Numerical and Analytical Studies of Single and
Multiphase Starting Jets and Plumes
by
Ruo-Qian Wang
Submitted to the Department of Civil and Environmental Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@ 2014 Massachusetts Institute of Technology. All rights reserved.
-,OHVCLOGY
Author ........
Signature redacted
JUN 13 201
....
LiBRARIES
Department of Civil and Environmental Engineering
May 18, 2014
Signature redacted
Certified by ........
E. Eric Adams
Senior Lecturer and Senior Research Engineer of Civil and Environmental
Engineering
Thesis Supervisor
14
Accepted by...................
A1
Signature redacted
Heidi M. AIepf
Chair, Departmental Committee for Graduate Students
Numerical and Analytical Studies of Single and Multiphase Starting
Jets and Plumes
by
Ruo-Qian Wang
Submitted to the Department of Civil and Environmental Engineering
on May 18, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in the field of Environmental Fluid Mechanics
Abstract
Multiphase starting jets and plumes are widely observed in nature and engineering systems.
An environmental engineering example is open-water disposal of sediments. The present
study numerically simulates such starting jets/plumes using Large Eddy Simulations. The
numerical scheme is first validated for single phase plumes, and the relationship between
buoyancy and penetration rate is revealed. Then, the trailing stem behind the main cloud
is identified, and the the formation number (critical ratio UAt/D, where U, D and At
are discharge velocity, diameter and duration) that determines its presence is determined
as a function of plume buoyancy. A unified relationship for starting plumes is developed
to describe behaviors from negative to positive buoyancy. In multiphase simulations, twophase phenomena are clarified including phase separation and the effect of particle release
conditions. The most popular similarity law to scale up from the lab to the field (Cloud
number scaling) is validated by a series of simulations. Finally, an example of sediment
disposal in the field is given based on the present study. In related theoretical analysis, an
analytical model on the vortex ring is developed and found to agree well with the direct
numerical simulation results.
Thesis Supervisor: E. Eric Adams
Title: Senior Lecturer and Senior Research Engineer of Civil and Environmental Engineering
Acknowledgments
I feel very fortunate to have my respectful advisors, warm-hearted colleagues, sincere
friends and the best family to accompany my Ph.D. endeavor. This thesis would not have
been accomplished without your steady supports, endless help and remarkable advice.
At first, I would like to thank my principle supervisor, Dr. E. Eric Adams. His guidance and devotion gave me huge encouragement to fumble in my research. I am always
impressed by his sharp research intuition and rigorous scientific attitude. His half-opened
door is like a lighthouse, always signaling a welcoming message and luring a fruitful discussion.
My great thanks should also be given to my co-advisor, Prof. Adrian Wing-Keung
Law. His enthusiasm and inspiration always stimulates me to solve the most challenging
problem. His comments in every draft are always insightful and significantly enhanced my
research quality.
I am also indebted to my committee members, Professors Ole Madsen, Heidi Nepf,
and Roman Stocker. Their perceptions in engineering and science have shed light on
my research exploration. My research also received generous help from Professors Oliver
Fringer, Zhenhua Huang, and Rex Britter, and Dr. Junwei Su.
I would like to extend my gratitude to the group members including Bing Zhao, Adrian
Chun-Hin Lai, Dai Chin, Dongdong Shao, Jenn Wei Er on the Singapore side, and Godine
Chan, James Gensheimer III, Mariana Rodriguez Buno, Jia Wang, and Cindy Wang on the
MIT side. I am grateful for supports from Danya Xu, Xianxiang Li, Peifeng Ma, Chi-Hao
Chang, Haoliang Chen and Kian Yew Lim in CENSAM, and many more friends from EFM
group, Parsons, MIT, Singapore and China.
Finally, I sincerely acknowledge my wonderful family who keep offering energy to my
research voyage. I often called my wife, Yuan Wang, the "Queen of Common Sense".
Every time I explained my research to her, she smells the unusual place against her sense,
which often indicates a mistake to avoid or a significant new discovery to explore. She is
5
also the one who bears my numerous presentation rehearsals and offers the most honest
suggestions. I have to thank my parents, who are always the exhaustless spring of my
courage and confidence. No matter up or down, they steadily stand on my side cheering
every step I moved forward.
6
Contents
1
2
3
11
Introduction
1.1
Research motivations
. . . . . . . . . . . . .
11
1.2
State of knowledge
. . . . . . . . . . . . .
12
1.3
Numerical methods
. . . . . . . . . . . . .
13
1.4
Research objectives
. . . . . . . . . . . . .
15
1.5
Thesis outline . . . .
. . . . . . . . . . . . .
15
Large-Eddy Simulation of Starting Buoyant Jets
23
2.1
Introduction . . . . . . . . . . . . . . . . . . .
24
2.2
LES approach . . . . . . . . . . . . . . . . . .
26
2.3
Validation . . . . . . . . . . . . . . . . . . . .
29
2.4
Buoyant jet . . . . . . . . . . . . . . . . . . .
33
2.5
Summary and conclusions
. . . . . . . . . . .
37
Buoyant Formation Number of A Starting Buoyant Jet
53
3.1
Introduction . . . . . . . . . . . . . . . . . . .
54
3.2
Numerical model . . . . . . . . . . . .
. . . .
58
3.3
Formation number for starting pure jets
. . . .
64
3.4
Buoyant formation number for buoyant jets
. .
65
3.5
Summary and conclusions
. . . . . . . . . . .
71
7
4
5
6
7
Pinch-off and Formation Number of Negatively Buoyant Jets
87
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2
Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3
Penetration characteristics
4.4
Formation number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5
Summery and conclusions
. . . . . . . . . . . . . . . . . . . . . . . . . . 93
. . . . . . . . . . . . . . . . . . . . . . . . . . 105
LES Study of Settling Particle Cloud Dynamics
121
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.2
Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3
Validation of numerical method . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4
Phase separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.5
Penetration and growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.6
Entrainment and deposition . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.7
Polydispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.8
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Scaling Particle Cloud Dynamics - From Lab to Field
153
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2
Numerical method
6.3
Results and discussion
6.4
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
An Axisymmetric Steady State Vortex Ring Model
163
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.2
A particular solution
7.3
Vortex ring model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.4
Properties of the vortex ring family . . . . . . . . . . . . . . . . . . . . . . 173
7.5
Comparison to numerical simulations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
8
. . . . . . . . . . . . . . . . . . . . 174
7.6
8
Summary and conclusions
. . . . . . . . . . . . . . . . . . . . . . . . . . 178
Summary and Example for Sediment-Disposal Field Operations
187
8.1
Sum m ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.2
Suggestions for sediment disposal in field operations
9
. . . . . . . . . . . . 190
10
Chapter 1
Introduction
1.1
Research motivations
Multiphase plumes, formed when a continuous phase fluid is mixed with immiscible buoyant particles, droplets or bubbles, are common in natural and engineered systems and span
multiple time and length scales, e.g. volcanic eruptions, piston engine fuel injection, and oil
spills in the deep ocean. The present study aims to track the fate of sediment released to the
ocean during land reclamation and open-water disposal of dredged materials to minimize
sand loss and associated turbidity and environmental impact in the water column.
Dredged material disposal in open water involves residual and re-suspension phenomena, which can significantly degrade coastal water quality (figure 1-1). With larger scale
operations, this problem raises global concerns. One example is in Boston harbor where
navigational dredging is performed to accommodate shipping. The dredged contaminated
sediments are dumped into confined aquatic disposal cells and then covered by clean sands.
Without careful attention, the disposal material may be washed away polluting the ambient
marine environment. An Asian example is Singapore where the local government continues
to reclaim new land to satisfy the increasing population. Sediment resources are scarce, so
local operators have to track the fate of sediment dumping to carefully control the operation
11
cost. The goal is to improve mathematical models to guide such engineering applications.
1.2
State of knowledge
In the short term after release, the particle cloud of sediment disposal experiences three
phases [1]: "convective descent", in which gravity is the dominant driving force to settle
the particle clouds; "dynamic collapse", when the cloud impacts the bottom or reaches
neutral buoyancy level; and "passive diffusion", in which ambient advection and diffusion
dominates the transport of sediments.
The present study focuses on the phase of convective descent, which can be further
classified into three stages [2] (see figure 1-2 [3]): initial acceleration, self-preserving
(also called thermal stage), and dispersive stages. These stages describe near-field behavior of the particle cloud. Specifically, after release, the particles rapidly mix with water
and quickly reorganize themselves, which is called the initial acceleration stage. Then,
in the self-preserving stage, the particle cloud exhibits a strong self-circulation; the cloud
shape and distribution of particles are maintained during the settling process, while cloud
size grows and its speed slows down. In the dispersive stage, the particles escape from the
cloud, and settle independently. More details can be found in chapter 5.
In engineering practice, the most popular model is that developed by Johnson and Fong
[4] of the US Army Corps of Engineers. They use an integral model to simulate the settling
particle cloud, and involve conservation equations of mass, momentum, and buoyancy.
This model is coded as a computational tool called STFATE, which is included in a more
complete package called "the automated dredging disposal alternative management system" (ADDAMS). This model has been compared against the on-site survies such as that
described in [5].
Recently, advances in experiments and numerical simulations have allowed further
studies on the details of settling particle cloud dynamics. Ruggaber [6] was among the
12
first to systematically vary all the basic parameters, including the particle size, the water
content, and the initial momentum, in a generalized integral model. In addition, he also
quantified the mass in trailing stems. Subsequently, additional ambient and initial conditions were studied, including ambient stratification [7], currents [8], waves [9], and release
heights [10]. Recently, the fluid phase characteristics were quantified by Lai et al. [11],
with an approximate flow field that is based on the Hill's vortex model.
