesis represents a small step towards meeting a part of this need.
Modeling Approach
In principle, gas-liquid flows can be modeled using the d e d complete local
description or exact formulation, as has been discus& by Drew and Lahey 1381, and
Stewart and Wendroff [I@], among others. In this approach, the dynamics of each
phase and the interface are dealt with on the basis of first principles. In each phase,
the appropriate governing equations are solved, and kinematic and dynamic balance
equations are imposed at the interface. Such a complete local description would very
accurately model the flows of interest. However, simulations that are based on such
complete local descriptions place enormous demands on computer resources, and are,
therefore, impractical.
Practical models of bubbly gas-liquid flows can be obtained by introducing the
notion of averaging. Often, averaging is done in the context of the concept of super-
imposed continua: each phase is treated as a continuum, occupying simultaneously
the same region in space. Detailed d i i i o n s of averaging techniques for multiphase
systems are available in the works of Ishii [69], Drew [41], Drew and Wallis 1391, Drew
and Lahey [38], and Rocco [150].
In this work, the time-averaging procedure of Ishii [69] was used. The resulting
governing equations fall into the category that is referred to as two-fluid models.
The details are presented and d i d in Chapter 3. It should be noted that only
statistically stationary (fullydeveloped) flows are considered in this thesis, and for
such flows, the time and ensemble averaging approaches should produce identical sets
of equations (the ergodic hypothesis).
Turbulence Models
Following the rationale presented earlier in section 15.1, in the context of single-
phase turbulent flows, two-cquation linear and nonlinear eddy-viscosity models (EVMs)
and an expliat algebraic stress model (EASM) were adapted in this work for the sim-
ulation of fullydeveloped, bubbly, two-phase turbulent flows.
Extensions of the single-phase twoquation k-c model for the simulation of dilute,
bubbly, *phase turbulent flows have been proposed by Lopez de Bertodano e t al.
[110], Kataoka and Serizaua 1741, and Issa et al. [?I], among others. After reviewing
these models, the two-time-scale k - c turbulence model of Lopez de Bertodano e t al.
[I101 was chosen for implementation in this work. The basic assumption in this model
is that shear-induced and bubble-induced turbulence may be lineariy superpod .
This model produces results that match well the homogeneous *phase turbulence
data of Lance and Bataille [go], and the pipe flow data of Serizaua et al. [168].
Another useful feature of this model is that its key ideas can be readily incorporated
into the nonlinear EVhls and the EASM considered in the single-phase flow portion
of this work, thereby extending their applicability to the bubbly, two-phase, turbulent
flows of interest. Details of the resulting turbulence models are presented in Chapter
3.
Wall-Function Peatment
In the literature, there are many examples of turbulent, bubbly, gas-liquid, two-
phase flow computations that were done using wall-function treatments specifically
designed for single-phase turbulent flows: Examples may be found in the works of
Lopez de Bertodano et al. [110], Marie [116]; and Oliveria [135]; and in the reviews
put forward by Lahey [85] and Lance and Lopez de Bertodano [89].
CHAPTER 1. INTRODUCTION 15
Over the last few years, there have been serious efforts to understand the near-wall
region of gas-liquid, bubbly, turbulent flows and to propase wall-functions specifically
designed for these flows. Examples include the works of Sato and Sekoguchi [160],
Lopez de Bertodauo [108], Marie et al. [114,115], and Lance et al. [91]. In this work,
an effort was made to enhance the available proposals (Afshar and Baliga [I]). The
main difference between the approach used in [I] and that of Marie et al. [I14 is the
speci6c accounting of the drag force in the logarithmic law of the wall put fomard
in [I]. In the formulation of Marii et al. [114], which is based on the earlier work of
Sato and Sekoguchi [160], there is no accounting of interfacial forces.
Despite the aforementioned efforts [I, 91,108,114, 1151, theavailable ud-functions
specifically designed for turbulent, two-phase, bubbly gas-liquid flows are not quite
ready for application to the flows of interest. The logarithmic laws of the wall proposed
by Marie et al. [I141 and Afshar and Baliga [I] were developed using the experimental
data of [I151 (The author would like to take this opportunity to express his gratitude
to Professor J. Bataille and Dr. J.L. Marie for providing this experimental data):
However, this experimental data and the proposed wall-functions apply specifically
to dilute, bubbly, gas liquid two-phase flow over a vertical flat plate Thus, they must
be enhanced in order to make them applicable to the flows of interest in this work.
Furthermore, such enhanced laws of the wall would then haw to be checked and fine-
tuned against experimental da tx Unfortunately, detailed and reliable experimental
data required for such checks and fine tuning tasks are not available for these flows
in ducts with irregular cross-sections, and even for pipes of circular cross-section, this
type of data are scarce (Marie et al., [114]; Nakoryakov et al., [128]; Nakoryakov and
Kashinsky, [127]).
In this work, therefore, it has been assumed, based on the aforementioned initial
efforts of Marie et al. [114,115,116] and Afshar and Baliga [I], that suitable logarith-
mic laws of the wall for upward, dilute, bubbly, gas-liquid two-phase flows in vertical
ducts, when they are eventually formulated, could be cast into a jonn akin to that of
single-phase logarithmic law of the u d l (Launder and Spaldiig [loo]). Then, starting
with this assumed form for a logarithmic law of the wall for the t u ~ p h a s e flows of
CHAPTER 1. JXIRODUCTION 16
interest, a wall-function treatment, compatible with the propased extension of the
-time-scale k - c model of Lopez de Bertodano e t al. [110], has been formulated.
The details of this wall-function treatment are gim in Chapter 3. This wall-function
treatment was then incorporated into the proposed CVFEM. Rgults obtained for
a demonstration problem, involving upward, fully-developed, turbulent, dilute, bub-
bly, gas-liquid flows in a vertical duct of triangular -section, are prese.nted and
discussed in Chapter 5.
Numerical Method
A rwiew of numerical methods for bubbly, -liquid, t u ~ p h a s e flows is given in
Chapter 2. In this work, following the arguments given earlier in section 1.2.1 for
single-phase flow computations, control-volume b i t e element methods (CVFEMs)
(Baliga, [Ill) were adapted for predictions of the bubbly gas-liquid flows of interest.
Special efforts were made to properly incorporate the aforementioned two-time-scale
turbulence model, the related wall-function treatment, and the interfacial forces be-
tween the two phases.
1.3 Outline of the Thesis
In the previous sections, the objectives of this thesis, an overview of the methods
and procedures proposed and/or used in this work, the background, rationale and
motivation behind this work, and a review of some of the relevant published works
in the literature were presented. In the next chapter, some additional published
literature pertinent to this work is reviewed. Mathematical models of the problems
considered in this work are presented and d i i in Chapter 3. Details of the
O/FEM used in this work are presented concisely in Chapter 4. Procedures for the
implementation of wall-function treatments in the context of the CVFEM are put
f o m d in Chapter 5.
In Chapter 6, the results obtained from numerical simulations of fullydeveloped,
turbulent, single-phase flows are presented and discussed. In Chapter 7, the proposed
Ch%PTER 1. INTRODUCTION 17
numerical procedures for -phase, dilute, gas-liquid, bubbly flows are demonstrated
by presenting results obtained for fullydeveloped, turbulent, bubbly, -phase flow
in a triangular duct. F i y , in Chapter 8, the main contributions of this thesis are
noted, and some suggestions for extensions of this mrk are put forward.
t Chum now
t Annular flow
GAS FLOW INCREASING - Figure 1.1: Gas-Liquid Two-Phase Flow Patterns In a Vertical Circular Pipe (Whalley
[1911).
a wavy now
GAS FLOW INCREASING
GAS FLOW INCREASING
Annular flow
Figure 1.2: Gas-Liquid Two-Phase Flow Patterns In a Horizontal Circular Pipe
(U'halley [191]).
Chapter 2
Synopsis of Relevant Publications
In Chapter 1, sections 1.2.1 and 1.2.2, the scope, background, rationale, and
motivation of the work presented in this thesis were d i d . In that context, some
of the published literature related to the investigations undertaken in this m r k were
reviewed. In this chapter, additional published literature pertaining to this work is
revieud.
Fi, succinct discussions of some of the main published works on gasliquid two-
phase flows and single-phase turbulence modeling are presented in sections 2.1 and 2.2,
respe :ively. The intent in these sections is not a detailed review of the major works in
these subjects, for that nnuld be an overly lengthy undertaking. Furthermore, there
are several excellent comprehensive reviews of these subjects by the pioneers in these
fields. These review articles are listed here. Following these sections, published works
germaw to the topics of fully-developed, turbulent, single- and bubbly two-phase
flows in straight ducts are reviewed in detail.
2.1 Gas-Liquid Two-Phase Flow
Significant progress has been made in the last half century in the ability to math-
ematically model two-phase flow systems, but much of the engineering design work
in the fields of two-phase flow and heat transfer is empirically based. Today, how-
ever, mathematicd modeling approaches are anilable and used in the design process,
CH.4FTER 2. SYNOPSIS OF RELEKCW PUBLICATZONS 20
but much remains t o be done to enhance the &ectivenes and reliabiity of these
approaches [83, 63, 61, 1501.
The review of the subject of *phase flow and heat transfer by Bergles [IS] is a
good start for a historical overview of the subject since the 18th century. I t was not
until about half a century ago, in 1930's and 1940's, that chemical engineers, faced
with many pressing needs of the chemical and petroleum industries, made many sig-
nificant contributions to the field of two-phase flow. They utilized a design approach,
which relied mainly on experiments and the development of empirical correlations.
.k with most empirically-based approaches, the uncertainty in predictions was large,
and extrapolations outside of the t t i t data range were prone to severe inaccuraati.
In the 1950's and 1960's the development of commercial nuclear power plants
provided the impetus for intensive research efforts in the fields of two-phase flow and
heat transfer. The so-called driftflu. models (Wallis [191]) were developed in response
to demands for more accurate predictions of two-phase flow and heat transfer. The
homogeneous and drift-flu. models of two-phase flows are well explained in the books
by Wallis [188], Soo [IT], and Lahey and hloody [ST]. These models account for
the relative velocity between phases, by employing algebraic expressions which are
based on the assumption that the flow is mainly onedimensional. Although these
models have their limitations, their use in industry is wide spread, and they provided
a useful technical basis for many design purposes. However, these models are not able
to accurately predict a wide range of transient flous and multidimensional two-phase
flow phenomena
By the early 1970's, the nuclear power industry had become a significant world-
wide industry. .& newer light water reactors (LWRs), having higher power density,
were developed, more accurate safety analysis tools became a priority. Achieving
this goal required a more thorough understanding of the physics of two-phase flows
than was available or required in previous models. The first step in this direction
was to develop a rigorous mathematical basis for the modeling of multidimensional
two-phase flows.
-4s uas discussed in Chapter 1, complete local descriptions of t w p h a s e flow are
CHAPTER 2 SYXOPSIS OF RELEVAhT PUBLICATIONS 21
impractical. Practical approaches to the mathematical modeling of two-phase flows
are all based on some form of averaging.
The modeling of m ~ p h a s e , bubbly, gas-liquid flows is usually based either on an
averaged Eulerian description for both phases, the so-called two-fluid model, or an
Eulerian formulation for the continuous phase and a Lagraagian one for the dispersed
phase, the so-ulled trajectory mofd The merits and drawbacks of each of these
approaches have been d i d by Crowe [33]. Two-fluid models [69,86,39,16,54],
have the a d w t a g e of uskg similar set of averaged governing equations for both
phases, hence, their implementation in computer codes is simpler than the other
approach. Trajectory models [33, 25, 441 enable a better formulation of the physics
of bubble motion and includes the history of their motion, however, their demands
on computer resources are much more than those of two-fluid models [33].
.As was mentioned in Chapter 1, the aim is this work is to provide a numerical
simulation tool for engineering investigations of gas-liquid, bubbly, two-phase flows.
Considering the fact that in such investigations, the motion of the individual gas
bubbles is usually not the main data of interest, a tn-fluid model, which is relatively
simple to implement and whose demands on computer resources are affordable, has
been used in this work
Averaging procedures applicable to two-phase systems are used in both the Eulerian-
Eulerian and the Eulerian-Lagrangian approaches. As was stated in Chapter 1, such
procedures are classified as time- , volume- , and ensemble-avemging techniques. The
choice of averaging technique and the selection of instrumentation are closely related,
since, in general, experimentally measured quantities represent some kind of mean
d u e s themselves. The various averaging techniques produce equations that are not
different in a fundamental way, other than in the interpretation of the averaged vari-
ables [83, 90, 1891. Ensemble averaging [39, 831 is the most fundamental averaging
procedure, but, in practice, it is difficult to choose a group of similar samples to per-
form this type of averaging. If the process is statistically stationary, then the time
averaging procedure [69], which samples a large number of different observations,
should approximate the ensemble averaging (the ergodic hypothesis). Similarly, vol-
CILiPTER 2. SYNOPSIS OF R E L E V M PUBLICATIONS 22
ume averaging 1861 is appropriate for spatially homogeneous flow. In this work, the
time-avtmging procedure, as rigorously derived by Ishii [69], is used. T i e averag-
ing has the enormous advantage of being dmely related to what is usually m d
in experiments. Again, considering that in this work, only statistically stationary
flous are considered, the time and ensemble averaging should produce identical sets
of averaged equations.
To dose the set of averaged equations, models for interfacial and fluctuating terms
have to be formulated. It is imposible, with the available knowledge, to establish
rigorous models which are applicable for all two-phase flows [29]. In addition, different
approaches in modeling two-phase flows require different constitutional relations for
these terms. A review of all such models is not attempted in this thesis. Instead, the
interested readers are referred to the authoritative and comprehensive works compiled
by Hetsroni 1611 and Rocco [150]. In asubsequent section, the modeling of these terms
relevant to gas-liquid, bubbly two-phase flows, in the context of two-fluid models, will
be renewed. Reviews of other gas-liquid two-phase flows are available, for example,
in the works of Buell et al., 1211; Crowe, 1331; Dobran [36]; Hewitt, [64, 631.
2.2 Turbulence Modeling for Single-Phase Flows
As %as stated in Chapter 1, the exact equations describing the turbulent motion
of single-phase Newtonian flows are the full, three-dimensional, unsteady form of the
Navier-Stokes equations. Indeed, Direct Numerical Simulations (DNS) of the full
three-dimensional unsteady Navier-Stokes equations have been used successfully to
study turbulent flows [75, 761. However, for any practically relevant turbulent flow,
DNS u-ould be impractical at present. This difficulty with the DNS approach a r k
because of the wide spectrum of length and time scales intrinsic to turbulent flous.
Typically, the sue of the small length scales in a turbulent flow are of the order of
lo3 times smaller than the larger scales of the flow 11521. The smallest scales are
established by the viscosity of the fluid, while the largest scales are limited by the
sue of the domain. The time scales exhibit the same variations as the length scales,
CHAPTER 2. SMVOPSI.5 OF RELEKCW PUBLICATIONS 23
in their order of magnitude [152,98,94]. Thus, to accurately resolve all temporal and
spatial scales of motion, a DNS would require at least 109 grid points, and sufXaently
small time steps (Launder 1941). Storing the flow dependent variables a t so many
grid points is still beyond the capacity of most modern computers, and, in addition,
the number of arithmetic operations which muld be required is so large that the
computing time would also be enormous For the problems of interest in this thesis,
DNS would be prohibitively expensive and impractical.
In most simulations of turbulent flous in engineering, the idonnation desired lies
generally in the large-scale events of the flow and not in the small eddies. Since the
1970's, an approach to the simulation of turbulent flows in which the large eddies are
computed, while the small eddies are modeled, has been developed and applied by
many researchers. This approach is called Large Eddy S i a t i o n (LES). Its rationale
lies in the fact that the characteristics of the large eddies in turbulent flow tend to be
particular to the problem being studied, while the small eddies are more universal in
nature, and, therefore, are amenable to general models 126, 104,147, 1641. Since LES
involves modeling of the'small eddies, the smallest cells in the computational grid can
be much larger than the Kolmogorov length scale, and much larger time steps can
be taken than are possible in a DNS. Therefore, uith LES, it is possible to simulate
turbulent flows of much higher Reynolds number than those possible with DNS, for
a given computing cost [193].
Although LES is more economical and practical than DNS, typically requiring 5
to 10% of the computer time needed for DNS in simulations of comparable flous, the
method still exerts very Iarge demands on computer resources, for it requires three-
dimensional unsteady simulations. Thus, LES is not yet suitable for engineering
design calculations, and this situation is unlikely to change over the next 5 to 10
years 1941. For these reasons, LES u w not used in this work
As was stated in Chapter 1, in spite of considerable advances in (DNS) 175, 761
and (LES) [26, 104, 147, 1641, the only economically feasible way to solve practical
turbulent flow problems is still based on some form of averaging of the governing
equations [151,193]. Today, a time-averaging approach initially suggested by Osborne
CH.4iTER 2. SYNOPSIS OF RELEKWT PUBLICATIONS 24
Reynolds [I481 in 1895, is still the most widely used way of simulating turbulent flows
in engineering.
In general, in Reynolds averaging, all instantaneous quantities are expressed as
the sum of time-mean and fluctuating parts, and the governing equations are time-
averaged over a time period which is long compared with the time-scale of turbu-
lent motion. Once the governing equations, namely, the continuity, momentum, and
energy equations, are time-averaged, the resulting equations - generally known as
Reynolds-avemged equations - mer from those describing a laminar flow only by
the presence of the terms containing averaged products of fluctuating portions of the
instantaneous dependent variables. The components of this so-called turbulent stress
or Reynolds sstnss tensor and turbulent fluxes are unknown elements in the time-
averaged governing equations. Thus, the process of averaging introduces additional
unknowns, and the resulting equations no longer constitute a closed system, since
they contain unknown terms representing the transport of mean momentum, heat
and mass by the turbulent motion. Therefore, a turbulence model is nemsary to
determine these turbulent transport terms. A turbulence model is defined as a set of
equations (algebraic or differential) which determines the turbulence traasport terms
in the mean-flow equations (1521. Turbulence models are based on hypotheses about
turbulent procgses and require empirical inputs in the form of constants or functions.
The pioneering works of Reynolds (1895) [148], on expressing all instantaneous
quantities as the sum of time-mean and fluctuating parts, and Boussinesq (1877)
[19], on introducing the eddy viscosity concept, are two important milestones in the
history of turbulence modeling. The mizing-length hypothesis of Prandtl (1925) [144],
which is closely related to the eddy-viscosity concept, formed the basis of virtually all
turbulence-modeling research for about the next twenty years until 1945. In modem
terminology, we refer to models with an algebraic prescription of the mixing-length
as zem-equation models of turbulence. By definition, an n-equation model signifies
a model that require solutions of n differential transport equations to model tur-
bulence terms, in addition to the continuity, momentum and energy equations [193].
Important early contributions uwe made by several researchers, most notably by Von
CHAPTER 2. SYNOPSIS OF RELEVANT PUBLICATIONS 25
Karman (1930) [187]. Prandtl (1945) pustulated a model in which the eddy viscosity
depends upon the kinetic energy of the turbulent fluctuations, k. He proposed a mod-
eled differential equation approximating the exact equation for k. This improvement,
on a conceptual level, takes account of the fact that the eddy viscosity is dected
by jw hishy. Thus, the concept of the so-called one-equation model of turbulence
was born. For this improved model to be m p I d e , a turbulent length scale must be
stipulated.
Kolmogorov (1942) [i9] introduced the first complete model of turbulence. In
addition to having a modeled diifferential equation for the kinetic energy of turbulence,
k, he introduced a second parameter, w, that he referred to as "the rate of dissipation
of energy in unit volume and timen, which satisfies a differential equation similar to
the equation for k. This model is thus termed a two-equation model of turbulence
While, this model offered great promise, it went with virtually no applications for the
next quarter century because of the unavailability of suitable computers to solve its
nonlinear differential equations.
Rotta (1951) [I541 laid the foundation for second-order or second-moment dosun
models of turbulence. He devised a plausible model for the differential equations that
govern the evolution of the independent elements of the Reynolds-stress tensor.
Thus, by the early 1950's, four main categories of turbulent models had evolved
as follows:
1. Algebraic (or Zero-Equation) Models
2. One-Equation Models
3. Two-Equation Models
4. Second-Order Closure Models
It was, however, after 1960's that the solution of the time-averaged Navier-Stokes
equations, along with turbulence models based on diifferential transport equations,
became feasible, as digital computer techno log^ advanced. Since then, numerous
developments of turbulent models ha\,e been reported for different turbulent flows.
CHAPTER 2. SYh'OPSIS OF RELEVANT PUBLICAnONS
These development have been comprehensively reviewed in a book by Wdaa [193],
and in the works of Rodi [Ijl], Launder [95], and Hanjalic [56].
Twoquation eddy viscosity models (EVMs), in particular, k-e high- and low-
Reynolds number turbulence models, have performed very well in the prediction of
many two-dimensional single-phase flow problems of engineering interest [151]. In
their basic form, these models yield satisfactory predictions of two-dimensional thin
shear-flows, some recirculating flous, and even some flows with streamline curvature
and body forces, when these effects are ureak 1561. However, these models, which are
based on the concept of an isotropic eddy viscosity (Boussinesq approximation), are
unable to predict the turbulence-induced secondary flows (secondary motion in the
plane perpendicular to the streamwise direction) in ducts of non-circular aoss-sections
11931. With these models, it is also difficult to systematically indude the efTects of
the streamline curvature, and the influence of body forces on turbulence [193]. One
approach to achieving a more accurate prediction of Reynolds-stress tensor anisotropy
and streamline curvature, uithout introducing any additional diierential equations is
through non-linear eddy viscosity models (NLEVMs) 1113, 183, 31, 1251. In addition
to the standard linear terms, NLEVMs include terms that are quadratic (or higher)
in mean velocities or mean velocity gradients. While NLEVhIs perform quite u d l in
predicting the secondary flows in turbulent flows in ducts [183], however, according
to Craft et al. 1311 quadratic turbulent stress-mean rate of strain relations offer only
slightly greater generality than the linear EVMs. In particular, the various effects of
streamline cunature on the turbulent stresses can not be adequately accounted for
at this level. Very recently, a Cubic EVM (CEVI11) has been proposed by Craft et al.
[31], which appears to do much better than h e a r EVMs and quadratic NLEVhls in
capturing the effects of streamline curvature over a range of flous.
In order to overcome some of the shortcomings of turbulent models based on the
isotiopic eddy viscosity concept, higher order turbulence models, such as Reynolds
Stress Models (RSMs), have been developed [99, 172, 53, 56, 1821. In these models,
the six independent components of the Reynolds stress tensor are obtained from the
solution of full differential transport equations. In addition, they require a length-
CK-IPTER 2. SYXOPSlS OF RELEK4NT PUBLICATIONS 27
scale-supplying equation, for which the great majority of models ernplay the tur-
bulence dissipation rate equation. Since, in these models, both the convection and
diffusion of st- are accounted for automatically, the turbulence history effects are
rea l i s t idy represented. Also, these equations contain convection, production, and
(optionally) body-force terms that respond automatically to effects such as stream-
line cunature, system rotation, and stratification, at least qualitatively [193]. These
equations are also appropriate for flows with sudden c h a w in strain rate, because
the values of normal st- depend upon initial conditions and other flow processes,
hence, there is no r e m n for the normal stresses to be equal even when the mean
strain rate vanishes. Nevertheless, there is a significant price to be paid in complex-
ity and computational difficulty for these gains [193]. There appear triple products
of velocity fluctuations, as well as products of fluctuating pressure and fluctuating
strain in the aforementioned stress transport equations, which have to be modeled
to close the mathematical equations. Computationally, there are six extra highly
non-linear, coupled, and source-term dominated stress equations which have to be
solved simultaneously with the mean-flow equations. Inevitably, this necessitates an
itemtive sequence, whose stability behavior responds sensi t idy to spurious features
in the approximation of convection and to the precise numerical practices adopted for
treating the model's non-linear and coupled source terms 11031. Needless to say, the
computer time and memory required to produce reliable results with RSMs is much
higher than EVMs.
In efforts to reduce the significant demands on computer resources by R S k , the
so-called Algebraic Stress Models (AShls) [35, 491 have been developed. In S h , l s ,
algebraic expressions for stresses are derived from the parent Reynolds Stresses equa-
tions by invoking simplifying assumptions. Gatski and Spaiale [49] regard such
models as straindependent generalization of NLEVkls. Compared to EVMs, X M s
have been reported to produce better predictions of turbulence induced secondary
velocities in turbulent fullydeveloped flows in ducts of non-circular crosssections,
and also in some flows with rotation and curvature, provided that these flows are
not far from equilibrium [56]. In wall-bounded flows in which the transport effects
CKWTER 2. SMVOPSIS OF RELEVAhT PUBLICATIONS 28
are of secondary importance, the ASM approach gives results very similar to those
produced by a full second-moment dosure, for about 60% of the computational dart
[94]. However, .&SMs are also reputed to pase numerical instabilities or lead to very
slow convergence [56]. Therefore, in many problems, no savings in computer resources
are gained if they are solved by XShk rather than RSMs
These Reynolds-averaged models have been s u d y used for the prediction of
numerous turbulent flows encountered in nature and in industry. The published works
d a t e d +th turbulent, fully-developed, single- and gas-liquid, bubbly, two-phase
flows in straight ducts, in the context of Reynolds-averaged models, are discussed in
subsequent sections.
2.3 Fully-Developed Turbulent Flows in Straight
Ducts
2.3.1 Single-Phase Flow
Turbulent flows in ducts or panages of circular and non-circular crosssections are
often encountered in engineering practice and in nature. Examples have already been
given in Chapter 1. Turbulent flows in ducts of noncircular cross-sections are accom-
panied by secondary motions in the plane perpendicular to the streamwise direction,
and this secondary motion can be caused by two dierent mechanisms. In curved
ducts, where centrifugal forces acts a t right angle to the main flow direction, this
motion is pressure-induced and is said to be of Prandtl's first kind. Thii kind of sec-
ondary motion also exists in curved arcular cros-section ducts and for laminar flows,
and the secondary velocities could be of the order of 20-30% of the bulk streamwise
velocity. Secondary motion of Prandtl's second kind is encountered in non-drcular
straight ducts, and, since, this secondary motion is present also under fullydeveloped
conditions, in contrast, such motion is absent in corresponding laminar flows, it is
caused by anisotropy in the turbulence field.
In this work, fully-developed turbulent flows in stmight ducts are considered, and,
CHAPTER 2. SYNOPSIS OF RELEVANT PUBLICATIONS 29
hence, it is concerned only with turbulent-driven secondary motion which has been
observed in non-drcular ducts A schematic illustration of such flows in a square duct,
b o m d from Speziale et al. [181], is shown In Fig. 2.1. Althoug!~ the magnitudes
of these secondary velocities are of the order 23% of that of the streamwise bulk ve-
locity, this secondary motion can have important consequences [35) By transporting
high-momentum fluid towards the corners, it causes a bulging of the velocity contours
tom& the corners, Fig. 2.2. Furthermore, the secondary motion produces an In-
crease of the wall shear-stress toward corners, an effect which is of great importance
for sediment-transport and erosion problems. For these reasons, it is important to
understand and to accurately predict turbulencedriven secondary motion in straight
ducts.
A considerable number of experimental investigations have been carried out to
understand better the origin of the secondary motion and its effect on turbulent flow
in straight ducts. Based on the results and findings of experimental investigations,
researchers have developed, or modified available, turbulence models to enable them
to predict the secondary velocities. Demuren and Rodi [35] have reviewed the exper-
iments on and calculation methods for turbulent single-phase Newtonian fluid flows
in straight ducts. The origin of the secondary motion and the shortcomings of ear-
lier calculation models are discussed in their paper. They d i the available
experimental results, such as those obtained by Launder and Ying [loll, Ggsner and
Emery [SO], and Klein [78], and summarized the experimental findings on the gen-
eration of secondary-motion in straight ducts as follows: "The anisotropy of normal
stresses, as well as the gradients of the shear stress terms, are the dominant factors
in secondary-motion generation. The viscous terms are negligible except very close
to corner.". Therefore, it is important to represent properly both these processes in
any mathematical model for turbulent flow in ducts.
