Numerical Investigation using RANS Equations of Two ......Kareem Akhtar Abstract This thesis...

Numerical Investigation using RANS Equations of Two-dimensional Turbulent Jets

and Bubbly Mixing Layers

Kareem Akhtar

Thesis submitted to the Faculty of Virginia Polytechnic Institute and State

University in partial fulfillment of the requirements for the degree of

Master of Science

in

Engineering Science and Mechanics

Saad A. Ragab, Chairman

Muhammad R. Hajj

Mark Cramer

July 26, 2010

Blacksburg, Virginia

Keywords:(confined jets, bubbly mixing layers)

Numerical Investigation using RANS Equations of Two-dimensional Turbulent Jets

and Bubbly Mixing Layers

Kareem Akhtar

Abstract

This thesis presents numerical investigations of two-dimensional single-phase turbulent

jets and bubbly mixing layers using Reynolds-Averaged Navier-Stokes (RANS) equations.

The behavior of a turbulent jet confined in a channel depends on the Reynolds number

and geometry of the channel which is given by the expansion ratio (channel width to jet

thickness) and offset ratio (eccentricity of the jet entrance). Steady solutions to the RANS

equations for a two-dimensional turbulent jet injected in the middle of a channel have been

obtained. When no entrainment from the channel base is allowed, the flow is asymmetric

for a wide range of expansion ratio at high Reynolds number. The jet attaches to one of

the channel side walls. The attachment length increases linearly with the channel width for

fixed value of Reynolds number. The attachment length is also found to be independent of

the (turbulent) jet Reynolds number for fixed expansion ratio. By simulating half of the

channel and imposing symmetry, we can construct a steady symmetric solution to the RANS

equations. This implies that there are possibly two solutions to the steady RANS equations,

one is symmetric but unstable, and the other solution is asymmetric (the jet attaches to one

of the side walls) but stable. A symmetric solution is also obtained if entrainment from jet

exit plane is permitted. Fearn et al. (Journal of Fluid Mechanics, vol. 121, 1990) studied the

laminar problem, and showed that the flow asymmetry of a symmetric expansion arises at

a symmetry-breaking bifurcation as the jet Reynolds number is increased from zero. In the

present study the Reynolds number is high and the jet is turbulent. Therefore, a symmetry-

breaking bifurcation parameter might be the level of entrainment or expansion ratio.

The two-dimensional turbulent bubbly mixing layer, which is a multiphase problem, is

investigated using RANS based models. Available experimental data show that the spreading

rate of turbulent bubbly mixing layers is greater than that of the corresponding single phase

flow. The presence of bubbles also increases the turbulence level. The global structure of the

flow proved to be sensitive to the void fraction. The present RANS simulations predict this

behavior, but different turbulence models give different spreading rates. There is a significant

difference in turbulence kinetic energy between numerical predictions and experimental data.

The models tested include k− �, shear-stress transport (SST), and Reynolds stress transport

(SSG) models. All tested turbulence models under predict the spreading rate of the bubbly

mixing layer, even though they accurately predict the spreading rate for single phase flow.

The best predictions are obtained by using SST model.

iii

Dedication

Dedicated to my Parents.

iv

Acknowledgments

First of all, I would like to thank my parents for their support and prayers which they made

for my success. I would like to express the deepest appreciation to my friend and academic

adviser Dr. Ragab, whose kind guidancemade me only possible to accomplish this work.

Without his guidance and persistent help this thesis would not have been possible. He was

always there at his office even on weekends to answer my questions and review my work. I

simply have no words to explain his help and support. I would also like to thank Dr. Salem

Said who helped me how to use ANSYS and CFX. I would also like to thank Dr. Imran

Akhtar for his generous help. I would then like to thank Tim Tomlin. Tim spent hours and

hours with me to show how to use lcc system. I am also very thankful to Dr. Hajj for his

kindness and support during my studies. I am also grateful to Dr. Cramer for his input and

taking the time out of their busy schedule to review my work. I would also like to thank to

my close friends for all their support.

v

Contents

Abstract ii

Dedication iv

Acknowledgments v

Contents vi

List of Figures ix

List of Tables xii

1 Two-Dimensional Turbulent Jets in a Symmetrical Channel 1

1.1 Introduction: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Governing Equations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Channel Geometry: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Boundary Conditions and Mesh Details: . . . . . . . . . . . . . . . . . . . . . 11

1.5 Effects of Expansion Ratio on the Turbulent Jet: . . . . . . . . . . . . . . . . 13

1.6 Effects of Channel Length: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

vi

Kareem Akhtar Contents

1.7 Effects of Entrainment: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.8 Effect of Numerical Treatment of the Outlet Boundary Condition: . . . . . . 23

1.9 Effects of Reynolds Number: . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.10 Conclusions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Numerical Simulations of Turbulent Bubbly Mixing Layers 31

2.1 Introduction: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Governing Equations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3 Mixing Layer Geometric Details . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Boundary Conditions and Grid Description: . . . . . . . . . . . . . . . . . . . 39

2.5 Numerical Simulations Parameters: . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 Liquid (Water) Superficial Velocity Profiles: . . . . . . . . . . . . . . . . . . . 43

2.7 Mixing Layer Growth Rate: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.8 Gas Superficial Velocity Profiles: . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.9 Void Fraction: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.10 Fine Grid Simulations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.11 Comparison Between Different Turbulence Models: . . . . . . . . . . . . . . . 54

2.12 Comparison with Experimental Data: . . . . . . . . . . . . . . . . . . . . . . 55

2.13 Conclusions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 A Numerical Study of Gas Hold Up in a Water Tank Supplied by a Dual

Jet of Air and Water 62

vii

Kareem Akhtar Contents

3.1 Geometric Details: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Grid Generation and Geometric Details: . . . . . . . . . . . . . . . . . . . . . 64

3.3 Numerical Simulations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Results: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Comparison and Conclusion: . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

viii

List of Figures

1.1 Two-dimensional computational domain . . . . . . . . . . . . . . . . . . . . . 10

1.2 Two-dimension grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 (a) Velocity contours (b) Streamlines, W=75mm, Re=30000 . . . . . . . . . . 13

1.5 Velocity contours for different width (W) . . . . . . . . . . . . . . . . . . . . 14

1.6 Streamlines for different width (W) . . . . . . . . . . . . . . . . . . . . . . . . 15

1.7 Attachment length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.8 Attachment length as function of expansion ratio . . . . . . . . . . . . . . . . 17

1.9 Non-dimensional attachment length vs channel width . . . . . . . . . . . . . . 18

1.10 Velocity contours for different channel length, W=110mm,Re=30000 . . . . . 20

1.11 Velocity streamlines for different channel length, W=110mm,Re=30000 . . . 20

1.12 Attachment length vs channel length with fixed width, W=110mm, Re=30000 21

1.13 Vertical velocity at section y=80mm for different channels with different lengths 21

1.14 Velocity contours and streamlines for different channel length, width=110mm 22

1.15 Special domain (a) boundary conditions (b) velocity contours . . . . . . . . . 24

1.16 Special domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

ix

Kareem Akhtar List of Figures

1.17 Velocity contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.18 Velocity contours for different Reynolds numbers . . . . . . . . . . . . . . . . 26

1.19 Attachment length vs Reynolds number . . . . . . . . . . . . . . . . . . . . . 26

2.1 Mixing layer geometry and boundary conditions . . . . . . . . . . . . . . . . . 39

2.2 (a)coarse grid (b)coarse grid (c) coarse grid . . . . . . . . . . . . . . . . . . . 40

2.3 (a)fine grid (b) trailing edge of splitter . . . . . . . . . . . . . . . . . . . . . 41

2.4 Velocity contours for single phase flow run 2-1 . . . . . . . . . . . . . . . . . . 44

2.5 Water superficial velocity for run 2-2 . . . . . . . . . . . . . . . . . . . . . . . 44

2.6 Water superficial velocity for run 2-4 . . . . . . . . . . . . . . . . . . . . . . . 45

2.7 Water velocity profiles for single phase run 2-1 . . . . . . . . . . . . . . . . . 46

2.8 Water superficial velocity profiles for run 2-2 . . . . . . . . . . . . . . . . . . 46

2.9 Water superficial velocity profiles for run 2-4 . . . . . . . . . . . . . . . . . . 47

2.10 Mixing layer thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.11 Mean velocity profile for single phase run 2-1 . . . . . . . . . . . . . . . . . . 49

2.12 Mean water superficial velocity for two-phase run 2-2 . . . . . . . . . . . . . 50

2.13 Mean water superficial velocity for two-phase flow run 2-4 . . . . . . . . . . . 50

2.14 Mean superficial air velocity profiles for run 2-4 . . . . . . . . . . . . . . . . 51

2.15 Mean superficial air velocity profiles for run 2-2 . . . . . . . . . . . . . . . . 52

2.16 Void fraction run 2-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.17 Mean velocity in the liquid phase run 2-4 . . . . . . . . . . . . . . . . . . . . 53