On the other hand, newly developed numerical tools have improved our understanding of the dynamics in details. Li [12] performed a two-fluid simulation with a mixing
length model as turbulence closure. Gu et al. [13] simulated two-phase flows directly.
The fluid phase motion was solved by the Reynolds Averaged Navier-Stokes method with
k - c turbulence model and the solid phase motion was computed with Lagrangian tracking.
Harada et al. [14] replaced the fluid phase computational approach with the Large Eddy
Simulations (LES), and obtained more detailed turbulence structures and better agreement
with experimental data. Available numerical methods are still under development and in
studies to-date, generally a small number of particles have been used to reduce computational resources. Furthermore, particle-particle collisions have been ignored, which might
cause errors near the point of release. In this study, a more advanced numerical scheme is
developed, and applied to the sediment disposal problem in the following chapters.
1.3
Numerical methods
Large Eddy Simulation (LES) has become very popular in turbulence modeling over the
last decade. It is an intermediate numerical method between Reynolds Average NaviorStokes (RANS) and Direct Numerical Simulation (DNS). The former only shows mean
characteristics, and needs empirical modeling equations to complete the closure of equations. And the latter requires the grid space to be in the Kolmogrovs inertial range, which
is prohibitively expensive for engineering applications. As a compromise between these
13
two methods, LES computes the large scale flow by DNS and approximately models the
sub-grid scale (SGS) stress tensor. Thus, it requires less computation load compared to
DNS but is much more accurate than RANS.
The SGS model has experienced a series of developments. The first SGS model was
derived by Smagorinsky [15]. Due to its simplicity, robustness, and stability, it is still one
of the most popular SGS models in LES. However, its drawback is also obvious, e.g. its
overwhelming dissipation prevents turbulence transition. To overcome the drawback, especially the inaccuracy in inhomogeneous flow simulations, the dynamic model was developed, which locally calculates the SGS coefficients of Smagorinskys model with a dynamic
adjustment. The most famous dynamic SGS model is developed by Germano et al. [16].
The SGS model used in chapters 2 to 4 is called Dynamic Mixed Model, which was
developed by Zang et al. [17]. This model divides the SGS stress tensor into a Leonard
part, which can be calculated explicitly from the resolved information, and a residual part,
which can be modeled by empirical equations. It also involves a double filter to define a
test scale in order to calculates the SGS coefficients locally. The model has been applied
to some engineering problems, e.g. a lid-driven cavity simulation [17], a round jet in cross
flow [18], etc.
In chapter 5 and 6, a multiphase LES model is implemented, which is realized in a numerical package called CFDEM [19]. This software couples the OpenFOAM CFD (Computational Fluid Dynamics) package and the DEM (Discrete Element Method) LIGGGHTS
toolbox to study particle clouds. More details can be found in chapter 5. All the simulations were practiced in a High Performance Computer cluster in Nanyang Technological
University, Singapore.
14
Research objectives
1.4
The objective of the present research is to further our understanding on the settling particle cloud and eventually offer suggestions to field operations. As a member of a team in
the Center for Environmental Sensing and Modeling (CENSAM), I contributed to the sediment disposal project by numerical simulations with comparisons to the other member's
experimental results.
In lab experiments conducted by the team [8], a sediment cloud has been reproduced
to demonstrate the characteristics of the particle cloud. A captured photo is shown in
figure 1-3. The particles are released together with dye, which represents the dissolved
contaminants encapsulated in the dredged materials. The particles are blue and the dye is
red in the laser sheet. Therefore, both solid and fluid phases can be clearly observed and
analyzed.
Comparing this lab photo and the physics described by the STFATE model, we find two
major characteristics that the latter fails to adequately capture. First, a trailing stem is developed behind the main cloud if sufficient sediment is released. Although the developers
[4] recognized this issue and added a "stripping" mechanism to allow the particles to leave
the main cloud, their model is still ad hoc and not able to fully predict the "trailing stem"
formation. Second, there is a solid-fluid phase separation in the main cloud. After separation, the fluid phase will remain in the water column, and thus threatens water quality.
The STFATE has no model to track the fluid phase, and the separation process takes place
instantaneously rather than continuously.
1.5
Thesis outline
In accordance with the above literature review and objectives, the thesis is divided into six
major chapters, each corresponding to a paper that has been published or submitted to a
peer-reviewed journal. Specifically, chapter 2 validates the numerical scheme for starting
15
buoyant jets and the physics of its penetration rate for a wide range of buoyancy. Chapter 3 characterizes the "trailing stem" formation mechanism, and finds that the formation
number, the critical value that determines the presence of the trailing stem, increases with
stronger buoyancy. Chapter 4 extends chapter 3's results to negatively buoyant jets (also
called fountains), and also outlines a dynamics map of the buoyant jet. Chapter 5 aims to
quantify the solid-fluid two-phase dynamics and the effects of the initial release condition
on the particle cloud behavior, including the distribution of the particle size, the total buoyancy, and the aspect ratio. Chapter 6 validates that a commonly used scaling law - cloud
number scaling - offers a method to scale up lab-scale observation to field scale operation.
As vortex rings are at the heart of cloud formation, a more accurate theoretical model of
vortex rings is developed in chapter 7. All results are summarized in chapter 8, which
includes an example that reflects many of the newly developed principles.
16
Bibliography
[1] Maynard G Brandsma and David J Divoky. Development of Models for Prediction
of Short-Term Fate of Dredged Material Discharged in the Estuarine Environment.
Technical report, DTIC Document, 1976.
[2] Hamid Rahimipour and David Wilkinson. Dynamic Behavior of Particle Clouds. In
11th AustralasianFluid Mechanics Conference, Hobart, Australia, 1992.
[3] B. Zhao, A.W.K. Law, E.E. Adams, and J.W. Er. Formation of particle clouds. Journal
of Fluid Mechanics, 746:193-213, March 2014.
[4] Billy H Johnson and Moira T Fong. Development and Verification of Numerical
Models for Predicting the Initial Fate of Dredged Material Disposed in Open Water.
Report 2. Theoretical Developments and Verification Results. Technical report, U.S.
Army Engineer Waterways Experiment Station, Vicksburg, Miss., 1995.
[5] Henry J Bokuniewicz, Jeffrey Gebert, Robert B Gordon, Jane L Higgins, and Peter
Kaminsky. Field Study of the Mechanics of the Placement of Dredged Material at
Open-Water Disposal Sites. Volume I. Main Text and Appendices A-1.
Technical
report, DTIC Document, 1978.
[6] Gordon J Ruggaber. Dynamics of particle clouds related to open-water sediment
disposal. PhD thesis, Department of Civil and Environmental Engineering, MIT,
2000.
17
[7] John W. M. Bush, B. A. Thurber, and F. Blanchette. Particle clouds in homogeneous
and stratified environments. Journalof FluidMechanics, 489:29-54, July 2003.
[8] RJ Gensheimer, EE Adams, and AWK Law. Dynamics of particle clouds in ambient currents with application to open-water sediment disposal. Journal of Hydraulic
Engineering, 139(2):114-123, February 2012.
[9] Bing Zhao, Adrian W. K. Law, Zhenhua Huang, E. Eric Adams, and Adrian C. H.
Lai. Behavior of Sediment Clouds in Waves. Journalof Waterway, Port, Coastal, and
Ocean Engineering, 139(1):24-33, January 2013.
[10] Bing Zhao, Adrian W. K. Law, E. E. Adams, Dongdong Shao, and Zhenhua Huang.
Effect of air release height on the formation of sediment thermals in water. Journal
of Hydraulic Research, 50(5):532-540, October 2012.
[11] Adrian C. H. Lai, Bing Zhao, Adrian Wing-Keung Law, and E. Eric Adams. Twophase modeling of sediment clouds. Environmental Fluid Mechanics, pages 1-29,
February 2013.
[12] CW Li. Convection of particle thermals. Journalof Hydraulic Research, 35(3):363376, May 1997.
[13] J Gu and CW Li. Modeling instantaneous discharge of unsorted particle cloud in
ambient water by an EulerianLagrangian method. Journal of Hydraulic Research,
42(4):399-405, 2004.
[14] Eiji Harada, Naoki Tsuruta, and Hitoshi Gotoh. Two-phase flow LES of the sedimentation process of a particle cloud. Journal of Hydraulic Research, 51(2):186-194,
April 2013.
[15] J. Smagorinsky.
General circulation experiments with the primitive equations.
Monthly Weather Review, 91(3):99-164, March 1963.
18
[16] Massimo Germano, Ugo Piomelli, Parviz Moin, and William H. Cabot. A dynamic
subgrid-scale eddy viscosity model. Physics of Fluids A: Fluid Dynamics, 3(7):1760,
1991.
[17] Yan Zang, Robert L Street, and Jeffrey R Koseff. A dynamic mixed subgrid-scale
model and its application to turbulent recirculating flows. Physics of Fluids A: Fluid
Dynamics (1989-1993), 5(12):3186-3196, 1993.
[18] L L Yuan and R L Street. Trajectory and entrainment of a round jet in crossflow.
Physics of Fluids, 10(9):2323-2335, 1998.
[19] Christoph Kloss, Christoph Goniva, Alice Hager, Stefan Amberger, and Stefan Pirker.
Models, algorithms and validation for opensource DEM and CFD-DEM. Progressin
ComputationalFluid Dynamics, an InternationalJournal, 12(2):140-152, 2012.
19
Figure 1-1: Sketch of a sediment disposal (illustration courtesy: Yuan Wang).