The first calculation of secondary-motion in straight ducts was carried out by
Launder and Ying [101]. They recognized that a model using an isotropic eddy
viscosity for calculating the turbulent stresses can not produce any secondary-motion
in ducts, due to its inherent isotropic characteristics. Hence, by simplifying the
CKWTER 2. SYNOPSIS OF RELEVXNT PUBLICATIONS 30
RSM of Hanjalic and Launder 1571, they derived algebraic expressions for turbulent
stresses. Naot and Rodi [I291 simplified the RSM of Launder et al. [99] to an ASM
and used it to calculate the secondary motion in fullydeveloped flows in ducts and
open channels. Similar to Launder and =ng, they neglected the convection and
diffusion terms in the Re-ynolds stress equations (assumption of local equilibrium),
and calculated the primary stress with the standard EVhl. Demuren and Rodi [35],
once they realized the importance of the secondary-velocity gradients in generating the
secondary-motions, proposed an AShl which retained these gradients in the algebraic
expressions. As was stated earlier in this chapter, even though AShls solve for st-
through algebraic expresions, since they often face convergence difficulties, they may
require significant CPU time to produce reliable results.
Another class of turbulence models, which has became quite popular in the last
decade, is twwquation k - 6 models based on NLEVMs. In these models, quadratic
and, occasionally, higher-order products of the strain-rate tensor and velocity gra-
dients, have been conjugated to linear EVMs (LEVMs) to enable them to capture
eifects of turbulence induced secondary-motions or streamline curvature. Some of
these models are tabulated in a recent paper by Craft e t al. 1311, and include the
proposals of Speziale [183], Nizima and Youshiiwa 11311, Rubinstein and Barton
[155], Myong and Kasagi 11261 and Shih et al. [171]. Craft et al. asserted that at
the quadratic level, only slightly greater generality is achievable than with LEVMs,
and they proposed a cubic stress-rate of mean strain relation which is presumed to
provide more generality. In this work, the linear EVM of Launder and Spalding, the
quadratic NLEVM model of Rubinstein and Barton [155], and the cubic NLEVM
model of Craft et al. [31] are cast in a general format, as in 1311, and are investigated
in the context of turbulent, fullydewloped flows in noncircular ducts. The model of
Myong and Kasagi 11261, however, is not investigated, because it involves wall-normal
distances.
RSMs have been fully d i e d in review papers by Hanjalic [56], Launder 194,951,
and So et al. [li5]. It is redundant to repeat those reviews here. Nevertheless, a brief
review of main contributions to these models is presented here, with an emphasis on
CIiAPTER 2. mOPSLS OF R E L E V M PUBLICATIONS 31
the surface topography needs in these models. In complex geometries, the wd-nomud
distance must not be used in the turbulence models, since it cannot be uniquely
defined for all geometries [30]. This was the problem with the proposal of Launder,
Reece and Rodi (LRR) 1991, and variants of it [97,172,170]. Even for fully-developed
flow in a pipe, the wall-distance term can not be dearly de6ned. Very m t l y ,
Craft and Launder 1301 have proposed a new model in which the wall-distance term
is eliminated. This model seems attractive, however, it has not been investigated
adequately, and, hence, has not been implemented in this work
The Speziale, Sarkar and Gatski RSM model (SSG) [I821 which mas originally
proposed for free-shear flows, and, therefore, did not contain the wall-distance term,
has performed satisfactorily in predictions of plane-channel and duct flows [34,17]. In
their systematic investigation of RShls for plane-channel flow , Demuren and Sarkar
[34] reported that the LRR and SSG models of the pressurestrain term performed
much better than the other four models investigated in their paper, Fig 2.3. Also,
Basara and Younis [17] reported essentially similar performance of LRR and SSG
models in prediction of flow over a backward facing step, Fig 2.4.
In this work, for the reasons already given in Chapter 1, attention is limited to
simplified forms of RSMs, the so-called algebraic stress models (ASMs). In particu-
lar, the recent explicit algebraic stress model (EASBI) of Gatski and Speziale [49] is
implemented in this work. In this model, they extended the original methodology of
Pope [141] for obtaining an explicit relation for the Reynolds stress tensor from the
implicit algebraic models. Pope's proposal for two-dimensional turbulent flows was
extended to three-dimensional turbulent flows by Gatski and Speziale [49], and uas
regulated for non-equilibrium effects . If their approach is applied to the pressure
strain-correlation model of Spaiale et al. (SSG model) [182], the resulting model
calculates the Reynolds stress terms through explicit algebraic expressions which in-
volve the timemean strain rate terms. Since, in the EASM, each Reynolds stress
component is obtained explicitly, it is expected that it should not be as prone to
convergence difficulties as uas reported in the earlier .Uhls.
CHAPTER 2. SYNOPSIS OF RELEVWiT PUBLICATIONS . 32
2.3.2 Bubbly Two-Phase Flow
The majority of the published works on the structure of u p d , wall-bounded, gas-
liquid, bubbly, **phase flows have been focused on the pipe geometry. Most of these
works, experimental and analytical, concentrate on the prediction of bubble distribu-
tion because of its importance in momentum transfer and heat transfer- The bubble
distribution is linked with many flow parameters, such as phase velocities, turbulent
st-, and interfadal transport terms. There have been numerous experimental
and numerical investigations of bubbly gas-liquid flows in pipes, and also other con-
duits, in which the role of these parameters has been studied. Comprehensive reviews
of this subject have been done by Spalding [179], Serizawa and Kataoka 11691, Lahey
[82] and Lance and Lopez de Bertodano [89]. In this section, some of the important
published worked are discussed.
Serizawa et al. [I681 and Michiphi et al. [I221 have meanwd pronounced wall
peaking of the local void fraction for turbulent bubbly air/water two-phase upflow in
a pipe. These results were later confirmed in a study by Wang et al. (1901, and w e
extended to show that, in contrast to the bubbly upflow results, void wring (i.e. void
concentration near the pipe's centerline) occurred for turbulent bubbly twc-phase
airlu-ater downflow in a pipe, Fig. 2.5.
Recently, the development of these lateral phase distribution profiles has been
studied by Class et al. [27] and Liu [105], where it was found that bubble size effects
are important. The importance of bubble size on lateral phase distribution has also
been recognized by other investigators, including Sekoguchi et al. [167] and Zun [196].
Similar lateral phase distribution phenomena have been observed in conduits of
more complex geometry. Sadatomi et al. [I591 have found pronounced wall peaking
for turbulent, bubbly, air/water upward twwphase flows in vertical triangular and
rectangular ducts. The results obtained by Lopez de Bertodano [I091 confirms the
occurrence of void-packing in triangular ducts, Fig. 2.6. Ohkawa et al. [I331 made
measurements of upward, bubbly airlwater two-phae in ducts with eccentric annular
cross-sections. They too obsemed significant lateral phase distribution. In the con-
CILMTER 2. Sk?vOPSIS OF RELEKkVT PUBLICATIONS 33
text of a simpler geometry, the migration of bubbles towards the wall has also been
obsenwi for airlwater upward flow past a fiat plate by Marie et aL [114].
Experimental evidences dearly indicate the existence of a strong lateral force on
the dispersed (vapor) phase, which leads to the observed phase distribution. As shown
by Drew and Lahey [I34 431, the interaction of turbulent st- and viscous terms
determine the void distribution in pipes and other ducts. Similar m l t s have also
been found by Kataoka et al. [74)
The interactions between bubbles and turbulent eddies in the liquid phase are
not fully understood yet. Even though measurements of Lance and Batadle [go] on
two-phase grid-generated turbulence, and of Theofanous and Sullivan [I851 on bub-
bly flow in a pipe with low liquid flow rates, indicate that singlephase grid-generated
and bubbleinduced turbulence can be linearly superimposed, there are several mech-
anisms which are expected to cause nonlinear coupling between them. These mecha-
nisms are discused by Lance and Lopez de Benodano [89]. The experimental results
of Serizawa [168,169] and \ h u g et al. [I901 indicate that the liquid-phase turbulence
can even be suppressed by the existence of bubbles. Lance et al. [91,90] observed that
in linear shear flow, the turbulence proved to be more isotropic a t low void fractions
than in single-phase flow. Thii effect apparently results from a competition between
the distortion of turbulent structures by the mean shear, and their random stretching
by the velocity field around the bubbles.
Nevertheless, in models for turbulent bubbly two-phase flow, the superimposing
ideas have been used [160, 109, 1101. Serizawa and Kataoka [I691 derived a k - c
model for bubbly two-phase flow in which they had included extra source terms
in k and c equations to account for the production of turbulence by the bubbles.
-4 two-time-scale k - c model was proposed by Lopa de Bertodano et al. [I101 for
bubbly two-phase flow. They introduced an additional differential transport equation
for the bubbleinduced turbulence kinetic energy, which plays a role similar to that
of the additional source terms defined in the model of Serizau-a and Kataoka [i4].
Hence, the shear-induced k - c equations do not contain or need any extra sources for
bubble interactions with flow turbulence. Thii replacement is important when more
CHAPTER 2. SYNOPSIS OF RELEK4NT PUBLICATIONS 34
sophisticated turbulence models such as NLEVMs are adapted for the prediction of
turbulent, fullydeveloped flows in nondrcular ducts. I t permits more sophisticated
turbulent models to be extended to two-phase flow, without requiring extra source
terms for tun-phase flows. Indeed, Lopez de Bertodano e t al. [I101 used this feature
in extending the ASM of Naot and Rodi [I291 for the simulation of bubbly two-phase
flow in triangular ducts, using the commercial PHOENICS computer code [48]. Their
model works well in reproducing experimental air-water data for homogeneous decay
of turbulence and turbulent flow in a pipe. Their d t s for triangular ducts, however,
are hard to evaluate because of the uncertainties related to the convergence of the
solution and grid-independency of results. Oliveria [I351 extended the single-phase
k - c turbulence model to a mixture of tupphasg. This model reduces to the shear-
induced k - < model of Lopez et al. [I101 for low void-fraction bubbly two-phase
flow.
in this work, the ideas borrowed from two-time scale k - c model of Lopez de
Bertodano et al. [I101 are incorporated in an explicit algebraic stress model (EASM).
This resulting model is adequate, within the scope of the objectives stated in C h a p
ter 1, for the fullydeveloped, turbulent, bubbly, two-phase flows in complex geom-
etry. The expectation h e x is that such turbulence models, along with appropriate
interfacial-term correlations, should enable prediction of the multidimensional effects
in two-phase flows in complex geometry. S p d c a l l y , following an approach similar to
that proposed by Lopez de Bertodano et al. [110], the EASM of Gatski and Speziale
[49] is extended for the solution of fullydeveloped, turbulent, bubbly, two-phase flows
in a triangular duct.
To model turbulent eddy viscosity, Sato et al. [I611 proposed a linear superposition
of a shear-induced and bubble-induced turbulent viscosity. ~ h i k form is also adopted
in this work.
CHrlPTER 2. SYNOPSIS OF RELEVANT PUBLIC.4TIONS 35
2.4 Two-Phase Numerical Methods
As was outlined in the previous section, the considerable amount of re sea . on
turbulence and on bubbly, two-phase f lms has provided several d-posed two-fluid
models of these flows The governing equations, however, are highly nonlinear and
coupled, and thus can not be solved analytically even for the relatively simple ows
A significant breakthrough in the solution of these equations has been obtained by the
development of multidimensional Computational Fluid D-ynamics (CFD) techniques
for -phase flow. Indeed, it appears that the numerical prediction of bubbly flows
in complex geometry may become a practical tool for industrial design in the near
future.
.%XI extensive survey of numerical schemes used for two-phase flows is unnecgsary
here, since it has been done, for example, by Stewart and Wendroff [MI, and Ellul
[45]. Authoritative reviews are also available in works compiled by Hetsroni [61] and
Rocco [150]. Thus, it suffices here to point out some of the salient points of these
reviews, which are relevant to this work.
-411 existing numerical procedures to solve the hvo-fluid model equations are some
form of extension of single-phase numerical procedures. The first effort in this direc-
tion was made by Harlow and .4msdem [58], and was later somewhat simplied by
them [59]. Their procedure is based on the single-phase, semi-implicit ICE method
[60], and includes techniques that have been later utilized by many other workers, for
example Spalding [178]. The next generation of methods were based on the Semi-
Implicit Method for Pressure Liked Equations (SIMPLE) procedure of Patankar
and Spalding [138]: Examples of the application of SIMPLE to two-fluid models are
atailable in the works of Spalding [180], Issa and Gosman [ill, and Carver [22, 231.
Initially, these methods presented some difference in the details of the solution prc-
cedure, however, they have evolved (Spalding [178], Looney e t al. [106]) towards a
similarly structured and successful sequence of operations in the overall iterative loop.
Another method is based on the single-phase PIS0 procedure of Isa [70], examples
indude the work of Looney et al. [106], which, owing to an extended number of
Ch3PTER 2. SYIiOPSIS OF RELEVANT PUBLIC4TlONS 36
corrector stags, allows for a more impliat (but complex) treatment of quantities.
Oliveria [135] pointed out that the PIS0 model is quite complex and, therefore, the
SMPLEC procedure has been used in their work and also in the work of V h g e t al.
[1S9].
Most of the numerical predictions of two-phase flows have been limited to rela-
tively simple, regular-shaped geometries. Problems involving multidimensional com-
plex geometries have typically been solved using codes based on finite volume methods
(FVMs). To utilize FVMs for the solution of problems in complex geometry, body fit-
ted coordinate (BFC) procedures are required to transform the problem information
from the physical domain to a computational domain and vise-versa. These proce-
dure. usually produce complex sourcedominated d imt ized equations, which could
lead to serious convergence difficulties [Ill. Therefore, it appears that the predic-
tion of tm-phase flow in ducts of nondrcular cross-sections requires computational
techniques more advanced than have been used previously. %ice the mathematical
models of these models are still developing, robust computational methods which can
be easily applied for the solution of these problems for complex geometry are desired.
Such developments would, in turn, help investigations of new modeling proposals for
these flows, with adequate confidence in the accuracy of the numerical predictions,
and with acceptable computing cost.
As was stated in Chapter 1, control-volume finite element methods (CVFEMs) is
a class of methods that combine ideas from finite volume methods (FVMs) and finite
element methods (FEMs) [ l l , 118, 121. This class of methods, established originally
by Baliga and Patankar [IS, 131, has already been used for the prediction of single-
phase turbulent flows in three-dimensional complex geometry [158]. Saabas [I581 used
a standard k - c model for the closure of turbulent transport terms in the governing
equations, however, as stated before, this model is not adequate for modeling the
Reynolds stress terms in the governing equations of fully-developed, turbulent flows
in ducts of non-circular cross-sections. Also, as \vas mentioned in Chapter 1, the
procedure for implementation of the logarithmic law of the wall in the work of Saabas
[I581 does not necessarily satisfy masconservation in the near-wall areas.
CHAPTER 2 SMVOPSlS OF RELEVANT PUBLIC.4TIONS 37
More recently, Masson [119], in his Ph.D. work, developed a CVFEM for the
simulation of dilute and dense gas-particle flows in complex geometries. His model is
based on the volume-averaging approach. This approach is very useful in the modeling
of dense two-phase flows, and also laminar dilute two-phase flows [117]. Furthermore,
attempts have been made to combine volume- and time-averaging techniques to der ix
governing equations for turbulent, dilute -phase flows (Elgobashi and Abou-Arab
[44]), but such approaches seem cumbersome and unnecessarily complicated for the
solution of turbulent, dilute, -phase flow problems. Both Saabas [ I S ] and Masson
[I191 have used a modified form of the SIMPLERevked algorithm [137, 118, 121 for
the solution of the governing equations of the problems under investigation. A similar
approach is used in this work
2.5 Summary
The review of the previous works indicates that there is a need to admnce atail-
able numerical methods and models for the prediction of turbulent twwphase flows
in complex geometry. Control volume finite element methods (CVFEMs) [ l l ] are an
elegant candidate for the solution of such complex flow problems, however, as indi-
cated before, the available CVFEMs need to be extended to make them capable of
predicting these flows. In this work, the available CVFEMs [118,12,11], are adapted
and extended in order to predict turbulent, single-phase and bubbly two-phase flows
in straight ducts of non-circular aoss sections. A new procedure for implementing
the logarithmic law of the wall to bridge the near-udl regions for wall-bounded tur-
bulent flows in the context of CVFEM is proposed, implemented, and tested. Also,
more sophisticated turbulence models, such as NLEVMs, have been incorporated in
CVFEMs for the first time, in this work. The solution of single-phase turbulent, fully-
developed flow in straight ducts in the context of CVFEMs has also been attempted
for the first time in this work, and related problems are discussed and resolved.
In gas-liquid, turbulent, bubbly two-phase flows, using guidance from the work
of Marie et al. [I141 and the effort of Afshar and Baliga [I], it is assumed that in
C m E R 2. SYNOPSIS OF RELEVrVvT PUBLICATIONS 38
the near-wall region, a law of the wall akin to the logarithmic law of the d can be
obtained. Based on this assumption, a wall-funaion treatment is proposed for the
mephase flows considered here. .a, to capture the secondary flows, the CZSM
of Gatski and Speziale [49] is extended and used for modeling the turbulent terms,
using ideas borrowed from the am-time-scale k - c model of Lopez de Bertodano et
al. [IlO].
Figure 2.1: Fully-Developed, Single-Phase Flow in a Square-Duct: Secondary Flows
in the Cross-Section, as presented by Speziale et al. [181].
CHriPTER 2. SYNOPSIS OF RELEVANT PUBLICATIONS
0
Figure 2.2: Fully-Developed, Single-Phase Flow in a Square-Duct: Bulging of the
Axial-Velocity Contours, as presented by Demuren & Rodi [%I.
CHAPTER 2. SYNOPSIS OF RELEVrWT PUBLICATIONS
Figure 2.3: Plane-Channel Turbulent Flow: Comparison of LRR and SSG models, as
reported by Demuren & Sarkar [34].
Figure 2.4: Distribution of Wall Static Pressure Coefficient: Taken from the Work of
Basara and Younis [IT].
CHAPTER 2. SYNOPSIS OF R E L E V W PUBLICATIONS
Figure 2.5: Void Coring in Tawphase, Bubbly, Downward Flow in a Pipe: Results
of Wang et al. [NO].
Figure 2.6: Void Packing in TwePhase, Bubbly, Upward Flow in a Triangular Duct:
Rgults of Lopez de Bertodano [Ill].
Chapter 3
Mat hematical Models
The mathematical models used in this work for computer simulations of fully-
developed, turbulent, single- and two-phase flow are presented in this chapter.
First, in section 3.1, fullydeveloped, turbulent, single-phase flows in straight ducts
of uniform, but arbitrary, crosssection are considered. Following that, in section 3.2,
attention in focused on fully-developed, turbulent, vertical, upward, dilute, bubbly,
adiabatic, essentially isothermal, two-phase flows, without phase change, in straight
ducts of uniform, but arbitrary, aosssection. As was stated in Chapter 2, there
are many comprehensive d i i i o n s of these topics in the published literature. In
this chapter, therefore, the emphasis is on concise descriptions and diiussions of
the salient features of the mathematical models of the flow of interest. For detailed
descriptions of these mathematical models, the interested reader will be provided with
appropriate references.
3.1 Fully-developed, Turbulent, Single-Phase Flows
3.1.1 Scope
Attention here is limited to fully-developed, statistically steady, turbulent, single-
phase flows of constant-property Newtonian fluids in straight ducts of uniform, but
arbitrary, cross section. A typical duct is illustrated in Fig. 3.1.
CHAPTER 3. MATHE&IATICAL MODELS
3.1.2 Governing Equation
h the fullydeveloped region, the time-mean velodty distribution in the duct aoss
sectioo does not change with the axial coordinate, z, and the gradient of the time-
mean reduced pressure in the axial direction is a constant. Thus, with respect to the
Cartesian coordinate system (shown in Fig. 3.1), the time-mean velocity components
u, v, w, in the x, y, z directions, rrspectively, ou? be exp& as follows:
The reduced pressure, p, is related to the static pressure by the following equation:
where p is the mass density of the fluid; and g is the acceleration due to gravity, di-
rected in the negative z direction. In the fully-developed region, the reduced pressure
can be expressed as follows:
where dpr T = -- = constant &
An overall force balance over the duct in the fully-developed region can be used
to relate f to the average wall shear stress, T-. The result is:
Here, Tw is the average wall shear stress (directed in the negative z direction) exerted
by the duct wall on the fluid, and Dh is the hydraulic diameter of the duct.
The Reynolds-averaged Navier-Stokes equations [I001 that govern thefully-developed
flows of interest can be expressed in the following Cartesian tensor form [6, 521.
C W T E R 3. MATHEM4TICAL MODELS 44
where the Einstein suf6x notation is implied. Here, i = 1, 2, 3 corresponds to the
z, y, and z directions, respectively; and ul, uz, u3 correspond to the Cartgi velocity
components u, v, and w, respectively.
The Reynoldsaveraged continuity equation reduces to:
In this equation, the density, p, has been formally retained, even though it could be
canceled out because it is assumed to be constant, in order to retain the relationship
of the continuity equation to the prinaple of macis conservation. -
The components of Reynolds stress tensor in the momentum equations, -pu;u;,
are calculated using suitable turbulence models, which are presented in the next
subsection (section 3.1.3).
At the wall of the duct, the boundary conditions are the following: u = 0, v =
0, and w = 0, because of no-slip and impermeability restrictions; and - p u x =
0, because turbulence is damped out at the wall. These boundary conditions are
mentioned here only for completing the mathematical model. They are not used
directly in the numerical solution procedure. As was d i u s s e d in Chapter 1, the
aim is this thesis is to propose and test numerical solution techniques that would be
practical in engineering applications, both for single- and tawphase flows. Thus, the
governing equations are not integrated numerically through the buffer and viscous
sublayers adjacent to the duct wall [162]. Rather, these sublayers are bridged by
suitable semi-empirical wall-functions, which will be presented in section 3.1.4.
3.1.3 Turbulence Models
As was discussed in Chapters 1 and 2, fully-developed turbulent flows in straight
ducts of noncircular cross-sections are characterized by secondary motions in the plane
normal to the mainstream direction (the y - z plane in Fig. 3.1). These secondary
motions, often referred to as Prandtl's second kind of secondary motions [145, 351,
are caused entirely by the anisotropy of the turbulence fields: Such secondary mo-
tions do not occur in laminar fully-developed flows in straight ducts of noncircular
CIUPTER 3. MATHEBZ-ITIC-rlL MODELS 45
massection. For detailed discusdons of these secondary motions and their origins,
the interested reader is referred to the works of Prandtl [145], E i e i n and Li [47],
Brundett and Baines [20], Gesmer and Jones [52], Perkins [140], Launder and Ying
[loll, iUy et al. [6], Gosman and Rapley [55], and Dernuren and Rodi [35].
As u-as discused earlier in the thesis, the aforementioned secondary motions are
only of the order of 2-3 % of the mainstream bulk velocity, but they can have im-
portant consequences on the velocity distribution in the duct crosssection, the wall
shear stress and heat transfer variations, and the lateral spreading rates of scalars
introduced into the flow [Xi]. Therefore, it follows that accurate predictions of fully-
developed turbulent flous in straight ducts of noncircular cross-section require turbu-
lence models which provide the capabiity to capture the an iso t ro~l of the turbulent
field and the consequent secondary motions in the duct cross-section. .4n authoritative
d i i i o n of turbulence models that provide this capabiity is available in a review
article by Hanjalic [56]. Briefly, this desired capability is provided by full Werential
Reynolds stress models (RSMs), also referred to as full differential second-moment
closure models, algebraic Reynolds stress models (AS&), and non-linear versions of
eddy-viscosity models, with rarying degree of generality, accuracy, and computational
cost [56]. Linear eddy-viscosity based models, such as the standard hm-equztion k-c
model [loo], fail completely in this respect.
As stated earlier in Chapter 1, in this work, the computer simulations were carried
out using the following models: the standard linear two-equation k - c model of
Launder and Spalding [IOO]; the nonlinear eddy viscosity model of Rubinstein and
Barton [155]; the Cubic Eddy Vkcosity model (CEVM) of Craft et al. [31]; and the
Explicit Algebraic Stress model (EXSM) of Gatski and Speziale [49]. The objectives
were to propose an effective numerical solution technique (here, appropriate adoption
of the control-volume finite element method [Ill), and a comparative evaluation of the
aforementioned turbulence models. Concise descriptions of these chosen turbulence
models are provided in the reminder of this subsection.
In the linear and nonlinear eddy-viscosity models, the a terms are calculated
CHAPTER 3. AIATHE.WTICAL MODELS 46
from the following expression:
Here, k is the kinetic energy of turbulence k = , and bij is the Kronedrer del ta ( 7 By retaining terms up to the cubic level in the relationship between stress and
the time-mean rate of strain, a general fonn of the btj term can be written as follows
[31]:
where v, = c,:; and S,, = !j (e + 2) and nJ = (2 - 2) are the symmetric
and asymmetric parts of strain-rate tensor, respectively.
In the linear eddy-viscosity models, the coeffiaents cl to c; are set equal to zero in
Eq. 3.9. The quadratic eddy-viscosity model investigated in this work, namely, the
R u b i i e i n and Barton model [155], involves, only, three coefficient cl - CS. While,
the cubic eddy viscosity model (CEVM) of Craft et al. [31] involves cl - c;, but with
c5 = 0 and c, = -a. A compact representation of the coefficients in Eq. 3.9 is
presented in Table 3.1 for these models.
It should be mentioned that the original CEVBI [31] is designed for modeling the
turbulent flows all the way to the wall. In this work, this model is implemented in
the fully-turbulent region, only, and the near-wall area is bridged by suitable wall-
functions. Therefore, instead of the expression proposed by Craft et al. 1311 for c,,
it has been given a constant d u e of 0.09. Also, the values of k and c are obtained
from differential equations similar to those employed in the standard model [loo]. The other turbulence model investigated in this work is the Explicit Algebraic
Stress Model (EAShl) proposed by Gatski and Speziale [49]. They extended the orig-
inal methodology of Pope [I411 for obtaining an esplicit relation for the Reynolds
CHAPTER 3. M.4THEbIATICAL MODELS 47
Table 3.1: Coefficient Values in the Non-Linear Eddy-Viscosity Models and the Ex-
plicit Algebraic Stress Model Expressed Via Eqs. 3.9 and 3.8.
Gatski &
stress tensor from the implicit algebraic models. Pope's proposal for two-dimensional
turbulent flows was extended to three-dimensional turbulent flows by Gatski and
Speziale [49], and was regulated for non-equilibrium effects. The proposed model cal-
culates the Reynolds stress terms through explicit algebraic expressions which involve
the time-mean strain-rate terms. If their approach is applied to the Speziale et al.
presnue-strain-correlation model (SSG model) [182], the resultant explicit algebraic
expressions can be cast into form similar to Eqs. 3.8 and 3.9. The values of the
related coefficients are shoun in Table 3.1.
The c; value in Table 3.1 is obtained from the follou<ng expression:
where
I Speziale [49] c;
and L = 0.1137.
In all turbulence models used in the single-phase part of this work, k and r values
were calculated using differential equations similar to those used by Launder and
0.0876 0.0935 0 0 0
CHAPTER 3. MATHEM.4TICA.L MODELS
Spalding [loo] in the standard k - c model:
where 41 = 1.44.42 = 1.83, uk = 1.0, ue = 1.3. P is the volumetric rate of production
of turbulent kinetic energy:
= -&5 (3.14) ' azj The turbulent viscosity, I+, is obtained from the following expression:
3.1.4 Wall-Function Treatment
The standard logarithmic law of the n d l and the associated wall-functions are used
in this work to bridge the near wall regions for fullydeveloped, turbulent, singlephase
flow in ducts. In this section, the related formulations are presented for a near-wall
point located at a normal distance 6 from the wall. For planar two-dimensional
turbulent flow along a flat wall oriented along the z axis, the velocity component
u, in the z-direction, is parallel to the d, and the y-direction is normal to the
wall. The particular forms of these relations for fullydeveloped, turbulent flow in
ducts with complex cross-sections will be presented later in Chapter 5, along with the
proposed procedure for implementation of these functions in CVFEMs.