2.18 Mean velocity in the liquid phase run 2-2 . . . . . . . . . . . . . . . . . . . . 54

2.19 Liquid superficial velocity profiles for different turbulence models . . . . . . . 55

x

Kareem Akhtar List of Figures

2.20 Comparison with the experiments for single phase flow . . . . . . . . . . . . . 56

2.21 Comparison with the experiments two-phase run 2-4 . . . . . . . . . . . . . . 57

2.22 Comparison with the experiments two-phase run 2-2 . . . . . . . . . . . . . . 57

2.23 Turbulent energy comparison with experimental data for run 2-4 . . . . . . . 58

3.1 Geometry details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 Two-dimensional grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 Air volume fraction contours for case 1: air jet below water jet . . . . . . . . 68

3.5 Air volume fraction contours for case 2: air jet above water jet . . . . . . . . 69

3.6 Air volume (%) in water tank for air jet below water jet . . . . . . . . . . . . 70

3.7 Air volume (%) in water tank for air jet above water jet . . . . . . . . . . . . 70

3.8 Water volume (%) in air tank for air jet below water jet . . . . . . . . . . . . 71

3.9 Water volume (%) in air tank for air jet above water jet . . . . . . . . . . . . 71

3.10 Air volume fraction comparison between case1 and case2 . . . . . . . . . . . 73

3.11 Water volume fraction comparison between case1 and case2 . . . . . . . . . . 73

xi

List of Tables

2.1 Fine and coarse grid details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Inlet conditions for different runs . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Numerical and experimental vlaues for σL . . . . . . . . . . . . . . . . . . . . 56

xii

Chapter 1

Two-Dimensional Turbulent Jets in

a Symmetrical Channel

This chapter presents a numerical study of the behavior of a two-dimensional turbulent jet

issuing from a slit in a wall and flowing midway between two parallel plates. The flow

geometry may also be considered to be a sudden expansion of a symmetric channel where the

expansion ratio is very high. Even though the boundary conditions support the possibility of

a flow that is symmetric about the channel mid-plane, experimental studies show asymmetric

flow; except at a very low Reynolds number (laminar flow) where the flow may be symmetric.

The behavior of the confined turbulent jet depends on the Reynolds number and geometry

of the channel given by expansion ratio (channel width to jet thickness) and offset ratio

(eccentricity of the jet entrance).

A numerical study is conducted to determine the effects of the expansion ratio (ratio of

channel width to jet thickness) on the jet behavior. The jet simulations are performed using

1

Kareem Akhtar Chapter 1 2

ANSYS Fluent-6. For a wide range of expansion ratio and high Reynolds number (turbu-

lent flow) the channel flow is asymmetrical and the jet attaches to one of the side walls. In

our numerical simulations we studied the behavior of a turbulent jet at Reynolds number

(Re=30000) for different values of expansion ratio. The attachment length increases linearly

with expansion ratio. The effects of the channel length and inflow/outflow boundary condi-

tions on this phenomenon are investigated. The channel length has no effect provided that it

is greater than the expected attachment length. The study also shows that the asymmetric

flow phenomenon is independent of the numerical treatment of the outflow boundary con-

dition, but dependent on the inflow condition. Numerical simulations show that the flow is

symmetrical only when entrainment is allowed at the inflow boundary.

1.1 Introduction:

Jets have numerous applications such as environmental discharges, burners, injectors, mixing

and flotation cells, automobiles etc. Confined jets exhibit a wide range of flow features

such as oscillations in a symmetric channel, asymmetric flow under symmetric conditions,

deflection towards the boundary, attachment with the solid boundary and change of flow

pattern under entrainment condition. All these features are important and can affect device

efficiency and performance. Confined laminar jet at low Reynolds number and moderate

expansion ratio (duct width/jet width) is studied and well documented by Sarma et al.

(2000). Confined symmetric and offset jet was studied numerically by Ouwa et al. (1986)

but for fixed expansion ratio and Reynolds number from 400 to 8000. Experimental and

numerical studies show that the symmetrical confined jet behavior depends on Reynolds


number as well as on expansion ratio. The effects of Reynolds number on confined symmetric

jet is studied by many authors, mostly laminar flow, no significant study is available on the

effects of expansion ratio on turbulent jet. The purpose of the present study is to investigate

the behavior of symmetrically confined turbulent jet under different expansion ratios at high

Reynolds numbers.

Due to its fundamental nature of the problem and its applications, jet flow is studied

from time to time. New features and properties are explored each time. Different aspects

of laminar and turbulent, free and confined jets with different geometric conditions (offset

ratios), have been studied. Sato (1960) studied the stability and transition of a two dimen-

sional jet. He deduced that, the oscillations in two-dimensional jet observed in the vicinity

of shear layer are caused by small velocity fluctuations, originated near the nozzle exit plan.

Fully developed nozzle velocity profiles led to anti-symmetric shedding, whereas undeveloped

profiles to symmetric shedding. However, he was unable to find the critical Reynolds number

for anti-symmetric flow. Sato and Sakao (1964) were able to distinguish the flow on the basis

of Reynolds number by investigating experimentally the stability of a two-dimensional jet

flow at low Reynolds number. They found that the whole jet is laminar if the jet Reynolds

number (based on the slit width and the maximum velocity of the jet) is less than 10. There

are some periodic fluctuations when Reynolds number is between 10 and 50. When Reynolds

number is greater than 50 they found that the periodic fluctuations develop into irregular

periodic fluctuations.

Durst et al. (1974) and Cherdron et al. (1978) demonstrated that symmetric flows can

exist in two-dimensional (plane) symmetric sudden expansion ducts for only a limited range


of Reynolds numbers. At higher Reynolds numbers, the small disturbances generated at

the lip of the sudden expansion are amplified in the shear layers formed between the main

flow and the recirculation flow in the corners. The result is a shedding of eddy-like patterns

which alternate from one side to the other with consequent asymmetry of the mean flow,

particularly of the sizes of the two regions of re-circulations. The anti-symmetric shedding

causes asymmetric mean flow patterns.

Mitsunaga and Hirose (1979) studied flow patterns in a rectangular channel. They found

that the critical Reynolds number ranges from 40 to 60 for transition from symmetric to

asymmetric flow. Ouwa et al. (1981) also studied two-dimensional water jet flow in a rectan-

gular channel and found that the critical Reynolds number (based on nozzle width) is of the

order of 30. The asymmetric flow remains steady up to Reynolds number of the order 60. The

main flow becomes unsteady for Reynolds number higher than of 60. The two experiments

by Mitsunaga et al. and Ouwa et al. have different offset ratio.

Sobey (1985) investigated steady and oscillatory flow through a two-dimensional channel

expansion. He reported vortex wave (due to shear layer instability) during steady flow past

a moving indentation in a channel. Ouwa et al. (1986) studied the characteristics of confined

symmetric, asymmetric and wall jets at low Reynolds number in a rectangular channel. They

found that, the transition from laminar to turbulent flow occurs when AR ≤ cRem, regardless

of the configuration of jet, where AR is the aspect ratio (height/length); c and m are empirical

constants.

Fearn et al. (1990) studied the asymmetric flow in a symmetric channel using bifurcation

theory and numerical simulations. They found numerically that the asymmetry rises at the


critical Reynolds number Rc > 40±0.17%, and a third recirculation region with the wall was

observed at Reynolds number 125. Mataoui and Schiestel (2009) also found the oscillation

of jet flow inside a channel cavity (one end closed) both experimentally and numerically.

Steady flow (non-oscillatory flow) was found at high value of aspect ratio (height/length),

however oscillatory flow (periodic oscillation) was found for several geometric and dynamical

parameters. They found that the jet flapping frequency increases linearly with Reynolds

number and decreases a lot with height of the cavity.

Along with experimental studies, numerical studies were also carried out from time to

time, under different geometric and boundary conditions. Ouwa et al. (1986) numerically

simulated two-dimensional plane jet in confined symmetric and asymmetric channel by means

of successive over-relaxation upwind schemes. Both symmetric and asymmetric flow patterns

were obtained for symmetric channel with critical Reynolds number Rec = 30. Different

flow patterns were reported for asymmetric channel with Reynolds number from 100 to 4000.

Umeda et al. (1990) simulated under-expanded slab jets by solving two-dimensional Eulerian

equations using second order-accurate explicit Osher scheme, with high aspect ratio i.e. 8 and

with a pressure ratio of the jet 4.48. They found that two- dimensional slab jet is symmetric

when symmetric boundary condition is applied. The asymmetric self sustained oscillations

occur when the asymmetric boundary condition is applied. The oscillations continue even if

the asymmetric boundary condition is replaced by symmetric boundary condition.