20
t
initial acceleration phase
self-preserving phase
0
0
00000
04000 0 0 0O00 000 )0)
0
0 00 0 0
0 000
0 000000
0000
0000
0
0
0
000
dispersive phase
I
Figure 1-2: A particle cloud descending through homogeneous stagnant water; (a) a
schematic diagram (dashed lines indicate the self-similar growth path of the particle thermal); (b) images of a typical laboratory particle cloud (reproduced from [3]).
21
"
MAj ,
'M ft%'
M 0
5
P Ir 0
r
. IIrI- - - __
--
Figure 1-3: Sketch of a two-phase particle cloud (photo courtesy: James Gensheimer III).
22
Chapter 2
Large-Eddy Simulation of Starting
Buoyant Jets
Abstract
A series of Large Eddy Simulations (LES) are performed to investigate the penetration of
starting buoyant jets. The LES code is first validated by comparing simulation results with
existing experimental data for both steady and starting pure jets and lazy plumes. The
centerline decay and the growth rate of the velocity and concentration fields for steady jets
and plumes, as well as the simulated transient penetration rate of a starting pure jet and a
starting lazy plume, are found to compare well with the experiments. After validation, the
LES code is used to study the penetration of starting buoyant jets with three different
Reynolds numbers from 2000 to 3000, and with a wide range of buoyancy fluxes from pure
jets to lazy plumes. The penetration rate is found to increase with an increasing buoyancy
flux. It is also observed that, in the initial Period of Flow Development, the two penetrative
mechanisms driven by the initial buoyancy and momentum fluxes are uncoupled; therefore
the total penetration rate can be resolved as the linear addition of these two effects. A fitting
equation is proposed to predict the penetration rate by combining the two independent
mechanisms.
Published as Wang, R.-Q., Law, A. W.-K., Adams, E. E., and Fringer, 0. B. (2010). "Large-eddy simulation of
starting buoyant jets." Environmental Fluid Mechanics, 11(6), 591-609.
23
2.1 Introduction
Buoyant jets are discharges of a volume of fluid having a density that differs from the
surrounding ambient. They are widely observed in natural phenomena, such as volcanic
eruptions and hydrothermal vents, and in engineering applications, such as jet propulsion and
the discharge of waste water to a receiving aquatic environment. The density of a buoyant jet
can be less than or greater than the ambient density, and the direction of initial momentum
can be in any direction relative to the force of buoyancy.
In this study we examine the
behavior of starting buoyant jets in which the direction of buoyancy is aligned with the
direction of initial momentum, i.e., either jets discharging a heavier fluid vertically downward
or a lighter fluid vertically upward. We refer to either as "positively buoyant jets" or simply
"buoyant jets".
The primary characteristics of the buoyant jet can be defined by the following parameters as
per Fischer et al. [2]:
Q= I D2U
4
M =
rCD2U2,
4
B = I
4
Apo D 2 U
(2.1)
po
where Qo, MO, and BO are initial volume flux, kinematic momentum flux, and kinematic
buoyancy flux, respectively; g' is reduced gravity (g Ap /p 0 ). With these parameters, the
Richardson number, Ro, which represents the ratio of buoyancy flux to initial momentum flux
can also be defined as follow:
R=-=J
01
_-QOB
M01/4
g'D
4
12
(2.2)
U02)
We adopt the naming convention whereby the range of the buoyant jet extends from a pure
jet (Ro = 0), through a forced plume (O<Ro<0.27), a pure plume (Ro = 0.27), a lazy plume (Ro
> 0.27), and a thermal (R 0 = oo). Most buoyant jet studies in the literature have focused on
steady state behavior [2][3][4]. However, many buoyant jet applications are inherently shortterm (e.g., the marine disposal of sediments from dredging or land reclamation), and even
24
continuous discharges begin as starting jets or plumes. Hence, it is important to study and
better understand the initial formation phase of buoyant jets.
When a buoyant jet is discharged from an orifice, a starting vortex forms immediately due to
the rollup of the source fluid against the stationary ambient. This starting vortex grows until it
reaches a maximum circulation state and then detaches from the trailing stem, a phenomenon
commonly referred to as "pinch off' [5]. Thereafter, the starting vortex ceases to grow and is
eventually engulfed by the regenerated head vortex leading the trailing stem. The analysis of
the continuous process has been discussed by Law et al. [6]. The initial period before the
engulfment is referred to as the Period of Flow Development (PFD), while the subsequent
period is referred to as the Period of Developed Flow (PDF). The naming is to differentiate
the two periods of development, whereby in the PFD the behavior is significantly affected by
the source conditions whereas in the PDF the behavior bears similarity and can be analyzed
by means of the gross discharge characteristics.
As implied above, a starting buoyant jet has two asymptotic states: (a) a starting pure jet with
no buoyancy flux (R, = 0) and (b) a thermal or starting lazy plume with negligible initial
momentum flux (large R,). The starting pure jet is a configuration of critical importance for
many engineering applications (in combustion for example) and has been studied by a
number of investigators. Generally, the jet is found to penetrate linearly with time in the PFD
[7][8], and with the square root of time in the downstream self-similar phase [9][10][11][12]
of the PDF. The other extreme of a starting lazy plume has been studied by Lundgren et al.
[13], Alahyari and Longmire [14], and Pottebaum and Gharib [15].
The first two studies
focused on the scaling comparison between laboratory results and the phenomenon of
microburst, and showed that an appropriate scaling can significantly simplify the analysis.
Pottebaum and Gharib [15] demonstrated quantitatively that the leading vortex has indeed
achieved a maximum circulation. In addition, Bond and Johari [16] studied the effects of the
nozzle geometry and found that the penetration rate can be divided into two periods: initial
acceleration and thermal-like phases (i.e. analogous to PFD and PDF respectively). The
penetration is proportional to the 1.4 power of time for the former, and the 0.5 power for the
latter.
Diez et al. [17] used video techniques to study the characteristics of starting buoyant jets over
a wide range of source buoyancy. Within the self-preserving phase, the plume was found to
25
In Section 2, the LES model used in this study is briefly described. This is followed in
Section 3 by a validation effort for buoyant jets, in which we use the LES model to simulate
steady pure jets and lazy plumes, whose limiting behaviors are well understood (e.g. Fischer
et al. [2]; Wang and Law [23]). It will be shown that the present LES model can simulate the
main features of these flows including the decay in the centerline velocity and concentrations,
as well as the velocity and concentration widths. Following the validation, we perform a set
of numerical experiments over a range of starting buoyant jets in Section 4.
Section 5
provides the summary and conclusions.
2.2 LES Approach
2.2.1 Governing Equations
The governing equations for the LES approach are the spatially-filtered continuity, NavierStokes and scalar transport equations with the Boussinesq approximation as follows:
axi
at
-+
at
=0,
+
(2.3)
=Si,
(2.4)
=0,
(2.5)
axi
ax
where
F,, 2 +r,,(2.6)
ii,-
S1 = -g#l(T - T
Rk = -T--
+
v
),3
(2.7)
+X
(2.8)
where the over-bar represents a spatially-filtered quantity. ui (i=1,2,3) are the Cartesian
velocity components in the direction of xi. Other quantities are as follows: t is time, p is
pressure, g is the gravitational acceleration, T is a scalar (e.g., density or temperature), To is
26
the background scalar, 8p=
viscosity, and
K
To JT
is the coefficient of thermal expansion, v is the kinematic
is the scalar diffusivity. Note that all the equations are subjected to the
Einstein rule of summation.
The subfilter-scale terms r, and Z, are specified as:
T, =nu - uW,
(2.9)
X, =uT -iT
(2.10)
These two terms are modeled with the dynamic mixed subgrid-scale model, details of which
can be found in Zang et al. [24].
2.2.2 Numerical Methods
The governing equations are transformed to generalized curvilinear coordinates and
discretized with a finite-volume formulation on a non-staggered grid [25]. The discretization
includes: 1) a semi-implicit scheme with Crank-Nicholson for the diagonal viscous and
diffusive terms and Adams-Bashforth for the other terms; 2) accurate upwind-difference
schemes on the convective terms; and 3) second-order accurate central differences on all the
other spatial differential terms. The convective terms of the momentum equations (Eq. 2.4)
are discretized using the QUICK scheme, whereas the convective terms for the scalar
transport equation (Eq. 2.5) are discretized using the SHARP scheme to avoid spurious
oscillations [26][27].
The numerical code has previously been validated by a series of comparisons with standard
experiments. A 3D lid-driven cavity flow was reproduced by Zang et al. [24] as the first
validation of this model. Subsequently, LES was performed on a range of round jets in cross
flows with low and moderate Reynolds numbers [28][29]. The trajectory and entrainment
characteristics were shown to agree with the experiments very well. In addition, upwelling
flows were simulated by Zang and Street [30] and Cui and Street [31], and the turbulent
current beneath nonlinear free-surface waves [32] and the flow over a wavy boundary [33]
27
were also convincingly reproduced by the present code. Recently, Zedler and Street [34] and
Chou and Fringer [35] further developed the code to incorporate bed-load sediment transport.
In summary, the LES algorithm of Zang et al. [24][25] has been demonstrated to be a
practical and efficient LES scheme that is suitable for a range of turbulent flows. In the
present study we extend the validation to include jets and plumes discharging into quiescent
environments. The parallel version of the code that we used here was developed by Cui and
Street [31].
2.2.3 Flow configuration
The computational domain used in this study is a rectangular volume with a square horizontal
cross-section that extends to 1.2 m x 1.2 mx 1.5 m for validation, and 0.5 m x 0.5 m x 1.5 m
for the remaining production runs in the Cartesian coordinates of x, y and z respectively as
shown in Figure 2-1(a). Note that the domain size is reduced for the prediction runs so as to
control the computational expense and the production results of penetration rate is free of
impact (which will be proved later). At t=0, a buoyant jet with positive buoyancy (i.e. the
buoyancy force is along the jet direction) is issued at a uniform velocity Uo into a
homogeneous and stationary ambient fluid with density p0 . The relative density difference
between the jet flow and the ambient is Apo /p.