The so-called universal logarithmic law of the wall 11621 is used to calculate the
velocity component parallel to the wall, u, at the near-wall point, 6, for turbulent,
singlephase flows: 1
u+= - l ~ ( ~ + ) + B (3.15)
where u+ = 3. Following the recommendations of Launder and Spalding [IOO],
CIWPTER 3. MciTHETHEUTICriL MODELS 49
the Von Karman constant is assigned the value rc = 0.435, and B = 5.2. The
nondimensional wall-normal distance, y+, is given by:
The near-wall node should be located such that 30 5 6+ 5 120. The d e d friction
velocity, f, is obtained from the following equation,
where r, is the magnitude of the wall shear stress.
The values of k and r a t the near-wall points (located at wall-normal distance 6;
with 30 5 6+ 5 120) are obtained from the following equations [162]:
and
where C, = 0.09. These expressions provided the basis of the wall-function treatment
for all single-phase turbulence models considered in this work.
The Reynolds stress terms in the momentum equations are obtained from algebraic
expressions in all turbulence models considered in this work, hence, they do not
require any additional treatment related to the boundary conditions. The effect of
the wall is projected in these expressions through mean velocity components and their
gradients, and the k and c values, at the near-wall nodes.
In fully-developed turbulent flo1r.5, in straight ducts of arbitrary cross-sections,
the velocity vector at near-wall nodes is not necessarily parallel to the adjacent duct
wall. Usually, the velocity vector has a component which is normal to the adjacent
\ d l in addition to the component which is parallel to the wall. The universal wall-
functions formulated in this section do not provide any expression for the velocity
component normal to the wall. A new procedure is proposed in t h i thesis for proper
implementation of these wall-functions for fully-developed turbulent flows in ducts,
which is presented in detail in Chapter 5.
C m E R 3. I\MTHE.WTICAL MODELS 50
3.2 Fully-Developed, Turbulent, Two-Phase Flows
3.2.1 Scope
This work is restricted to fullydeveloped, turbulent, upward (with respect to Fig.
3.1, in the positive z d i i i o n ) , dilute, bubbly, tawphase flows, in which there is no
phase change.
It is assumed that the properties of the liquid and gas phases remain constant in
the fully-developed region. Since attention is limited here to essentially isothermal,
two-phase flows, this constant-property amunption applies very well to the density
and dynamic viscosity of the liquid phase, and also to the dynamic viscosity of the
gas phase. For the density of the gas phase to be essentially constant, however, the
changes in static pressure in the vertical, upward, flows of interest must be much
less than the absolute d u e of the static pressure: This condition is often well a p
proximated in pressurized two-phase flow loops, such as pressurized water reactors
(PWRs) (Drew [37]), in which the average absolute static pressure is a t least 10-100
times the ambient atmospheric pressure. Two-phase flows akin to those considered
in this thesis may not be commonly encountered in engineering practice, but they
provide a very convenient framework for the development and comparative evalua-
tion of mathematical models and cumerical solution procedures. It is mainly for this
reason that fully-developed flows, similar to those considered here, have been the
subject of theoretical, computational, and evperimental investigations over the last
twenty years: E.wples include the published works of Lahey and Lee [Sl], Hewitt
and Owen [62, 641, Drew and Lahey [42], Drew [37], and Lopez De Bertodano [Ill].
3.2.2 Governing Equations
The time-averaged conservation equations as derived by Ishii [69] in the context of
the so-called two-fluid model of two-phase flow are used in this work. The derivations
of these equations are very well documented in the book by Ishii [69], so they will not
be repeated here. Also, the details of interfacial and turbulent terms in the governing
CHAPTER 3. U-ITHEMTICAL MODELS 51
equations are presented and discussed in the works of Lahey [85], Lahey and Drew
[83], Lopez de Bertodano e t al. [log], Lance and Lopez de Bertodano [89], Ellul and
Issa [46], and Gosman et al. 1541.
The full, three-dimensional, time-averaged equations expressing the consenation
of mass and momentum for phase k can be expressed in the following forms [69]:
Continuity Equation: %A - at + ~ . ( a k h v k ) ' r k (3.21)
Mommtwn Equation:
- where A, v k , &, Tk, g, r k , Mk, $* and r: = -pkV;V; are the density, the velocity,
the pressure, the viscous stress tensor, the gravitational acceleration, the rate of
interfacial mass transfer, the resultant interfacial force on phase k, the pr- at
the interface on phase k, and the turbulent st- tensor, respectively. The tilda and
overbar indicate the phasic and mas-weighted average values [69], respectively, and
a * stands for the averaged local presence mtio of phase k, which will be referred to
as the void froction in this thesis. The diffa-ent averaging procedures used in the
derivation of these equations are discussed in the work of Ishii [69]. I'k is zero when
there is no phase change, as for the airfwater bubbly twc-phase flow considered in
this work.
In order to simplify the presentation of the equations used in this work, all nota-
tions for averaging, such as tilda and overbar, are dropped in subsequent discussions,
except where such notations are essential.
For statistically steady, vertical, upward, turbulent, isothermal, gas-liquid bubbly
two-phase flows, without phase change, the aforementioned continuity and momentum
equations can be simplified and written in the following forms:
Gas-Phase Continuity Equation:
CHAPTER 3. MATHEMATICAL MODELS
Liquid-Phase Continuity Equation:
Gas-Phase Momentwn Equation:
Liquid-Phase Momentwn Equation:
where subscripts 1 and g refer to the liquid and gas phases, respectively, and, for
convenience, a is used to denote the gas-phase void fraction, and (I -a) is the liquid-
phase void fraction. Here, considering the relatively small values of the *phase
density and viscosity in comparison to the cormponding values for the liquid phase,
the advection terms V.(apgVgVg), the stress tensor terms (r, + <) and, also, the
(pig -pg) terms have been neglected in the gas phase momentum equation, compared
to the pressure gradient and the interfacial forces, following the suggestion of Lopez
de Bertodano et al. [log], and Ellul and Issa [46]. From here on, the subscript I in the
stress terms will be dropped, with the understanding that these terms are retained
only in the momentum equation for the liquid phase.
Furthermore, as was stated earlier in this chapter, it is assumed that both the
liquid and gas phase densities are constant. It should also be noted that the interfacial
forces on the liquid and gas phases assumed to be equal in magnitude and opposite
in direction: M1 = -M,.
In order to complete the two-fluid model, expressions or models to calculate the
interfacial forces and the Reynolds stress terms are needed. A significant technical
hurdle to overcome before the two-fluid model can accurately predict a wide range of
two-phase flows is the development of such "closuren models. As was noted in Chapter
1, these models must appropriately account for the small-scale physics, which is lost
during the averaging process.
Such dosure models are presented in the next two subsections. In reviewing
these models, it should be noted that the following additional m p t i o n s have been
invoked: The bubbles are essentially spherical in shape; the diameter of the bubbles
is essentially uniform; and, since the attention here is restricted to dilute tn-phase
5ous, the bubbles do not coalesce, and they do not break up.
3.2.3 Expressions for interfacial momentum transfer
The interfacial force is customarily divided into several components; however, for
the bubbly two-phase Bow considered in this work, many components have a negligible
effect on the establishment of the phase distribution, as has been discussed by Lahey
et al. [SO] and Oliveira [135]. The dominant interfacial forces for a free-field, dilute,
gasliquid bubbly t-phase 5ow are the drag and lift forces [SO, 1351. Thus,
In this work, the drag force (per unit volume) is obtained from the following expression
proposed by Ishii and Mihima [68]:
where V, = V, - Vl is the relative velocity, & is the bubble radius, and CD is the
drag coefficient which is defined by the following expression [68]:
where
is the bubble Reynolds number,
is the mixture viscosity, and Db is the bubble diameter.
C H A P T E R 3. MATHEhZ-lTICAL MODELS 54
The average lift force exerted by a rotational inviscid flow on a sphere as derived
by Drew and Lahey [do], is used here for Mf:
where C L is the li coefficient and ranges from about 0.05 to 0.5 [SO]. This lift force
is caused by inertia and the proposed expression is based on potential flow theory.
A viscous lift force could also be included. However, for the bubbly flows of interest
here, Reb - 1000; thus, inertia predominates and the viscous component of the lift
force can be neglected.
The interfadal pressure term, pit - pt, in the liquid-phase momentum equation
must also be modeled. Both the turbulence and the relative velocity atfect the in-
stantaneous pressure at the interface, pd. The &ect of turbulence is not yet fully
understood. The effect of the average relative velocity is clearer. The simplest possi-
ble case is for potential flow around a sphere 1881:
where the pressure coefficient for non-interacting spherical bubbles is C p = 0.25. For
real bubbles, however, the value of Cp may be expected to be greater than 0.25,
as explained by Lopez de Bertodano [ill]. It should be noted that this interfacial
pressure term was assumed to be zero in the calculations performed by Lance et al. a t
Ecole Centrale de Lyon [91,89], pd -p; = 0. This approach has also been adopted in
this work. Furthermore, noting that pg << pt and neglecting surface tension, pg = p i .
The forces presented above have been expressed as a function of the time-mean
gas and liquid velocities. Therefore, the role of turbulence on the motion of bubbles
was neglected. To consider this effect, it umuld be necessary to include fluctuating
drag and lift forces. A simpler approach, proposed by Lopez de Bertodano [ill],
consists of modeling the effect of turbulence by a turbulent dispersion force, which
can be expressed as:
MTD = C r ~ p ; k s , V a (3.34)
where ks, is the shear-induced kinetic energy of turbulence in the liquid phase, and
CHAPTER 3. nlUTHEJZ4TICIU. MODELS 55
GD = 0.1 [Ill]. In thii work, the interfadal force is calculated as folloas: M1 =
MP +M:.+W~.
3.2.4 Turbulence Models for Bubbly Two-Phase Flow
Several extensions of the singlephase tm-quation k-r models for the simulation
of dilute, turbulent, bubbly two-phase flows are available in the literature: Evamples
indude the models proposed by Lopez de Bertodano et al. 1109, 1101, Serizawa and
Kataoka [169], and Issa et al. [71]. After reviewing these models, as was mentioned
in Chapter 2, the two-time scale k - c turbulence model of Lopez de M o d a n o et
al. [I101 was chcsen for implementation in this work. The details of this k - c model
are well documented [110]. h important feature of this model is that its key ideas
can be readily incorporated into the nonlinear k - r models, including the M M ,
in order to extend their applicability to the two-phase flows considered here. As
was stated earlier, such extended nonlinear k - r models are vital in capturing the
secondary motions in fullydeveloped turbulent flow in straight ducts of noncircular
CTOSS-section.
In the k - r model of Lopez de Bertodano et al. [110], the shear-induced and
bubbleinduced turbulent kinetic energy terms are calculated separately and super-
imposed linearly. A detailed derivation of this model is available in Refs. [110, 1111,
hence, it is not repeated here. Rather, only the final forms of the relevant equations
are presented and discussed concisely in thii subsection. Furthermore, the effect of
turbulence in the gas phase is negligible. Thus, only the modeling of turbulence in
the liquid-phase is discussed here.
Following Lopez de Bertodano e t al. [110], the total turbulent kinetic energy of
bubbly two-phase flow is expressed as:
where ksr is the shear-induced turbulence and ksr is the bubbleinduced turbulence.
CHAPTER 3. .MATHEM4TICrZL MODELS 56
The transport equation for k s ~ is ':
w h m the terms and coefficients are similar to those in the corresponding single-phase
flow k equation [Eq. (3.12)) In dilute liquid-gas bubbly flows, for most practical
purposes, kBr can be set equal to its asymptotic \due, k ~ ~ , , , which applies stictly
when the relative velocity V, reaches its terminal or fullydeveloped value [110]. In
this work, therefore. 1
~ B I = b r a = -QCVM IvAZ 2 (3.37)
For potential flow around a sphere, the virtual mass coefficient CVM is equal to 0.5
[IIO]. These developments imply a bubble relaxation time, rb, is the second time
constant in this model: It is obtained from the following expression [110]:
The c s ~ values are obtained from a modified equation (relative to the c equation
for single-phase flows) for bubbly two-phase flows:
The coefficients in this equation are the same as the corresponding coefficients in the
single-phase flow c-equation [Eq. (3.13)]. Following Lopez de Bertodano e t al. [110],
the rate of diiipation of k can be obtained as follows:
The principle of linear superposition is also used for the two-phase Reynolds s t r e s
and turbulent viscosity [110]. Thus,
'Here, i , j and k are indices used in the C;mesian tensorial notation and should not be confused
with phase or interkdd indices.
CHAPTER 3. MATHEbL-iTICAL MODELS 57
where the shear-induced (SI) components are given by Eqs. 3.8 and 3.9 (with k and
c replaced by k s ~ and csI), and the bubble-induced components are obtained from
the following expresior: for pseudo-turbulence around a g o u p of spheres in potential
f!ow ( h o l d [9]): - 1 3
( ~ U ~ I B , = ff [,vrvr + - 20 ~vr~~ij] (3.42)
This expression may be rewritten in matrix form as follows:
The turbulent viscosity, which is used to calculate ( a ) s r from the modi6ed Eqs.
3.8 and 3.9, is expressed as [110]:
where vt,, , as proposed by Sato et al. [161], is equal to:
and I+,, is calculated according to the expression used in the standard k - c model,
but with ksl replacing k [110], e-1 vt,, = c,-. CSI
3.2.5 Specialization of the Momentum Equations
Liquid-Phase Momentum Equations
For the bubbly twc-phase flows of interest here. a reduced pressure, p,, may be
defined by the following equation:
CHAPTER 3. MATHEhUTICrlL MODELS 58
In the fullydeveloped region, this reduced presnve can be expressed as follows
where
f = -& = m t &
(3.49)
The value of T, in the fullydeveloped region, is related to the average shear stress at
the wall (in the negative r d i i t i o n ) , ?,,,, by the following expression:
where < a > is the average void-fraction of the gas phase over the duct cr05~-section.
Using the expressions introduced earlier for the interfacial forces, Reynolds stress
tenns, and the reduced. pressure, the liquid-phase momentum equations in the fully-
developed region can be recast as follows:
Gas-Phase Momentum Equations
As was d i e d earlier in this section, for the two-phase flows considered in this
work, the gasphase momentum equations can be reduced to an algebraic balance
beht-een the different forces acting on the bubbles. These algebraic equations can
be cast in the following forms, once the models presented for interfacial forces are
incorporated in Eq. (3.25).
2-momentum
3 CD aT + a(pt - p,)g - gapt-(w)lV,l + aplC~
5
y-momentum
CHJWTER 3. MATHETHEMATCcU MODELS
Eqs (3.52 ) to (3.54) can be rearranged, by gathering together terms related to each
of the three components of relative velocity, G, v,, and w7, and expressed in matrix
form:
5 7 5
[: ; 1 [;I = [;I (3.55)
Therefore, the values of relative velocity components in the z, y and z d i i i o n s can
be calculated by solving this set of algebraic equations. It must be remembered that
these are nonlinear equations. Thus, the co&cients in Eq. (3.55) depend on w,
vr, and w,. In this work, in each cycle of an iterative solution procedure, the most
recent d u e s of +, v,, and w, uwe used to calculate the coefficients in these equations
(successive-substitution procedure), and the resulting nominally-linear equations were
solved simultaneously. The complete expressions for the coefficients and the right
hand side terms are given in Appendix A.
3.3 Boundary Conditions
As was discussed in Chapter 1, due to the complexity of the physical phenomena
in the near-udl region in turbulent, bubbly, twc-phase gas-liquid flows, it is not
practical to numerically solve differential mathematical models of these flows all the
way to the ud. Therefore, in this work, the near-wall area is bridged by appropriate
udl-functions for turbulent, bubbly *phase flows.
The near-wall region in bubbly, turbulent, gas-liquid tuwphase flows is very com-
plicated. The interactions between the bubbles and the turbulent eddies, in the
vicinity of the wall, is not yet clearly understood. -4s uas observed by Marie e t al.
[115], close to the ud l , because of the finite size of the bubbles and the steep velocity
CH=IPTER 3. hWTHE.4t.iTIC.U MODELS 60
gradients, an additional drag force acts on the bubbles. The standard drag-force coef-
fiaent expression, Eq. 3.29, does not account for this additional drag force. Also, as
discused by htal [7], there are evidences that an additional lift force is also present
dose to wall, which prevents the bubbles from coming into contact with the wall.
As was mentioned in Chapter 1, recently, there have been several efforts to propose
uall functions specif idy designed for turbulent, dilute, gas-liquid, bubbly -phase
flows. The works of Sato and Sekoguchi [160], Lopez de Bertodano [108], Marie et
al. [114, 115], and Lance et al. [91] a;e examples of such efforts. In this work, an
attempt was made to enhance the anilable wall functions for the flows of interest,
with some success (AEjhar and Baliga [I]). However, as was noted earlier, the wall-
functions proposed in the works of Marie e t al. [114] and Afshar and Baliga [l] are
designed and fine-tuned with reference to experimental data for turbulent, dilute,
bubbly two-phase flows over a vertical flat plate, which make them inapplicable to
the flows considered in this thesis. Furthermore, experimental data that is essential to
extend the available wall-functions to the flows considered here are not available in the
published literature, a t this time. Nevertheless, based on the aforementioned works
[I, 91, 108, 114, 115, 1601, and considering that the near-uaU structure of upward,
turbulent, bubbly two-phase flows is similar to that of single-phase turbulent flow
(Marie et al. [114]), it seems reasonable to assume that the desired mall-functions,
when they are eventually formulated, could be cast into forms that are akin to those of
wall-functions for turbulent single-phase flows, but with parameters that are functions
of the properties and characteristics of the two-phase flows of interest.
In the spirit of the discussions presented in the previous paragraph and in Chapter
1, it uas assumed in this work that for the liquid phase, the time averaged resultant
velocity component tangert to the d l , at a suitably located near-wall node, is gov-
erned by a logarithmic law of the wall. As w a s done in Section 3.1.4 in the context of
the uall-function treatment for single-phase flows, for convenience in the presentation,
the logarithmic law of the ud l , and the related wall functions, for the turbulent, two-
phase flow of interest are presented in this section for a near-wall point located at a
wall-normal distance 6, with the liquid-phase velocity component in the z direction,
C I L W E R 3. iUATHGAfATICAL MODELS 61
ul, parallel to the wall, and the y-direction normal to the d L The particular forms
of these relations for fullydeveloped, turbulent, dilute, bubbly two-phase f l m in
vertical ducts with complex crosssections will be pnsented later in Chapter 5, along
with the proposed procedure for implementation of these functions in CVFEMs
For low void fraction, bubbly, tao-phase flows, as suggested by Marii et al. [114].
the friction velocity may be approximated as:
where < a >, and < a >, are average void-fractions values in the near-wall region
and in an area far from the vds , respectively, and Db(= 2%) is the average bubble
diameter. The following nondimensional variables are now introduced:
where &(= u, - u,) is the relative velocity component in the z direction.
The expression for the liquid-phase, time-mean, resultant velocity component tan-
gent to the wall (represented by ul in this subsection) in the logarithmic region is
assumed to have the following form:
1 u: = - In(y+) + Btp (3.58)
%
where r;,, and B,, would be determined b+g the parameters and characteristics of
the turbulent, dilute, bubbly two-phase flows of interest [I, 1141.
In near wall regions, assuming lccal equilibrium, the rate of production of turbu-
lent kinetic energy is equal to its rate of dissipation. Thus,
The rate of production of turbulent kinetic energy is modeled by the following ex-
pression,
Using Eq. 3.56, 7, is equal to:
CHAPTER 3. MATHEhfATICAL MODELS 62
then,
Calculate % by differentiating Eq. 3.58 with respect to y,
Here, q, and Be are 2sumed to be dependent on tuc-phase flow parameters, but
not on y, so their differentiations with respect to y are zero. Therefore,
Substituting for T, and 3 in Eq. 3.60, and using Eq. 3.59, the value of 6 a t a
near-d l node ( y = 6) is obtained from the following expression:
At this stage, it is assumed that:
Using Eq. 3.67, Eqs. 3.59 and 3.60 can be combined and recast as follows:
Substituting ior rW from Eq. 3.63,
For dilute, bubbly, tu-o-phase flow, using Eq. 3.44 for vt,
C H . W E R 3. MATHEBf.4TIC4L MODELS 63
Substituting for u t ~ ~ and utsr from Eqs. 3.45 and 3.46, respectively,
Substituting for c from Eq. 3.66, and rearrange the resulting equation, an expression
for k a t near-wall nodes is obtained:
Values of (i@)sr, kBr, (mBI and relative velocity component are calculated
from algebraic expressions, hence, they do not require the speci6cation of any addi-
tional boundary conditions.
Finally, it must be noted that the void fraction, a, of the gas phase is governed
by the corresponding continuity equation, Eq. 3.23. This equation involves only
first-order derivatives. Thus, it is necessary to 6x the value of a a t only one point
in the duct crosssection of interest. In the fullydeveloped flows considered in this
work, in each cycle of the overall iterative solution procedure, details of which will be
presented in Chapter 4, the d i e t i z e d gas-phase continuity equation was iteratively,
solved without fixing the value of a a t any grid point in each iteration; however, after
each sweep of the iterative solver, the calculated a distribution was adjusted to obtain
a desired average value of a. Additional details are given in Chapter 7.
The continuity equation for the liquid-phase is used to obtain the distribution
of pgl. Again, with the constantproperty assumption, the absolute value of p*, is
arbitrary, only its distribution is important. Thus, the value of p*, can be fmed to an
arbitrary value at any suitable point.
Again, it is noted that the aforementioned %-all functions are based on the ns-
sumption that a logarithmic law of the d l , expressed in Eq. 3.58, for the turbulent,
dilute, bubbly two-phase flows in ducts can be formulated. I t is emphasized, again,
that such a law of the u d is not currently available. In addition, detailed and reliable
experimental data that is necessary to formulate such a law of the wall are also not
available a t this time. In Chapter 8, the formulation of an appropriate law of the
CHriPTER 3. MX17-ECI14TICAL MODELS 64
a d 1 for turbulent, dilute, bubbly two-phase flows in ducts is suggested as a posible
extension/continuation of the work reported in this thesis.
In Chapter 5, the implementation of these wall functions into the proposed CVFEM
(Chapter 4) is discussed. Following that, in Chapter 7, the application of the pro-
posed wall functions to a demonstration problem, involving fullydeveloped, turbu-
lent, dilute, bubbly two-phase flows in a vertical duct of triangular cross-section, is
illustrated by using the particular, simplified, forms obtained when < cr >+ 0: for
this condition, the proposed d formulations take on forms identical to those used
for single-phase flows.
In this chapter, the mathematical models used in this work for computer simu-
lations of fullydeveloped, turbulent, single-phase and dilute, gasliquid, bubbly two-
phase flows were presented concisely. For the benefit of the interested reader, suitable
references, that give detailed derivations and/or discussions of the models, were also
given.
As was stated and discussed in this chapter, for both the single- and the t u ~
phase flows considered here, it is impractical to numerically integrate the governing
differential equations all the way to the zd. Instead, suitable wall functions are used
to bridge the near-wall regions. Such uall-functions for single-phase turbulent, and
dilute, bubbly, tuwphase fullydeveloped flows were presented in this chapter.
CI-LWTER 3. U4mhLATICrlL MODELS
I
Figure 3.1: Euample of a Straight Duct of Uniform Cross-Section.
Chapter 4
Numerical Model
The formulation of a control-volume-based finite element method (CVFEM) for
the computer simulation of steady, tuPdimensiona1, planar, fluid flow and heat trans-
fer phenomena is presented in this chapter. CVFEMs combine concepts that are na-
tive to F i i t e Volume Methods (FVMs) and Fiite Element hlethods (FEMs). The
formulation of CVFEMs is amenable to easy physical interpretation, and their solu-
tions satisfy both local and global conversation requirements, wen on coarse grids. In
addition, they provide the geometric flexibility that is traditionally associated with
FEMs. The CVFEM was used to solve the mathematical models of fullydeveloped,
turbulent, single-phase and bubbly *phase flows in straight ducts, descriptions of
which were presented in Chapter 3.
The mTEi~ pmented in this chapter is based on a primitivevariable, co-located,
equal-order formulation: It works directly with the velocity components, pressure,
void fraction, and turbulence kinetic energy and its dissipation rate; these dependent
variables are stored at the same nodes in the finite element mesh, and are interpolated
over the same elements. This CVFEM is based on the CVFEMs proposed earlier by
Baliga and Patankar [13, 141, P r b h and Patankar [142], Schneider and Raw [163],
Saabas and Baliga [L56,157], and hlasson et al[117,118]. Recent detailed descriptions
of the underlying ideas are available in the Ph.D. theses of Saabas [I581 and Masson
[119], and a mview article by Baliga [ll].
As was discused in Chapters 1 and 2, the treatment of udl-functions in the
CHAPTER 4. NU:MERICAL .MODEL 67
turbulent flow simulations of Saabas [I581 and Saabas and Baliga [156,157] does not
ensure mas consenation. Thus, a new procedure is required to properly incorporate
the wall-function approach into the CVFEM. Considering the importance of this new
procedure for accurate simulations of the flows of interest, and because the proposed
procedure is one of the main original contributions of this work, it will be presented
separately in Chapter 5. .As m a s stated earlier, detailed description of the formulation
of CVFEMs are available in the published literature [ l l , 119, 1581. Nevertheless, for
the sake of completeness of the thesis, a concise presentation of the key ideas of the
CVFEhl for the simulations of the flows of interest are introduced in this chapter.
I t should also be noted that in this work, for the first time, CVFEMs have been
used for the solution of fullydeveloped, turbulent flows in straight ducts, using non-
linear k - e models and an aplicit algebraic s trgs model. Simulations of fully-
developed, turbulent, single-phase and dilute, bubbly taPphase flows in ducts ne-
cessitated some augmentations of the atailable CVFEMs. These new features in the
proposed CVFEM will be discussed in subsequent sections of this chapter.
4.1 General Form of Governing Equations
Mathematical models for fullydeveloped, turbulent, single-phase and dilute, bub-
bly two-phase flows were presented in Chapter 3. The single-phase flow model is made
up of six differential equations: a continuity equation, three momentum equations,
and differential transport equations for turbulent kinetic energy and its dissipation
rate. The two-phase flow model consists of seven differential equations: a continuity
equation for each of two phases, three momentum equations for the liquid phase, and
differential transport equations for the shear-induced turbulent kinetic energy of the
liquid-phase and its dissipation rate. As was discussed in Chapter 3, the three mo-
mentum equations of the gas phase can be simplified to algebraic expressions, which
are used to calculate the gas-liquid relative velocities in three directions. These set
of equations are completed by six algebraic expressions for turbulent stresses, using
the chosen non-linear k - e models and also an expliat algebraic stress model, and
CHAPTER 4. NUMERICAL MODEL 68
suitable models for interfacial forces, bubble induced kinetic energy, Reynolds stress
terms, and wall-functions for near-wall regions as discused in Chapter 3.
The differential equations can be cast in the following general form:
The appropriate governing equations can be obtained from Eq. (4.1) by defining the
dependent vaiable, 0, the void fraction, /3, the diffusion co&cient, ??, the mas
density, p, the secondary velocities (y-mmponent, v , and z-component, w) , and the
volumetric source term, S*, according to Table 4.1. In this table, the terms m f , mi,
and m:, which appear in the S* expressions for bubbly hvo-phase flows, are given by
the following equations:
The turbulent stress terms, -~JLVY', are obtained from algebraic expressions (Eqs.
3.8, 3.9) and appropriate values of the associated coefficients (Table 3.1).
The relative velocity components in Eqs. (4.2) to (4.4) are obtained by the simul-
taneous solution of the reduced, algebraic, versions of the gas-momentum equations,
Eq. (3.25), in each iteration, as was discussed in Chapter 3. Numerical details uill
be presented later in this chapter.