Gu (1996) simulated two-dimensional turbulent offset jets by solving two-dimensional

unsteady Navier-stokes equations for velocity components along with k− � turbulence model.

The two-dimensional model predicted that attachment of non-buoyant offset jet with offset


ratio 25 to a boundary did not occur in a case of symmetric geometry and external forces

about the initial jet centerline. The symmetric flow becomes stable if the discharge is located

at a sufficiently large distance from boundaries and if no disturbance is introduced.

Sarma et al. (2000) simulated laminar jet flow in a confined channel. They found that at

low expansion ratio (channel width/jet width) and Reynolds number the jet is symmetric, but

for higher expansion ratio and Reynolds number the jet flow at steady state becomes asym-

metric. For still higher values it becomes oscillatory with respect to time. When entrainment

is introduced the asymmetric instabilities and temporal oscillations occur at higher critical

Reynolds number for given expansion ratio. For fixed expansion ratio when Reynolds num-

ber is increased the flow development first becomes asymmetric and for still higher Reynolds

number it develops temporal oscillations.

Deniskhina et al. (2005) simulated plan turbulent jet flow into a rectangular cavity i.e.

with one end closed (dead end). The calculations were performed for two flow modes of which

the first one is statistically steady and second is self-oscillatory. Three different approaches

were used for numerical simulations i.e. large eddy simulations (LES) with subgrid model of

Smagorinsky and steady and unsteady Reynolds averaged Navier-Stokes equations (SRANS

and URANDS). For the first flow mode all the three approaches gave close results, but for

second flow mode (self- oscillatory) best results were obtained by LES and three dimensional

URANS.

Kana and Das (2005) simulated two-dimensional incompressible non-buoyant offset jet by

stream functions and vorticity formulation considering the problem as asymptotic solution to

the transient equation. They studied the behavior of the jet with respect to offset ratio and


Reynolds number. It was found that the reattachment length depends on Reynolds number

and offset ratio. Reattachment length increases with the increase of Reynolds number for

fixed offset ratio. Also at high offset ratio the decay of the horizontal velocity component

depends on Reynolds number only.

1.2 Governing Equations:

The Reynolds-averaged continuity and momentum equations (RANS equations) are given by

∇ • U = 0 (1.1)

∂ρU

∂t+∇ • (ρU ⊗ U) = ∇ • {τ − ρu′ ⊗ u′} (1.2)

where the instantaneous velocity is decomposed into mean and fluctuating components,

u = U + u′ (1.3)

and

U =1

4t

∫ t+4tt

u dt (1.4)

ρ=Density, u=velocity vector, U =average velocity,u′=fluctuating velocity τ = stress tensor.

Term ρu′ ⊗ u′ is called Reynolds stress tensor. The k − � which is a two-equation model

uses the gradient diffusion hypothesis to relate the Reynolds stresses to the mean velocity

gradients and the turbulent viscosity. The turbulent viscosity is modeled as the product of a

turbulent velocity and turbulent length scale. The turbulence velocity scale is computed from

the turbulent kinetic energy, which is provided from the solution of its transport equation.

The turbulent length scale is estimated from two properties of the turbulence field, usually


the turbulent kinetic energy and its dissipation rate. The dissipation rate of the turbulent

kinetic energy is provided from the solution of its transport equation. The k − � model

introduces two new variables into the system of equations. The continuity equation remain

the same, and momentum equation becomes,

∂ρU

∂t+∇ • (ρU ⊗ U) = −∇p̃+∇ •

(µeff

(∇U +∇UT

))(1.5)

µeff is the effective viscosity accounting for turbulence, and is given by

µeff = µ+ µt (1.6)

p̃ is the modified pressure and is given by defined by

p̃ = p+2

3ρk +

2

3µt∇ • U (1.7)

where µt is the turbulent viscosity. For k − � model, turbulent viscosity is given by

µt = Cµρk2

�(1.8)

Cµ is a constant with value 0.09. k is the turbulent kinetic energy and is defined as the

variance of the fluctuations in velocity. � is the turbulence eddy dissipation (rate at which

turbulence kinetic energy is dissipated). The values of k and � come directly from the dif-

ferential transport equations for the turbulence kinetic energy and turbulence dissipation

rate:

∂(ρk)

∂t+∇ • (ρUk) = ∇ • [

(µ+

µtσk

)∇k] + Pk − ρ� (1.9)

∂(ρ�)

∂t+∇ • (ρU�) = ∇ • [

(µ+

µtσ�

)∇�] + �

k(C�1Pk − C�2ρ�) (1.10)


Where C�1, C�2, σk and σ� are constants with values 1.44, 1.92, 1 and 1.3, respectively. Pk is

the turbulence production due to viscous and buoyancy forces given by

Pk = µt∇U •(∇U +∇UT

)− 2

3∇ • U (3µt∇ • U + ρk) + Pkb (1.11)

Pkb is buoyancy production term and depends on the buoyancy turbulence, which are given

in CFX manual. Based on k−ω, SST model accounts for the transport of the turbulent shear

stress and gives highly accurate predictions of the onset and the amount of flow separation

under adverse pressure gradients. Although, the Baseline (BSL) k − ω model combines the

advantages of theWilcox k − ω and the k − � model, but still fails to properly predict the

onset and amount of flow separation from smooth surfaces. The main reason is that both

models do not account for the transport of the turbulent shear stress. This results in an over

prediction of the eddy-viscosity. The proper transport behavior can be obtained by a limiter

to the formulation of the eddy-viscosity given by

νt =a1k

max(a1ω, SF2)(1.12)

where S is an invariant measure of the strain rate. F2 is a blending function, which restricts

the limiter to the wall boundary layer, as the underlying assumptions are not correct for free

shear flows. The blending function is given by

F2 = tanh(ζ2

2)

(1.13)

ζ2 = max

(2√k

β′ωy,500v

y2ω

)(1.14)

Where y is the distance to the nearest wall, ν is the kinematic viscosity,ω is the turbulent

frequency and β′ is constant having value 0.09.


1.3 Channel Geometry:

The baseline two-dimensional geometry used in simulations is shown in figure 1.1. The domain

width (W) is 75mm and height (H) is 180mm. Although we changed the width and height

for different runs but we started each series from this baseline geometry. The jet nozzle exit

is 3mm. The jet thickness and jet velocity are kept constant in each simulation. Also in all

simulations b1 = b2 (see figure 1.3), so the offset ratio in all runs is equal to 1. The entrance

channel length of the jet is extended in the upstream direction up to 60mm in order to get

a fully developed profile. Our simulations show that the jet is sensitive to boundary layer

profile at the exit. For some cases different results were observed for with and without the

extended entrance channel of jet. The extended entrance channel profile is used in all runs.

k − � turbulent model is used for all presented simulations.

Figure 1.1: Two-dimensional computational domain


1.4 Boundary Conditions and Mesh Details:

ANSYS ICEM CFD 12.0 is used for grid generation. To minimize truncation error, very

dense grid is used near all walls. Different sections of the grid are shown in figure 1.2. The

boundary conditions are labeled in figure 1.3. We have three types of boundary conditions:

wall, inlet and opening. At the bottom we have velocity inlet for water jet, at the top opening

boundary condition, where static pressure is prescribed, is used. All the rest of the boundaries

are no-slip walls. The results show that there is no difference between the boundary condition

“opening” and “outlet”. We keep the “opening” condition in all runs which is more physical.

Figure 1.2: Two-dimension grid


The boundaries labeled as side “b1” and “b2” are changed to “pressure inlets” for en-

trainment case, discussed in section 1.6 in detail. The velocity of the jet is kept at 10(m/s)

in each run. So, with the jet thickness of 3mm, and kinematic viscosity of 1 × 10−6(m2/s)

the Reynolds number is 30000 in all simulations.

Figure 1.3: Boundary conditions

Results:

Results are arranged into five different sections: effects of expansion ratio on turbulent

jet, effects of channel length, effects of entrainment, effects of numerical treatment of the

outlet boundary condition and effects of Reynolds number.


1.5 Effects of Expansion Ratio on the Turbulent Jet:

In this section, we discuss the effects of expansion ratio. We define expansion ratio as the

ratio of the channel width to the jet thickness. The velocity contours and streamlines are

shown in figures 1.4. It is evident that the flow is asymmetrical. In fact the jet deflects

towards the boundary wall, either left or right, and attaches to it. The jet remains attached

all the way to the exit boundary. There are three circulation regions, one in the middle of

channel and one in each corner. The size of circulation region changes with change in width

of the channel.

Figure 1.4: (a) Velocity contours (b) Streamlines, W=75mm, Re=30000

To study the effects of distance between the walls on this phenomenon, different simula-

tions are run by changing the width of the channel, but keeping the Reynolds number and

the channel length constant. The purpose of these runs is to see whether the flow becomes

symmetrical after certain width, second to determine the changes in the attachment distance

of the jet. The velocity contours and streamlines for different width or expansion ratio are

plotted in figures 1.5 and 1.6, respectively.