The buoyant jet is discharged at the top of
the domain through a circular nozzle which has a diameter D=5 cm. Thus, the size of the
domain can also be expressed as 24D x 24D x 30D and 6D x 6D x 30D for validation and
production respectively, which shall limit the domain of investigation of the penetrative
behavior. The computational domain is discretized into a stretched mesh with increased
refinement along the vertical axis (see Figure 2-1(b)).
The boundary conditions are also presented in Figure 2-1(a). The velocity field is specified at
the top boundary as an incoming uniform jet with a velocity field given by w((x-3D) 2+(y3D) 2<D 2/4, z=O)=Uo, while the other boundaries are all outflow boundaries. Following the
recommendation of Yuan [36], the outflow boundary condition is essentially the "no
gradient" condition, which has limited effect on the starting results before the flow structure
penetrates beyond the domain. A constant volume flux, determined from the prescribed
inflow velocity Uo, is enforced at the inlet.
28
2.3Validation
In this section, the LES model is validated specifically for the study of buoyant jets
discharging to quiescent environments by comparing the simulation results to experiments for
the asymptotic cases of a pure jet and a lazy plume, the behaviors of which have been well
studied [2][19][23]. Two types of validations are performed: (a) steady state and (b) starting
phenomenon. In addition to the validation, the numerical simulations serve to optimize the
required grid spacing.
It is necessary to pinpoint the tip of the penetrative front in order to determine the penetration
rate of a starting buoyant jet. We search for the tip at a particular time by scanning the
concentration field layer by layer from the bottom towards the top of the computational
domain. The first vertical position at which the threshold concentration was exceeded was
determined to be the tip front. We chose a threshold concentration equal to 10% of the initial
concentration, but because the front is characterized by steep scalar gradients, its location is
quite distinct and not very sensitive to small variations of the chosen threshold concentration.
An example of the concentration field is shown in Figure 2-2.
2.3.1 Jet
For a pure jet, the controlling characteristic is the initial momentum flux. Strictly speaking, a
pure jet does not possess any density difference with the ambient, i.e. Apo /p =0 .
However, in order to visualize the scalar structure, the jet density is increased slightly to
Apo /p = 1099 in the validation simulations so that the density can act as a tracer to
illustrate the scalar distribution but without introducing significant buoyancy.
The initial
velocity of the jet, Uo, is 0.05 m/s. The corresponding Reynolds number is Re=UoD/v=2500,
i.e. the jet is turbulent at the source.
2.3.1.1 Steady state
Since the present simulation using the LES approach is transient, an averaging period which
is large enough to average the instantaneous variations of the flow is required to show the
29
steady state characteristics of the mean velocity and concentration fields. For this reason, the
mean characteristics of the flow averaged over different periods with a sampling frequency of
1 Hz are drawn in Figure 2-3 for the pure jet case (and later in Figure 2-6 for the lazy plume
case).
The velocity and concentration decay rates along the center line of the pure jet are compared
with the experimental results from Wang and Law [23] in Figure 2-3(a). From the figure, the
axial velocity and concentration retain their initial values within a potential core of about 6D.
Downstream in the Zone of Established Flow (ZEF), the axial velocity and concentration
decay continuously with a rate that decreases with penetration distance. It can be observed
that the agreement with the experimental results is good with a sampling duration of 40s
covering 60-100s. From Figure 2-3(a), more scatter can be noted for the concentration decay,
but the agreement remains satisfactory. The equivalent axial velocity and concentration
radius of the jet are shown in Figure 2-3(b). To evaluate the jet expansion, the boundary is
defined by the locations within a particular horizontal plane where the axial velocity (or
concentration) is 37% (1/e) of its maximum value. The growth of the velocity width (b") and
the concentration width (b,) defined in this manner indicates, respectively, the spreading rate
of the axial velocity and concentration spreading in the vertical direction. Simulations
indicate that both widths increase with penetration distance, in a manner consistent with
experiments in the ZEF (i.e. z/D>6). Note that a consistent radius is observed in the potential
core region.
2.3.1.2 Starting purejet
The penetration rate of a starting pure jet represented by the temporal rate of change of the
vertical position of the tip is shown for different grid sizes in Figure 2-4(a). From the figure,
the penetration rate with 64x64x384 grid cells is similar to the simulation with 80x80x480
grid cells, thus implying that the simulation with 64x64x384 grid cells has sufficient grid
resolution. To demonstrate the absence of boundary effects, penetration rates in different
domain sizes are compared in Figure 2-4(b). The penetration rates overlap each other, which
indicates that in the early stage of development, the penetration rate is not impacted by the
domain size. Therefore, the small domain size of 0.3m*0.3m* 1.5m is used for the production
results.
30
The dimensionless penetration rate, i.e. the slope of q versus the square root of time, is shown
4
in Figure 2-5, where q = z, /(M 0 )" = z, /(
4
D2UI2)4and zt is the position of the front. As
discussed before, the initial period of varying penetration rate can be referred to as the Period
of Flow Development by Ai et al. [19]. After the flow is developed with self-similar profiles,
the penetration rate possesses a constant power law relationship with time, and the slope on a
log-log scale would become nearly a constant. From the simulation results, the slope of the
last section for the simulation results is found to be equal to 4.0, which matches almost
exactly the result of Ai et al. [8]. Hereto, the simulations successfully reproduce the
penetration of a starting pure jet.
2.3.2 Plume
Following the pure jet validation above, a further study is conducted for the validation of a
lazy plume whose characteristics are controlled by the initial buoyancy flux Bo.
For these
simulations, the initial velocity Uo is taken to be a small value of 0.05 m/s, while the density
2
difference is set to be Apo /po =7x10- . The corresponding Richardson number R, is 3.49
which lies within the lazy plume regime [1].
2.3.2.1 Steady state
Figs. 6(a) and 6(b) show that, using an averaging period duration of 40 s and a sampling
frequency of 1 Hz, the axial velocity and concentration decay along the centerline become
consistent beyond t-40 s. The results agree the experimental measurements by Wang and
Law [23] beyond z=6D. One interesting aspect of the lazy plume is that the axial velocity
increases very quickly in the laminar region near the source before transition into turbulence
following the decay in the self-similar region. The concentration decay is also well predicted
with virtual origin correction, which assumes that a point source at z=2.5D can properly
represents the nozzle of a lazy plume [37]. Note that the initial concentration remains
constant in the laminar region until the velocity peaks. For the growth rate, Figure 2-6 shows
that the velocity and concentration (virtual origin corrected) spreading also agree with the
31
proposed values by Wang and Law [23], despite the equivalent radius narrows slightly in the
potential core. Summarizing the steady state results in Sections 3.1.1 and 3.2.1, it appears that
the centerline flow characteristics approach steady state earlier than the spreading width
characteristics.
This is somewhat expected, since axial flow development near the centerline
of a jet or plume is dominant while the lateral development depends on the transverse
turbulent shear dispersion and thus a larger time scale.
Besides the mean flow characteristics, comparisons were also made between simulated
turbulence characteristics and experimental data reported by Wang and Law [23]. The
simulated longitudinal turbulence intensities along the centerline,
w12
/w for both a non-
buoyant jet and a pure plume were about 25% less than the corresponding experimental
values of 0.26 and 0.27, while the simulated results for U
2
/wc were about 10% below the
experimental values of 0.19 for both cases. The simulated transverse profiles of
of
-/
-2
U'/W
were similar to the experiments in both magnitude and shape, but showed somewhat more
variability.
2.3.2.2 Starting lazy plume
A numerical study on the dependence of grid size is first conducted. The results, presented in
Figure 2-7, suggest that a grid mesh finer than 64x64x384 is required for the penetration
analysis. According to Figure 2-8 the non-dimensional penetration rate has a time power of
0.78 in the PDF, which is very close to the results of 3/4 in Diez et al. [17] and Ai et al. [19].
This comparison between the numerical and experimental studies validates the code for the
simulations of the starting lazy plume. According to these validations and balancing the
computational expense, the grid mesh of 64x64x384 is used for the following simulations.
In summary, the present LES model performs credibly for the asymptotic cases of a pure jet
and a lazy plume. Its transient performance is particularly convincing when the simulation
results are compared with previous experimental data. The steady state results are also
reliable, and can be further improved if longer simulations are performed.
32
2.4 Buoyant jet
2.4.1 Penetration rate with starting buoyant jets
The penetration rates of buoyant jets with different initial buoyant fluxes (hence, different
values of Ro) are shown in Figure 2-9. Initially all jet fronts penetrate at the same rate until
tUO/D=0.8. Afterwards, the pure jet penetrates linearly with time, whereas jets with higher
buoyancy penetrate faster. When the jets reach the self-similar phases (e.g. tU0 /D=3.5-4 for a
buoyant jet with Ro=2.95), the penetration rates of the buoyant jets approach an asymptotic
value as described above, i.e. 3/4 of the square root of time.
According to the results, the development of the penetration rate can be divided into three
phases: the initial overlapped phases, the accelerated phase, and finally the asymptotic phase.
To isolate the effects of buoyancy flux, the penetration distance by the pure jet is subtracted
from the total penetration distance, and the excess penetration distances
zB
are shown versus
time in Figure 2-10. For comparison, the excess penetration distances of jets with different
initial momentum fluxes (i.e. Reynolds numbers) are shown in the same figure.
The three phases of penetration rate can be interpreted by the relationship of the two driving
mechanisms, i.e. the initial momentum flux and the buoyancy inducement. In the initial
overlapped phase, buoyancy does not have time to significantly accelerate the penetration.