4.2 Domain Discretization
In this work, the calculation domains are the txvodiiensional (y - z ) planar aoss-
sections of the straight ducts, in the fullydeveloped region, Fig. 4.1. The calculation
domain is fiat divided into three-node triangular elem-ents. Then, the centroids of
CHAPTER 4. NUMERIC-U MODEL
Single-Phase Flow
r-pii'd-piiii
'-momentum
-3L - @ij- ~7
Continuity
k-equation k 1 p v w p + 5 P(P - 4 c-equation c 1 w p + p (&lip - G&) p 1 v
Bubbly Twc+Phase Flow
I I I
Gas Phase
Continuity 1 a 1 I 1 pg
v l w l re I S*
Table 4.1: Specific Forms of the General Differential Equation.
CHAP7ER 4. NUMERICAL MODEL 70
the elements are joined to the midpoints of the corresponding sides. This creates
polygonal control volumes around each node in the finite element mesh. A sample
domain discretization is shown in Fig. 4.2: The solid lines denote the domain and
element boundaries '; the dashed lines represent the control-volume facg; and the
shaded ares show the control volumes aESOdated with one internal node and one
boundary node.
4.3 Integral Conservation Equation
Consider a typical node i in the calculation domain: it could be an internal node,
such as the one shown in Fig. 4.3% or a h u n k node, similar to the one shown
in Fig. 4.3b. An integral formulation corresponding to Eq. (4.1) can be obtained by
applying the appropriate consenation principle for the dependent variable, 4, to a
suitably chosen control volume. The resulting integral conservation equation, when
applied to the polygonal control volume surrounding node i in Fig. 4.3, can be written
as follows:
+ [similar contributions from other elements associated with node i]
+ [boundary contributions, if applicable] = 0
where n is a unit outward vector normal to the differential length element, ds, and J
is the combined convection and diffusion flux of 4:
Jc = P V ~ (4-8)
'Modifications required for proper incorporation of the ad-function treatments will tx discusd
in Chapra 5.
C W E R 4. NUMERICAL MODEL n
The form of Eq. (4.5) emphasizes that it can be assembled by using an element-by-
element procedure akin to thzt used in FEMs.
4.4 Interpolation Functions
The derivation of algebraic approximations to the integral conservation equations
requires the s p d c a t i o n of element-based interpolation functions for the dependent
variable, 6, dSusion coefficient, r6, source terms, So, mid fraction, a, and mass
density, p. The interpolation functions are specific to each element.
In each element, a local (y, t) coordinate system is d e h e d such that the origin is
a t the centroid of the triangular element. A typical three-node triangular element 123
and the local y-z coordinate system are shown in Fig. 4.4a Some of the interpolation
functions will be e x p d with respect to this local coordinate system.
4.4.1 Diffusion Coefficients, Density, and Sources
In each triangular element, the centroidal values of ro and pare assumed to prevail
over the corresponding element.
The source term, So, is linearized, if required, and expressed in the f o l l o h g
general form [137]:
s, = sc + Sp6 (4.9)
In each element, the values of Sc and Sp are computed at the nodes and assumed
to prevail over the portions of the corresponding control volumes that lie within the
element. Thus, with respect to the element 123 shown in Fig. 4.4% three sets of Sc
and Sp are s t o d Scl, Sa, SC~, SPI, SPZ. and SP~.
4.4.2 Mass Flow Rates
In the calculation of mass flow rates across the control-volume faces, the velocity
is denoted by:
V" = vmj + wmk (4.10)
CIUPTER 4. NU-UERIC.4.L MODEL 72
When mass flow rates of the gas phase are considered, vm = v i , and wm = e, and
urn and wm are the elemental dues . However, when the mass flow rates of the liquid
phase are considered, a special treatment, borrowed from the works of Prakash and
Patankar [142] and Saabas and Baliga[l56, 1571, is used to prevent the orrurrence of
spurious presure o d a t i o n s in the proposed co-located CVFEM. The development
of this interpolation is based on the discretized momentum consenation equations.
It will, therefore, be presented later in this chapter.
4.4.3 q3 in Diffusion Terms
In the derivation of algebraic approximations to surface integral of difhsion fluxes,
Eqs. (4.5) and (4.7), the dependent variable 4 is interpolated linearly in each element:
Referring to Fig. 4.4q the constants A, B, and C can be uniquely determined in terms
of the y, r coordinates of three nodes, and the corresponding values of 6. Thus, with
reference to the element 123 and the local y - r coordinate %%em shown in Fig. 4 . 4 ~
CHAPTER 4. JWMERIC~U MODEL 73
4.4.4 4 in Convection Terms
In the derivation of algebraic approximations to surface integrals of the convective
fluxes, Eip. (4.5) and (4.8), the MAS Weighted upwind scheme (MAW) was used.
The MAW scheme is based on the work of Masson et al. [118], which is an adaptation
of the positiw-coeflident schemes of Schneider and Raw [I631 and Saabas and Baliga
[156, 1571. It ensures, at the element level, that the extent to which the dependent
variable at a node exterior to a control volume contribnts to the convective outflow
is less than or equal to its contribution to the infiow by convection. Thus, it is a
d c i e n t condition to ensure that the algebraic approximations to the convective
terms in Eq. (4.5) add positively to the d i i t i z e d equation. Furthermore, the
MAW scheme takes better account of the influence of the d i i i o n of the flow than
the donor-cell scheme of Prakash [143]. Thus, the MAW scheme produces less false
diffusion than the donorsell scheme [156, 157,158].
The MAW scheme defines a masweighted average of 6 at each of the three control
surfaces of a triangular element (Fi. 4.4b), namely, &, +,, and &, in the following
manner:
Let
where 4, n,, and nc are the unit normals shown in Fig. 4.4b.
f+O, + ( 1 - f+)& where f+ = min [max (-%,o) , I ] i f & > 0
f-6.+(1- f-)&where f - = m i n [ m m ( - % , 0 ) , 1 ] i f + < O (4.17)
f+dt + (1 - f+)& where f+ = min [maz ( %,0) , I ] i f m, > 0 4, = {
f - & + ( I - f-)&where f -=min[mnz( -%,0) ,1] i f m.<O (4.18)
f +& + (1 - f +)dl where f + = min [m (- e, 0) , I ] i f mt > 0 (4.19)
f -+,+(I- f-)&where f - = m i n [ m ( %,0).1] i f mt<O These mawweighted averages of 4 are assumed to prevail over each control surface
when the surface integrals of the convection terms, Eqs. (4.5) and (4.8), are evaluated.
CHAPTER 4. NUMERICAL MODEL 74
The algebraic approximations of the m s flow r a t s in Eq. (4.16) will be discussed
later in this chapter. It should be noted that in this scheme, to obtain expressions
for Or. $,. and 4, in terms of 01, &, and 63, a 3x3 matrix of element-interpolation
coefTicients must be inverted. Further details are available in the M.Eng. thesis of
Akhar [S], and the Ph.D. theses of Saabas [IS], and Masson [119].
4.4.5 Reduced Pressures
The reduced pressure, p,, is expressed as:
where T is the constant axial reduced pressure gradient ( - dp ld z ) and p0(y,:) is a
cross-sectional pressure. Since it is assumed that the static pressures for liquid and
gas phases are equal, hence, p'(y,r) is equal for both phases. More details of the
associated approximations were given in Chapter 3.
The cross-sectional pressure, pg, is interpolated linearly in each element. With
respect to the local (y , : ) coordinate system in Fig. 4 . 4 ~
The constants d, e, and f can be obtained using procedures similar to those used to
calculate A, B, and C in Eqs. (4.12) - (4.14).
The value of T is obtained in each iteration from Eq. 3.47. The average u d l shear
force is calculated by adding up the corresponding near-wall nodal \dues. Since these
local values are calculated in the new procedure for the implementation of the wall-
functions in CVFEMs, which will be d i e d in the next cLpter, the calculation of
T value will also be elaborated in the next chapter.
4.4.6 Void Fraction
The function used to interpolate the void fraction, a, in most of the available
finite volume methods for twc-phase flow is based on the upwind scheme [23,58]. The
CHriPTER 4. NVnIERICAL MODEL 75
donor-cell scheme of Prakash [143] is one way of implementing this idea in CVFEMs
In this work, the MAW scheme described previously, and as proposed by Masson
[119], has been implemented. As was discused in the last section, the MAW scheme
takes better account of the influence of the direction of the flow than the donor-cell
scheme.
The modified M.4W scheme defines a material masfweighted average of a at the
integration points on each of the three control surfaces of a triangular element (Fig.
4.4b), namely, a,, a,, and at, in the following manner:
Let
where G, n,, and n, are the unit normals shown in Fig. 4.4b.
Then
f + a t + ( l - f+)n,where f + = m i n
f -a , + (1 - f -)az where f - = m i n [muz (-%,O), I ] i f M! < o (4.23)
ftat + (I - f t ) w where f t = rnin [muz ( %,O) , l] i f M: > 0
f -a , + (1 - f -)a2 where f - = min [maz (-$,o), I ] i f M: < 0 (4.24)
f tar + (1 - f+)al where f* = m i n maz -$$.0), 1 if M! > 0
a + ( - a w h e f = M n ~ ~ [ B,O),l/ i f @ < o (4.25)
These material mass-weighted averages of a are assumed to prevail over each
con:rol surface when the mass flow rates in the integral continuity and momentum
equations are evaluated.
4.5 Discretized Equations
The d i i t i z e d equations are obtained by first deriving algebraic approximations
to the element contributions and the boundary contributiow, if applicable, and then
assembling these contributions appropriately.
CHAPTER 4. NUMERICAL MODEL
4.5.1 Discretization Equations for 4
The following discussions pertain to node 1 of the element shown in Fig. 4.4. The
total dement contribution is the sum of the diffusion, convection, and source contri-
butions. The derivation of algebraic approximations to each of these contributions is
presented separately.
D i i i o n Contribution
In each element, the *ion flux of 9, JD, can be expresed in terms of its
components in the y and z directions:
where j and k are unit vectors in y and z directions, respectively. The linear interpo-
lation function given in Eq. (4.11) is used to appmximate JD, and JD,. Thus, with
reference to element 123 and the local coordinate system (y - z) in Fig. 4.4% the
di&lsion contributions are approximated as follows:
[ J D . ~ = [rulyr - re&] (4.28)
where A and B are given by Eqs. (4.12) and (4.13), respectively, and n in both of
these equations is a unit normal pointing out of the control volume associated with
node I.
Convection Contribution
In each element, the convection (or strictly, advection) flux of 6, Jc, can be
expressed in terms of its components in the y and : directions:
where 9 is given by Eqs. (4.17) - (4.19) when the M.4W scheme is used. It should be
CH..IpTER 4. NUMERICAL MODEL - I I
noted here again that P and w"' denote components of the velocity vector, V"', in
the masfflux terms.
Here, with reference to dement 123 in Fi. 4.4% the convection contribution is
exp&=
& B J ~ . ~ = -A&& [ B J ~ ~ = a- ( 4 3 1
It should be noted that &A& and ah& are the mass flow rates lir, and mt, respec-
tive?v; which are the mass flow rates across the corresponding coutrol surfacg, in the
direction of the normals n, and nt, respectively (see Fig. 4.4b), as expressed in Eq.
(4.16).
Source T e r m
With respect to the contribution of element 123 to the control volume surrounding
node 1 (Fig. 4.4a), the volume integral involving the source terms Se is approximated
as follows:
where Scl and Spl are the stored nodal values within each element, and
is the volume defined by points 1, a, o, and c, with DET given by Eq. (4.15).
Discretized 4 Equation
Adding up the diffusion, convection, and source contributions, the total contri-
bution of element 123 to the conservation equation for node 1 is obtained. The
algebraic approximation to this element contribution can be compactly expresed as
follows [118, 156, 1571:
CHAPTER 4. NUllfER1CA.L MODEL T8
Expresions similar to Eq. (433) can be derived for the contributions of all ele-
ments assodated with the node i shown in Fig. 4.3. Such expressions, when substi-
tuted into Eq. (4.5), yield the complete discretized equation for 4 associated with
node i. A general representation of this equation can be cast in the following form:
4.5.2 Discretized Liquid-Phase Momentum Equations
Except for the presence of the integrals of the p m gradient, the liquid-phase
integral momentum conservation equations are identical in form to the integral con-
ser-ation equation for +. Therefore, only the treatments of the pressure gradient
terms are discussed in this section.
The axial reduced pressure gradient term (-dp,/&) in the axial momentum equa-
tion has a c o w a t value, T, for turbulent fullydeveloped 5ows considered in this
work Thus:
The cros-sectional pressure, p', as expressed in Eq. (4.21), is assumed to vary
linearly within an element. Therefore, the gradients of p' are constant within an
element, and the corresponding volume integrations are handled as follows:
The pressure gradients in these equations are computed using Eq. (4.21).
The d i re t ized liquid-phase momentum equations are derived and assembled us-
ing element-by-element procedures akin to those used to obtain the discretized equa-
tion for 6. The resulting u', v', and w' diret ized equations for a node i can be cast
CHAPTER 4. NUMERICAL MODEL
in the following general forms
- - The t- (-%)- and (-%Im are volume-averaged Pressure gradients d t e d
with the control volume, V,.
4.5.3 Discretized Equations for Cross-Sectional Pressure and
Void Fraction
Denoting the velocity in the -flux terms by Vm, the integral mas consenation
equation, when applied to the control volume surrounding node i in Fig. 4.3, can be
written as follous: [r flPvm.Jds + / - ~ P v . h ] (4.41)
+ [similar contributions from other elements asociated with node i]
+ [boundary contributions, if applicable] = 0
Discretized Cross-Sectional p' Equation
In each element, the velocity V" can be expressed in terms of its components in
the y and z d i t i o n s , um and wm, respectively, as shown in Eq. (4.10). Interpolation
functions for um and urn have to be prescribed in order to approximate the mass flux
integrals in Eq. (4.41). First, the liquid-phase diret ized momentum equations, Eqs.
(4.39) and (4.40), are written in the following uay:
CHAPTER 4. IiUhfERICriL MODEL 80
and
@=V"- ~ p l = L. 4' e1 '
(4.44)
For the evaluation of the mass luxes on the facg a - o and o- c in element 123 (Fig.
4.4) the liquid-phase velocity components are written as:
where ir', 6' are interpolated linearly from the corresponding values a t the vertices of
the element. The centmidaldues of 8' and d"' are obtained by taking the arithmetic
mean of their d u e s a t the vertices of the triangular elements, and these centmidd
d u e s are then arnuned to prevail over the element. This interpolation for vm and
wm is adapted from the work of Prakash and Patankar [I421 and Masson e t al. [118].
It prevents the occurrence of spurious presure oscillations in the proposed CVFEM,
and ensures that the coefficients in the disaetized p' equations are positive.
In the derivation of algebraic approximations to integrals of m s fluxes in Eq.
(4.41), vm and wm are interpolated in each element by the functions given in Eq.
(4.45). The same functions are also used to approximate integrals that represent the
mass Bow rates in the momentum equations. Using these interpolation functions to
approximate the integrals in Eq. (4.41), the contributions of element 123 (Fig. 4.4),
to the liquid-phase mass consenation equation for the node 1 can be expressed as:
where v,", w,", v,", w2, up, and w," are obtained using Eq. (4.45), and (1 - a),
and (1 - a)t are obtained using the modified MAW scheme given by Eqs. (4.23) and
(4.25), respectively. Algebraic approximations to the mass flow rates in Eq. (4.16)
are obtained analogously.
CHriPTER I. NUMERlCAL MODEL 81
Using expressions similar to Eqs. (4-12) and (4.13) to evaluate the ~oss-seaional
p- gradients in Eq. (4.45). and adding similar contributions of the other ele-
ments snrmunding node i, the complete discretization equation for the ~psfsectional
premue is obtained. A compact representation of this equation for a typical node i
is the fol!owi~g.
Diswtized cr Equation
In the evaluation of the gas-phase mass fluxes on the faces a - o and o - c (Fig.
4.4). the mass-flux wloaty components are taken to be the same as the velocity
components, v,9 and e, at the element centroid. The elemental v,9 and w,9 values are
obtained by adding the -dues of relative velocity components, v, and w,, over each
element, to the corresponding liquid veloaty components a t the centroid, 4 and do,
over each element. The centroidal values 3, and wj, are obtained by averaging the
corresponding values of liquid velocity a t the vertices of the element. Using these
values to approximate the integrals in Eq. (4.41), the contributions of element 123
(Fig. 4.4) to the gas-phase integral mass consenation equation for the node 1 are
expressed as:
[ c r p ~ g . n h = a7p [-yaw: + (4.49)
Here, a, and at are obtained using the modied MAW scheme giwn by Eqs. (4.23)
and (4.25), respectively.
Addition of contributions of the other elements surrounding node i yields the
complete diret ized gas-phase continuity equation. The set of gafphase diret ized
continuity equations are used to compute the gas-phase concentration, a. A compact
CHrlPTER 4. h'UhfERlCAL MODEL 82
representation of this equation for a typical node i is the following:
<a; = z ~ - + b - d,
(4.51)
For important additional details of these equations, and the rational behind the var-
ious assodated parameters, the interested reader is referred to the works of Saabas
and Baliga 1156, 1571 and Mason and Baliga [117].
4.6 Complementary Algebraic Equations
As was mentioned in the beginning of this chapter, the governing equations for
upward, turbulent, fullydeveloped, bubbly nmphase flow in straight ducts includes
seven differential equations and nine algebraic equations (three gas-momentum equa-
tions and six equations for the Reynolds stress components). In this section, the
details of the calculations related to these algebraic equations are discused.
4.6.1 Gas-Momentum Equations
The gas-momentum equations are simplified to algebraic equations for the calcu-
lations of relative velocity values, as discussed in Chapter 3. The results of the volume
integration of the various terms in these equations are presented in this section. These
equations are solved simultaneously, as was explained in Chapter 3, to obtain the rel-
ative velocity components. The relative velocity components are calculated over each
element, and are asumed to prevail over the element.
It sbould be noted that these results are also valid for the corresponding tenns in
the liquid-phase momentum equations, but with an opposite sign.
Drag Force
1- ilifdv = -0.75 ( a ~ c ~ v v l v r l ) ~ ~ (4.52)
where the elemental value of cr is the average of the correspondiig nodal values on
an element. The CD value is calculated from the elemental Re* value.
CHAPTER 4. IW?dERiC.rlL MODEL
Lift F o e
The liquid-phase velocity terms are interpolated linearly in each element for the cal-
culation of their gradients in this expression. The other terms in this expression
are calculated in a manner similar to that used for the corresponding terms in the
drag-force expression.
Turbulent Dispersion Force
l_ M F ~ = - ( G ~ p t k ~ , v a ) ~ ~ (4.54)
ksr term is the average of the corresponding nodal \dues on an element. The a terms
are interpolated linearly in each eleaent for the calculations of their gradients in this
expression.
4.6.2 Algebraic Equations for the Reynolds Stresses
These expressions, Eq. (3.9), indude linear, quadratic, and cubic terms of symmet-
ric and asymmetric parts of the shear strain rate tensor, namely, S,, and Oij. -4s
was mentioned earlier, in this work, for the first time, these terms were included in
the proposed CVFEM. After exploring different approaches, the following procedure,
which is consistent with the proposed CVFEM for turbulent flow modeling, is used
to add the contribution of different parts of the algebraic expressions for Reynolds
stress terms to the diret ized momentum equations.
The contribution of the linear part to the momentum equations is handled through
the introduction of a turbulent eddy viscosity tmn, PI, in a similar fashion to the
diffusion term. This practice is similar to the common treatment of turbulent terms
in the standard linear two-equations k - c models. The elemental pt value is obtained
from the corresponding averaged d u e s of k and c over each element. More discussions
and details can be found in the Ph.D. thesis of Saabas[l58].
The contribution of the non-linear terms are added appropriately to the source
terms. These non-linear parts are obtained by assuming that the liquid-phase velocity
ChXPTER 4. NU.MERIC.-U MODEL &2
terms are ii terpolated linearly in each element. Then, the components of Si, d Q, are calcukted, and the non-linear parts of the algebraic expressions for the Fkyzolds
stress components are calculated over each element, and are asumed to p d over
the element. Later, the contribution of each element to the constant terns of the
disaptized momentum equations are asembled appropriately. For the standard linear
tapequation k - c models, the contributions due to the non-linear parts set to
z w .
4.6.3 Bubble-induced Kinetic Energy of Turbulence and Reynolds
Stresses
The k ~ , values are calculated over each element, and assumed to premil over that
element. The expression for ksr indudes the void fraction and the relative velocity
terms (Eq. 3.37). The relative velocities, as .was explained earlier, sere calculated over
each element and asumed to prevail over that element. The void-irction value over
each element was obtained by averaging the corresponding nodal values. Therefore,
with respect to element laoc in Fig. 4.4a:
k~rm = 0.5 ( ~ C V M f ~ ~ / ~ ) ~ ~ (4.55)
The contribution of bubble-induced Reynolds stress terms are calculated similarly.
4.7 Boundary Conditions
In this work, it is assumed that the domain boundaries remain at fixed spatial 10-
cations, and they could coincide with solid walls, or symmetry surfaces. As was
discussed earlier, since, in practice, for the turbulent flous considered here, the com-
putations with diretized equations uwe not carried out all the way to the walls,
the first grid point in the internal calculation domaio (total cross-section area mi-
nus the near wall region) is not on the u d , but it is located at an appropriate
distance from the udl, inside the domain of interest. The near-wall region is then
CHriPTER 4. NUMERICAL MODEL 85
bridged by suitable wall-functions, and the boundary conditions are prescribed at the
near-wall nodes The new procedure for appropriate implementation of the cbcsen
wall-functions in the context of the propased CVFEM, will be discused in the next
chapter.
Bridy, in a general formulation, two typg of boundary conditions are consid-
e d specified value or given gradient of 4. The following discusion pertains to the
discretized Oequation for node 1 of the element 123 shown in Fig. (4.4b). The link
between the points 1 and 2 is asumed to coinade with the boundary of the domain
of interest.
Specified Value
When the value of the dependent variable, 6, is given at the boundary node,
and denoted by q b , the discretized equation assodated with that node is written as
f0Uom:
acf = 1, ac$ = 0, bf = ~ S P (4.56)
Specified Gradient
When the gradient of the dependent variable normal to the boundary is given, say
(g)SP, the combined convection-diffusion flux of 6 normal to the boundary is given
by:
When Vn is the velocity component normal to the boundary. With reference to node
1 in Fig. 4.4b, the values of D, p, T, and (g)SP are assumed to be constant on the
link 1 - a. Thus, the contribution of the boundary link 1 - a to the consenation
equation for the control volume associated with node 1 is given by:
where
CHAPTER 4. NUM2ELlC.U MODEL 86
Convection Contribution
The convection contribution is evaluated using Sipson's rule The variations of
Vn and p over the link 1-2 are approximated with linear and piecewise prr\miling
interpolations, rrspeaively. The Qiterpolation has to be consistent with the inter-
polation scheme used in the convection terms. When the MAW scheme is used, the
convection contribution is given by
1 ppvn& = PI&+ LPfvn~) + 4 ~ ( V n m ) + ~ ( V r ) l (4.60)
This derimtion has been done for the general 6 equation. The same treatment is
also applicable to the momentum equations
For the continuity equation, only boundaries having mas flow crossing have non-
zero contributions With respect to the link 1 - a in Fig. 4.4b, this mass flow can be
expressed as:
[ p P v n ~ = A$ MV.J + ~P(v.-) + ~ ( v n a ) ] (4.61)
It should be noted that the mass flow rates across the boundary edges, such as
1 - a in Fig. 4.3, are calculated using the latest available d u e s of the nodal velocity
V , not Vm. Only the mass flow rates across control-volume faces in the interior of
the domain are calculated using Vm.
4.7.1 Special treatments
In the proposed CVFEM, special treatments are needed for $, G', d"' and d"' on
the boundaries with prescribed liquid-phase velocities, such as near-wall nodes. At
these nodes, d"' and d"' are set to zero, and, therefore,
d' = u;, G' = wiP (4.62)
CHAPTER 4. NUiZlERIC.IL MODEL
4.8 Solution Algorithm
Only the solution procedure that uas used to solve the -phase flow problems is
presented here. The solution procedure for the single-phase flow problems can be
obtained by suitable simplifications of the procedure given in this section.
The iterative sequential variable adjustment scheme proposed by Saabas and
Baliga [156, 1571 was modified here to solve the nonlinear-coupled sets of discretized
eqaations. The steps in this solution procedure are the follming.
1. Start with guessed or available veloaties, reduced pressure (axial and moss-
sectional), and void-fraction values.
2. Calculate the shear induced and bubble induced turbulent eddy viscosities, and
obtain the equivalent diffusion coetlicients.
3. Calculate the relative veloaties in the z, y, and r directions, and calculate the
various source terms in the liquid-phase diret ized momentum equations in
these three directions, and also in the k a d c equations.
4. Calculate the Reynolds stress terms and add their contributions to the source
terms of the momentum equations appropriately.
5. Calculate coefficients in the liquid-phase diret ized y-momentum equations
without including contributions of the cross-sectional pressure gradient terms.
6. Add the flux-boundary conditions through near-wall areas to the liquid-phase
y-momentum equations . This will be d i u s s e d in the next chapter.
7. Store the < and b" terms, add the contributions of cross-sectional pressure-
gradient terms, under-relax, implement specified-value boundary condition, if
required, and solve these equations.
CIiAP'TER 4. IVUMEEUC& MODEL 88
8. Use the stored values of and be tenns to calculate d and a' terms Imple-
ment special treatment of thge terms, if required.
9. Repeat steps 4-8 for the liquid-phase z-momentum equations
10. Calculate the coe5aents of the ~ ~ ~ ~ s e a i o n a l discretized prrsnue equation,
and solve these equations
11. Calculate the reduced pressure gradient in the axial direction and add i t to the
source term of the u-momentum equation
12. Calculate the coefficients of the liquid-phase axial-momentum equations (x-
direction), implement boundary conditions, under-relax, and solve these equa-
tions.
13. Calculate the d c i e n t s in the discretized k and c equations, implement bound-
ary conditions, under-relax, and solve them.
14. Calculate the coefficients in the void-fraction equations, and solve these equa-
tions.
15. Return to step 2, and repeat until convergence.
16. Calculate and solve d i t i z e d equations for other 4 variables, if required.
In this chapter, a control-volume finite element method (CVFEM) for the solution
of the mathematical models of the flowvs of interest xvas presented concisely. The modi-
fications to the formulations of the earlier CWEMs were discussed for fully-developed
flow in ducts. The implementation of non-linear k-c models in the proposed CVFEM
was explained. A new procedure for the incorporation of wall-functions in the pro-
posed CVFEM will be d i e d in detail in the next chapter.
CHAPTER 4. NUMERICAL MODEL
Figure 4.1: Example of a Straight Duct of Uniform Cross-Section, and the Global
Coordinate System.
Figure 4.2: Discretization of a Calculation Domain into Three-Node Triangular Ele-
ments and Polygonal Control Volumes.
CHAF TER 4. NUMERICAL MODEL
Figure 4.3: Typical Control Volumes Surrounding (a) an Internal Node; and (b)a
Boundary Node.
Figure 4.4: Typical Triangular Element and .4ssociated Nomenclature: (a) Local
(y - z ) Coordinate System; (b) an Interior Element: (c) and a Boundary Element.
Chapter 5
Fully-Developed Turbulent Flows
in Ducts: Implementation of
Wall-Functions in CVFEMs
As was d i earlier in this thesis, in computer simulations of turbulent flows
in ducts significant reductions in computer CPU times can be achieved by using
CVFEMs (or any other numerical method) to solve the governing differential equa-
tions from the central region to suitably located near-udl nodes, and wall-functions
to bridge the region from the near-udl nodes to the u-all. In this chapter, a neu-
procedure is proposed for the implementation of the appropriate u-all-functions in
CVFEMs. The wall-functions used in this work for modeling the near-uall regions in
turbulent single-phase and dilute, bubbly, tua-phase flows were presented in Chapter
3.