Figure 1.5: Velocity contours for different width (W)


Figure 1.6: Streamlines for different width (W)


We observe that the jet attaches to one of the walls. The jet attaches randomly to one

of the walls if the channel width is less than 180mm, which the length of the computational

domain. By further increasing the channel width, we find that the jet does not attach to

either wall in the given length, but it oscillates in the domain of length 180mm. Even for

very high expansion ratio and width of 250mm, the jet oscillates indicating that it would

attach if the length is increased. We define the attachment point as the point where the

wall shear stress is zero as sketched in figure 1.7. Below this point we have the entrapped

fluid in circulation region, and above this point the jet moves up along the wall and exits the

channel. The point is determined by visual interrogation of the streamlines in the channel.

The distance of the attachment point from the bottom of the channel is the attachment

length. By changing the width and keeping the length and thickness of the jet constant,

we determined the attachment length for a range of expansion ratio. It is found that the

attachment length increases linearly with increase of expansion ratio as shown in figure 1.8.

However, the ratio of the attachment length to the width of the channel is approximately

equals 0.7; hence it is nearly independent of the channel width as shown in figure 1.9.


Figure 1.7: Attachment length

Figure 1.8: Attachment length as function of expansion ratio


Figure 1.9: Non-dimensional attachment length vs channel width

1.6 Effects of Channel Length:

In the previous section, we discussed the effects of expension ratio on the two-dimensional

jet by changing the width of the symmetrical channel at constant Reynolds number. The

purpose of this section is to investigate the effects of streamwise channel length on velocity

field, jet flapping, and the attachment length. We expect that for very large channel length,

the outer boundary will be far away from the inlet, and its location should not influence the

flow development in the channel. This section will demonstrate that the results we obtained

are independent of the outflow boundary location.

Different simulations are run by changing the length of channel in steps of 180mm, but

with constant width (110mm) and jet Reynolds number equal to 30000. The velocity contours

and streamlines are plotted in figures 1.10 and 1.11, respectively. The velocity contours and


streamlines show that a two-dimensional jet will attach to a side wall provided that the

channel length is greater than a critical value; which is approximately equal to the channel

width. The numerical simulations performed by Sarma et al. (2000)for a two-dimensional

laminar jet show further jet oscillations along the channel. We do not see such oscillations

for a turbulent jet in our domain with maximum channel length 720mm.

Figure 1.12 shows the attachment length as function of the channel length. The attach-

ment length of the jet is almost constant. The attachment length decreases slightly with the

increase in channel length, but remains constant with further increase. Figure 1.13 shows a

comparison of the vertical velocity profiles at a section near the attachment point (y=80mm)

for different channel lengths. The changes in velocity profiles due to increasing channel length

become negligible for channel length greater than or equal 360mm. Hence, the velocity field

is independent of the location of the outflow boundary (channel length). Since, there is no

difference in the attachment length and the velocity profile for different channel lengths, we

conclude that the results presented are independent of the outer boundary location.


Figure 1.10: Velocity contours for different channel length, W=110mm,Re=30000

Figure 1.11: Velocity streamlines for different channel length, W=110mm,Re=30000


Figure 1.12: Attachment length vs channel length with fixed width, W=110mm, Re=30000

Figure 1.13: Vertical velocity at section y=80mm for different channels with different lengths


1.7 Effects of Entrainment:

In all simulations presented, impermeable walls were assumed on both sides of the jet at

its exit plane. In this section, the effects of entrainment from the jet exit plane will be

investigated. To allow entrainment, we changed the boundary condition of sides b1 and b2

(figure 1.3) to pressure inlet; where atmospheric pressure is specified.

Simulations are run on channel with a large length of 720mm and width 110mm. The jet

Reynolds number is still the same i.e. Re=30000. Symmetric flow is predicted for the two

dimensional jet as shown in figure 1.14. This indicates that when entrainment at the jet exit

plane is allowed, it is possible to obtain symmetric flow. The critical limit of entrianment for

symmetric solution has not been determined. Sarma et al.(2000) found that for a laminar jet

at Re=200 and ER=20, the flow in the channel becomes asymmetric even in the presence

of entrainment. Further investigation is needed to study the effects of entrainment on the

turbulent jet case for different Reynolds number and expansion ratio.

Figure 1.14: Velocity contours and streamlines for different channel length, width=110mm


1.8 Effect of Numerical Treatment of the Outlet Boundary

Condition:

This section deals with the discussion of different measurements taken to make sure that

flow is independent of the type of boundary condition imposed at the channel exit. Both

“opening” and “outlet” boundary conditions are tested for the upper edge of the boundary.

No difference is found in the results, and will not be shown here.

In most of the results reported here, “opening” is used as a boundary condition for the

upper edge as shown in figure 1.3. “Opening” boundary condition allows reversed flow at

the boundary as shown by the streamline depicted figure 1.11. A special type of domain

is created as shown in 1.15(a). Now, we change the boundary condition at the top of the

channel to a solid wall, and allow the flow to exit the channel from two symmetrical holes

on the side walls. The thickness of each hole is equal to half of the jet thickness. It is found

that the jet still attaches to one of the side walls as shown in figure 1.15(b). So, no change

is found as a result of changing the outer boundary condition type.

For further verification that the jet attachment phenomenon is independent of the numer-

ical treatment of the outflow boundary condition, we simulate the flow in the domain shown

in figure 1.16. We removed the outer edge and put the entire channel in a larger square room

having an edge length of 540mm. The entire domain in initialized with zero velocity. All

surfaces are treated as solid walls, except the outer bottom surface is treated as “opening” to

allow the flow to exit the large room. The velocity contours are shown in figure 1.17. Results

show that jet still attaches to one of the side walls; which confirms that the jet attachment

phenomenon is independent of the numerical treatment of the outflow boundary condition.


Figure 1.15: Special domain (a) boundary conditions (b) velocity contours

Figure 1.16: Special domain


Figure 1.17: Velocity contours

1.9 Effects of Reynolds Number:

In this section, we discuss the effects of Reynolds number on the two-dimensional turbulent

jet. We change the jet velocity, and fix the expansion ratio. Figure 1.18 shows that the velocity

contours for three different velocity of the jet i.e. 10(m/s), 50(m/s) and 100(m/s). The

attachment length is calculated for each Reynolds number. It is found that the attachment

length remain constant with the increase in Reynolds number as shown in figure 1.19.


Figure 1.18: Velocity contours for different Reynolds numbers

Figure 1.19: Attachment length vs Reynolds number


1.10 Conclusions:

We conducted a numerical study of a two-dimensional incompressible turbulent jet in a

symmetrical channel. We investigated the jet flow as function of expansion ratio (channel

width to jet thickness), channel length, Reynolds number and entrainment condition. The

resulting flow in the channel is asymmetric for a wide range of expansion ratio at high

Reynolds number. The jet deflects and attaches to one of the channel walls. The attachment

length increases linearly with the channel width for fixed value of Reynolds number. The

ratio of the attachment length to channel width is approximately 0.7. The attachment length

is also found to be independent of the (turbulent) jet Reynolds number for fixed expansion

ratio.

Results indicate that the attachment length of the jet and the velocity field are indepen-

dent of the location or numerical details of the outflow boundary; hence we conclude that

the phenomenon of turbulent jet attachment is not a result of the numerical treatment of the

outer boundary condition.

By simulating half of the channel and imposing symmetry, we can construct a steady

symmetric solution to the RANS equations. This implies that there are possibly two solutions

to the steady RANS equations, one is symmetric but unstable, and the other solution is

asymmetric (the jet attaches to one of the side walls) but stable. A symmetric solution

is also obtained if entrainment from jet exit plane is permitted. Fearn et al. (Journal of

Fluid Mechanics, vol. 121, 1990) studied the laminar problem, and showed that the flow

asymmetry of a symmetric expansion arises at a symmetry-breaking bifurcation as the jet

Reynolds number is increased from zero. In the present study the Reynolds number is high


and the jet is turbulent. Therefore, a symmetry-breaking bifurcation parameter might be the

level of entrainment or expansion ratio.


References:

Cherdron,W., Durst, F., and Whitelaw, J.H., “Asymmetric flows and instabilities in sym-

metric ducts with sudden expansions” Journal of Fluid Mechanics (1978), 84:1:13-31 Cam-

bridge University Press.