Therefore, the momentum flux dominates the driving force and the penetration distances
overlap each other. During the accelerated phase, the potential energy contained in the
buoyancy flux is transformed by gravitational acceleration to kinetic energy. Thus, the
penetration rate differs for different buoyancy fluxes as shown in Figures 2-9 and 2-10.
Figure 2-10 isolates the buoyancy effect from the initial momentum in terms of excess
penetration distance: for the same density difference but different initial momentum,
penetration distances are found to overlap each other. This suggests that the total penetration
distance can be resolved as the sum of the separate effects of initial momentum and
buoyancy. Because these two factors are uncoupled, the relationship between them appears to
be linear.
33
At the final self-similar phase (or PDF), the total penetration rate decreases due to the greater
entrainment of ambient fluid. At the same time, the momentum flux and buoyancy
inducement interact nonlinearly with each other, resulting in the front advancing with an
asymptotic limit of t 314 .
2.4.2 Penetration equation
A key objective of the present study is to examine the time-dependent penetration of buoyant
jets in the PFD. As discussed in the introduction, the penetration in the PFD is more complex
than in the PDF due to a lack of self similarity.
The source conditions, in particular,
significantly affect the penetration behavior in PFD. These conditions include the velocity
profile and history at the source, nozzle geometry, laminarity, and the presence of
overpressure. The LES results obtained in the study are relevant for the penetration of a round
turbulent jet with a piston-driven type uniform velocity profile without overpressure. The
following shows how a proper choice of scaling can yield a general fitting equation for the
penetration rate within the PFD.
First, we explore non-dimensionally the analysis of the excess penetration, zB. Generally, for
a turbulent flow we expect that
zB
==
(t, B,,
Q,
M). However, since
zB
distance beyond that of a pure jet, M, may be insignificant, and B0 and
is the penetration
Q. can
be used as
repeating variables to yield normalizing time, penetration distance and momentum scales:
4/5__
5
/5_
t' QO
z'= B
B0
B
MI'=
/
(2.11)
4/B
Noting that MIMO'= RO-4 5, Figure 2-11(a) plots zB/z' versus t/t' for various Ro. For Ro < 1.2
(solid lines), a reasonable fit is provided by
ln(zB /Z')
(2.12)
=21nQt/t')-2.3
which results in
ZB
/D =0.10 B 0
QOD
2
orzB=O.1Ogt 2
34
(2.13)
For
objects
free-falling
in
a gravitational
field
without
significant
resistance,
proportionality coefficient in Eq. 2.13 would be 0.5. The present coefficient of -0.1
the
is
substantially less. This implies that strong resistance, probably in the form of a drag force and
momentum sharing by entrained ambient stationary fluid, is acting on the starting vortex by
The fact that R, = 1.2 is well on the lazy plume side of a pure plume,
the ambient fluid.
suggests that for most buoyant jets,
ZB
is indeed independent of M, as hypothesized.
An alternative relationship covering the full range of Richardson numbers used in our model
simulations (0 < Ro < 3) is provided by normalizing penetration by a different scale; thus
"=
-'
Q
4/
5
___115__112_
and z"=
B 35
BJ25
(2.14)
0
Figure 2-11(b) plots non-dimensional penetration versus time using Equation 2.14, from
which
(2.15)
ln(z, / z") = 2.41n(t/t")-3,
or
ZB
=0.05
0 5 B .04
AJ
M1 B1L
I t
172
(2.17)
Q
Because it has a stronger physical basis and is valid for most buoyant jets, Equation (2.13) is
used to predict the following total penetration distance.
Second, the penetration distance driven by the initial momentum flux (i.e., the penetration of
a pure jet) can be determined from Figure 2-9 as
(2.18)
z, = 0.53Uet
or
M32
ZA / D = 0.47
02
(2.19)
t,
go
where, as expected from physical arguments, zM is proportional to Uot. The coefficient of 0.53
reflects the roll-up of the discharged fluid to form the starting vortex.
It is close to the
theoretical value of 0.50 expected in the potential core region of a non-buoyant jet if,
35
following arguments of Prandtl [38], we assume that the stagnation pressure is the same on
either side of the front. Summing up the penetration due to momentum and buoyancy, the
total penetration distance can be expressed as
z,/ ID = zB /D+ z. / D =0.10
0
Q0 D
(2+0.47 M02
t
(2.20)
Again this equation is applicable only in the PFD region when the penetration is led by the
starting vortex. After the pinch-off and when the jet stem engulfs the starting vortex and
regenerates a leading vortex, the penetrative behavior would evolve into the PDF behavior
described in Ai et al. [19].
2.4.3 Penetration measured by different flow parameters
For buoyant jets, the flow characteristics can be represented by a number of parameters
including the scalar concentration and the spatial components of velocity and vorticity. While
the results in the previous sections are established based on the concentration front, it is
pertinent to examine whether the penetration is similar for the other parameters. For this
purpose, we consider a buoyant jet having a uniform initial velocity UO = 0.05 m/s, and initial
relative density difference Apo /po = 4 x 10-'. The threshold tests described in Section 3 are
then repeated by replacing the value of concentration with the values of velocity and vorticity
components, i.e. u, v, w and o, coy, a). The penetration distances extracted from these
different fields are shown in Figure 2-12, which shows that the penetration rates are almost
identical, although the concentration field penetrates slightly faster than the velocity field.
This suggests that there is no significant discrepancy between the various fields regarding the
penetration analysis. In other words, the intrusion can be taken as a shock front in the PFD.
36
2.5 Summary and Conclusions
A numerical study using the LES approach has been conducted to investigate the penetration
behavior of a starting buoyant jet during the PFD. The behavior of the two asymptotic cases
of a pure jet and a lazy plume are first reproduced to validate the numerical code. The steadystate results of the centerline decay and the growth rate of concentration and velocity fields
compare favorably with the experimental data reported in the literature. The corresponding
transient simulations are also consistent with the experiments reported previously. These
validations show that the present numerical model is effective and sufficiently accurate for
the analysis.
After the validation, the model is used to simulate starting turbulent buoyant jets with three
different Reynolds numbers from 2000 to 3000, and a wide range of buoyancy effects from
pure jets to lazy plumes. The penetration front generally advances faster with higher
buoyancy. More importantly, the penetrative distances induced by the initial buoyancy fluxes
and by the initial momentum fluxes are found to be independent; therefore, the total
penetration distance can be treated as a linear combination of these two parts. An equation is
proposed to approximate the penetration behavior of a starting buoyant jet in the PFD by
performing curve fitting to the numerical results. Future experiments would be desirable to
verify the present conclusions.
37
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40
buoyant jet
free-slip
outflow
2,
A*
0 8,
10,
12,
14,
y
z
y/D
(a)
x/D
(b)
Figure 2-1 Computational domain: (a) boundary conditions (not to scale); (b) a typical
grid mesh (only every 4th mesh point shown in each direction; the center of the jet
outlet is at x=3D, y=3D, Z=O)
41
=1
(s)
2
3
4
0.5
5
6
7
8
9
10
0
1
2
3
x/D
4
5
C/C
0
Figure 2-2 Penetrative front of the buoyant jet (Re=2500, Ro=0.186). The dash line
denotes the location of the jet front.
42
+
*SsseS~ge*
1
40-80s
50-90s
o
60-1l00s
Wang and Law (2002)
0.8wC/UO
0.6-
*
C /C
0.4-
++++++
00 0 0
OC
0.2-.
0
++
'
2
6
4
8
12
10
16
14
18
20
z/D
(a)
2bc/D-----.
+
6 *+
1.5-+0
b /D
1.
0
+
M
0.5 NS00
*
--
0
(b)
2
4
6
8
40-80s
50-90s
60-1 00s
Wang and Law (2002)
10
12
14
16
z/D
Figure 2-3 (a) Decay of the mean axial velocity and concentration of a pure jet with
different averaging periods; (b) Growth of the mean velocity and concentration width
of a pure jet with different averaging periods.
43
80
480 x 80
384 x 64
48 x 192 x 48
-------------. 32 x 96 x 32
0.5 -
-64
0.4-
x
-
x
..
~0.30.2-
0.1
0
0
5
10
15
20
time (s)
(a)
10987654-
3e0.5*0.5*1.5
0.75*0.75*1.5
1.0*1.0*1.5
S1.25*1.25*1.5
32-*
10
(b)
'
1
3
5
7
9
11
time (s)
13
15
17
Figure 2-4 Penetration rate of a pure jet with different grid sizes (domain size length
unit in m).
44
20 r
slope=4.0
15 [
CD
N
10
r5'
I
I
5.5
5
4.5
4
3.5
t1/2 (s1/2
Figure 2-5 Dimensionless penetration rate of a pure jet
4.5
+
4
o
++
*~
3.5 F
*
,*
*
3
40-80s
50-90s
60-100s
Wang and Law (2002)
+4
CC
2.512
5
'a
Cc/CO
1
0.5
0
0
(a)
5
15
10
z/D
45
20
21.8-
+
40-80s
1.6-
o
50-90s
60-100s
Wang and Law (2002)
*
1.4-
*
1.2-*'
0.8
0.6
b,/D
0.4
0.2 01
0
(b)
*...
.*.*
5
10
15
20
z/D
Figure 2-6 Validation for lazy plume: (a) Decay of the mean axial velocity and
concentration of a lazy plume and (b) growth of the mean velocity and concentration
width of a lazy plume with different averaging periods
46
1.5 r
E
0.580
x 480 x 80
/-64
384 x 64
-- 48 x 192 x 48
32 x 96 x 32
0
0
-
2
4
6
8
x
10
t (s)
Figure 2-7 Penetration rate of a lazy plume with different grid sizes
47
12
3.3
3.23.1 3o2.9-
2.8-
slope=0.75
2.72.62.5
2.4
2.4
2.6
2.8
3
4 /B ) 1/3
In t/(D
Figure 2-8 Dimensionless penetration rate of a lazy plume
48
3.2
3.4
16 -+-R 0 =0.000
SR 0 =0.373
14 -
R=0.510
<
R=0.722
D.
x
xx
0
X
0
12 R =1.103
000
12 -
-- R0=1.768
.