An example of wall-functions for single-phase turbulent flows is the so-called log-
arithmic law of the wall, originally proposed for boundary-layer flow over a flat plate
[162], a relativel, simple geometry and flow. Examples of the application of the loga-
rithmic law of the u d l to more complicated flows and other geometries are also in the
literature [121, 1931. One of the first implementations of the logarithmic law of the
u-all and related wall-functions in CVFESls was done by Saabas [158], but, as men-
CHAPTER 5. Jinplementation of Wall-Functions i n CVFEMs 92
tioned earlier, his procedure does not ensure the consmation of mass in the near-aall
regions. The new implementation procedure presented in this chapter overcomes this
difficulty.
The discussions in the next section are devoted to the implementation of aall-
functions for single-phase, turbulent, fullydeveloped flom in straight ducts. Fol-
lowing that, the implementation of wall-functions for dilute, bubbly, fullydeveloped,
tv.-c-phase flows is disc& concisely, with emphasis on aspects that are different
from, or in addition to, that involved in the modeling of single-phase flous.
5.1 Near-Wall Treatment for Single-Phase Fluid
Flows
5.1.1 Discretization of Near-Wall Regions
The discretization of the near-wall regions in complex geometries has to be done
with care for consistent implementation of the wall-functions for single-phase turbu-
lent fluid flows. Here, attention is limited to the fullydeveloped region of flous in
straight ducts of uniform, but not necessarily circular, cross section. It is also assumed
that there is no reversal of the resultant (or total) wall-tangential component (v,) of
the velocity vector a t near-wall nodes, though the wall-tangential component of the
velocity vector in the mss-section of interest could, and generally would, experience
reversal.
In this work, the CVFEhsl discretization of the governing daffmtial equations
start from suitably located near-wall nodes, not a t the walls, and proceeds into the
interior region of the dud. The usual practice in Finite Volume Methods (FVMs),
for fluid flow problems, to start the FVM diietizations of the governing differential
equations of the velocity components parallel to the \wall (in the computational space),
from the near-wall node, and for the other velocity components and pressure from
the wall, was also examined in this work. This alternative approach require consider-
ation of two different discretization domains in CVFEhls, which makes this approach
CH.4PTER 5. Implementation of IVd-Functions in CVFEMS 93
quite complicated, particularly in complex geometries, where it become prohibitively
0bm:ring.
The wall-normal distances (6) of the near-wall nodes are chosen (or adjusted) so
that their non-dimensional d u e s (6+ = p6v;lp) lie in the range 30 5 6+ 5 120.
Wall functions, based on the logarithmic law of the wall and related expressions ob-
tained by invoking assumptions of equilibrium turbulence, and appropriate numerical
integrations, are used to bridge the region from the near-wall nodes to the wall.
Starting the CVFEM d i i t i za t ion of the governing differential equations from
the near-wall nodes require specification of boundary conditions for the velocity com-
ponents and pressure a t these nodes. These boundary conditions should be appro-
priately specified, in conjunction with the wall-functions that are used to bridge the
near-wall region. In this work, a new procedure is developed to discretize the near-wall
region, which enables proper treatment of this region in the proposed CVFEM.
In the proposed procedure for discretization of the near-wall regions, and in the
asociated procedure for the implementation of the uall-functions, a distinction is
made between smooth portions of the udl and corners. With reference to the notation
introduced in Figs. 4.1 and 4.2, suppose the duct wall in the aossection of interest
is described by the function y = /(t). Then, if at a point along the wall, y and
d y l k are both continuous, wall is considered to be smooth If, on the other hand, y
is continuous, but dy/d= is discontinuous, then the point of interest lies on a corner.
Furthermore, if the dnrlation domain is bounded by ualls and symmetry l ies, then
the point of intersection of a wall and a symmetry line is also considered as a corner.
Discretization of Near-Smooth-Wall Regions
The discretization of the near-wall region adjacent to a smooth wall is illustrated
in Fig. 5.1. Consider a typical near-wall node adjacent to a smooth wall, say i =
3, and its two immediate near-wall neighbor-nodes, 2 and 5, as illustrated in Fig.
5.1. The discretization of the region in the interior of the duct, up to the near-
wall nodes, is done using triangular elements, as mas d i d in Chapter 4. With
reference to Fig. 5.1, the region in between the near-uall nodes, 2, 3, and 5, and
C m 5. Implementation of Wall-hctions in CITE.& 94
the smooth wall is discretized as follows: (i) from the near-wall nodes (nod= 2, 3,
and 5, for example) normals are dropped to the walls, and the intersection points are
marked (points 1, 4, and 6 in Fig. 5.1); (i) the smooth wall is approximated by a
piecewise linear curve, if necesary, joining the aforementioned points of intersection
(1 - 4 - 6 in Fig. 5.1); and ( i ) normals are also dropped from the midpoint of
the links between near-ud nodes (points a and e in Fig. 5.1) and their intersection
points with the piecewiilinear approximation of the wall are noted (points f and g
ia Fig. 5.1). This procedure creates quadrilateral elements adjacent to the piecewise
linear approximation of the wall (for example, quadrilateral elements 1234 and 3456
in Fig. 5.1), and huff polygonal control volumes surrounding the near-wall nodes (for
exa.mp!e, the shaded region iabcde assodated with the near-dl node, i = 3, in Fig.
5.1).
Discretization of Near-Corner Regions
The discretization procedure for the near-uall region adjacent to a re-entmnt
comer (a comer that protrudes tou-ards the interior, or towards the central region of
the duct) is illustrated in Fig. 5.2. F i t , the near-ud node is located along a line
that bisects the angle, Om, of the comer, and the distance from the tip of the comer
to the near-wall node along thii b i i t o r line (4 - i) is assumed to be the wall-normal
distance. The near-wall node (i) is located (adjusted) along the b i i t o r line in a
manner that ensures 30 5 6+ 5 120. The other aspects of the discretization of the
near-wall region adjacent to a re-entrant comer closely parallel that for a smooth
wall, described in the previous subsection.
The discretization procedure for an outward comer that protrudes towards the
exterior, or away from the central region, of the duct is illustrated in Fig. 5.3, and
that for a comer created by the intersection of a symmetry surface and a smooth wall
is shown in Fig. 5.4. The steps involved in these discretizations are relatively easy to
deduce by examining these figures, because they are akin to those d i e d earlier
in thii section, so they will not be elaborated further.
CHAPTER 5. Implementation of Wall-Functions in CVFEhG 95
5.1.2 Specialization of WaU Functions
The wall functions used in the modeling of turbulent single-phase fluid flows were
introduced and discussed in Chapter 3. H a , a specialized form of these functions is
provided that facilitates the presentation of the proposed procedure for incorporating
them into CVFEMs.
The logarithmic law of the wall can be expressed as 11521:
where y, is the mall-normal coordinate, ut is the resultant (or total) component of the
velocity vector that is tangent to the wall, K is the Von Karman constant, and B is
a constant: Here, following Launder and Spaldiig [IOO], the values of K = 0.435 and
B = 5.2 are used for smooth walls. The friction velocity (v:) and the dimensionless
wall normal coordinate (y:) are given by:
,; =nu:
P (5.3)
where r, is the magnitude of the wall shear stress. Again, it is desirable to locate
near-wall nodes so that their dimensionless wall-normal distance,
Pm 6+ = -, P
(5.4)
is in the range 30 < 6+ 5 120.
In the deriwtion of the boundary conditions for the d i ie t ized momentum bal-
ances on the half polygonal control volumes associated with the near-wall nodes (for
example, the n e a r - d l node i in Fig. 5.1), i t is necessary to obtain the wall shear
stress. TW, using the previously calculated, or guessed, value of the corresponding
tangential velocity component, s. Here, the value of r, is found using the following
procedure. First, consider:
CH..IpTER 5. Implementation of Wall-Functions in CVFEhis 96
where 6 is the d - n o r m a l distance for the near-wall node, and & is a m o l d s
number based on vt and 6.
If 6+ _< 11, tben v: = 6+ (Laundes and Spalding IlOO]), and
Thus, Grst, using the previously calculated, or guessed, d u e of vt, &(= 9) is
obtained. If Red 5 121, then T, is obtained by ~0hking.
On the other hand, if R e > 121, the logarithmic law of the wall is used:
This equation is solved by trial and enur, to obtain 6+, and, then, the d u e of r, is
obtained (Eqs. 5.2 and 5.4).
The d u e s of k and c at the near-wall nodes are obtained using the arguments and
equations introduced earlier in Chapter 3. Thus, at the near-wall nodes, following
the calculation of rW, V; is obtained, and:
and
5.1.3 Implementation Procedure
The procedure for the implementation of the logarithmic law of the wall and
related wall functions, as formulated in section 5.1.2, is presented in this section
for fully-developed, single-phase, turbulent flow in straight ducts of complex cross-
section.
CHAPTER 5. ImpIementation of IWl-Functions in CVFEMs 97
Near-Wall Nodes Adjacent to Smooth Walls
Consider a typical near-wall node adjacent to a smooth wall (Fig. 5.5) and its
half control-volume (iabcde). The associated near-uall elements are 1234 and 3456.
The velocity vector ,V, a t node i has three orthogonal components u, the axial
velocity component, and v and w, the \docity components in the y and : direction%
respectively. This velocity vector can also be expressed i3 terms of two orthogonal
components, one parallel or tangential to the wall, vt, and the other normal to the
wall, v,, as shoun in Fig. 5.5.
It should be mentioned that the half-control-volume face eia has txo parts, half-
face ei in element 1234 and half-face ia in element 3456, and each half-face could have
a different angle of indination to the horizontal or :-direction, 81 and 82, as shown in
Fig. 5.1. Therefore, there k a problem of what is the appropriate angle between the
horizontal and the wall for the half control volume iabcde. In this work, this angle is
taken to be 0, the angle between the line i4 (the wall normal line from the near-wall
node, i, to the smooth uall) and the vertical (please see Fig. 5.1). It should be noted
here that as the grid is refined, the angles 01.82 and 0, all approach the same value.
With respect to the notation in Fig. 5.6, the area of the near-wall face, eia, of
half control volume iabcde, is denoted by A,,, = (6sl + 6.~~) x 1; the duct length is
assumed to be unity. Once the stresses on the face eia are multiplied by this area, the
resulting force can be divided into components in each Cartesian coordinate direction,
F,, F,. F,. This force can also be divided into components parallel and normal to the
uall, F, and F,, as shown in Fig. 5.6.
Once the coefficients of diret ized equations are calculated for the axial or z-
direction velocity component, u, with the most recent available values of v and w ,
the implementation procedure for the wall functions proceeds as follows for the near-
wall node i , adjacent to a smooth wall. The following procedure pertains to a wall
with an inclination angle 0 5 4S0, which implies that the inclination angle of the
velocity component in the z direction, w, to the smooth wall is !gs than the corre-
sponding angle between the other cross-section velocity component, v , and the uall.
CHAPTER 5. Implementation of Wall-Functions in CVFE,% 98
If 0 2 45'. then, this condition is reversed, and the terms involving the w and the
z direction in the procedure should be appropriately replaced by the corresponding
terms with v and the y direction: These modifications are briefly noted a t steps where
these replacements are necessary. The implementation procedure is described by the
following sequence of steps:
1. Calculate the veloaty component parallel to the wall in the wsr-secrionaI
plane, vb, using the v and w velocity components, and the angle 0 (see Fig.
5.5):
vb = d + u s i d (5.12)
2. Calculate the overall (axial + cross-sectional tangential) doci ty component
parallel to the wall, vt:
3. Using Eq. 5.5, calculate the Reynolds number, Red, based on wall distance
(6 = i - 4 in Fig. 5.1) and the calculated vt d u e .
4. Calculate the 6+ d u e from the Red value, by using Eq. 5.6 if Red 5 121, or
iteratively using Eq. 5.9 if R e d > 121.
5. Calcu!ate the friction velocity value, v;, using Eq. 5.4.
6. Calculate the wall shear stress, r,, using Eq. 5.2.
7. Calculate the turbulent kinetic energy, k, and its dissipation rate , c, at the
near-wall node, i, using Eqs. 5.10 and 5.11, respectively.
8. Multiply the % d l shear stress (r,) by the corresponding half control volume
face area, An,, to obtain the magnitude of the shear force parallel to the ndl,
Ft .
CHAPTER 5. Implementation of Wal-Functions in CVFEWs 99
9. Divide Ft into two components, one in the avial direction, F,, and the other in
the crass-section plane, F,, (Fig. 5.6). This is done by projecting the Ft vector
in these two-directions:
F, = VF: (5.14) vt
10. F, is then appropriately incorporated into the d i i t i z e d equation for the a d
velocity, u, associated with the near-wall node, i, and the corresponding half
control volume, iabcde. Once the other discretized axial veloaty equations are
properly completed, this set of discretized equations ( nominally linearized and
decoupled, as described in the sequential solution procedure given in section
4.8) are solved, and an updated value of u is obtained a t node i.
11. Obtain the wall shear force component in the y direction, Fp, from the d i i
cretized v-veloaty equation associated with node i and the corresponding half
control volume, iabcde (note that the value of vi is obtained from continuity
considerations, as will be described later in this section).
In this equation, the b" term in the value obtained by the assembly of only the
element contributionsin the interior domain (total duct cross-section minus the
near-wall regions).
If 0 2 45", follotving s. similar way, the corresponding shear force in the r
direction is obtained as follows:
CHAPTER 5. Implementation of IWJ-Functions in CVFEhk 100
12. Use the F, and F, values to calculate the wall shear-force in the z-direction,
F,, for the half control volume iabcde:
F= = ,- F" F,tanB (5.18)
If 0 2 45 ", the corresponding Fu is obtained as follows
13. Incorporate the F, value into the d i e t i z e d :-direction momentum equation
associated with the near-wall node i. Appropriately complete all other d i
cretized :dimtion momentum equations, an2 solve this set of equations (see
procedure described in section 4.8) to obtain an updated value of w a t node i.
If 0 2 45 ", an updated u value at node i is obtained in a similar routine.
14. Calculate the magnitude of the mass flow rate through the sides of near-wall
elements, 1 - 2, 3 - 4 and 5 - 6, by full integration of the logarithmic law of the
wall along these l i es . These lines are normal to the wall. Note that the flow
is assumed to be fully-developed, and the axial length of the duct is considered
to be unity.
FLOW = 11 [60.5 * K + 6+ + ln(E6+) - llln(99) - (6+ - l l ) ] (5.20)
if 6+ > 11, and
FLOW = 0.5/.16+'
15. Divide this mass flow rate, FLOW, into two components, one in the axial di-
rection, FLOW,, and the other in the cross-sectional plane, FLOW,. This is
done by projecting the FLOW vector in these two-directions.
ChQP'I'ER 5. Implementation of Wall-Functions in CVFE1Cf.s
FLOW. = %FLOW ut
16. The F W W , values are used to calculate the axial mass flow rates in the near-
wall areas, and then to appropriately adjust the axial velocities, so as to aa tch
with the desired Reynolds number based on the cnsssection average axial ve-
locity and the hydraulic diameter.
17. Calculate the mass flow rate which enters through the links 2 - 3 and 3 - 5 into the computational domain. This is done by using mass conservation
requirements in the near-wall elements. Since fullydeveloped flow is considered
here, the axial mass flow rates into and out of the near-mall elements are equal.
There is, also, no flow through the near-wall element sides on the wall, because
only impermeable duct walls are considered in this work. Hence,
(FL0WLK)z-3 = (FLOWs)l-z - (FLOW.)34 (5.24)
(FLOWLK)3-s = (FLOW,)3-4 - (FLOW.),, (5.25)
18. Calculate the mass flow rate entering into each half control volume by linear
interpolation of the mass flow rate through links 2 - 3 and 3 - 5.
19. Add the mass flow rates into the half control volumes, FLOWCK, to the con-
stant terms of diicretized pressure equations associated with the corresponding
near-wall nodes, to complete these equations. Following the procedure described
in section 4.8, solve these equations, and obtain an updated value of pressure
a t the near-wall node. i.
CHAPTER 5. Implementation of W d - h c t i o n s in CVFEMs 102
20. Assuming that the nonnd velocity to the wall, (L& at node i pmails over the
half control volume face, eia, its value is obtained from this equation:
21. The y-direction velocity component, v, is calculated by assumkg that its value
a t i is prevails over the half control volume face, ein.
This is the value of u-velocity component a t near-wall node, i. Its value is
iteratively updated using Eq. 5.28 until overall convergence is achieved. If
6 2 45 ", the w velocity component is calculated from a similar equation which
relates v, w, and v, \dues.
Near-Wall Node Adjacent t o Re-entrant C o m e r
Consider a node i adjacent to a re-entrant comer and the near-uall elements
associated with this node, 1234 and 3456, as shown in Fig 5.2. The angle between
two adjacent sides on these two elements (sides 1 - 4 and 4 - 6 ) a denoted as 26 (Fig.
5.2). For the smooth dls , this angle is very close to 180' (n radians), and as the
grid is refined, it become exactly 180"; but for a reentrant corner, this angle is not
180' and should be properly accounted for in the implementation procedure.
The wall-functions implementation procedure for a re-entrant comer follows closely
the corresponding procedure for a smooth wall, except for the modifications which
should be made to consider the angle 26 in the procedure. The velocity and force
components are defined analogously to those which were used in corresponding terms
for a smooth wall (Figs. 5.5,5.6). The tangential direction is defined to be normal to
the lime i-4 in Fig. 5.2: The Line i -4 is the bisector of the re-entrant comer, and the
normal distance from the comer to the near-comer node i is 6. The implementation
procedure proceeds in the following sequence of steps:
C K m R 5. Implementation of IVd-Functions in C V F E E ~ 103
1. Repeat steps 1-7 in the implementation procedure for smooth walls
2. Multiply the d l shear stress (r,) by the corrsponding half-control-volume
face area [A,,- = (bsl + 6 4 x 11, Fig. 5.2, to obtain the magnitude of the
shear force parallel to the wall, Ft.
3. Repeat steps 9-19 in the implementation procedure for smooth walls
4. .Assuming that and (v&, the velocity components normal to the faces
f - i and i - a of half control volume, along f - g and a - h lines, respectively,
p r e d over these faces, FLOWW, can be expresed as:
(vn)1 and (us)= are related to the v and w as follows (see Fig. 5.2)
5. The ydirection velocity component, v, is calculated by assuming that its value
a t i prevails over the half control volume face, f ia.
This is the d u e of v-velocity component a t near-wall node, i. Its value is
iteratively updated using Eq. 5.32, until overall convergence is achieved.
Near-Wall Node Adjacent t o Outward Corner
Consider a node i adjacent to an outward comer and the near-wall element asso-
ciated with this node, 1234, as shown in Fig 5.3. The normal d i c e s of node i to
the two adjacent walls, lines 2 - 3 and 3 - 4, are made equal in this work. The other
two sides of the element 1234, 1 - 2 and 1 - 4 are assumed to be impermeable, so
there is no mass flow rate through them. 4h, because of the flow symmetry across
the diagonal 1-3, there is no mass flow rate across this line. These flow conditions in
element 1234, Fig 5.3, imply that the net m flow rate through sides 2 - 3 and 3 - 4
should also be zero. To fulfill this condition, in this work, it is assumed that at node
i, the secondary vdoaty components, v, and wi, are both zero. This assumption is
acceptable for the flows of interest in this work: For low Reynolds number flows in
ducts, the secondary velocities are quite weak, and in the vicinity of the corner, their
values are negligible; and for high Reynolds number flows in ducts the near-dl nor-
mal distances (2 - 3 and 2 -4 ) are so small that the secondary flow rates across them
are insigni6cant. Presuming vi and w, are zero at node i, the following simplifications
can be adopted in the implementation procedure presented earlier for near-wall nodes
adjacent to smooth walls.
The implementation procedure of wall-functions for an outward corner, therefore, is
much simpler than the corresponding procedure for a smooth wall which was presented
earlier. It is as follows:
1. Calculate the Reynolds number, Rea, based on the ud-normal d i c e (6 =
2 - 3 or 4 - 3 in Fig. 5.3) and vt = u, using Eq. 5.5.
2. Repeat steps 3 to 7 in the implementation procedure for smooth walls.
3. Multiply the wall shear stress (r,) by the corresponding half-control-volume
face area, A,, = (3 - d + 3 - a ) x 1 , to obtain the shear force parallel to the
ud1, Fz = Ft.
4. Repeat steps 10, 14, and 16 in the implementation procedure for smooth walls,
retaining FLOW, = FLOW.
5. Repeat steps 17 - 19 in the implementation procedure for smooth walls, with
the understanding that (FLOLV,)2-3 and (FLOW,)3-4 are zero.
6. Set ui = w, = 0
CHAPTER 5. ImpIementation of Wd-Functions in CVFEhIs 105
Near-Wall Node Adjacent to a Corner Between a Wall and a Symmetry
Plane
Consider a node i adjacent to a corner between a wall and a symmetry line,
for example, the node i shown in Fig 5.4. The half control volume iW touches the
symmetry surface. The veloaty component normal to the symmetry surface is always
zero. Using this information, the wall functions presented earlier in this chapter, a d
suitable adaption of the implementation procedure given earlier in this section, the
velocity components, ui, v,, and w, at node i are computed.
Concluding Remarks
During the course of developing these procedures, several alternative approaches were
also investigated. The proposed procedures and the alternatives were tested for flow
in ducts with different crosssections to examine each step, the order of steps, and the
overall capabilities of the procedures. The proposed procedures were the best ones
in all these tests. The final results of t3ese test problems will be presented in the
next chapter. The details of the discarded alternative procedures are not important
enough to be covered in the thesis, however, there are some remarks which are worth
noting. These remarks are noted in the following sub section.
5.1.4 Comments on an Alternative Approach
In the proposed procedure, with reference to Fig. 5.1, the axial velocity component
u, and w (if 0 5 45 ') or v (if 0 2 45 ") are obtained by specif>+ng the wall shear-stress
conditions on the appropriate faces of the near-udl half control volumes, and solving
the corresponding d i t i z e d equations, as outlined in the procedure presented in the
previous subsection.
An alternative approach was tried in which the contributions from internal el-
ements to the coefficients in the diret ized equation for velocity components were
first calculated, and, then, using the most recent values of the velocity components,
the values of all shear-stresses at half-control-volume faces (eia) were obtained us-
CHAPTER 5. Implementation of Wd-hc t ions in CVFEMs 106
ing equations similar to Eq. 5.16. Then, the shear-stres parallel to the wall, T,,
was calculated at each near-wall node from the shear-stress components in the three
Cartesian directions Using this r, value, the friction velocity was calculated, Eq. 5.2,
and the value of the near-wall velocity parallel to the wall, ut, was obtained from Eq.
5.1. The values of near-wall velocity components, 4, u, and wi were then updated,
using appropriate combinations of vt and v,.
This alternative procedure, however, did not work well, since the assodated iter-
ative solution sequence was highly unstable. In the beginning of the overall iterative
solution procedure, the guessed values for the dependent variables are far from the
h a l converged values, and with the aforementioned alternative procedure, the iter-
ative solution sequence is unable to adjust the dependent variables correctly. The
alternative procedw also has some adverse effects on the disaetized pressure equa-
tion: With this procedure, in the discretized pressure equation, the connmtion of
near-wall nodes to the rest of domain is through pre-specified values of ir, 6, du and
dw, and the last two are both zero; this weakens the connection between pressure
values at internal nodes, and those at the near-wall nodes, and makes the iterative
solution procedure even more unstable.
In the proposed approach, only one of the velocity components is fixed cin the
near-wall nodes. Thus, the corresponding connections in the discretized pressure
equations are stronger.
5.2 Near-Wall Treatment for Dilute, Bubbly , Two-
Phase Flows
In this section, the implementation of the near-mall treatment for dilute, bubbly,
two-phase flows is dixussed. The discretization of the near-wall region for the flows
considered in this section is similar to the corresponding discretization of the near-
wall region for single-phase flows discussed in the last section, for smooth walls and
comes, and, therefore, its description is not repeated here.
CHAPTER 5. Implementation of \W-Functions in CVFEMs 107
The implementation procedure of the logarithmic law of the wall for dilute, bubbly,
-phase flows in CVFEMs closely follows the corresponding procedure for single-
phase flows discused in the previous section. As was mentioned in Chapter 3, the
turbulence in the gas-phase is assumed to be negligible for the -phase flows con-
sidered in this work, and it has been established that the bubbles slip along the duct
ualls. Furthermore, since the bubbles are of the finite size in the vicinity of the wall,
and they occupy most of the whole near-wall region [115, 1071, it is assumed that
the void fraction is uniform over this region and is equal to its value, a,,, a t the
near-wall node. The boundary condition for the void-fraction dXerential equation
are specified as d i in Chapter 3, and the near -ad values of void-fraction are
updated in each iteration. Thes presence of the second phase, houwer, implits that
the whole near-wall region is not occupied by the liquid phase (in a temporal averag-
ing approach). Thus, proposed procedure for implementation of wall functions in the
CVFEM for single-phase flows must be modiied, in order to account for the presence
of the bubbles.
In particular, the following additional aspects should be appropriately considered
in the implementation procedure of wall-functions in near-wall regions for dilute,
bubbly, two-phase flows:
r The mass consenation of the gas phase in the near-wall area.
0 The presence of the liquid phase in (1 - %,)V of the near-wall region.
The implementation procedure for the logarithmic law of the wall and the associ-
ated uall-functions for dilute, bubbly, tuvo-phase flows proceeds as follo\vs:
1. Repeat steps 1 and 2 in the implementation procedure for single-phase floas
over smooth ualls for both the liquid-phase and the gas-phase. The gas-phase
velocity components are obtained by the addition of the corresponding liquid-
phase and relative velocity components, as discussed in Chapter 4.
2. For the liquid phase, repeat steps 3-10 in the implementation procedure for
single-phase flous over smooth walls.
CHriPTER 5. Implementation of Wall-Functions in CVFEAL 108
3. Repeat step 11 in the implementation procedure for single-phase flows over
smooth walls, but with the following modified form of the Eq. 5.16:
4. Again for the liquid phase, repeat step 14 in the implementation procedure for
singlephase flows over smooth nalls. The FLOW values are obtained from the
following equations:
FLOW1 = 5 [60.5 * rr + 6+ + ln(E6f) - llln(99) - (6+ - l l ) ] (1 - an,)
(5.35)
if 6+ > 11, and
The corresponding value for the gas-phase is obtained from this equation:
FLOW, = a,,(ct),6 (5.37)
5. For both liquid- and gas-phases, repeat steps 15-18 in the implementation p m
cedure for singlephase flows over smooth walls, to obtain (FLOWCV,), and
(FLOWCV,),.
6. For the liquid- phase, repeat step 19 in the implementation procedure for single
phase flows over smooth walls for the liquid-phase. &, add the corresponding
(FLOWCV,), values to the constant terms of diret ized void-fraction equa-
tions associated with the corresponding near-wall nodes, to complete these equa-
tions. Following the procedure described in section 4.8, solve these equations,
and obtain an updated d u e of void-fraction at the near-\.all node, i.
CHAPTER 5. Implementation of Wd-Functions in CVFELk 109
7. For the liquid-phase, repeat steps 20-21 in the implementation procedure for
single-phase flows over smooth a&.
The implementation procedure of the logarithmic law of the wall and the asso-
ciated wall-functions for dilute, bubbly -phase flows a t corners is done similarly,
by using the corresponding procedures for single-phase flows, but with modifications
akin to those d i d in this subsection for the corresponding flows over smooth
walls.
In this chapter, new procedures for the implementation of the standard logarithmic
law of the wall and the associated a d 1 functions for fullydeveloped, turbulent, single-
phase flows, and a logarithmic law of the wall and the associate d l functions for
fullydeveloped, turbulent, dilute, bubbly, two-phase flows in the proposed CVFEMs
were presented. A new domain discretization for near-wall regions was proposed, and
this facilitated the appropriate implementation of the aforementioned wall-function
in the CVFEMs.
The results produced by following the proposed implementation procedures for
the near-wall regions, in conjunction with the mathematical and numerical models
presented in Chapters 3, and 4, for fullydeveloped, turbulent, single-phase flows
and fullydeveloped, turbulent, dilute, bubbly, *+phase flows in straight ducts of
arbitrary cross-sections are presented in Chapters 6 and 7, respectively.
CHAPTER 5. Implementation of IW-Functions in CVFE3fi 110
Figure 5.1: Typical Elements and Control Volumes .4ssociated With a Near-\Val1
Node Adjacent to a Smooth Wall.