Denisikhina, D.M., Bassina, I.A., Nikulin, D. A., and Strelets,M.Kh., ”Numerical simu-

lation of self-excited oscillation of a turbulent jet flowing into a rectangular cavity “ High

Temperature. Vol. 43, No. 4, 2005, pp. 568-579. Translated from Teplofizika Vysokikh

Temperatur, Vol. 43, No. 4, 2005, pp. 568-579. Original Russian Text Copyright 2005 by

D. M. Denisikhina, I. A. Bassina, D. A. Nikulin, and M. Kh. Strelets.

Durst, F., Melling, A., and Whitelaw, J.H., “Low Reynolds number flow over a plane

symmetric sudden expansion” Journal of Fluid Mechanics (1974), 64:1:111-128 Cambridge

University Press.

Fearn, R. M., Mullin, T., and Cliffe, K. A. “Nonlinear flow phenomena in a symmetric

sudden expansion” Journal of Fluid Mechanics (1990), 211:595-608 Cambridge University

Press.

Kanna, P.R, and Das, M.K., “Numerical simulation of two-dimensional laminar incom-

pressible offset jet flows” International Journal for Numerical Methods in Fluids Volume 49

Issue 4, Pages 439 - 464 Published Online: 8 Jun 2005.

Mataoui, A. and Schiestel, R., “Unsteady phenomena of an oscillating turbulent jet flow

inside a cavity: Effect of aspect ratio” Journal of Fluids and Structures Volume 25, Issue 1,

January 2009, Pages 60-79.

Mitsunaga, S., and Hirose, T., : Trans. Jpn. Soc. Mech. Eng. 42 (1976) No. 364,3889[in


Japanese]

Ouwa,Y., Watanabe, M., and Matsuoka, Y., “Behavior of a confined plane jet in a rect-

angular channel at low Reynolds number I. general flow characteristics” Jpn. J. Appl. Phys.

25 (1986) pp. 754-761.

Ouwa, Y., Watanabe, M., and Asawo,H., “Flow visualization of a two-dimensional water

jet in a rectangular channel” Japanese Journal of Applied Physics, Volume 20, Issue 1, pp.

243 (1981).

GU, R., “Modeling two-dimensional turbulent offset jets” Journal of hydraulic engineering

1996, vol. 122, no11, pp. 617-624.

Sarma, A.S.R., Sundararajan, T., and Ramjee, V., “Numerical simulation of confined

laminar jet flows” International journal for numerical methods in fluids 2000, vol. 33, no5,

pp. 609-626.

Sato “The stability and transition of a two-dimensional jet” Journal of Fluid Mechanics

(1960), 7:1:53-80 Cambridge University Press.

Sato, H., and Sakao, F., “An experimental investigation of instability of a two-dimensional

jet at low Reynolds numbers” J. Fluid Mech. (1964), vol.20, part 2, pp. 337-352

Sobey, I.J., “Observation of waves during oscillatory channel flow” Journal of Fluid Me-

chanics (1985), 151:395-426 Cambridge University Press.

Umeda, Y., Ishii, R., Matsuda, T., Yasuda, A., Sawada, K., and Shima, E., “Instability

of astrophysical jets.II. Numerical simulation of two-dimensional choked underexpanded slab

jets” Progess of Theoretical Physics, vol. 84, no.5, November 1990.

Chapter 2

Numerical Simulations of Turbulent

Bubbly Mixing Layers

This chapter presents numerical simulations of two-dimensional turbulent bubbly mixing

layers. The purpose of this study is to evaluate the effects of turbulence models on the sim-

ulations of bubbly mixing layers using Reynolds-Averaged Navier-Stokes (RANS) equations.

The models tested include k − �, shear-stress transport (SST), and Reynolds stress trans-

port (SSG) models. Mean liquid and gas velocity profiles, expressed in similarity variables,

are compared to experimental data for single and two phase flows. It is found that RANS

simulations show that the spreading rates of turbulent bubbly mixing layers are greater than

that of the corresponding single phase flow, but different turbulence models give different

spreading rates. The increase in spreading rate is consistent with experimental observations.

All tested turbulence models under predict the spreading rate of the bubbly mixing layer,

even though they accurately predicted the spreading rate for single phase flow. The global

31


structure of the flow proved to be sensitive to the gas volume fraction (as also concluded by

Roig 1997), and the best predictions are obtained by using SST model.

2.1 Introduction:

Bubbly flow is widely used in industrial processes like mineral processing and bubbly reactors

for heat and mass transfer. In mineral processing, such as flotation cells, bubbly flow is used

as a technique to separate solid-solid particles and hence to extract minerals. In mechanically

agitated flotation cells, the raw ore is mixed with water in a large tank, and air is injected

and dispersed by the action of a rotor into small bubbles. The desired mineral particles are

chemically treated to make them hydrophobic. Upon collision with bubbles, particles attach

to the bubbles and float to the top of the tank. In this way minerals are collected and removed

from the tank. The process depends on bubbles generation, bubbles size distributions, bubble-

particle interaction, and bubbles dispersion within the flotation cell. Similarly, the bubbly

reactors have many applications ranging from chemical processes to water treatment. The

efficiency of the reactors depends on efficient mixing, interfacial transfer rates, and bubbles

dispersion, etc. Performance improvements of these industrial processes require a correct

understanding of mechanisms involving multiphase flow dynamics. Computational Fluid

Dynamics (CFD) is a promising tool for understanding and investing these complex processes.

The mixing layer is a simple, yet fundamental, shear flow that has been proven useful

for testing of turbulence models and understanding basic mechanism of fluid mixing and

dispersion. Both experimental and numerical research have been carried out in the last two

decades for the prediction of bubbly mixing layers. These experiments give the structure


of the turbulent bubbly flow, knowledge about bubble size distributions and the turbulence

kinetic energy in the bubbly flow.

Roig et al. (1997) found that turbulent mixing-layer structure is sensitive to void fraction

in their experiments on single phase and bubbly flow with low void fraction and low liquid

velocities. The bubbly mixing layer spreading rate was found to be more than that of single

phase flow. Also, the turbulent kinetic energy of bubbly flow was found to be higher than that

of single phase flow. Rightley et al. (2000) studied bubbly mixing layer flow in a horizontal

channel with very low void fraction on one side only. They studied the interaction of micro

bubbles with free shear layer and investigated the coherent large scales of the flow on the

bubble dispersion field and the energy redistribution within the carrier phase (water). Coline

et al. (2001) found the bubble distribution in a vertical upward flow, a vertical downward

flow and upward flow under micro gravity condition in a turbulent pipe flow. They found

that in the vertical upward flow, the bubbles move radially towards the pipe wall under the

action of lift force. However, bubbles do not move towards the wall in microgravity condition.

In downward flow the bubbles move towards the center of the pipe.

When air is injected in water the bubble size varies in shape and size due to break up and

coalescence. Martnez-Bazn et al. (2002) injected an air jet inside a water jet, and found that

the bubble size distribution depend on dissipation rate of turbulent kinetic energy, global

air void fraction and on the ratio between the residence time and the break-up time. Neto

et al. (2007) injected air in a water tank together with a water jet and found a critical

nozzle Reynolds number value of Re=8000 for large bubbles to break up into small bubbles.

They also found that a Webber number of Wew = 25 is necessary to produce strong bubble


deformation and breakup away from the nozzles. Qi and Shuli (2008) also found that the

presence of bubbles increases the level of turbulence in the flow. Tournemine and Roig (2010)

analyzed the primary instability of buoyant confined bubbly mixing layers. They found that

the induced buoyancy effect generates longitudinal accelerations which are at the origin of self

excited large scale oscillation under certain conditions. Experimental evidence was provided

that these oscillations are global modes.

Numerical work on single phase mixing layer started earlier and is well developed. But Nu-

merical work on two phase flow became as a matter of interest in the past decade. Reynolds-

Averaged Navier-Stokes (RANS) and Large-Eddy Simulations (LES) in conjunction with the

Euler-Euler model are being used most of the time for the prediction of multiphase flow.

Lakehal et al. (2002) performed LES of turbulent shear flows. They simulated a mixing layer

with the same conditions as that experimentally investigated by Roig et al. (1999). They

found that there is no difference between two-dimensional and three dimensional simulation

for the tested conditions. Close comparison with experimental work was obtained with bub-

ble lift coefficient of CL = 1.25. Results obtained by using Smagorinsky model were closer

to experimental data than those obtained by Germano’s dynamic model. Also the Dynamic

Smagorinsky Model (DSM) strategy didn’t work well. Dhotre et al. (2007) also found that

the dynamic approach of Germano does not perform better than the Smagorinsky model

in their LES of square cross-sectioned bubble column. Close agreements with experiments

were found with Smagorinsky model for a model constant of Cs = 0.12. Ayed et al. (2007)

reported new results for a turbulent buoyant bubbly shear layer experiment. The results were

simulated using Euler-Euler two-fluid model. The concentration of dissolved oxygen in the


liquid was reproduced by the model. The profile of dissolved oxygen were not satisfactory

for y > 0.12 (transverse horizontal axis) and x > 0.2 (streamwise vertical axis) due to the

effect of wall. They reported that the effect of turbulence, bubble deformation, and of im-

purities are not yet completely explored even though some formulations based on turbulent

time scales were proposed for stirred gas-liquid systems and validated against experimental

investigations. Hu et al. (2007) simulated gas-liquid bubbly flow in a flat bubble column

using Eulerian-Lagrangian based large-eddy simulation. A concept of particle-source-in-cell

(PSI-ball) is formulated to map a Lagrangian quantity to Eulerian reference frame. High

prediction accuracy was achieved in an extensive comparison with the experimental data. A

detailed second order statistics related to pseudo-turbulent fluctuations (missing in RANS)

is reported.