100
R =2.947
8-
60
2 0-
0
246
810
12
tU0 /D
Figure 2-9 Penetration of buoyant jets in the PFD under different buoyancy fluxes
49
1.8x10-2
0.3-
APOPO=5x1 0-2
4x
+
+
X
Re=2000
Re=2500
Re=3000
y
x
+
7x03
+
x
0.25-
+
x+
+
+
0.2-
3x10;-3
+
+
N
+
x
+
0.15-
+
+
+
-x
+ +X
+
*
0.1
A
X
S+
+
+
xl
1.5x10-3
++
Xx4-
+-
+
0. C5 -
x
x
8x±
+
t+
+
+
-x44
+
x
Xf -4
030
0
1
2
3
5
4
6
7
t (s)
Figure 2-10 Excessive penetration induced by various buoyancy fluxes
50
8
9
3
2
XIX
slope=2
x0
Re=2500 R=0.510
-Re=2500
X0
,N
N
Re=2500 R0 =0.132
-
1
0
R=1.103
Re=2500 R0 =2.283
x
Re=2500 R,=2.947
8'
-
-1
X
-
Re=2000 R=0.466
-
Re=2000 R0 =0.638
-4 0
~1
-2
-4
Re=2000 R=0.233
-
Re=2000 R0 =0.902
Re=3000 R0 =0.155
-
-
Re=3000 R=0.425
Re=3000 R=1.474
0
0.5
1
1.5
2
2.5
3
log t/t"
(a)
Re=2500 R0 =0.132
-
4-
Re=2500 R=0.510
e
3-
2-
N1 0-
-
-
Re=2500 R=2.283
Re=2500 R=2,947
-e-
Re=2000 R0=0.233
-
Re=2000 R0=0.466
---
Re=2000 R0=0.638
-v- Re=2000 R0=0.902
-4Re=3000 R =0.155
-+-
Re=3000 R =0.425
---
Re=3000 R =1.474
slope = 2.4
-1
-2-
-3-
-5
0
0.5
1
1.5
2
2.5
3
log t/t"
(b)
Figure 2-11 Dimensionless excess penetration of buoyant jets induced by buoyancy
fluxes with dimensionless time
51
0.4
0.350.3-
0.25-
o
0
o
concentration
velocity u
velocity v
A
+
velocity w
vorticity o
x
vorticity o
O0.2-
vorticity oz
0.150.10.050
2
4
6
8
t (s)
Figure 2-12 Penetration distance with different flow characteristics
52
10
12
Chapter 3
Buoyant Formation Number of
A Starting Buoyant Jet
Abstract
Understanding the influence of buoyancy on the formation number is important for analyzing the
development of a starting buoyant jet and the interactions between its vortex ring and trailing
stem. Numerical simulations with a large-eddy simulation model are performed to reproduce the
starting buoyant jet in conditions ranging from pure jet to lazy plume. From the results, an
improved method to determine the formation number is proposed based on the occurrence of a
step jump in the vortex ring circulation. A comparison of the numerical results with the
experimental data for a starting pure jet is first performed. The widely accepted formation
number (~4.0) is obtained, which implies that the method is satisfactory. The effect of buoyancy
on the formation number is then investigated for two turbulent discharge conditions of Re=2000
and 2500 and with a wide range of buoyancy flux. Best-fit results are obtained that correlate the
formation number with the Richardson number. Finally, a slug model that incorporates buoyancy
is developed to allow prediction of the "buoyant formation number" for the starting buoyant jet
using a limiting value of 0.33 for the dimensionless energy, which is the same value for a pure
jet.
*
Published as Wang, R.-Q., Law, A. W.-K., Adams, E. E., and Fringer, 0. B. (2009). "Buoyant formation number
of a starting buoyant jet." Physics of Fluids, 21(12), 125104.
53
3.1 Introduction
Starting buoyant intrusions are common in natural and engineered systems and span multiple
time and length scales.
Examples include volcanic eruptions, piston engine fuel injection,
delivery of scent to a theatre through an air canon [1], and micro scale heat exchange in MicroElectro-Mechanical Systems (MEMS). The significance of such intrusions has drawn numerous
studies in the past to understand and quantify their behavior (Gharib et al. [2], etc).
A significant aspect of the starting buoyant intrusion is the relationship between the starting head
vortex and the trailing stem formation. The behavior of the starting head vortex closely
resembles a discrete vortex ring, which has been extensively studied (a good recent summary can
be found in Shariff and Leonard [3]). Yet the phenomenon cannot be examined solely by the
starting vortex, as the trailing stem can continue to feed mass, circulation, and energy to the
vortex. A well known illustrative example is provided by Gharib et al. [2] who performed an
experimental study on the vortex ring generated by a piston/cylinder arrangement. They showed
that two distinct states exist depending on the "formation time", tf, which is defined as the ratio
of the stroke distance over the nozzle diameter, L/D. Note that tj is referred to as a nondimensional time because it is the ratio of the release time, L/U, where U is the release velocity,
divided by the time, D/U, required for the released fluid to travel a distance D. With a short
formation time of 2, an isolated vortex ring was formed without the development of a trailing
stem (i.e. the intrusion was totally absorbed in the starting vortex), whereas with a longer
formation time of 14.5, the trailing stem was observed following the front vortex ring. The
critical value of the formation time was called the "formation number", above which a trailing
stem is present. They found that the formation number is approximately equal to 4.0, and
54
relatively invariant through various nozzle conditions and velocity history.
The formation
number was proposed as the criterion for the "pinch-off' occurrence (i.e. detachment of the
starting head vortex from trailing stem). This is because the starting vortex could not absorb the
additional intrusion from the piston if the formation time were higher than the formation number,
and would physically separate from the trailing stem. In another words, when the formation
number is reached, the vortex ring has reached the upper limit of circulation by the driving
mechanism [2].
The existence of the formation number has also been confirmed by numerical simulations.
Rosenfeld et al. [4] simulated the formation of non-buoyant vortex rings by symmetrical laminar
flow configurations, and observed the different flow structures below and above the formation
number. They reported that the formation number was strongly affected by the initial velocity
profile but less by the velocity history, while Zhao et al. [5] clearly illustrated the circulation
absorption process between the vortex ring and the trailing stem. Mohseni et al. [6] extended the
simulation to include a variety of non-conservative body forces and concluded that a varying
nozzle diameter can delay the pinch-off. Their work stimulated an interest to further examine
nozzles with changing diameters, such as those studied by Dabiri and Gharib [7] and Allen and
Naitoh [8].
Taking advantage of the observations and understandings from these numerical and experimental
studies, quantitative models were developed to describe the initial starting behavior. They
include Mohseni and Gharib [9], Linden and Turner [10], Mohseni [11], Kaplanski and Rudi
[12],
and
Shusser et al.
[13].
Gharib
et al.
55
[2]
defined a dimensionless
energy
End
=
E/
=
n /2 (D / L), where E, F7, and I are the kinetic energy, circulation and impulse
of the starting jet (detailed further in Section 4.3). They calculated a limiting value of End ,
or
Elim, to be 0.33 (referred to as a in their paper), below which pinch-off would occur. Since End=
0.33 corresponds to L/D = 3.8, pinch-off would thus occur for L/D greater than about 4.
Analytically, they demonstrated that Elim represents a level beyond which the piston cannot
deliver more energy, circulation, and impulse to the starting vortex steadily according to the
Kelvin-Benjamin variational principle [14].
Mohseni and Gharib [9] formulated a model to
describe the phenomenon of formation number based on the Norbury's family of vortices and the
slug-model, which was extended to various piston velocity programs by Shusser et al. [13].
Similarly, Linden and Turner [10] matched the properties of the injected plug of fluid and the
Norbury's family of vortices to show that Eim reflects the constraint due to the volume limitation
of the vortex ring core. Kaplanski and Rudi [12] refined the model of Linden and Turner [11] by
incorporating the viscosity.
Most studies on the starting phenomenon so far have focused on non-buoyant intrusions with the
piston-cylindrical configuration. The equivalent situation of a starting buoyant jet with a strong
buoyancy component (in cases such as the eruption of a volcano and open-water disposal of a
sediment mass whereby the buoyancy effect is prevalent due to the large density difference
between the intrusion and ambient fluid) has not been well investigated. For the asymptotic case
of a thermal ((lazy plume without initial momentum), Pottebaum and Gharib [15] confirmed the
existence of the maximum circulation by performing experiments with plumes generated by a
heat disc; and Shusser and Gharib [16] suggested based on an analytical model that the
equivalent formation number would be t=4.73, where r is the duration of the release divided by
56
the characteristic time To =
required for an effective gravitational force of g' to accelerate
the mass a distance equal to the cylinder diameter D.
Neither the starting pure jet models (e.g Mohseni and Gharib [9]) nor the starting thermal models
(e.g Shusser and Gharib [16]) are able to fully address the dynamics of a starting buoyant jet with
a combination of initial momentum and buoyancy fluxes. As far as the formation number is
concerned, it is not clear a priori whether the initial buoyancy would delay or speed up the pinchoff process after the turbulent intrusion occurs. In addition, at present the formats of the
formation time and formation number are distinctly different between the existing pure jets and
thermal models. A suitable analytical model that can bridge the two asymptotic cases and cover
the buoyant jet situation would be highly desirable. It is noted that such unified analyses have
been conducted for steady state buoyant jets (e.g Wang and Law [17]), but a similar investigation
for starting buoyant jets has not been reported in the literature as far as the authors are aware.