CHrlPTER 5. Implementation of Wall-Functions in CVFEMS
Figure 5.2: Typical Elements and Control Volumes Associated With a Near-Wall
Node Adjacent to a Re-entrant Comer.
7
Figure 5.3: Typical Elements and Control Volumes Associated With a Near-Wall
Node Adjacent to an Outward Comer.
CHrlPTER 5. ImpIementation of Wall-Functions in CVFEMs
I
Figure 5.4: Typical Elements and Control Volumes .Wciated With a Near-Wall
Node Adjacent to a Comer Between a Wall and a Symmetry L i e .
Figure 5.5: Velocity Components in the Cartesian Coordinate Directions and in the
Tangential and Normal Directions to a Smooth Wall.
CHAPTER 5. ImpIementation of Wall-Functions in CVFEMs
a
Figure 5.6: Wall Shear Force Components in the Cartesian Coordinate Directions
and in the Tangential and Normal Directions to a Smooth Wall.
Chapter 6
Applications to Single-Phase
Fully-Developed Turbulent Flows
In this chapter, the results obtained by employing the proposed CVFEM along
with the new procedure for the implementation of wall-functions are presented for
fullydeveloped, single-phase, turbulent flow in straight ducts of four different cross-
sections. The duct crosssections are:
1. square (full domain and a quarter domain);
2. tilted square (full domain);
3. circular sector; and
4. triangle (full domain).
The ducts mentioned above, collectively, require the use of all of the various features
of the CVFEM (single-phase part), discussed in Chapter 4, including the various
procedures for including the boundaries of the domain (Chapter 5). The results
obtained are compared with available experimental data and numerical results.
All tests were performed for fullydeveloped, turbulent flows of an incompress-
ible, constant-property, Newtonian fluid, in a two-dimensional Cartesian coordinate
system. The governing equations and turbulent models (Chapter 3). along with the
CHAPTER 6. Single-Phase Rrlly-Developed Turbulent Flows 115
CVFEM and the new wall boundary treatment procedure (Chapters 4 and 5), were in-
corporated in computer programs which were completely developed in this work All
programs are written in the standard F O m - 7 7 language, and were executed on
a single-processor (R-8000) of a SiconGraphics Power Challenge computer. Before
use in this work, the computer programs were thoroughly tested, using test problems
and procedures similar to those used by Mason [I191 and Saabas [158].
Appropriate convergence criteria were dwised to ensure consistency and accuaq
of the results: Details given later in this chapter. The iterative solution was stopped,
when these criteria were satisfied. For each problem, grid-independent results were es
timated by using an extended Richardson's extrapolation procedure, details of which
are available in the works of Sebben [166] and Sebben and Baliga [165].
The problem statement and numerical details particular to each problem, are
explained separately, for each test-case, in the subsequent sections.
6.1 Fully-Developed, Turbulent Flow in a Straight
Duct of Square Cross-Section
Fullydeveloped, turbulent flow in a straight duct of square cros-section has been
investigated by many researchers, with the specific objective of understanding the
physics of secondary flows in non-circular ducts. Ewmples include the works of
Launder and Ying [loll, Demuren and Rodi [35], and Myong and Kobayashi [125].
It has also been used as a test problem to evaluate different proposals for turbulence
modeling.
The coordinate system and pertinent variables for this problem are shown in Fig.
6.1. Since only fullydeveloped flows are considered here, there is no change of the
veloaty components, turbulence variables, and the pressure gradient in the axial
direction, and the flow can be solved in a two-dimensional planar cross-section, as
shown in Fig. 6.2. Furthermore, symmetry surfaces in the geometry and the flow
allow solution of the problem in a quarter or one-eight of the duct cross-section,
CHAPTER 6. Single-Phase My-Developed lbrbdent Flows 116
rather than the full crass-section. In this work, the quarter duct crcsssection was
used in the computations This decreasg the number of grid nodes by a factor of
four, which substantially reduces the memory and CPU time requirements to obtain
a converged solution. The boundary conditions indude two symmetry lines and two
solid walls for a quarter of a square duct. Computations were also done in a full
square aos-section, in order to establish that the expected symmetries were indeed
produced by the proposed numerical solution procedure: The symmetry of the d t s
is a good check-point for the d ida t ion of the proposed numerical procedure and the
computer program.
The computations were done with the following turbulence models: the standard
linear k - c model of Launder and Spaldiig [loo]; the Cubic Eddy Viscosity Model
(CEVM) of Craft et al. [30]; the Rubiitein and Barton non-linear (quadratic) k - c
model (denoted here as RUB) [155]; and the Explicit Algebraic Stress Model (EASM)
of Gatski and Speziale [49]. The results are checked against the experimental data
obtained by Gessner et al. [51] and Gessner and Emery [50]. These experimental
data were adopted for testing turbulent flow predictions a t the 1980-81 Stanford
Conference [n, 1251. Most of the results are presented here for flows with Reynolds
number of 250000, based on the bulk average velocity and hydraulic diameter. This
is one of the lagest values of the Reynolds number for which reliable experimental
results are available. In addition, the predicted d u e s of the friction coefficient, f ,
are shown for different Reynolds numbers, and are compared with experimental data
from the aforementioned and other researchers.
Numerical predictions for these flows, recently obtained by Myong and Kobayashi
[125], using an anisotropic low-Reynolds number turbulence model are also shown
here. This low-Reynolds number model, with its specially tuned constants, is expected
to perform very well for this particular problem. Thus, the results in Ref. 11251
indicate the level of accuracy that may be expected from numerical simulations of
this problem, with the aforementioned high-Reynolds number turbulence models in
conjunction with wall functions.
CEt.IPTER 6. Single-Phase Fullullv-Developed Zhbulent Floas 117
6.1.1 Domain Discretization
Ln this work, the domain discretization starts with the establishment of the physi-
cal nonnal distance of the near-wall nodes from the wall, using the following approach.
Fi, a guessed axial gradient of the reduced pressure is obtained from Jones's corre-
lation [I621 for friction kctor. Then, the average wall shear stres value is calculated
as:
from which a guessed average friction veloaty, u', is obtained. Based on a desired
6+ value, the average nonnal distance between the walls and the near-wall nodes is
obtained: v6+ 6=- u' (6.2)
The near-mall nodes are positioned with a normal distance of 6 to the wall. For
near-mall nodes next to outward corners (Fig. 5.3), the normal distance to the two
adjacent walls were made equal. The converged results were checked to see if the h a l
6+ values were in the desired range of 30-120, and if not, the procedure was repeated
with an appropriate adjc tted guessed initial 6+ value.
Once the locations of the near-wall nodes were fixed, the internal domain (entire
domain minus the near-wall regions) was divided into elements with a power-law
expansion in the spacing of the nodes away from the uall. This allows, if required,
to have denser grid in ryions next to the near-wall nodes. If power in this grid-
line spacing procedure is set equal to one, it produces a uniform grid in the internal
domain. After the construction of this finite-element mesh, control-volumes were
constructed about each node, as explained in Chapter 4. A typical uniform finite-
element mesh is shown in Fig. 6.2.
CH.WER 6. Single-Phase My-Developed Turbulent Flora
6.1.2 Numerical Details
Boundary Conditions
Boundary conditions were prescribed along symmetry planes and at near-wall
nodes. At symmetry planes, the veloaty component normal to the symmetry plane
is set equal to zero, while for all other quantities, the gradients normal to this plane
are taken to be zero. The logarithmic law of the wall and associated wall-functions
were implemented to bridge the region in between the wall and the near wall nodes,
as was explained in Chapter 5.
Solution a n d Convergence Cri ter ia
The governing equations are highly nonlinear and coupled, and, therefore, it is not
easy to obtain a converged solution. To decrease the CPU time required for solving
these problems, a continuation procedure was used in this work: The results obtained
from one test m e were used as guessed values for the next test case, which involved
the same general physical problem, but with a different mesh size, Reynolds number,
or turbulence model. Nevertheless, results in each iteration were under-relaxed in
order to ensure convergence of the overall iterative solution procedure. The under-
relaxation parameters were in the range of 0.1-0.5 for the various dependent variables
and for the different test cases.
The iterative solution procedure was stopped when the sums of the absolute values
of the normalized residues in the w and p sets of diicretized equations, as well as the
relative change in the value of the duct-center axial velocity, were all individually less
than
Grid-Independent Results
To estimate the error in the numerical results, grid independent results were ob-
tained for each test case using an extended Richardson's extrapolation procedure,
proposed by Sebben and Baliga [165]. The results obtained for this problem using
the cubic eddy viscosity turbulence model, are shown in Table 6.1. The values of
CH.4PTER 6. Single-Phase Fully-Developed Tbrbulent Flows 119
the axial gradient of the reduced pressure, 2, and the corresponding average Fan-
ning friction -or, f = &, are tabulated, as obtained from five Werent mesh
sizes. The extrapolated @-independent d t s as calculated from these d u e s , are
also shown in this table: Ext-1 represents experimental values obtained using the
results of the 41 x 41, 51 x 51, and 61 x 61 grids, Ext-2 values are obtained using
the results of the 51 x 51,61 x 61, and 7l x 71 grids; and the results of the 61 x 61,
71 x 71, and 81 x 81 grids were used to obtain the extrapolated values denoted by
E&. The relative changes in second and third extrapolated values (R-Change 2-3)
are just over 0.001. Therefore, t\e thii extrapolated value is asamed to correspond
to the grid-independent result. The results obtained with a grid size of 81 x 81 nodes
show a relative error of approximately 4% in the axial reduced pressure gradient,
2, and average Fanning friction factor, f , values, compared to the corrrspondiig
grid-independent values. Regarding the nature of this turbulent flow problem, and
the intrinsic uncertainties in the turbulent models, this order of error was considered
acceptable, and, therefore, all final runs awe executed on a 81 x 81 mesh: The results
are presented in section 6.1.3.
6.1.3 Results
The results presented in this section all pertain to a Reynolds number of Re = =
250000, where us is the bulk or average axial-velocity in the duct CTOSS-section, and
Dh is the hydraulic diameter of the duct. Furthermore, these resulcs were all obtained
with a uniform finite element mesh of 80 x80 nodes in the internal calculation domain
(complete domain minus near-wall regions).
Axial Velocity Contours and Secondary-Velocity Vectors
The axial-velocity contours along with vector plots of secondary velocities, ob-
tained using the CEVM turbulence model, are shown in Figs. 6.3 and 6.4, for a full
square cross section and a quarter of the square cross-section, respectively. From
the results for the full square cross-section, Fig. 6.3, it is evident that the numerical
CHAPTER 6. Single-Phase Fully-Developed Turbulent Flows
Table 6.1: Flow in a Quarter Square Duct, Results for Re = 250000. The grid sizes
correspond to the number of nodes in uniformly spaced finite element meshes in the
internal domain.
51 x 51
61 x 61
n x n 81 x 81
simulations have succgsfully predicted the perfect symmetry in both the axial veloc-
ity contoun and the eight symmetric recirculation zones for the secondary velocities.
When this problem is solved aver a quarter of a square crosssection, the numerical
simulation succgsfully predicted the expected symmetry about the diagonal of the
square, as shonn in Fig. 6.4.
The axial velocity contours as predicted by the different turbulence models con-
sidered here are shown in Fig. 6.5. Because of the relatively wide scatter in the
experimental results 1501, they are not shown here. Most of the plots in Fig. 6.5,
show the expected bulging of the velocity contours towards the corner. The turbu-
lence driven secondary flows in non-circular ducts, which are caused by the difference
in lateral normal stresses [35], distorts the streamwise mean velocity contoun toward
the comes. The linear k - c model, however, has no built-in mechanism for the devel-
1.615127
1.632260
1.644864
1.654453
3.6405596
3.6791775
3 . 7 0 ~ s . a
3.7292023
Grid Indp. 1.724849 3.8880109
CHAPTER 6. Single-Phase My-Developed Tbrbulent Flow 121
opment of secondary flow, due to its inherent isotropic characteristics, and, therefore,
it is unable of predicting any bulging toward the corner for this flow. Among the
other models, the ELGM d t s show the largest bulging toward the comer.
Fanning Friction Factor
The average Fanning friction factor, f , values, defined as
are compared with the experimental data taken from Ggmer and Emery 1501, in Fig.
6.6. As can be seen, the predicted \dues for flows with Reynolds number values of
50000, 150000, and 250000 are all in very good agreement with the corresponding
experimental values. These values are obtained using the CEVM of Craft et al. 1311.
The values obtained for Re = 250000, with different turbulence models are compared
with the corresponding experimental d u e s in Table 6.2. Clearly, all models, quite
successfully predict the average friction factor for these flows.
11 Gesmer et al., &p. I k - c I CEVM I EASM I RUB 11
Table 6.2: Flow in a Quarter Square Cross-Section: Average fiction Factor x103 for
Re = 250000.
Wall Shear Stress Distribution
The distribution of wall shear stresses, normalized with respect to the average
wall shear stress, 7,. = T, along the bottom wall of the square cross-section is shown
in Fig. 6.7. EASM predictions are closer to experimental results, though, it does
not show the maximum value (slightly bulging) of the experimental da ta All four
turbulence models used here lead to overprediction of the wall shear stress in the
central region of the -dl, and underpredict values in the near-wall region: Similar
CHAPTER 6. Single-Phase Fdy-Developed Tt~bulent Flows 122
results are reported in the earlier work of Ggmer and Emery [SO]. The linear k - r
model, though it predicts the average fiction fz.?x quite satkktorily, performs
relatively poorly in predicting the local properties.
Axial Veloaty Profiles
Axial veloaty profiles along the wall bisector and the comer b i i r are presented
in Fi. 6.8 and 6.9, respectively. In general, the profiles predicted by aU four models
are acceptable. The predictions obtained by using the EXSM, though, are in better
agreement with experimental results; this is because of its capability to capture sec-
ondary motions more accurately. The predictions obtained by Myong and Kobayashi
[I251 are in better agreement with experimental data
Turbulent Kinetic Energy
Turbulent kinetic energy proses are shown in Figs. 6.10 and 6.11, along the
wall and comer b i i o r s , respectively. All four models show the plateau-like behav-
ior, but overpredict the k values in the center of the duct. Near the wall, however,
the performance of the RUB model is quite satisfactory, and the EASM shows bet-
ter performance than the CEVM, in predicting the k values. The local equilibrium
assumption inherent in the CEVM, the EASM, and the RUB models, is not valid
here far from the wall, which could explain the overprediction of k values in this
area. Again, the predictions of Myong and Kobalahi [I251 compare best with the
experircental data
Secondary Velocity Distributions
The predictions of secondary, u, veloaty in Figs. 6.12 and 6.13, show the same
variation as the experimental data, though, they are not in good quantitative agrre-
ment, especially in areas close to the comer for all models (including predictions by
Myong and Kobayashi). The reasons for these discrepancies are not very clear to
the author. EASM predictions, however, are in the best overall agreement with the
experimental data
CHAPTER 6. Single-Phase Fully-Detlooped lkrbulent Flows 123
Shear Stress Profiles
Figs. 6.14 and 6.15 show the primary shear stress profiles along the wall and the
corner bisectors, respeaively. The Reynolds shear stress value, (-~3, near the
corner region changes rapidly in the transverse direction, hence, the experimental
data in this region may haw large errors [SO]. This may explain overprediction of
turbulent shear stress values dose to the corner by all models. Here, results produced
by Myong and Kobayashi [I251 and the E S M compare best with the experimental
data.
6.2 Fully-Developed, Turbulent Flow in a Straight
Tilted Duct of Square Cross-Section
In this test problem, the square crcs-section used in the previous section (6.1)
is tilted by an angle 0 = (30") with respect to the horizontal direction, Fig. 6.16.
Phys idy , this problem is identical to that considered in Section 6.1. However, the
rotation, 8, results in boundaries which are not parallel to the y and z Cartesian
coordinate directions, and, hence, the secondary veloaty components, u and w, are
not parallel to the walls. Thus, this is a good test for the boundary treatment outlined
in Chapter 5.
The results obtained are again compared against the experimental data of Gessner
and Emery [50], and also against the corresponding r d t s discussed in the prwious
section, for flow in the square cross-section duct with the walls parallel to the Carte-
sian coordinate axes (0 = 0).
6.2.1 Numerical Details
The problem was solved for similar conditions as those used in the first test-
case (section 6.1), that is, Re = 250000, based on Dh and bulk or average axial
velocity. The domain discretization, solution procedure, and convergence criteria
were all simiiar to those used in the first test-case. This problem, however, was only
CHAPTER 6. Single-Phase Fully-Dendoped 'Ikubulent Flows 124
solved over a full square crosssection. Based on the experience with grid independent
results for the first test case, a grid size of 121 x 121 was chasm for all final runs of
the computer code for this test problem.
Boundary Conditions
W boundaries are solid walls, none of which are parallel to the Cartesian coordinate
directions. The procedure for the implementation of logarithmic law of the waU and
related wall-functions described in Chapter 5, were utilized to bridge the near wall
regions.
6.2.2 Results
Axial Velocity Contours and Secondary-Velocity Vectors
Axial velocity contours obtained with the four different turbulent models consid-
ered here are presented in Fig. 6.16. All contours are symmetric around the duct
center. Also, a plot of the predicted secondary velocity vectors is presented in Fig.
6.17. The simulations have been successful in producing eight symmetric redrtula-
tion zones in the duct cmsssection. These two figures indicate the ability of the
numerical procedures and computer codes developed in this work to capture the right
flow pattern of this flow in a tilted square geometry.
Similar to the corresponding flow in a non-tilted square duct (section 6.1), the axial
velocity contours are bulged towad the corners, because of the turbulent secondary
velocity in the d u a cross-section. The linear k - e model, of course, is unable to
predict these secondary flows, and the EXSM contours show the maximum bulging
amongst the other models.
Based on the results obtained for the non-tilted square duct problem (section
6.1), from now on, ody, the results obtained with two turbulence models, namely,
the linear k - c model and E-GM, will be presented. The linear k - c model, as a
popular baseline model, and the E-GM, for its better performance, were chosen.
CHAPTER 6. Single-Phase Fully-Developed Turbulent FIous
Axial Velocity Proses
The axial velocity profiles along the wall and comer S i n are presented in F i
6.18 and 6.19, mspectively. The results obtained for this t eskase are compared with
the experimental data, as well as the corresponding results obtained with 0 = 0. -4s
was expected, the results are in excellent agreement with the corresponding results
for the first tgt-case: The small d i p a n c i e s are caused by the slightly different
grids used in Seaions 6.1 and 6.2.
6.3 Fully-Developed, Turbulent Flow in a Sector
of a Straight Duct of Circular Cross-Section
Fullydeveloped, turbulent flow in a straight duct of circular cross-section can be
solved in a one-dimensional radial direction, along the radius of the circular cross-
section. The velocity distribution and turbulence variables do not change in the
aid direction (fully-developed flow), nor in the circumferential direction (axisym-
metric flow). This physical situation is, however, a tawdimensional problem in the
Cvtesian coordinate system. Nevertheless, because the flow is physically axisymmet-
ric, the Cartesian formulation can be solved in a sector of the circular crosssection,
rather than full duct cross-section, ~ h i c h reduces CPU time and computer memory
requirements considerably.
The problem geometric configuration is shown in Fig. 6.20. The computational
domain is a sector with an angle 0, uith two symmetry boundaries, and a solid %-all
arc a t the bottom. This is a challenging irregular geometry, with a smooth curved
bottom wall, a sharp comer at the top, and two straight symmetry boundaries that
are not parallel to any of the chosen Cartesian coordinate directions.
Strictly, there is no secondary flow in the cross-section in this problem. However,
since the problem is solved on a finite number of elements, and the curved arc wall
boundary is approximated by piecewise straight l i e s connecting the boundary nodes,
it is expected that some secondary flows would be predicted in the cross-section.
C m E R 6. Single-Phase Fully-Developed Thbulent Flows 126
Nevertheless, as the grid becomes h e r , this false secondary flow should vanish. The
linear k-c turbulence model, on the other hand, is unable of capturing any secondary
flows, and, therefore, the simulations with this model do cot face this problem. The
results obtained with the other models can be compared with the results produced
by the linear k - c model.
The numerical predictions produced in this work are compared to the experimental
data of Laufer [92], and the numerical predictions of So et al. [176], who solved this
problem in a one-dimensional radial direction, in a cylindrical coordinate system, with
a very h e grid (250 grid points), and using a low-Reynolds number version of the
Speziale, Sarkar, and Gatski (SSG) model for the presure-strain term in the Reynolds
stress transport equation [182]. Therefore, their numerical predictions are amongst
the most accurate predictions available in the literature. They solved this problem
for four different values of Reynolds number in the range 7500500000. Here, the
numerical simulations were performed for a flow with Re = 420000, which corresponds
to the experiments of Laufer at a Re, = 8758 (based on friction veloaty and tube
diameter). This case appears to be the pipe flow with the bighest Reynolds number
for which reliable experimental results of turbulence variables are available.
6.3.1 Domain Discretization
Here again, the domain discretization starts with finding the wall normal distance
of the near-wall nodes. This was done using a procedure akin to that already explained
for the first test-case (section 6.1.1). Then, the internal domain (complete domain
minus the near-wall region) is divided into elements. One of the great advantages of
triangular elements is the flexibility they provide in the discretization of domains such
as sectors or triangles. If quadrilateral elements were used for the sector, it would
have been impossible to obtain a uniform grid distribution at all locations.
Once the wall-normal distances of the near-wall nodes are obtained, the coordi-
nates of the near-wall nodes on both symmetry boundaries are fixed. Then, nodes
along both symmetry boundaries are fixed using a power-law expansion, as in the
first testcase. This allows non-uniform distribution of nodes, if required. The cw
CHAPTER 6. S i n g l e - P h Fdy-Developed Turbulent Flows m
ordinates of the internal nodes are obtained by intersecting lines parallel to the left
side, which originate from the right side nodes, and lines parallel to the bottom arc.
The near-wall nodes are located on radii which pas through the duct center. This
facilitates the generation of perpendicular lines (radii) which connect the near-wall
nodes to the bottom solid h u n k .
As can be seen in Fig. 6.21, the domain is divided into a line-structured ar-
rangement of triangular elements, which are uniform over the internal domain. The
particular form of the near-mall elements generated here are needed for the proper
treatment of the logarithmic law of the wall and the assodated wall-functions, through
the procedure outlined in Chapter 5.
6.3.2 Numerid Details
Boundary Conditions
Boundary conditions are prescribed at symmetry planes(sides) and at near-wall
nodes(bottom). At symmetry planes, the velocity component normal to the symmetry
plane is set equal to zero, while for all other quantities, the gradient normal to this
plane are taken to be zero. The logarithmic law of the wall and related wall-functions
were implemented to bridge the region between the uall and the near-wall nodes, as
am explained in Chapter 5.
Solution and Convergence Criteria
The solution of the sets of the discretized equations were performed iteratively as
explained in Chapter 4. However, in the solver, the line-by-line sweeps are performed
from each of the three boundaries toward the opposite side. The convergence criteria
were simiiar to those specified for test-case one (section 6.1). The same range of
under-relaxation values were employed, and a monotonic solution convergence was
achieved.
CHAPTER 6. Single-Phase my-Developed Thbulent FIom
Grid-Independent Results
Grid-independent results were obtained for this problem in a manner similar to
that used in the first test case (section 6.1.2), and are tabulated in Table 6.3. Thge
computations were done with the cubic eddy viscosity turbulence mod& -4s can
be seen, the relative change between the second and the third extrapolated d u e
is 0.00198 for the axial pressure gradient, -2, and 0.00357 for the average Darcy
friction factor. Therefore, the third set of extrapolated values are assumed to be the
grid-independent d t s The numerical error involved in solving this problem on
a mesh with 6083 eIements in the internal domain is 6%, which, as wis mentioned
earlier, was considered acceptable for these problems. rU1 final runs are executed on
a 6083 element mesh in the internal domain.
E e m e 0 . : / -2 x lo2 I f x lb 11 Internal Domain
Table 6.3: Flow in a Sector: Results for Re = 420000.
Grid Indp. 4.640208 1.479933
CHAPTER 6. Single-Phase Fully-Dewloped 7kubdent Flows
6.3.3 Results
It should be noted that the results presented in this section correspond to Re =
420000, and were aU obtained with a 6083 element mesh.
Axial Velocity Contours
The d a l velocity contours as predicted by the CEVM are shown in F i 6.22,
and are compared against cormsponding results obtained with the other three models
in Fig. 6.23. It is evident that the numerical simulations suecgsfully predicted the
right pattern of these contours. Hmwer, as %as mentioned earlier, the predictions by
the k-c model are the most accurate results, because this model d o g not produce the
false secondary motions. The other models show some bulging in the contours, which
is because of the false secondary motion predicted by these models in the nondrcular
aoss-section that results because of the piecewise linear approximation of the drcular
wall boundary.
Axial Velocity Profiles
The axial velocity profiles along the radius (symmetry boundary), u+ = u/%, are
shown in Fig. 6.24, where y+ is the nundimensional distance from the wall to the
center. The predicted profiles are very dose to each other, the experimental data,
and the predictions by So et al 11761.
Average Darcy Friction Factor
The average Darcy friction factor values, as predicted by the four dierent turbulence
models are presented in Table 6.4, along with the corresponding value calculated from
the Colebrook formula 11921.
-4s can be seen, all models quite successfully predicted the average friction factor.
It shows that, as far as the general characteristics of this flow are concerned, all the
turbulence models tested here are able to produce reasonable results.
CHAPTER 6. Single-Phase Fudly-Developed Turbulent Flows
Table 6.4: Flow in a Sector, Average Riaion Factor x102, Re = 420000.
Turbulent Kinetic Energy Profiles
Colebrook ( k - c
Turbulent kinetic energy profiles, k+ = klu:, along the radius are presented in Fig.
6.25. Here, R is the radius of the pipe. The results are quite similar to predictions
by So et al [176]. All numerical results shown in this figure overpredict the turbulent
kinetic energy near the duct center (r = 0) and underpredict it near the duct wall (r =
R). None of the turbulence models considered here are able to correctly predict the
slope exhibited by the experimental results. This di5culty is though to be caused by
inaccuracies in the modeled form of the differential equation for the rate of dissipation
of turbulent kinetic energy, c.
1.385
Dissipation Rate Profiles
RUB
1.37 1.380
CEVM ( E S M
The predicted c+ = vclu: values, by all the four turbulence models, are in ex-
cellent agreement with the numerical results of So et al, Fig. 6.26. All four models
predict the rapid decrease in c+ with distance from pipe mall which is the behavior
expected for flow a t such a high Reynolds number. This is because as the Reynolds
number increases, the viscous sublayer becomes thinner and thinner, and the viscous
dissipation rate approaches zero faster with d i c e away from the wall.
1.382
Shear Stress Profiles
1.383
The nondiensionaliied shear stress values, - m / u : , along the radius are plotted in
Fig. 6.27. The results of So et al. [I761 are the best with respect to the experimental
d a t a
CH.4PTER 6. Single-Phase My-Developed Tbrbulent FIoas 131
6.4 Fully-Developed, Turbulent Flow in a Straight
duct of Triangular Cross-Section
Straight, triangular crosssection ducts have also been used extensively, along
with ducts of square czosssection, in investigations of the turbulent flow behavior in
non-drcular ducts 161. In addition, this is a geometry for which there are reasonable
useful experimental data for turbulent, gas-liquid, bubbly -phase flows [ill], which
will be used in the next chapter to check the numerical simulations of these flows
performed in this work. Hence, the triangular aos-section duct was chosen as a test-
case for single-phase turbulent flows and, also, as a pre-requisite to the numerical
simulations of turbulent, bubbly, gas-liquid, tapphase flows presented in the next
chapter.
The geometrical configuration is shown in Fig. 6.28. The duct crosssection is an
equilateral triangle. The program, however, was written such that i t could handle
equilateral as well as isosceles triangle cross-sections; the latter to be used in the
next chapter for turbulent, single-phase and bubbly -phase flows. The detailed
experimental data and corresponding numerical simulation results for fully-developed,
turbulent flows in equilateral triangular ducts obtained by Aly e t al. [6] were used to
e-.aluate the results obtained in this work.