Min et al. (2008) numerically simulated gas dispersion in an aerated stirred reactor with

multiple impellers. An Euler-Euler approach with constant single average bubble diameter

(SABD) and the population balance model (PBM) and multiple size groups (MUSIG) model

were used with commercial CFD (CFX) code to compute gas void fraction. When compared

with experimental data, the SABD does not predict the void fraction correctly. The prediction

of PBM and MUSIG model were found to be correct. The reason is that SABD ignores the

breakup and coalescence of bubbles while the PBM-MUSIG model considers these events.

Niceno et al. (2008) simulated square cross-sectional bubble column using Euler-Euler

large eddy simulation using Neptune CFD code. Results were also compared with results

obtained from simulations performed using the CFX-4 (one equation model) code. Results

from Smagorinsky SGS model in the Neptune CFD code and CFX were close to each other.


The liquid velocity profiles were under predicted but the gas profiles were predicted well by

both codes.

2.2 Governing Equations:

The governing equations for multiphase flow are the continuity and momentum RANS equa-

tions which are given by

∂

∂t(rαρα) +∇ • (rαραUα) = 0 (2.1)

∂

∂t(rαραUα) +∇ • (rα (ραUα ⊗ Uα)) = −rα∇pα +∇ •

(rαµα

(∇Uα + (∇Uα)T

))+ rαραg +Mα

(2.2)

where α refers to the phase, rα denotes the volume fraction of each phase, ρ is the density,g

is the gravitational acceleration and U is velocity vector. Mα is the interfacial forces acting

on phase due to the presence of other phases. The total force on phase α due to interaction

with other phases e.g. β is given by

Mα =∑β 6=α

Mαβ (2.3)

and

Mαβ = MDαβ +M

Lαβ +M

LUBαβ +M

TDαβ + .... (2.4)


Interphase Drag

The following general form is used to model interphase drag force acting on phase α due

to phase β :

Mα = c(d)αβ (Uβ − Uα) (2.5)

c(d)αβ =

3

4

CDdrβρα|Ubeta − Ualpha| (2.6)

For spherical particles the Schiller Naumann Drag Model the drag coefficient CD is given

by

CD =24

Re

(1 + 0.15Re0.687

)(2.7)

The multiphase versions of turbulence models are equivalent to the single phase version,

with all flux and volumetric source terms multiplied by volume fractions. The eddy viscosity

hypothesis is assumed to hold for each turbulent phase. Diffusion of Momentum in phase α

is governed by an effective viscosity:

µαeff = µα + µtα (2.8)

For the k − � model, the turbulent viscosity is modeled as:

µtα = cµρα

(kα

2

�α

)(2.9)


The transport equations for k and � in a turbulent phase are assumed to take a similar

form to the single-phase transport equations:

∂(rαραkα)

∂t+∇ •

(rα

(ραUαkα −

(µ+

µtασk

)∇kα

))= rα (Pα − ρα�α) + T (k)αβ (2.10)

∂(ρ�)

∂t+∇ • (ρU�) = ∇ • [

(µ+

µtσ�

)∇�] + �

k(C�1Pk − C�2ρ�) (2.11)

where C�1, C�2, σk and σ� are constants with values 1.44, 1.92, 1 and 1.3, respectively.

The multiphase versions of Reynolds stress models are equivalent to the single phase version,

with all flux and volumetric source terms multiplied by volume fractions. Sato successfully

modeled particle induced turbulence this for bubbly flow using an enhanced continuous phase

eddy viscosity:

µtc = µts + µtp (2.12)

2.3 Mixing Layer Geometric Details

The two-dimensional computational domain used in simulations is shown in figure 2.1(a).

It has the same dimensions as the test section of the mixing layer facility used by Roig et

al. (1997). It is a rectangular channel of width of (2b) = 400mm and height of H = 2.3m.

A splitter is inserted in the middle dividing the channel into two sides, side 1 to the right

and side 2 to the left of the plate. The geometric details of splitter plate trailing edge are

shown in figure 2.1(b). The plate height is 300mm, and its thickness is 2mm over most of

its length, except for the last 10mm where its thickness drops to 0.5mm at the trailing edge.


This design closely matches the trailing edge, which is described as a cusp, in the Roig et al.

(1997) experimental facility. The simulations reported here show that no vortices are shed

from the trailing edge of the splitter plate.

Figure 2.1: Mixing layer geometry and boundary conditions

2.4 Boundary Conditions and Grid Description:

The commercial software ANSYS ICEM CFD 12.0 is used for grid generation. The grid is

block structured as shown in figures 2.2 and 2.3. Two grids, denoted as coarse and fine grids,

are used to check the effects of grid resolution on the simulations accuracy; the number of

elements is shown in table 2.1. A Cartesian coordinate system is introduced with origin at the

trailing edge of the splitter plate, and the x-axis is vertically upward (streamwise direction)

and the y-axis is horizontal (transverse to the mixing layer). The gravitational acceleration


is downward. The boundary conditions are labeled in figure 2.1. We have three types of

boundary conditions: solid walls, inlets and outlet. The two vertical side walls and the sur-

face of splitter plate are assumed to be no-slip walls for both phases. At the bottom we have

two velocity inlets conditions; one on each side of the splitter plate, and at the top we impose

a pressure outlet condition. Subscripts L and G are used to denote liquid and gas phases,

respectively. Subscripts 1 and 2 refer to the right (y > 0) and left (y < 0) sides of the splitter

plate, respectively.

Figure 2.2: (a)coarse grid (b)coarse grid (c) coarse grid


Figure 2.3: (a)fine grid (b) trailing edge of splitter

Domain Nodes Elements

Coarse grid 274,400 136,145

Fine grid 940,800 468,476

Table 2.1: Fine and coarse grid details

2.5 Numerical Simulations Parameters:

Different numerical runs were performed by changing the void fraction (�) and magnitude of

inlet velocities. ANSYS CFX 12.0 is used to solve the RANS two-phase flow equations. The

parameters of different numerical runs are given in table 2.2; they correspond to conditions of

Roig et al. (1999) experiments. Experiment 2-1 is purely single phase (liquid water) because

the void fraction in both sides is zero. The void fraction (�) in each experiment is less than


Experiment 2-1 2-2 2-3 2-4

�10% 0.0 1.9 1.9 0.0

UL10(m/s) 0.615 0.53 0.51 0.58

�20% 0.0 1.9 0.0 1.5

UL20(m/s) 0.255 0.23 0.18 0.19

∆UL0 = UL10 − UL20 0.36 0.3 0.33 0.34

ULm0 =(UL10−UL20)

2 0.435 0.38 0.345 0.385

λ0 =∆UL02ULm0

0.414 0.395 0.478 0.506

σL 24 11 13 14

Table 2.2: Inlet conditions for different runs

2%. The liquid velocity in each experiment is less than 1 (m/s). The mean bubble diameter

is 2mm. The Reynolds number Re = ∆ULoxνL

varies from 1.9×104 to 4.4×105 indicating that

the flow is fully turbulent, where “x” is streamwise distance measured from the splitter plate

trailing edge, as shown in figure 2.1(a), and ∆ULo is velocity difference across the mixing

layer.

Results

Results in this chapter are divided into seven sections: liquid (water) superficial velocity

profiles, mixing layer growth rate, gas superficial velocity profiles, void fraction, fine grid sim-

ulations, comparison between different turbulence models and comparison with experimental

data.


2.6 Liquid (Water) Superficial Velocity Profiles:

The water volume fraction is greater than 98% in each case. So, the water flow gives us the

global structure of the flow. A single phase simulation was performed first. It is expected

that RNAS model works well for single flow as reported in the literature. The water velocity

contours for single phase flow run 2-1 are shown in figure 2.4. The red color shows the high

velocity side. For two-phase flows, the superficial velocity counters of liquid in runs 2-2 and

2-4 are shown in figures 2.5 and 2.6, respectively. The superficial velocity is the product of

the local velocity and local volume fraction. Contours for run 2-4 are significantly different

from those for run 2-2. This is due to the difference in the inlet values of void fraction. At

the inlet, the void fraction is zero in the high speed side for run 2-4, whereas for run 2-2, the

void fractions are equal on both sides of the mixing layer. The mixing layer drifts towards

the side of higher void fraction. This result is supported by experimental data. Even though

the void fraction in run 2-4 is only 1.5 %, yet it has a strong effect on the overall behavior of

the mixing layer, hence indicating how the flow is sensitive to the void fraction.