In this study, a series of numerical experiments are performed using the Large-Eddy Simulation
approach to examine the formation dynamics of a starting buoyant jet. In the following, the
numerical approach and the flow configuration are first introduced in Section 2. In Section 3, the
numerical model is verified with the asymptotic case of a starting pure jet, for which
experimental data regarding the formation process has been reported previously in the literature.
Subsequently, extensive simulations are performed covering a broad range between the pure jet
and the lazy plume in Section 4, and a slug model is developed to include the buoyancy effect on
the formation number. Finally, the conclusions from the current study are listed in Section 5.
57
3.2 Numerical Model
3.2.1
Governing Equations
The governing equations of the grid-filtered continuity, Navier-Stokes and temperature transport
equations with the Boussinesq approximation can be expressed as follow [18]:
=0,
(3.1)
axi
-ahi BF
+± "=S
1 ,
at ax,
+
at
(3.2)
=0,
(3.3)
axi
where
=
UUJ+
1j -V
xI
TV(34
j
Ri = Ujo - ro
+Xj,
(3.6)
In the above equations, the over-bar variables represent the grid-filtered quantities. t is time, u,
(i=1,2,3) are the Cartesian velocity components in the direction of x, y, and z. Other quantities
are defined as: p is the pressure, g is the gravity, 0 is temperature, q0 is the background
temperature,
K
P=
1
is the coefficient of thermal expansion, v is the kinematic viscosity, and
is the thermal diffusivity. Note that all the equations are subjected to the Einstein rule of
summation.
58
The subgrid terms rj and X. are specified as
1
*J
(3.7)
=uiu1 -uiu,
yX = uj-
(3.8)
E
These two terms can be modeled by the dynamic mixed subgrid-scale model, the details of which
can be found in Zang [19] and Salvetti et al. [20].
The governing equations are discretized by a finite volume formulation on a single nonstaggered grid [19]. Various discretization schemes are applied: 1) a semi-implicit scheme with
Crank-Nicholson method on the diagonal viscous and diffusive terms, and Adams-Bashforth
method on the other terms in time; 2) accurate upwind-difference schemes on the convective
terms; and 3) second-order central difference on all the other spatial differential terms. The
convective terms of the momentum equations (Equation 3.2) are discretized using the QUICK
scheme to minimize the expense of computation, whereas the convective terms of the scalar
transport equation (Equation 3.3) are discretized using the SHARP scheme to avoid spurious
oscillation [19][21].
The numerical model has been verified by a series of comparisons with standard experiments,
e.g., Cui and Street [22], and implemented in recent studies [23][24][25][26]. The parallel code
used in the present study was developed by Cui and Street [22].
3.2.2 Flow configuration
The computational domain is three-dimensional with a square horizontal cross-section which
extends 0.3 m x 0.3 mx 0.8 m in the Cartesian coordinates, x, y, and z respectively (Figure 3-la).
59
The starting buoyant jet is issued downward from the top boundary with a uniform velocity Uo
into a homogeneous and stationary ambient fluid through a circular nozzle with a diameter D=5
cm. Thus the computation domain is equivalent to a size of 6D x 6D x 16D, which has been
verified to be sufficient for the investigation of the near-field formation process [28].
The
buoyant jet is heavier than the ambient and with a relative density difference of Apo /po (thus
the buoyant action is in the same direction as the plume injection). The governing equations are
discretized into a stretched mesh which has denser grids at the center axis through the nozzle (as
shown in Figure 3-1b).
Before the simulation begins, the fluid in the computational domain is stagnant with a uniform
density, po. The boundary conditions are depicted in Figure 3-la. The top boundary is a free-slip
surface permitting no vertical through flow; while the other boundaries are "no-gradient" outflow
boundaries following the recommendation of Yuan [26]. A constant volume flux, determined
from the prescribed inflow velocity Uo, is then imposed at the inlet at t=O to initiate the
simulation.
To facilitate the analysis, some characteristic parameters are defined following Fischer et al.
[27].
Apo D2U,
1
B =7g
4
po
1 21
MO= -rD2Ue2,
Q0 =-rD2UO,
4
4
1
m314Q
=
B
m1
S1/2
R0
1"
(T)4
M\
0"
4
UO2
=/
4
Re= "(3.9)
Fd
V
where Qo, Mo, and BO are initial volume flux, momentum flux, and kinematic buoyancy flux,
respectively; g' is reduced gravity (g Apo /p ); Apo /p is the normalized difference between
discharge density and ambient density;
IM
and l
60
are the characteristic length scales of
momentum and volume flux, respectively; Fd and RO are the jet densimetric Froude number and
Richardson number, respectively; Re is the jet Reynolds number, and the kinematic viscosity v
10-6 m 2/s.
3.2.3 Vertical distribution of circulation and dynamics of buoyant jets
In the following two sections, a strategy is developed to determine the value of the formation
number, or the critical value of formation time above which a trailing stem is maintained behind
the leading vortex ring. The strategy to determine the formation number includes three steps: (a)
identifying the starting vortex ring, (b) locating the pinch-off if and when it occurs, and (c)
analyzing the circulations of both the whole computational domain and the vortex ring to
quantify the development process.
First, we define a cross-sectional circulation to distinguish the vortex ring from the trailing jetvortex ring system as follow:
(0*(z)=
(3.10)
co,(x,z)dx
HCP
where HCP represents the half central plane, i.e. either left central plane (O<x<3D,y=3D), or
right central plane (3D<x<6D,y=3D), and w, represents the vorticity in the y direction. Figure
3-2 shows the vorticity field of the vortex ring formation process coupled with the vertical
distribution of o)* versus the penetration depth, z, for the case of Re=2500, t/=L/D=12, and
RO= 0 .186. The peak of the absolute value of the cross circulation, lw*1, indicates the position of
the centre of the starting vortex ring. Note that Jwo*
also has a relative maximum at the edge of
the orifice generated by the free-slip boundary condition. At tUO/D=3 and 5, the head vortex is
well organized and followed by a trailing stem. At tUoID=8, the JC*J curve shows two minima
behind the center of the head vortex, which indicates that another new vortex is forming at the
61
leading edge of the trailing stem. At tUID=9, this "leading-trailing vortex" gathers towards the
central axis and is being merged into the head vortex. Meanwhile, a new leading-trailing vortex
is being produced behind. At tUID=10, the absorbed vortices have merged completely with the
head vortex ring. With the added circulation, the head vortex ring pushes forward and detaches
from the trailing stem at tUI/D=1 1 to complete the pinch-off process. Here, we consider the time
between tUID=9 and 10 to be the time of pinch-off, when the circulation of the absorbed
vortices is added into the head vortex ring. After the discharge ceases at tUID=12, the orifice
stops providing any additional momentum, and the trailing stem disintegrates into turbulent
patches of remant mass.
From Figure 3-2, a minimum of o* I can be found between the starting vortex and the trailing
stem throughout the development process until the full turbulent flow is established. Specifically,
this minimum always closely follows the center peak of the vortex ring. This minimum is used as
the criterion to divide the flow field into: (1) the starting vortex region (below) and (2) the
trailing stem region (above). Hereafter, the term "head vortex" refers to the starting vortex ring
in order to differentiate it from "leading vortex" which is the regenerated vortex ring formed
after the pinch-off (see Law et al. [28]). Note that a similar criterion has been used in Zhao et al.
[5] to show the onset of the jet instability and a radial vorticity distribution of the head vortex
ring has been used in Moseni et al. [6] to analyze the pinch-off dynamics.
3.2.4 Determination of buoyant formation number
Figure 3-3 shows the total and starting vortex circulations for the above case of Re=2500,
RO=0. 186 and t= 12, as a function of non-dimensional time tUID. (Note that because Uo and D
62
have the same magnitude (5 cm/s and 5 cm, respectively), tUID and t are numerically equal.)
Basically, the circulation progress can be categorized into two segments. When 0 < tUID <tj, the
total circulation in the computation domain continuously increases due to the jet discharge; when
tUID >t, the addition by the discharge ceases, but the circulation in the domain continues to
increase (though at a different rate) due to the buoyancy of the plume fluid already introduced.
Therefore, the slope of the total circulation changes at t = tf. The lower circulation curve in
Figure 3-3 illustrates the entire development of the starting head vortex ring. The circulation of
the starting vortex keeps increasing during the formation process fed by the trailing leading
vortices. As discussed previously, the circulation of the starting vortex then experiences a stepjump when the pinch-off occurs. After pinch-off, the vortex ring is separated from the trailing
stem. Subsequently, it would lose the supply from the trailing stem and thus the circulation
increase would slow to a magnitude corresponding to the buoyant action of the vortex itself.
Gharib et al. [2] proposed a method to determine the formation number of a starting jet by
defining it based on the circulation of the starting vortex attaining a maximum. This method has
been used by many subsequent studies (e.g. Rosenfeld et al. [4], Zhao et al. [5], etc.). For
buoyant jets, however, this method cannot be used directly since the total circulation as well as
the circulation of the head vortex ring continue to increase after pinch-off due to the buoyancy of
the plume fluids. Thus, we seek an alternative approach through the identification of the stepjump in circulation that enables the head vortex to pinch off from the trailing stem.
The present strategy to determine the formation number is illustrated by the two arrows in Figure
3-3. First, the circulation value after the step-jump in circulation is distinguished. The
63
intersection point on the total circulation line which has the same value is then noted. The
corresponding time, indicated by the vertical downward arrow, is finally obtained as the
formation number.