Aly et al. [6] conducted measurements a t Reynolds numbers, based on bulk ve-
locity and the hydraulic diameter of the duct, of 53000, 81000 and 107300. In this
work, numerical simulations were performed at all these Reynolds numbers, though,
most of the results are only presented for Re = 53000.
The secondary motion in an equilateral triangular duct consists of six symmetric
recirculation zones bounded by the corner b i i o r s , as shown in Fig. 6.31. Each
of these recirculation zones is identical. Thus, a knowledge of the flow properties in
any one cell is d c i e n t to describe the entire flow field. The results in this section , therefore, will be presented for just one recirculation cell (hatched area in Fig. 6.28).
CHAPTER 6. Single-Phase my-Developed Zkrbulent Flows
6.4.1 Domain Discretization
As for the other geometries discussed earlier in this chapter, the wall-normal
distance of the near-wall nodes was first calculated through the procedure explained
for the first testcase. Then, the y-coordinate of near-wall nodes was calculated by
starting from the base, followed by calculation of thii coordinate for internal nodes.
The x-coordinate of the nodes on the bottom walls uas calculated, then, the x-
coordinate of nodes on the left wall, and, also, corresponding near-udl nodes were
obtained. F i l y , the x-coordinate of other nodes \vzi calculated along the constant
y-lines. A typical mesh is shown in Fig. 6.29.
6.4.2 Numerical Details
Boundary Conditions
All boundaries for this problem are solid-wall boundaries. The wall functions and
implementation procedures explained in Chapter 5 were used to handle, boundary
conditions at all near-wall nodes.
Solution and Convergence Criteria
The same convergence criteria and solution procedure were used for thii t eskase
as those for test- one (section 6.1).
Grid-Independent Results
The grid-independent results were estimated for thii test-case using a similar
extrapolation procedure as that used in the other test-cases, and are shown in Table
6.5. These simulations were done with the cubic eddy viscosity turbulence model.
The relative change between the second and the third set of extrapolated values is
0.0004 for both the pressure gradient, -2, and the average Darcy friction factor, f.
Hence, the third set of extrapolated values are assumed to be the grid-independent
results. The values obtained for these two miables over a 5922 element mesh in
the internal domain (complete domain minus near-wall regions) have an error of 3%
CHAPTER 6. Single-Phase my-Developed Twbulent Flows 133
relative to the grid-independent results, which was considered acceptable for the flows
considered in this problem. Final runs vmre executed on a 5922 element mesh.
Internal Domain
2202
3242
Table 6.5: Flow in a Mangular Duct, Results for Re = 53000
R-Change 2-3
Grid Indp.
6.4.3 Results
All results presented in this section correspond to Re = 53000: The computations
were done using a 5922 element mesh.
0.0004
4.122878
Axial Velocity Contours a n d t h e Secondary-Velocity Vectors
0.0004
1.989185
The axial velocity contours and the secondary-velocity vector plots are shown in
Figs. 6.30, and 6.31, respectively. These results were obtained using cubic eddy
viscosity turbulence model. The numerical simulations successfully predicted the ex-
pected symmetry in flow field across the corner bisectors. The axial velocity contours
CHWTER 6. Single-Phw My-Developed Tbrbdent Flows 134
as predicted by the linear k - c and CEVM turbulence models are compared to the
corresponding experimental d t s of Aly et al. 161 in Fig. 632. As can be seen, the
linear k - e model, as Bcpected, does not show any bulging in the contours toward
the corner, while CEVM contours exhibit bulging toward the corners Nevertheless,
the bulging of the axial velocity contours in the experimental results is larger than
that predicted by CEVX
Average Darcy Friction Factor Values
The average Darcy friction factor values are plotted against Reynolds number
in Fig. 6.33. The predicted values (with CEVM) are in \pry good agreement with
experimental results. The average friction factor values for Re = 53000 as predicted
by the four different turbulence models are presented in Table 6.6.
Table 6.6: Flow in a Triangular Duct: Average Friction Factor xlOZ for Re = 53000.
It is evident that all models are able to predict the average friction factor values
satisfactorily, though, the EASM predicted value is closer to the experimental da ta
RUB
1.927
Aly et al. Exp.
1.940
Turbulent Kinetic Energy Profiles
The turbulent kinetic energy profiles along the wall b i i t o r are plotted in Fig.
6.34. EASM results show the best agreement with the experimental results, while the
RUB results overpredict the experimental results throughout. All models overpredict
the k values dose to the duct center; similar reasoning as for other test- could
be repeated here for this overprediction.
k - c
1.927
CEVM
1.928
EASM
1.937
CHriPTER 6. Single-Phase My-Developed Tbrbulent Flows 135
Wall Shear Stress Distr ibution
The distribution of local wall shear stress values, normalized with reference to the
average wall shear stress stress (r, = f ) , along the (half) triangle base are shown in
Fig. 6.35. EASM predictions are in very good agreement with experimental results,
and this performance is even better than the predictions by Aly et al. [6] in the middle
area The linear k - c model does not predict the right pattern of wall shear profile:
This defiaency, as v a s discused before, is because of its inherent stress isotropy
asumption.
Reynolds Normal Stress Distribution
The distribution of Reynolds normal st- along the wall b i i t o r are presented
in Fig. 6.36. The predictions are in acceptable agreement with experimental results.
The EASM nredictions, are closer to experimental results than the corresponding
CEVM and RUB model predictions. It should be remembered that the one of the
aims in turbulence modeling is to improve the abiity to predict the local and average
properties of the Eow, which are both important in engineering design. More so-
phisticated turbulence models, such as RSM, are expected to produce more accurate
predictions of Reynolds st-.
In this chapter, the proposed CVFEM, and the computer program, along with
the new procedure for the implementation of the logarithmic law of the wall and
related wall-functions, were tested for fully developed, turbulent, single-phase flow
in straight ducts of four different cross-sections. Four different turbulent models,
namely, the linear k - c model, CEVM, RUB, and EASM were compared for their
performance in predicting these flows.
The results indicate that the numerical simulations with the CEVM, RUB, and
EASM models of turbulence are quite satisfactory qualitatively in predicting the
correct flow patterns for all cases. Also, the predicted axial velocity profile and
CHAPTER 6. Single-Phase Fully-Developed llubulent Flows 136
average friction factor values are in good agreement with experimental data and
previous numerical predictions for similar conditions.
For local results, the performance of EASM was generally better than the other
three turbulence models tested here. The results obtained with the linear k - c
model, as expected, were not as good for ducts with non-cirdar cross-sections. This
is particularly important for local properties, such as local wall shear stress, or the
local Nuselt numbers on the ualls in case of heat-transfer problems To obtain decent
local results that exhibit at least correct qualitative behavior, it is important to use
non-linear k - c or algebraic st- models
Figure 6.1: Fully-Developed Flow in a Square Duct: Schematic Configuration and
Notation.
Figure 6.2: Fully-Developed Flow in a Square Duct: Typical Uniform Finite Element
Mcsh.
Figure 6.3: Fully-Developed Flow in a Full Square Cross-Section: Re = 250000,
Results Obtained Using CEVM Turbulence Model.
Figure 6.4: Fully-Developed Flow in a Quarter Square Cross-Section: Re = 250000,
Results Obtained Using CEVhl Turbulence Model.
Level: 1 2 3 4 5 6 7 8
Figure 6.5: Fully-Developed Flow in a Square Duct: .kid Velocity Contours: Linear
k - r (-), CEVM (- - - . EASM (- . - . -) , and RUB (- . . -1.
Figure 6.6: Fully-Developed Flow in a Square Duct: Variation of Average Friction
Factor: Results Obtained Using the CEVM [31].
Figure 6.7: Fully-Developed Flow in a Square Duct: \Wl Shear Stress Distribution.
Figure 6.8: my-Developed Flow in a Square Duct: kda l Velocity Variations Along
the Wall Bisector.
J
Figure 6.9: Fully-Developed Flow in a Square Duct: Axial Velocity Variations Along
the Comer Bisector.
Figure 6.10: my-Developed Flow in a S q m Duct: Turbulent Kinetic Energy
Distribution Along the Wall Bisector.
Figure 6.11: Fully-Developed Flow in a Square Duct: Turbulent Kinetic Energy
Distribution Along the Comer Bisector.
Figure 6.12: Fully-Developed Flow in a Square Duct: v-Veloaty Distribution Along
the Wall B i i r .
Figure 6.13: Fully-Developed Flow in a Square Duct: v-Velocity Distribution Along
the Comer B i i t o r .
Figure 6.14: Fully-Developed Flow in a Square Duct: -u'Y'/u: Distribution Along
the Wall B i r .
Figure 6.15: Fully-Developed Flow in a Square Duct: -u'Y'/u: Distribution Along
the Comer Bisector.
Figure 6.16: Fully-Developed Flow in a Tilted Square Duct: Axial Veloaty Contours:
k - c (-)0, CEVM (- - -), EASM (- . - . -) , and RUB (- . . -).
Figure 6.11: Fully-Developed Flow in a Tilted Square Duct: Secondary Veloaty
Vectors.
Figure 6.18: Fully-Developed Flow in a Tilted Square Duct: .&a1 Velocity Distribu-
tion Along the Wall Bisector.
Figure 6.19: Fully-Developed Flow in a Tilted Square Duct: Axial Velocity Distnbu-
tion Along the Corner Bisector
Figure 6.20: Fully-Developed Flow in a Sector: Geometric Configuration and Nota-
tion.
Figure 6.21: Fully-Developed Flow in a Sector: Typical Uniform Finite Element
Mesh.
Figure 6.22 Fully-Developed Flow in a Sector: Axial Velocity Contours.
Figure 6.23: Fully-Developed Flow in a Sector: Axial Velocity Contour: Linear k - c
( 1 , CEVM (- - -), EASM (.. . - . -) , and RUB (. . . . .).
Figure 6.24: Fully-Developed Flow in a Sector: Velocity Profiles Along the
Nondimensional Wall-Normal D i c e .
Figure 6.25: Flow in a Sector: lhrbulent Kinetic Energy Profiles Along the Radius.
Figure 6.26: Flow in a Sector: Rate of Turbulent Kinetic Energy Dissipation Profiles
Along the Radius.
Figure 6.27: Flow in a Sector: Shear Stress Profiles Along the Radius.
Figure 6.28: Fully-Developed Flow in a Triangular Duct: Geometrical Configuration
and Notation.
Figure 6.29: Fully-Developed Flow in a Triangular Duct: Typical Uniform Finite
Element Mesh.
Figure 6.30: Fully-Developed Flow in a ?tiangular Duct: Axial Velocity Contours for
Re = 53000.
Figure 6.31: Fully-Developed Flow in a Triangular Duct: Secondary Velocity Vectors.
---
Figure 6.32: Flow in a 'Aiangular Duct: &a1 Veloaty Contours, Comparison of
Predicted and Experimental Results.
I
Figure 6.33: Flow in a Triangular Duct: Average Friction Factor.
Figure 6.34: Flow in a Triangular Duct: Turbulent Kinetic Energy Distribution.
Figure 6.35: Flow in a Triangular Duct: Local Wall Shear Stress Distribution.
Figure 6.36: Flow in a Triangular Duct: Reynolds Normal Stresses Distribution.
Chapter 7
Applications to Fully-Developed,
Turbulent, Bubbly Two-Phase
Flows
In this chapter the results of the computer simulations of fullydeveloped, u p
ward, turbulent, dilute, bubbly, gasliquid -phase f l m in vertical ducts are pre-
sented. The governing equations were presented in Chapter 3. The CVFEM and
the new wall boundary treatment procedure (Chapters 4 and 5) uwe incorporated
in computer programs, which were completely developed in this umk. The proposed
CVFEM and related uall-function treatments for single-phase flows were d i d a t e d
in the last chapter. That activity has established a strong foundation and prepared
the stage for computer simulations of the fullydeveloped, dilute, bubbly wephase
flows of interest. The computer programs were written in standard FORTMN77,
and were executed on a single processor (R-8000) of a 8-processor Silicon Graphics
Power Challenge computer.
7.1 Background Notes on the Test Problem
Most of the available, published experimental data on the bubbly twephase flows
of interest are for developing or fully-developed flows in the vertical pipes of circu-
CHAPTER 7. Fully Developed, Turbulent, Bubbly Tzn-Phase Flows 157
lar aoss-section (Wang e t al., [190]; Serizawa et al., [168]; Liu and Bankoff, [loti];
Nakoryakov e t al., [I=]; Nako~+~v and Kashinsky, [127]). The mathematical mod-
els and numerical methods presented in Chapters 3 and 4, respectively, are sped-
cally established for fullydedoped, turbulent, -phase bubbly flows in nondrcular
ducts, which are known to produce secondary-flows in the duct crossections. The
particular form of the discretized wid-fraction equations in t h e models produce
non-vanishing results, if, and only if, these secondary motions esdst in the duct aos-
section. The fullydeveloped, turbulent flows in circular pipes do not exhibit such
secondary-motions, because of the axial symmetry in the pipe cross-section. There-
fore, the proposed approach can not be used for the predictions of the fullydeveloped
flows of interest in pipes of circular cros-section.
Experimental data for fullydeveloped, turbulent, dilute, bubbly two-phase flows
in the vertical ducts of noncircular cross-sections, with details that are needed for
the validation of numerical predictions of such flows, are very S i t e d . Lopez de
Bertodano [159,111] did experiments on turbulent, dilute, bubbly, two-phase (air and
water) flows in a vertical duct with an isosceles triangle cross-section (Fig. 7.1). He
also performed computer simulations of these flows, using the commercial PHOENIX
computer programs [48], and evaluated the numerical simulation d t s by comparing
them to the above-mentioned experimental data. Therefore, it was believed that his
experimental data, though not perfect, could be used for the evaluations of the results
produced in this work.
The ratio of the duct length (13) to the hydraulic diameter (4) in this experiment
u-as 13/01, = 73, and Lopez De Bertodano claimed that the flow uas fullydeveloped
close to the end of the duct [Ill]. Unfortunately, there is no reported experimental
data to demonstrate that the flow was, in fact, fully-developed a t the measurement
station (such as an examination of the axial pressure gradient along the duct). Never-
theless, the results obtained in this work are compared against the experimental data
of Lopez de Bertodano [Ill]: These are the most recent detailed experimental data,
and seem to be adequate to be used a qualitative checks, a t least, on the results
obtained in this work.
CHAPTER 7. Fully Developed, Turbulent, Bubbly TaPPhase Flows 158
Lopez de Bertodano's [111] experimental data are reported for three superficial
liquid velocities, JL, a t the inlet plane. For each JL, the data of experiments with
three superficial gas-velocities, Jc, a t the inlet plane were reported, and are given
in Table 7.1. Lopez de Bertodano [I l l ] reported values of the liquid axial velocity,
ul, the void fraction, a, and the Fteynolds stresses [I l l ] for each set of experiments
at different Locations in the duct cms-section at the meanuement station. For each
JL d u e , the first experiment was done with Jc = 0, which is a single-phase liquid
flow, and the second and the third experiments involved tvmphase flows. The third
experiment, a t each JL d u e , involved bubbly twc-phase flows which can not be
categorized as dilute, since the void fractions values are quite high, and hence, these
particular flows were not simulated here. Also, the computer simulations for the tesb
cases with the liquid velocity, JL = 0.3, were not simulated in this work, because
the wall-normal distances of the near-wall nodes become relatively large for this low
liquid flow rate, when the nondiensional wall-normal d i c e s of these nodes are
in the desired range of 30 < y+ < 120. The crass-section of the duct was an isosceles
triangle of dimensions 2L = 5 m , H = 1 0 m , and a vertex angle that is % 28' (see
Figs. 7.1 and 7.2).
Table 7.1: Liquid- and Gas-Phase Superficial Velocity Values in the Experiments of
Lopez de Bertodano [Ill].
Lopez de Bertodano [ I l l ] also performed computer simulations of hi experi-
ments on turbulent, bubbly, twc-phase flows in an isosceles triangular duct, using the
commercial computer program PHOENICS [48]. Since that program was based on
a lie-structured three-dimensional finite-volume method formulated for hexahedral
CHAPTER 7. F d y Developed, Turbulent, Bubbly Two-Phase Flows 159
cells, a rather peculiar domain discretization of the isosceles triangular c m s e c t i o n
was used in his simulationsr He used a seaor aoss-section, instead of the isosceles
triangular aoss-section, as the computational domain; the quadrilateral ems-section
of the cells generated near the vertex of the sector had a very large aspect ratio; and,
the mesh was very heavily dense in that region. In addition, since he solved this prob-
lem in a three-dimensional domain, even though he executed his programs on a Cray
supercomputer, he could not guarantee the convergence or the grid-independence of
his results [Ill]. Considering these defiaenaes, the results produced in this work were
not compared to the numerical predictions reported by Lopez de Bertodano [Ill].
In the numerical predictions of turbulent, single-phase flows, d i d in Chapter
6, the explicit algebraic stress model (EASM) produced the best overall results, in
comparison to the linear and nonlinear eddy viscosity models. Therefore, only the
results produced by the EASM for the shear-induced Reynolds stress terms in the
governing equations of fullydeveloped, turbulent, bubbly two-phase flows of inter&
are presented in the remainder of this chapter.
7.2 Domain Discretization
The isosceles triangle cross-section of the vertical duct (Figs. 7.1 and 7.2: H = 10
cm and 2L = 5 cm), u w diicretized into uniform three-node triangular elements
and polygonal control volumes in a way similar to that used in the discretization of
the equilateral triangular duct of the fourth testcase in Chapter 6. The near-wall
distances were calculated using the same procedure as in the singlephase flow sim-
ulations. Although this procedure not necessarily yield the best estimation of the
near-wall distances, since these are based on guessed values, the final converged solu-
tion is not adversely affected by these values: As in the singlephase flows simulations,
the final converged solutions were checked to ensure that the nondimensionalized uall-
normal distances of all near-wall nodes were between 30 5 y+ 5 120. A typical finite
element mesh used in these simulations is shown in Fig. 7.2.
CHAPTER 7. Fully Developed, Turbulent, Bubbly Tu-Phase Flows 160
7.3 Numerical Details
7.3.1 Grid-Refinement Checks
Numerous preliminary computations, including grid refinement checks, were per-
formed for the selected test problem. Based of the results of these preliminary com-
putations, and using the results of the singlephase flow computations presented in
Chapter 6 for additional guidance, it was concluded that a 5 9 Z triangular element
grid, akin to that shown in Fig. 7.2, was adequate for the final computations
7.3.2 Boundary Conditions
As was stated in Chapter 4, the governing equations of the fully-developed, tur-
bulent, dilute, bubbly two-phase flous consists of 7 differential transport equations (3
liquid-phase momentum equations, liquid- and gas-phase continuity equations, k, and
c equations) and nine algebraic equations (3 simplified gas-momentum equations and
6 algebraic equations for the 6 independent Reynolds stresses). Boundary conditions
must be specified for the 7 differential transport equations; the other 9 algebraic equa-
tions do not require specilications of boundary conditions. The wall-function treat-
ment for turbulent, dilute, bubbly, two-phase flow, as explained in Chapters 3 and
5, was used to specify near-udl boundary conditions for the 3 liquid-phase momen-
tum equations, and also the k and c equations. The liquid- and gas-phase continuity
equations, which are used to obtain the d i i t i z e d pressure and void fraction equa-
tions, respectively, are first order differential equations, and hence, only require one
boundary condition. The procedure that was used to specify the boundary condition
for the d i i t i z e d pressure equations is similar to the corresponding procedure used
for the single-phase discretized pressure equations (Chapter 4): So, it is not repeated
here. The overall continuity of the liquid-phase mass flow rate is also satisfied by
a procedure similar to the corresponding procedure employed for the singlephase
flows (Chapter 6). The boundary condition for the diret ized void fraction equation,
however, requires more attention, as is d i i in the next subsection.
CHAPTER 7. Fully Developed, Turbulent, Bubbly TripPhase Flows 161
7.3.3 Boundary Conditions for the Discretized Void Fraction
Equations
As was discused in Chapter 3, since the gas-continuity equation, which is used to
obtain the discretized void fraction equations, is a 6rst order differential equation, it
requires only one boundary condition, which could either be a specified void fraction
value a t a particular point in the domain, or an average void fraction value over the
duct cross-section.
Bubbly, gas-liquid, turn-phase flous in vertical, straight ducts are usually char-
acterized by the superficial velocities of the liquid- and the gas-phases, JL and Jc,
respectively, the bubble diameter, Db, and the duct cross-section geometry and a rea
As the flow develops in the duct, the bubble distribution over the duct cross-section
a t the duct inlet, changes, because of the pressure gradients and the associated bubble
density and volume changes, and also because of the lateral forces, such as lift forces,
exerted on the bubbles. Indeed, the relative velocity between the bubbles and the l i p
uid increases along the d u a , until it reaches its terminal d u e in the fullydeveloped
region. Once the flow is fullydeveloped, the average void fraction and gas superficial
velocity in the duct cross-section are not equal to the corresponding values a t the
duct inlet.
The aforementioned observation implies that the inlet conditions for the gas-phase,
such as the d u e s of the gas superficial velocity, Jc, or the average void fraction,
< cr >, can not be taken as the appropriate flow parameters in the fullydeveloped
region. These inlet conditions are adequate for specifying two-phase bubbly flow
problems, if the computer simulation starts from the duct inlet and reaches the fully-
developed section by marching through the developing region. As uas d i e d in
Chapter 1, this approach has its oun important drawbacks, which makes the develop
ing case less desireable as a test problem for computer simulations of two-phase flows.
Therefore, strictly, experimental values of the gas-phase flowrate and/or the average
void-fraction are needed in the fullydeveloped region, for appropriate sp&cation of
the fullydeveloped, dilute, bubbly, twwphase flous considered here. Since this work
C m E R 7. Fully Developed, Turbulent, Bubbly T ~ P ~ Flows 162
represents the first time that these flows have been simulated in this way, that is,
by undehaking simulations only in the fullydeveloped region, i t is not surprising to
note that such fullydeveloped data have not been specifically reported in the experi-
mental investig?tions of these flows, including the work of Lopez de Bertodano [Ill].
Considering these findings, it is suggested here and also in the next chapter, that in
future experimental investigations of these flows, this fullydeveloped infonnation be
obtained and adequately reported, in order to facilitate the formulation and testing
of the corresponding mathematical models and numerical solution methods.
In the absence of the desired experimental infonnation in the Nlydeveloped re-
gion, the attention here was focused on the specification of the value of void fraction a t
a particular point in the crosssection (calculation domain). In the experimental data
tabulated by Lopez de Bertodano [ill], the values of void fraaions are given at several
locations in the duct cros-section. The particular advective form of the gas-phase
continuity equation, however, must be considered in specifying this type of boundary
condition for this variable. This is because if, for example, the gas-phase secondary
velocity vectors in the duct cross-section follow the corresponding rrcirculating liquid-
phase velocity vectors, which are similar to the corresponding single-phase velocity
vectors (Fig. 6.31, but weaker [Ill]), then the specified void-fraction value would
serve to 6x the a distribution only in the particular recirculation zone in which that
point is located: Since closed secondary-flow streamling mark the boarders between
these recirculation zones, there is no advection from one recirculation zone to the
other. Hence, in the absence of any a priori infonnation on the gas-phase secondary
velocity vectors in the duct cros-section, specifying the void fraction value a t one
particular node could be misleading and may produce false or meaningless results.
Considering the fact that computer simulations of only :he fullydeveloped region,
and the demonstration of the importance of the gas-phase secondary-velocity vectors,
are being undertaken for the first time, in this work, again, it is not surprising that
there are no experimental data on the gas-phase secondary velocity vectors in the
published works, including those of Lopez de Bertodano [Ill].
Considering the above-mentioned problems with the available experimental data
CHAPTER 7. Fully Dewloped, Turbulent, Bubbly Tuv-Phase Flows 163
for the fullydeveloped region, and the corresponding boundary condition for the
gas-phase continuity equation, it was decided to numerically integrate, for each test
case simulated in this work, the available experimental void fraaion \dues over the
duct cross-section, and thus obtain an approximate value for the average void frac-
tion over the duct -section in the fully developed region. This could only be
an approximation of the actual average void fraction value, because of the limited
experimental values of the void fraction reported over the duct aassec t ion for each
test case. More importantly, the values of void fraction values dose to the comers
(in particular, dose to the triangle vertex) have not been experimentally measured:
This reduces the armracy of the numerical integration of the experimental local void
fraction values, since the highest void fractions are reported to be in the comers [lll].
In the computer simulations, for each liquid-phase superficial veloaty, the calcu-
lations were started with a zero average void-fraction value (single-phase flow), and
then, the bubbles were introduced into the flow. For each series of runs, the average
void-fraction value was increased, in four stages, from zero to the average void-fraction
value estimated from the experiment$ data of Lopez de Bertodano [Ill]. This strat-
egy allowed investigations of the behavior of the fuily-developed, bubbly two-phase
flow over a range of < a > \dues, from the single-phase flows (< a >= 0) to the
approximated experimental average void-fraction values: i t also allowed observations
of the evolution of the two-phase flows in this process.
7.3.4 Solution Strategy and Convergence Criteria
As was mentioned in the last paragraph, the computer simulations were started
with the single-phase flow (Jc = 0) cases, and the numerical predictions were com-
pared with the corresponding data of Lopez de Bertodano 11111. After this first step,
in which the performance of the proposed CVFEM is evaluated, the results produced
in this step are used as the guess values for the next test case, in which bubbles are
introduced into the fluid flow. This strategy helped to reduce the overall computer
CPU time demands substantially.
The solution procedure for each test- was explained in Chapter 4. The void-
CHAPTER 7. IWly Developed, Zbrbulent, Bubbly T I F D - P ~ ~ FJows 164
fraction discretized equations are assembled and solved after the calculations of the
liquid velocity components, the pressurereESUre and the Reynolds st-. In each triangular
element, the relative veloaty components are calculated and stored in the beginning
of each iteration, using the most recent values of the terms which appear in the
algebraic expressions for these veloaties (Chapter 3).
The iterative solution procedure was stopped when the stipulated convergence
criteria a w e satisfied. In these simulations, the convergence criteria were similar to
the convergence criteria for the singlephase flow simulations, plus an additional cri-
terion that the relative change in the calculated average void fraction values in two
successive iterations had to be less than, lo4. However, it should be mentioned that
the convergence criteria, which required that sums of the absolute values of the nor-
malized residues in each of the w and p sets of discretized equations, as well as the
relative change in the value of u at the duct centerline in tuv successive iterations, to
be less than in the single-phase flow simulations, uas increased to b e c a w
of challenges posed by the strong couplings and non-linearity of the governing equa-
tions, plus the fact that the specification of an average void-fraction value decreased
the convergence rate considerably. Implicit under-relaxations 11371 of the discretized
equations were also necessary to ensure the convergence of the numerical solutions of
the fullydeveloped two-phase bubbly two-phase flows: in general, somewhat smaller
d u e s of the under-relaxation parameters were required here than in the singlephase
flow simulations.
7.4 Results
The results presented in this section pertain to both the single and two-phase
bubbly, fully-developed, turbulent flows investigated in this work. The triangular
duct is the same as that used in the experiments of Lopa de Bertodano [Ill]: With
reference to Figs. 7.1 and 7.2,2L = 5 cm and H = 10 cm. The test cases are shown
in Table 7.2, which gives the liquid superficial velocity and the specified average void-
fraction values for each test cse . The Jt = 1.0 and Ji = 0.5 superficial liquid-velocity
CHAPTER 7. M y Developed, Turbulent, Bubbly Two-Phase Flows 165
d u e s correspond to the single-phase flows with Reynolds number of Re = 32430 and
Re = 16215, based on the JL and the duct hydraulic diameter, respeaively- As was
mentioned earlier, the last merage voidfmction value for each liquid-phase superficial
velocity shorn in Table 7.2 was obtained by an appmximate numerical integration
of the experimental void--ion data in the experiments of Lopez de Bertodano,
for each of the speci6ed JL values in the table, and the corresponding inlet value
of the gas-phase superficial veloaty, Jc. These Jc values are also given, in the last
column of the Table 7.2. It should be noted that Lopez de Bertodano performed two
experimental runs for each set of JL and Jc values [Ill]: Both these set of data are
used in the comparisons in this section, and are indicated by the symbols "I" and
"II" in the legends of Figs. 7.3 to 7.14.