Figure 2.4: Velocity contours for single phase flow run 2-1

Figure 2.5: Water superficial velocity for run 2-2


Figure 2.6: Water superficial velocity for run 2-4

The water velocity profiles for single-phase flow run 2-1 at different streamwise stations

x=-1cm, 20cm, 30cm, and 40cm are shown in figure 2.7. The profiles show the spreading

of the mixing layer with the streamwise distance. SST turbulence model is used here. The

superficial velocity profiles for two-phase flow runs 2-2 and 2-4 at different stations are plotted

in figures 2.8 and 2.9, respectively. An important observation can be seen that the velocities

to the left and right of the mixing layer almost remain constant and equal to inlet conditions;

only the mixing layer thickness increases. The same observation can be seen in the single

phase flow.


Figure 2.7: Water velocity profiles for single phase run 2-1

Figure 2.8: Water superficial velocity profiles for run 2-2


Figure 2.9: Water superficial velocity profiles for run 2-4

2.7 Mixing Layer Growth Rate:

In order to quantify mixing layer growth for single-phase and two-phase runs, we introduce

non-dimensional similarity variables U+L and ηL defined by

U+L =UL − UL2UL1 − UL2

(2.13)

ηL = σLy − y 1

2

x(2.14)

Where y1/2 is defined by UL(y1/2) = ULm and σL is a measure of mixing layer growth rate,

which will be compared with the experimental lateral expansion. If the spreading rate of the

mixing layer is measured by the angle θ as shown in figure 2.10, then we define σL as

σL =1

tan θL=

1

θL(2.15)


Figure 2.10: Mixing layer thickness

So a small value of σL means greater expansion rate of mixing layer. A good approximation

of the velocity profile of single phase mixing layer is the Gortler’s profile (Schlichting, 1979

seveth edition), which is given by

u(ηL) = 0.5[1 + erf(ηL)] (2.16)

where erf(ηL)is the error function given by

erf(η) =2

π

∫ η0e−µ

2dx (2.17)

In figures 2.11, 2.12, 2.13 show a comparison between the computed velocity profiles, ex-

pressed in terms of similarity variables U+L and ηL , and Gortler’s profile. For each run, the

value of (σL) is determined to obtain the best match between the two profiles. This was done

by varying σL and then visually inspecting the plots of the two profiles. The best value for

σL for single phase is 24. Extending the same concept for two phase flow, we present liquid

velocity profiles for two phase runs 2-2 and 2-4 in figures 2.12 and 2.13, respectively. For run

2-2 the value of σL was found to be 21 while for run 2-4 having bubbles only on one side the

value of σL was found to be 19. This indicates that the spreading rate measured by 1/σL is


different from single phase flow and depends on the void fraction. The difference is due to

the presence of bubbles which generates buoyancy effects in the flow and density difference

between the two sides. We note that the profile at the edges i.e. near |ηL| = 1 is not matching

well with Gortler’s profile. The comparison is more favorable as we refine the grid as will be

shown in section 2.10.

Figure 2.11: Mean velocity profile for single phase run 2-1


Figure 2.12: Mean water superficial velocity for two-phase run 2-2

Figure 2.13: Mean water superficial velocity for two-phase flow run 2-4


2.8 Gas Superficial Velocity Profiles:

The mean superficial air velocity profiles for run 2-2 and run 2-4 at different streamwise

locations x = 6cm, 20cm, 30cm, and 40cm are shown in figures 2.14 and 2.15, respectively.

As show in these figures the air velocity is high compared to liquid velocities. The slip

velocity (UG − UL) for run 2-4 (having air bubble only in one side) is 0.21 m/s. There is

a peak in the velocity profile for run 2-4, which corresponds to a peak in the void fraction.

The experimental work by Roig et al.(1999) also show some peaks of void fraction for some

sections but not in the entire flow.

Figure 2.14: Mean superficial air velocity profiles for run 2-4


Figure 2.15: Mean superficial air velocity profiles for run 2-2

2.9 Void Fraction:

Void fraction profiles are plotted in the figure 2.16. In the experimental work performed by

Roig et al. ( 1999 and 2006) peaks in void fraction that depends on the inlet conditions are

also observed. These peaks are due to the boundary layer developing on the splitter plate,

which depend on inlet conditions.

Figure 2.16: Void fraction run 2-4


2.10 Fine Grid Simulations:

Fine grid simulations are run to make sure that our calculations are independent of grids.

The computational domain and all boundary conditions are identical to that of the coarse

grid. The value of σL is calculated for fine grid. We find σL = 18 for run 2-4 (for coarse grid it

is 19) and σL = 20 for run 2-2 (for coarse grid the value is 21). The water superficial velocity

in similarity variables are plotted in figures 2.17 and 2.18 for run 2-4 and 2-2, respectively.

Better agreement between simulations and Geortler’s profile is evident.

Figure 2.17: Mean velocity in the liquid phase run 2-4


Figure 2.18: Mean velocity in the liquid phase run 2-2

2.11 Comparison Between Different Turbulence Models:

The effects of turbulence models on RANS predictions of two-phase mixing layers are reported

here. Three models known as SST model, k − � model and SSGR model are tested. Results

show that SST model gives better prediction in comparison to other models. The k−� model

prediction are close to SST model, but SSGR model predictions show large deviation in the

spreading rate of bubbly mixing layer. Figure 2.19 shows plots of water velocity profiles

for three turbulence models compared with Geartler’s law. The plots indicate that the SST

model gives better agreement with Geortler’s law for σL equal to 19. The k−� model deviates

slightly from SST but the SSGR model deviates a lot for the same value of σL.


Figure 2.19: Liquid superficial velocity profiles for different turbulence models

2.12 Comparison with Experimental Data:

Comparison is being made with the experimental work performed by the Roig et al. (1999).

The value of σL is for single phase flow in the numerical simulations was found to be 24 which

agrees with the experimental value reported by Roig et al. The velocity profiles in similarity

variables are compared in figure 2.20. For two-phase flow RANS simulations considerably

underestimate the growth rate of the mixing layer. The values of σL in the experiments

for run 2-4 and 2-2 are 14 and 11, respectively, as compared to 20 and 18. If we use the

experimental values for scaling the RANS model predictions, we obtain the profiles shown in

figures 2.21 and 2.22. Only if use the higher values of σL, we can bring the RANS models


Run Experimental Value Numerical Value ( fine Grid)

2-2 11 20

2-4 14 18

Table 2.3: Numerical and experimental vlaues for σL

simulations to match with the Gortler profile or experimental data profile.

Figure 2.20: Comparison with the experiments for single phase flow

To investigate the turbulence kinetic energy, we plot the longitudinal velocity fluctuations

of the liquid velocity in a non-dimensional form. u/12 =

u/L2

∆U2LThe profile u

/12 agianst ηL is

shown in the figure 2.23. The plot indicates that there is large difference in the turbulence

kinetic energy between the experimental results and numerical results. This increase in

velocity or turbulence kinetic energy is due to turbulence induced by the bubbles, which is


Figure 2.21: Comparison with the experiments two-phase run 2-4

Figure 2.22: Comparison with the experiments two-phase run 2-2


not captured by the RANS models.

Figure 2.23: Turbulent energy comparison with experimental data for run 2-4

2.13 Conclusions:

We tested RANS models for a multiphase flow given by a bubbly mixing layer. The simula-

tions were performed with two grid resolutions. RANS models including SST model, k − �

model and SSGR models were used in simulations for single phase (water only) and multi-

phase (air and water) flow. The mixing layer thickness was calculated, and the results are

compared with experiments. For single phase flow the numerical predictions agree with the

experimental data for mean velocity and spreading rate. For multiphase the RANS models

predictions do not agree with experimental work. Numerical results show that the SST model

gives slightly better predictions for multiphase flow than the k − � and SSGR models. It is


found that RANS model do not predict the turbulence kinetic energy correctly. There is a

significant difference in turbulence kinetic energy of numerical predictions and experimental

data. There are also differences in the liquid mean velocity profiles. The failure of local

turbulence models are also reported by Lakehal et al (2002). The possible reason may be

that RANS work on constant bubble diameter, but in reality the bubble diameter varies and

has a distribution. The constant diameter of bubble ignores the coalescence and break up of

bubbles that induce local turbulence in the flow and changes the kinetic energy of the carrier

phase. The argument is presented by Min et al (2008) that a model with constant single av-

erage bubble diameter (SABD) does not predict correctly void fraction. The RANS models

need further investigation to include turbulence generation due to breakup and coalescence.


References:

Ayed, H., Chahed, J., and Roig, V., “Hydrodynamics and mass transfer in a turbulent

buoyant bubbly shear layer” AIChE Journal Volume 53 Issue 11, Pages 2742 - 2753.

Colin, C., Legendre, D., and Fabre, J., “Bubble distribution in a turbulent pipe flow”

Microgravity Research and Applications in Physical Sciences and Biotechnology, Proceedings

of the first international symposium held 10-15 September, 2000 in Sorrento, Italy. edited by

Minster, O., and Schrmann, B., European Space Agency, ESASP − 454, 2001., p.91

Dhotre, M.T., Niceno, B., and Smith, B.L., “Large eddy simulation of a bubble column

using dynamic sub-grid scale model” Chemical Engineering Journal Volume 136, Issues 2-3,

1 March 2008,Pages 337-348.

Dr. Schlichting, H., “Boundary-Layer Theory” translated by Dr. J. Ketin seventh Edition

[pages 737 and 738]

Hu,G., and Celik, I., “Eulerian-Lagrangian based large-eddy simulation of a partially

aerated flat bubble column” Chemical Engineering Science Volume 63, Issue 1, January 2008,

Pages 253-271.

Lakehal, D., Smith, B.L., and Milelli, M., “Large-eddy simulation of bubbly turbulent

shear flows” Journal of Turbulence, Volume 3, N 25 May 2002.

Martnez-Bazn, C., Montas, J.L. and Lasheras,J.C., “Statistical description of the bubble

cloud resulting from the injection of air into a turbulent water jet” International Journal of

Multiphase Flow Volume 28, Issue 4, April 2002, Pages 597-615.

Min, J. Bao, Y., Chen, L., Gao, Z., and Smith, J.M., “Numerical simulation of gas

dispersion in an aerated stirred reactor with multiple impellers” Ind. Eng. Chem. Res.,


2008, 47 (18), pp 7112-7117.

Neto, I.E.L, Zhu, D.Z., and Rajaratnam, N., “Bubbly jets in stagnant water” International

Journal of Multiphase Flow Volume 34, Issue 12, December 2008, Pages 1130-1141.

Niceno, B., Boucker, M., and Smith, B.L., “Euler-Euler large eddy simulation of a square

cross-sectional bubble column using the Neptune CFD code” Hindawi Publishing Corporation

Science and Technology of Nuclear Installations Volume 2009, Article ID 410272.

Rightley, P.M., and Lasheras, J.C. “Bubble dispersion and interphase coupling in a free-

shear flow” J. Fluid Mech. (2000), vol. 412, pp. 21-59.

Roig, V., Suzanne, C., and Masbernat, L., “Experimental investigation of a turbulent

bubbly mixing layer” International Journal of Multiphase Flow Volume 24, Issue 1, February

1998, Pages 35-54.

Qi, S., and Shuli, W. “Measurement of turbulent of gas-liquid two-phase flow in a bubble

column with a laser velocitymenter” Advances in Natural Science ISSN 1715-7862 Canadian

Research Development Center of Sciences and Cultures.

Tournemine, A.L.D. and Roig, V. “Self-excited oscillations in buoyant confined bubbly

mixing layers” Phys. Fluids 22, 023301 (2010).

Chapter 3

A Numerical Study of Gas Hold Up

in a Water Tank Supplied by a

Dual Jet of Air and Water

This chapter deals with the numerical simulations of mixing of two parallel jets of air and

water injected into a tank initially filled with water. The purpose is to explore this flow

configuration as a new conceptual design of a mineral separation machine. In mechanically

agitated flotation machines, air is injected within an impeller placed in a tank of water and

minerals. Air break up into bubbles by a rotor installed in the tank. The injection point of

the air is important. It can affect the generation of bubbles and hence the separation process.

In this chapter, we simulate the mixing of an air jet with a parallel water jet. The effects

of the relative position of the two jets on gas hold up in the tank is an important design

parameter. In this study we determined the gas hold up in a tank for two cases by changing

62


the air entrance, in one case air jet is above water jet and in the other case air jet is below

the water jet.

It is found that when air jet is below the water jet, the gas hold up is more than the case

when the air jet is above the water jet. When air jet is below the water jet, air has more time

to travel and interact with the shear induced by the water jet, hence, gives more mixture.

3.1 Geometric Details:

The two-dimensional geometry is shown in figure 3.1. The domain width and height are

selected so as to make sure that the water and air jets do not deflect and attach to the

bottom boundary. The jet deflection and attachment are discussed in detail in chapter 1.

The total height of the domain is 1400mm and width of the domain is 600mm. The two jets

are supplied by two separate but parallel channels. Each jet width is 10mm. The main tank

is divided into two parts; and according to their initializations, we refer to them as a water

tank and an air tank, as shown in figure 3.2. The purpose of air tank is to provide a space

for the mixture generated due to the air entering into the water tank. So, we have four parts:

air tank, water tank, water jet and air jet.


Figure 3.1: Geometry details

3.2 Grid Generation and Geometric Details:

ANSYS ICEM CFD is used for grid generation. Different sections of the grid are shown in

figure 3.3. The different colors show the different parts of the domain discussed in section 3.1.

A fine grid is generated at the center of the grid, where the two jets interact, and a coarser

grid is generated away from the center. Each part of the domain is initialized separately. The

water tank is initialized with water with zero velocity; air tank is initialized with air with

zero velocity. The air jet is initialized with air and water jet is initialized with water. The

boundary conditions are labeled in figure 3.2. The boundary conditions are inlet, opening

and no-slip wall. At inlets, mass flow rate is specified instead of velocity. The upper edge of


the domain is an opening for air only with zero gauge pressure. The bottom edge is an outlet

for water only. Continuity is imposed on both air opening and water outlet. The boundary

between the air and water tanks is an internal boundary which is treated as interface. An

interface is a flow through boundary where mass, momentum and turbulence parameters are

conserved.

Figure 3.2: Boundary conditions


Figure 3.3: Two-dimensional grid

3.3 Numerical Simulations:

Numerical simulations are performed using ANSYS CFX. The mathematical modeling for

two-phase flow is discussed in chapter 2 section 2.2. Numerical simulations are performed for

two cases: Case 1: air jet is injected below the water jet, Case 2: air jet is injected above the

water jet. The boundary conditions for each case are the same. The air jet velocity in both


cases is 3.19(m/s), and water jet velocity in each case is 1.7(m/s).; the only difference in the

cases is the vertical stratification of two jets. The air bubble diameter is selected as 1mm.

3.4 Results:

Simulations are performed by solving the unsteady RANS equations. The air volume fraction

contours after different time steps for case 1 and case 2 are shown in figure 3.4 and figure

3.5, respectively. The air volume fraction contours provide visual evidence of better air

dispersion when air jet is injected below the water jet. Gas hold up is monitored in the water

tank (initially filled with water) as function of time for each case, and the results are shown

by figures 3.6 and 3.7. The gas holdup increases with time. The water volume fraction in

the air tank (initially filled with air) is also calculated after different time steps. Figures 3.8

and 3.9 show the water volume fraction for air tank for case 1 and case 2. As expected, the

water volume fraction in air tank also increases with time because the air volume fraction in

water tank increases. The total quantity of water and air in the domain (water tank + air

tank) remains constant.


Figure 3.4: Air volume fraction contours for case 1: air jet below water jet


Figure 3.5: Air volume fraction contours for case 2: air jet above water jet


Figure 3.6: Air volume (%) in water tank for air jet below water jet

Figure 3.7: Air volume (%) in water tank for air jet above water jet


Figure 3.8: Water volume (%) in air tank for air jet below water jet

Figure 3.9: Water volume (%) in air tank for air jet above water jet


3.5 Comparison and Conclusion:

Comparison is made between the gas hold up in the two cases. Figure 3.10 shows the air

volume fraction in water tank for cases 1 and 2. The figure shows that if the air jet is injected

below the water jet, enhanced air dispersion is achieved in a shorter time compared to the

case when the air jet is injected above the water jet. Furthermore, the enhanced gas hold up

for case 1 persists over time. Similarly, water volume fraction in air tank is compared in both

cases as shown in figure 3.11. Like the air volume fraction in water tank, the water volume

fraction in air tank is also greater in case 1 than case 2. Injection of air jet below the water

jet gives air more time to interact and be influenced by the shear of the water jet. Whereas

when the air jet is injected above the water jet, buoyancy accelerates the upward motion of

the air jet with less time to be influenced by the water jet. This offers a new concept for the

design of efficient flotation cells. Stratification of layers of air jets and water (or slurry) jets

can be explored to enhance particle-bubble collision for less power consumption. This will

make the flotation process more efficient and mechanically simpler than the case of impeller

agitated cells.


Figure 3.10: Air volume fraction comparison between case1 and case2

Figure 3.11: Water volume fraction comparison between case1 and case2

Numerical Investigation using RANS Equations of Two ......Kareem Akhtar Abstract This thesis...

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