JL m/s 1 Average Void Fraction < a > 1 Jc m/s -Experimental [ I l l ]
0.5 1 0.0 0.02 0.05 0.080 1 0.05 [for < a >Y 0.0801
Table 7.2: Test Cases Investigated: Liquid-Phase Superficial Veloaty and Average
Void Fraction Values.
1.0
All results were obtained with a uniform finite element mesh of 5922 triangular el-
ements in the internal calculation domain (complete domain minus near-wall regions):
The discretization pattern is illustrated in Fig. 7.2. The results of the fully-developed
single-phase flow simulations (Jc = 0) are presented first, and then the numerical
predictions of Nlydeveloped, bubbly two-phase flous are presented. In most of the
*phase flow simulations, it uas assumed that the bubbles are essentially spherical
with a uniform diameter of Db = 5 mm.
7.4.1 Numerical Predictions of Single-Phase Flows
0.0 0.02 0.05 0.067
The axial veloaty profiles along the duct cross-section center-line ( z = 0 in Fig.
7.1), as predicted in this work, along uith the experimental data of Lopa de Berto-
0.10 [for < a >Y 0.080j
CHAPTER 7. Fully Dereloped, lhrbulent, Bubbly Tm-Phase Flows 166
&no [111], are shown in Fig. 7.3; and the corresponding d t s along a line parallel
to ,- axis, at y = 3cm, are shown in Fig. 7.4. The results are in good agreement with
the experimental data, except near the triangle vertex (y/H = I), where the predicted
results are louw &an the experimental data. The near-wall regions were bridged by
using wall-functions in this work, thus the last node in the internal computational
domain is at y/H = 0.95 for the JL = 1 m/s test case, and at y /H = 0.90 for the
JL = 0.5 m/s test case. h can be seen in Fig. 7.3, the disagreement between the
computed and experimental resulrs for the JL = 0.5 m/s test c a ~ e is more than that
in the other case, JL = 1 m/s, which could be because larger near-wall regions were
modeled by wall-functions for the JL = 0.5 m/s test case, particularly in the vicinity
of the vertex of the triangular duct aoss-section. The results in Fig. 7.4, along the
line parallel to z-axis at y = 3 an, are in excellent agreement with the experimental
results of Lopez de Bertodano [Ill]: Along this particular l i e , the predicted veloc-
ities are not directly affected by the wall-function approximations in the vicinity of
the triangular cross-section vertex.
This problem persists in the two-phase flow simulations, which will be presented in
the next section, and it is made more serious (complicated) by the bubbles introduced
into the flow.
7.4.2 Numerical Predictions of Bubbly Two-Phase Flows
In this section, numerical predictions of the fully-developed, bubbly -phase
Bous are presented. h was mentioned earlier, for each value of the liquid superficial
velocity considered here, the specified average void-fraction values are increased in
four stages to the average void-fraction d u e which should approximate the exper-
imental data of Lopez de Bertodano [ill], for each set of the speci6ed JL and Jc
values shoun in the last column of Table 7.2.
CIWPTER 7. Fully Developed, Turbulent, Bubbly TwPhase Flm 167
Liquid-Phase Axial Velocity a n d Void -ion Profiles
The liquid-phase axial velocity and the void-fraction profiles along the center line
of the duct ( z = 0) are shown, respectidy, in Figs. 7.5 and 7.6 for JL = 1.0 m/s, for
three nominal average void-fraction d u e s , along with the experimental data of L o p
the Bertodano [Ill], as specified in Table 7.2; the corresponding profiles are shown
in Figs. 7.7 and 7.8 for JL = 0.50 m/s. The axial velocity profiles overpredict the
experimental data near the bottom d l of the triangle and underpredict them dwe to
the vertex of the triangular aoss-section. One of the causes of the latter discrepancy,
as was mentioned in the last subsection for the corresponding singlephase (< a >= 0) numerical predictions, is because of the relatively sharp acute vertex angle, and
the associated large distance of the last near-wall node from the vertex. Again,
as was explained in the context of singlephase flow results, the underpredictions
of the experimental data are larger for the results obtained for lower liquid-phase
superficial velocity (Fig. 7.7, JL = 0.5 m/s), since, for this case, the near-wall region
is larger than that for JL = 1 m/s. It should be noted that some part of these
disagreements between the numerical predictions and the experimental data could be
attributed to the fact that the predictions were made based on approximate average
void-fraction values, < a >, numerically estimated from the local experimental a
values. Furthermore, it has been assumed that the fullydeveloped conditions in the
isosceles triangular duct were achieved in the experiments of Lopez de Bertodano
[I l l ] , but, as was mentioned earlier, he has not provided explicit evidence to support
this assertion.
To understand the overprediction of the experimental data in regions close to the
bottom udl of the triangular cross-section (y = O), the corresponding void-fraction
profiles should be investigated. -4s is shown in Figs. 7.6 and 7.8, the experimental
data indicates that the void-fraction peaking occurs is in the vicinity of the bottom
wall, and, then, the void-fraction values decrease away from this wall, followed by
a region of almost constant void fraction in the central regions of the duct, and,
finally, the void-fraction values increase again, close to the vertex of the triangular
CHAPTER 7. Fully Developed, Turbulent, Bubbly Two-Phase Flows 168
moss-section. The numerically predicted void-fraction profiles follow the same general
pattern as that of the experimental data: peaking of the void fraction values occurs
in the region dose to the bottom uall, howwer, they deviate significantly from the
experimental data, in the location, m d also in the magnitude, of the peak (highest)
void-fraction in this region.
As was mentioned in Chapters 1 and 2, in the near-wall regions, there are extra
interfacial forces which d e c t , in particular, the bubble distributions in this area, as
indicated in the published works of r\fshar and Baliga [1], Marie e t al. [114], and Antal
et al. 181. However, available models of these forces are still in an embryonic stage:
Much more work is needed to establish reliable models for these extra interfacial
forces in the near-wall regions, particularly for bubbly two-phase flows in ducts. For
example, in the near-wall regions, an additional lift force (or the so-called dl-force)
has been proposed by Antal et al. [S], which could be of particular importance in
the predictions of the void fraction peaking in the near-mall regions. A model for
this force, with some fine-tuned coefficients, has been prop& by Antal et al. 181,
but when Lopez de Bertodano [ill] used this model in hi numerical predictions
of the bubbly, two phase flows of interest, he found that he had to double these
coefficients to match the experimental data [Ill]. In this work, the intension is
mainly to demonstrate the application of the proposed numerical procedures to the
flows of interest, not the fine-tuning of available models. Therefore, Antal's and other
models, which are not well established, were not tested here. Nevertheless, the d t s
obtained indicate the need for more research in developing more sophisticated models
for these extra interfacial forces, particularly for such flows in ducts. The author hopes
that the proposed numerical procedures would be useful in such model development
research.
The prediction of the smaller maximum void-fraction d u e in the vicinity of the
lower wall could also be partly attributed to the inaccuracies in the modified MAW
scheme, used in this work for the interpolation of the convection terms in the governing
equations, including that for the void fraction (Chapter 4). This scheme is known to
be first order accurate, and prone to false numerical diffusion [4], in the approximation
CHrlPTER 7. M y Developed, Turbulent, Bubbly Tm-Phase Flows 169
of the convection terms. Thus, the predicted void peaking is not as sharp as that
in the experimental data: rather, the predicted o! distribution is ilattened, and the
maximum value of a is lower than the corresponding experimental value.
The inaccurate predictions of the void peaking in a location which is displaced in
comparison to experimental data also atfects the liquid-phase axial velocity profiles
in this region. The drag force, between the liquid and bubbles, increasg with the
increase in void-fraction values in this region, and, hence, increases the numerically
predicted liquid-phase axial velocity, as shown in Figs. 7.5 and 7.7.
The numerical void-fraction values in the region near the vertex underpredict
the experimental results. Unfortunately, no experimental, tabulated, results for the
liquid- or gas-phase secondary velocities in the duct are reported by Lopez de Berto-
dano [Ill], which would have facilitated an analysis of this particular underprediction
of the experimental data. Neverthelgs, it seems reasonable to assume that modeling
of a relatively large near-mall region dose to vertex by the wall-functions has con-
tributed to this disagreement between numerical predictions and experimental d t s .
Effect of Assumed Void-Fraction Distribution in t h e Near-Wall Regions
-4s was mentioned in Chapter 3, the values of void fractions over the entire near-
wall region were assumed to be equal to their values at near-wall nodes, based on
the experimental evidence that, dose to the wall, the bubbles occupy most of the
wall region. Since the liquid-phase superficial velocities were relatively slow in these
simulations, relatively large n e a r - d l regions were bridged by the wall-functions
Therefore, any assumption for the void-fraction distribution in the near-wall regions
was expected to affect the h a l results. To investigate these effects, complementary
computer simulations were done, in which it was assumed that the void-fraction d u e s
over the entire near-wall region are zero. This is the second extreme l i t that can
be assumed for the void-fraction values in the near-wall region, compared to the first
extreme limit, in which it was assumed that the void-fraction values a t near-wall nodes
prevails over the entire near-wall region. The results obtained for the liquid-phase
axial velocity and void-fraction profiles along the duct center line (z = 0) with these
ChXPTER 7. FuUy Developed, Turbulent, Bubbly Tapphase Flows 170
mapproaches are compared in Figs. 7.9 and 7.10, respectively. As was expected, the
void-fraction and the liquid-phase axial velocity values obtained with zero values of
void fraction over the entire near-wall region are higher than the corresponding values
obtained with prevailing void-fraction (near-wall values model) over these regions. In
this test problem, the differences produced by &ese two extremes in the a ~ ~ l l ~ ~ l e d a
distributions in the near-wall regions are significant, but not dramatic.
Effect of Li-Force CoeEcient
The effect of the lift-force coefficient on the numerical predictions are shown in
Figs. 7.11 and 7.12 for the liquid-phase axial velocity and the void-fraction d u e s
along the center line of the duct -section (r = O), respectively. With rereference
to the lift-force expression given in Eq. 3.32, results are presented for CL = 0.1
and 0.2, with JL = 1.0 m/s, < Q >= 0.08. Even when the lift-force coefficient is
doubled in this range, it only slightly increases the liquid-phase axial velocity and the
void-fraction d u e s in the regions close to bottom wall (y = 0) and the vertex of the
triangular crosssection (y = H).
Effect of Bubble Diameter
The effect of the bubble diameter on the numerical predictions are shown in Figs.
7.13 and 7.14 for the liquid-phase axial velocity and the void-fraction d u e s along
the center line of the duct crosssection, respectively. For the case of JI. = 1.0 m/s,
< a >= 0.08, the numerical predictions were obtained for a nominal, uniform bubble
diameter of Db = 3mm, and are compared, in Figs. 7.13 and 7.14, with the earlier
numerical predictions, obtained, with a uniform bubble diameter of Db = 5mm. The
plots clearly indicate that the effect of the bubblediameter size is this range, on the
void-fraction and the liquid-phase axial velocity profiles are quite significant: similar
effects were also observed by Lopez de Bertodano [ l l l ] . In this work, it was asumed
that all bubbles are spherical and of uniform radius. The results presented in Figs.
7.13 and 7.14 indicate the importance of accurate experimental data on the bubble-
size distribution, and also the need for more sophisticated formulations that account
CU4PTER 7. - W y Developed, lbrbulent, Bubbly T - P k Flous in
for to involve the bubble-size distribution in the mathematical models of these flows.
7.5 Summary
In this chapter, numerical predictions of fullydeveloped, turbulent, dilute, b u b
bly two-phase flows in a vertical duct with an isosceles triangular aoss-section wexe
presented. The required experimental data for establishing appropriate inputs for the
proposed numerical method for these flows were discuaed, and the shortcomings of
the information in the available experimental data were elaborated.
The d t s obtained agree qualitatively with the experimental data of Lopez de
Bertodano [Ill]. However, this investigation shows dearly that much more d
is required to develop better models for these flows in nondrcular ducts. In particular,
the models of interfacial forces in the vicinity of the walls need much more attention:
it is also important to propose more sophisticated treatments of the near-wall regions.
The recent works of Professor J. Bataille, Dr. J.L. Marie, and their collaborators at
the Ecole Centrale de Lyon, France [90,91,123], are quite encouraging in addresing
these demands.
CH..IpTER 7. lkrbulent, Bubbly Tuv-Phase Flow Simulations
Figure 7.1: Fully-Developed, Tn-phase Bubbly Flows in a %angular Duct:
Schematic Configuration and Notation.
Figure 7.2: Fully-Developed, Two-phase Bubbly Flows in a Triangular Duct: Typical
Uniform Finite Element Mesh.
CHAPTER 7. Tkubulent, Bubbly TwPhase Flow Simulations
Figure 7.3: Wly-Developed. Two-phase Bubbly Flows in a ?tiangular Duct: Liquid-
Phase Axial Velocity Profiles Along the Line : = 0, for < a >= 0.
Figure 7.4: Fully-Developed, Ttvo-phase Bubbly Flou-s in a Triangular Duct: Liquid-
Phase Axial Velocity Profiles Along the L i e y = 3 cm, for < a >= 0.
CHAPTER 7. Turbulent, Bubbly TRV-Phase Flow Simulations
Figure 7.5: Fully-Developed, Two-phase Bubbly Flows in a Triangular Duct: Liquid-
Phase Axial Veloaty Profiles Along the L i e z = 0 for JL = 1.0.
Figure 7.6: Fully-Developed, Two-phase Bubbly Flows in a Triangular Duct: Void
Fraction Profiles Along the L i e y = 3 crn for JL = 1.0.
CHAPTER 7. Turbulent, Bubbly T m - P k Flow Simulations
Figure 7.7: Fully-Developed, Tawphase Bubbly Flows in a Triangular Duct: Liquid-
Phase Axial Velocity Profiles Along the L i e z = 0 for JL = 0.5.
Figure 7.8: Fully-Developed, Tw~phase Bubbly Flows in a Triangular Duct: Void
Fraction Profiles Along the Line z = 0 for JL = 0.5.
CHrlPTER 7. Turbdent, Bubbly Tm-Phase Flow Simulations
Figure 7.9: Fully-Developed, Tw*phase Bubbly Flom in a Triangular Duct: Liquid-
Phase Axial Velocity Profiles Along the L i e z = 0 for JL = 1.0, < a >= 0.08: Effect
of the rZssumed Void Fraction Distribution in the Xear-Wall Regions.
Figure 7.10: Fully-Developed, Twwphase Bubbly Flous in a 'Riangular Duct: Void
Fraction Profiles Along the Line z = 0 for JL = 1.0, < a >= 0.08: Effect of the
.Assumed Void Fraction Distribution in the Near-U'all Regions.
CH.WER 7. Turbulent, Bubbly T - P k Flow Simulations
Figure 7.11: Fully-Developed, Two-phase Bubbly Flows in a Triangular Duct: Liquid-
P h e Axial Velocity Profiles Along the L i e z = 0 for Jr, = 1.0, < a >= 0.08: Effect
of the Li t Force Coefficient.
Figure 7.12: Fully-Developed, TWO-phase Bubbly Flows in a Triangular Duct: Void
Fraction Profiles Along the Line z = 0 for JL = 1.0, < a >= 0.08: Effect of the Lift
Force Coefficient.
CHriPTER 7. Wbdent , Bubbly Two-Phase Flow Simulations
Figure 7.13: Fully-Developed, Two-phase Bubbly Flows in a lkiangular Duct: Liquid-
Phase Axial Velocity Profles Along the L i e z = 0 for JL = 1.0, < Q >= 0.08: Effect
of the Bubble D i e t e r .
Figure 7.14: Fully-Developed, Twc+phase Bubbly Flows in a Triangular Duct: Void
Fraction Profiles Along the L i e t = 0 for JL = 1.0, < a >= 0.08: Effect of the
Bubble Diameter.
Chapter 8
Conclusion
This chapter is divided into two sections h the Grst section, a review of the thesis
and its main contributions are presented. It is followed by a section in which some
recommendations for extensions of this work are made.
8.1 Review of the Thesis and Its Main Contribu-
tions
The work presented in this thesis was directed towards the formulation, imple-
mentation, and demonstration of numerical procedures for computer simulations of
fullydeveloped, turbulent, single-phase and d i k e , bubbly, two-phase flows in straight
ducts of arbitrary, but uniform, cros-sections. The available control-volume finite el-
ement methods (CVFEbIs) (Baliga, [Ill), were adapted, and modified appropriately,
for numerical predictions of the above-mentioned fullydeveloped flows. The fully-
developed, turbulent, single-phase flows were studied first, in order to resolve some
problems related to the numerical predictions of these flous in near-wall regions, and
also as a prerequisite to the numerical predictions of the fullydeveloped, turbulent,
dilute, bubbly, two-phase flows of interest. The various tasks that were completed in
this work are summarized in this section, along with succinct comments on the main
contributions of the thesis.
CHAPTER 8. Conclusion 180
r In Chapter 1, the main goals of the t h e and overviews of the investigations
that were undertaken to achieve them were presented. The underlying motin-
tion and rationale for this work was also duadated.
r A synopsis of published investigations relwant to the work described in this
thesis was presented in Chapter 2 The review of the literature pertaining to
single-phase, turbulent, fullydeveloped flows in nondrcular, straight ducts high-
lighted one of their distinguishingcharaaeristis Secondary redrculating flows,
in the -section of the duct, which are created solely because of anisotropy
in the flow turbulence. These secondary motions have a significant effect on the
axial velocity profiles in the duct crosssection, and also on the distribution of
the local mal l shear stress.
The posibiities offered by direct numerical simulations (DNS), large eddy sim-
ulations (LES), differential Reynolds stress models (RSMs), algebraic stress
models (ASMs), nonlinear twoquation eddy-viscosity models, and linear two-
equation eddy-viscosity madels of turbulent flows were examined, in the context
of engineering computer predictions of the flows of interest. It was concluded
that the non-linear eddy-viscosity and an explicit algebraic stress turbulence
models offer decisive admntages with regard to the objectives of this work:
These models are relative simple to incorporate into CVFEMs; they do not
impose overly large demands on computer resources; and they provide the ca-
pability to predict the aforementioned recirculating secondary flows in ducts of
noncircular cros-sections.
The review of the literature pertaining to dilute, bubbly, -phase flows showed
that the mathematical models of these flows are becoming increasingly well
established, even though there is much work to be done before quantitaively
accurate predictions of these flows become feasible. It was found that the so-
called twwfluid models have been used successfully to simulate such flows in
ducts with regular geometries, for example, in pipes of circular cross-section and
in ducts of rectangular cross-section, but there is a need for more sophisticated
CHAPTER 8. Conclusion 181
numerical methods that would fadlitate the prediction these flows in ducts with
complex, irregular geometries
The review of control-volume finite element methods (CVFEMs) indicated that
these methods were a good candidate for computer simulations of the flows con-
sidered in this work. Ho-r, it was argued that new procedures are required
for proper implementation of wall functions in CVFEhk, in order to affordably
and consistently bridge tke near-wall regions in the turbulent flows of interest.
0 In Chapter 3, the mathematical models used in this work for computer simu-
lations of the Nlydeveloped, turbulent single- and bubbly two-phase flows in
straight ducts were presented. The relwant differential equations that govern
the transport of mass and momentum, models and correlations for turbulence
terms and interfacial forces, and logarithmic laws of the wall and related wall
functions, were concisely discused. These equations and models were appropri-
ately simplitied, while respecting the essential physics of the flows considered,
in order to make the corresponding numerical predictions affordable for engi-
neering purposes. In particular, the six aeren t ia l transport equations for the
six independent Reynolds stress components, and the three differeatid trans-
port equations for the gas-phase momentum components in the three Cartesian
coordinate directions, were replaced by simplified algebraic equations. Also, the
fullydweloped flows of interest were solved in a two-dimensional domain over
the d u a crosssection, rather than starting the solution from the d u a inlet, and
advancing the computations along the length of the duct to the fullydeveloped
region. These simplifications and solution strategies significantly reduced the
computer CPU time and memory requirements in the simulations of the flows
of interest.
0 In Chapter 4, the formulation of a control-volume finite element method (CVFEM)
for computer simulations of steady, planar, two-dimensional, fluid flow was pre-
sented. This CVFEM is based on the CVFEMs of M s o n et al. [I181 and Baliga
[Ill. These CVFEhls were modified appropriately to make them suitable for
simulation of the flows of interest. In particular, tao nonlinear eddy-viscoSity
and an expliat algebraic stress turbulence models rnere appropriately imple-
mented u the proposed CVFEM: This is the &st time that such turbulence
models have been implemented in C V F E b k
In this work, a new procedure was dewloped for the implementation of wall
functions in CVFEM simulations of fullydeveloped, turbulent flows in ducts
with complex, irregular-shaped acs-sections In this procedure, described in
detail in Chapter 5, the near-wall regions are briiged by wall-function treat-
ments which ensue mass consenation; in contrast, similar procedures proposed
earlier in the published literature did not ensure mass consenation. The pro-
posed procedure uas &st developed for fullydeveloped single-phase turbulent
flows, and is applicable to smooth walls and also to walls with corners. An a d a p
tation of this procedure to make it appli&,ie to fullydeveloped, dilute, bubbly,
turbulent -phase flous in ducts with irregular cross-sections was also worked
out.
The results of computer simulations of fullydeveloped, turbulent, single-phase
flows in ducts were presented and discussed in Chapter 6. These simulations
were performed for flows in ducts uith four different cross-sections, namely, a
square, a tilted square, a sector, and an equilateral triangle. Results were ob-
tained with four turbulence models: a linear two-equation eddy-viscosity model;
a quadratic and a cubic two-equation nonlinear eddy-viscosity models; and an
evplidt algebraic stress model. Grid-independent results were estimated for
each test case using a modified Richardson's extrapolation procedure. The
errors in the resuks obtained uith the final recommended grid size, in com-
parison to the corresponding estimated grid-independent values, were of the
order of 1% - 3%. The results obtained using the E.GM are, overall, in better
agreement with the available experimental results, than those produced by the
other aforementioned turbulence models. The linear EVM, as uas expected,
failed to predict any secondary flows in the duct crosssection. It should also be
CHAPTER 8. h d l s i o n 183
noted that the comparative evaluation of the abovementioned four turbulence
models, in the context of fullydadoped turbulent flow in ducts, has been done
for the first time in this work.
In Chapter 7, the results of the computer simulations of Nlydeveloped, tur-
bulent, dilute, bubbly, two-phase flows in a duct with an isosceles triangular
crosssection were presented. They were intended to demonstrate the possi-
bilities offered by the proposed nmaical methods and procedures in the in-
vestigations of such flows. Some shortcomings of the available experimental
data, with regard to their usefulness in the validation of numerical predictions
of the turbulent, dilute, bubbly two-phase flows of interest, were also discussed.
The numerical rgults obtained were found to be are in acceptable qualitative
agreement with the anihble expemental data for the chosen test problem.
8.2 Recommendations for Extensions of This Work
A few recommendations for improvements and extensions of this work are pre-
sented in this section.
fillydeveloped, turbulent, single-phase flows:
- An extension of the proposed procedure for the implementation of wall
functions in twc-dimensional, planar, CVFEMs to twdimensional ax-
kymmetric and t h d m e n s i o d CVFEMs would be very useful from the
point of view of engineering applications.
- The utilization of more advanced solution algorithms to improve the con-
vergence rate of the numerical solution of these problems is highly desire-
able. The enhanced sequential solution algorithm of Afshar and Baliga
[I, 31 and algebraic multigrid methods 1671 are of particular interest in this
w d -
- The finite element meshes in this work were generated using structured
grids. It would be very useful to incorpolate the ability to construct finite
CHrlPTER 8. Conclusion 184
element meshes using unstructured or/and adaptive grid strategig. In
this context, the recent work of Venditti and Baliga [I861 is of special
interest.
FUydeveloped, turbulent, dilute, bubbly -phase flows
- Considering the challenges which exist in the modeling of interfacial and
turbulence terms in these flows, m y extensions and improvements of math-
d c a l models and numerical solution methods for the prediction of such
flows require more suitable experimental d a t a In this regard, the following
suggestions are made:
For the fullydeveloped flows of interest, reliable indicators are needed
to establish that the experimental data does, indeed, apply to the
fullydeveloped region. In particular, the changes in the axial pressure
gradient along the duct axis, the density of the gas phase, and the
average void-fraction values in the fullydeveloped region are critical
experimental data that are needed.
Experimental data for the flows of interest in ducts with a variety of
irregular-shaped cross-sections would be very useful.
* Detailed experimental data for fully-developed, turbulent, dilute, b u b
bly two-phase flows in straight ducts, in the vianity of walls, similar to
the results obtained by Lance et al. [91,90] are needed: They would
allow the development of an appropriate logarithmic law of the wall,
as well as related wall functions, for such flows.
- The mathematical models of the bubbly tuwphase flows of interest require
substantial improvements. Some comments in this regard follow:
* Improved models are needed for the additional drag and lift forces on
the bubbles in the vicinity of walls.
* In this work, it was assumed that the bubbles are uniform spheres.
There have been many efforts to develop models for predictions of
ClUlTER 8. Conclusion 185
the bubblesize distribution and bubble shapes, and bubblebubble
interactions in these flows The works of Serizawa e t al. [168], and
Nigmatulin et al. 1130) are interesting in this regard. Nevertheless,
more research is needed to establish reliable models of bubblebubble
interactions, and account for the effects of bubblesize distribution and
non-spherical bubble shapes.
* The superposition oi the shear- and bubbleinduced turbulence terms
should be improved by considering the posible nonlinearities and cou-
plings between these terms.
- Improvements are also needed in the numerical methods for solving the
mathematical models of these these flows. Examples include the following:
Improvements in solution algorithms and grid-generation techniques,
similar to those already suggested for singlephase flows, are highly
desireable.
* Regarding the purely advective nature of the differential equation that
governs the distribution of the void fraction in the flows of interest, im-
proved d i t i z a t i o n schemes (better than the -weighted scheme
used in this work) are required, in order to reduce the effects of false
(or numerical) di&sion on the results.
* In this work, considering that the bubble (air) density is much smaller
than the corresponding liquid (water) density, the gas-momentum
equations were simplified to algebraic balances between interfacial
forces, pressure, and buoyancy terms. Nevertheless, to have a more
general simulation tool, the extension of the proposed CVFEM to solve
the full differential equations that govern the gas-phase momentum is
suggested.
In conclusion, the author hopes that the methods and ideas presented in this
thesis would be useful to other researchers interested in turbulent singlephase and
gas-liquid tmphase floss. He also hopes that this work has contributed positively
CHAPTER 8- Condusion 186
to our understanding of these compliorted phenomena, at least in some small way.
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Appendix A
Calculations of Relat ive-Velocity
Components
In this appendix, the calculations of relative-veloaty components are explained.
As was discussed in Chapter 3, the \dues of relative velocity components in z, y and
z directions, namely, u,, v, and w,, could be obtained by solving the simplified forms
of the gasnomenturn equations (Egs 3.52-3.54). These equations, as represented in
the compact form of Eq. 3.55, are repeated here:
These sets of equations are nonlinear and coupled, and the coefficients in Eq. (3.55)
depend on u,, v,, and w,. Hence, in each cyde of an iterative solution procedure, the
most recent d u e s of u,, v,, and w, were used to calculate the coefficients in these
equations (successive-substitution procedure), and equations were solved simultane-
ously.
The expressions for the terms in the coefficient matrix on the left hand side of Eq.
A.l, and the terms on the right hand side of this equation are as follows:
MPEIVDLX A. Calculations of Relatit+Veloaty Components 207
< = apICL- az < = -aplC L --- L a,)
The values of relative velocities are obtained by solving Eq. A.l simultaneously for
w, v,, and w,.
where D m , is calculated from the following equation:
Numerical details for the calculation of these terms in the context of the proposed
CVFEM ware d i d in Section 4.6.1.
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