3.3 Formation Number for Starting Pure Jets
We perform numerical simulations with the configuration of a piston-driven pure jet for
verification purposes. The evolution of the total and starting vortex circulations with different
formation times is presented in Figure 3-4. In the beginning, both the total and head vortex
circulations increase linearly with time. Then, the total circulation becomes a constant after the
discharge ceases at tUID = tf, and decreases gradually due to viscous dissipation (which is
different from Figure 3-3 in which buoyancy is present); while the head vortex circulation
reaches another constant value earlier before a step-jump occurs. The formation number can be
identified to be 3.9 using the strategy discussed previously, which is consistent with the range of
3.6-4.5 found by Gharib et al. [2]. The value is also close to the previous numerical predictions
of 3.3 in Rosenfeld et al. [4], and 3.8 in Zhao et al. [5], although their studies were restricted to
the laminar range with symmetric vortex rings which differ significantly from the present
simulations.
Two different vortex-stem systems with ty=L/D=3 and 5 are shown in Figure 3-5 to illustrate the
absence and presence of the trailing stem, respectively. At tUID=3, the vorticity fields of both
cases have the same pattern in which the head vortex leads the stem at the same speed. At tUID
= 5, the trailing stem with tj=
3
huddles together to form a leading-trailing vortex, which is
subsequently absorbed into the starting vortex. In comparison, the starting vortex of tj=5 is still
64
supported by the trailing stem and keeps developing at tUID=5. At tU,/D =8, the vortex ring of
tj=3 has finished engulfing the trailing vortex and begun to decelerate due to viscous dissipation,
while the starting vortex of t1 =5 has initiated the pinch-off process. At last, all the circulation is
contained in the head vortex for t=3, whereas an obvious remnant is left for t=5. To show the
quantitative differences, the cross-sectional circulations of the stem region between these two
formation time cases are compared in Figure 3-6. The circulation of t=5 is clearly higher than
that of tf= 3 around z=3D in the remnant region. This again indicates that a significant circulation
is left in the remnant mass of the former, and reinforces the conclusion that the formation
number falls between 3 and 5, which is consistent with the result of 3.9 obtained above.
3.4 Buoyant Formation Number for Buoyant jets
The main objective of the current study is to investigate the effect of buoyancy on the formation
process of starting buoyant jets. In the following, the LES simulation results of starting buoyant
jets are presented. To begin, we shall first define the categories of buoyant jets based on the
initial conditions of the momentum and buoyancy fluxes at their source. Then, an example of the
development of the buoyant jet is described to demonstrate the simulation results and the
differences introduced by the buoyancy. Using the strategy detailed in Section 2.3 and 2.4, the
buoyant formation number, i.e. formation number incorporating the buoyancy effect, is
determined covering the entire range of Richardson number from pure jet to lazy plume. Finally,
a model is developed based on the slug model to predict the buoyancy influence on the formation
process.
65
3.4.1 Richardson Number and Buoyant jets
Morton [29] suggested the following dimensionless source parameter to classify buoyant jets:
A=
5Q2 B
0B
4aM5
=
12 .3
g'D
u2
=
12 .3
R2
0
4)
(3.11)
/2
where a (=0.09) represents the entrainment coefficient. Using this source parameter, a pure jet
(no buoyancy) can be defined with A=0, a buoyant jet for 0<A<1, a pure plume for A= 1, and
alazy plume for A>1,. Since this parameter relates directly to the Richardson number, RO, the
definitions can be represented alternatively as RO=O for pure jets, O<Ro<0.27 for buoyant jets, Ro
= 0.27 for pure plumes, and Ro>0.27 for lazy plumes.
3.4.2 Numerical Results
In the numerical simulations, the buoyancy flux is added into the discharge in an incremental
manner. The total and head vortex ring circulations are shown in Figure 3-7 based on the
numerical simulations. According to this figure, the circulation is generally increased by the
buoyancy flux, which is reflected in the steeper slope of the circulation lines. Consequently, the
curves with the buoyancy flux deviate from the linear relationship of the pure jet in Figure 3-7.
The pinch-off times are shown to be the same in this figure. This implies that the effects of
buoyancy and momentum fluxes are uncoupled, which has been observed in the analysis of the
buoyant jet penetration rate [25].
The effect of buoyancy on the formation number is shown in Figure 3-8, where it can be
observed that the formation number generally increases with the Richardson number. Note that
the Richardson number in the present simulations covers a wide range from pure jet (Ro=0 ) to
66
lazy plume (Ro>0.27). When RO reaches a large number of more than 1.0, the formation number
cannot be clearly determined because irregular fluctuations are present in the circulation curves,
flooding the step-jump and destroying the characteristic of a constant maximum.
Based on the numerical results with the two source conditions of Re = 2000 and 2500, the
following best fit curve relating the formation number to the Richardson number can be
obtained:
Nf =m l+er4RO-
j
(3.12)
+JNfO
where the amplitude m=1.3, the standard deviation o-0. 1, the median p=0.3, and the formation
number of the pure jet Nfo=4.
Drawing on the similarity between pure jets and thermals, Shusser and Gharib [16] argued that a
characteristic time scale, To, should be the time required for the jet or thermal to penetrate a
distance of one nozzle diameter based on the velocity acquired at the instant when the jet or
thermal has traveled one diameter (i.e., To e=D/UO for a pure jet; and T.'r =
D /2g' for a
thermal, the latter calculated by assuming that the flow has acquired a kinetic energy per unit
mass of g'D at a vertical displacement D). Analyzing the experimental data of Lundgren et al.
[30] for a laminar plume without initial momentum, they obtained the formation number as 4.73,
somewhat larger than the traditional number of about 4. They attributed the larger value to the
combined action of buoyancy inducement and momentum acceleration. Following the reasoning
of Shusser and Gharib [16], the time scale for a thermal can be established as To =
67
D
VUJ +2g'D
.
This is the time characteristic time required for the buoyant jet to penetrate one diameter, and is
obtained by equating the kinetic energy of the flow at a distance D to the sum of the kinetic
energies associated with the pure jet and the thermal. Consequently, a buoyant formation time
can be defined as follows:
Tiet -
Tther
ttU
-; t
1* t
T ther
t
=-
(3.13a)
D-,
-
Toje
D
-
tkg
t 2g
tVU2+2g'D
To
(3.13b)
v[D-
(3.13c)
,
D
The results for Re=2500 are transformed to the format of the buoyant formation number and
shown in Figure 3-9. It can be observed that the buoyant formation number is not a constant after
the scaling of Equation 3.13, and generally increases with larger buoyancy. Note that the present
condition pertains to turbulent discharge at the source, which is contrary to thermals which are
generally laminar near the origin; the difference between the present study and Shusser and
Gharib [16] can thus be attributed in part to the distinctly different source conditions.
3.4.3 Analytical model
As an analytical tool, the slug model proposed by Gharib et al. [2] and Mohseni and Gharib [8]
for pure jets can be further developed for buoyant jets. The kinetic energy, circulation, and
impulse of a pure jet can be approximated by the slug-model as
El = I 1TpD2LU2
8
(3.14)
1
F = -UOL
(3.15)
2
68
11
(3.16)
=!I pD2UoL
4
where E, is the kinetic energy, i, the circulation, l1 the impulse, and L =Uo tj the equivalent
stroke of the piston. The addition of buoyancy flux changes the flow behavior in three aspects: 1)
an excess kinetic energy of the buoyant jet can be generated from the transformation of the
potential energy; 2) an excess circulation can be induced by the difference between the additional
velocity due to buoyancy and the ambient fluid; and 3) the excess impulse is increased by the
gravity force. We assume that the discharge fluid rolls up to form a vortex ring and half of its
surface,
-
2
DL, is exposed to the ambient quiescent fluid, where the circulation inducement
occurs. The expression for kinetic energy, circulation, and impulse can thus be changed into
E=1 +,
1
1
(.7
1
2
)
TpD 2 L U2 +-irpD2Lg'(Uot+-g't
8I=4
2
E =E +E2
1
4
1
2
(3.17)
F'=Il' +12 =- UOL + - Lg't
(3.18)
1
2
1
I=1I + I 2 =-IpD2U
0 L+-vpDLg't
(3.19)
4
4
where the subscript "2" refers to the gravitational effects.
Likewise, the dimensionless energy can be expressed as
E
d
=
1/2
E3/2
_
D (I+g)3/2
L
(3.20)
where
5=
(3.21)
g't
g' t+2U
69
The coefficient 8 is a function of time, which indicates that the buoyancy effect continuously
increases the non-dimensional energy End. If we replace t by L/UO,
S=
L
2
(3.22)
g'L + 2U
and End becomes the non-dimensional energy at the end of a discharge. In this manner, End
would give the limiting value when L/D equals the formation number.
Based on the Kelvin-Benjamin variational principle, Gharib et al. [2] first proposed that the
formation number for pure jets corresponds to Ed-0.33. This is confirmed by both numerical
simulations and theoretical analysis [4][5][8]. Incorporating the buoyancy effect, we find that
this limiting value is also applicable to the formation processes of buoyant jets. The
dimensionless energy with different buoyancy and momentum fluxes is plotted against the
formation time, L/D, in Figure 3-10. The formation numbers determined by the numerical
simulations are marked with solid squares in the same figure, which fall in a narrow region
between Es=0.31 and 0.35, with a median of 0.33. The results imply that the formation number
can be predicted by the intercepts of End=0.33 and the curves for different buoyancy fluxes. In
another words, Ed=pO.33 is still valid for starting buoyant jets.
In the asymptotic case of a pure jet with g'=0 and 8=0, End is consistent with the model used
previously in the literature for pure jets, which corresponds to L/D ~4 as expected. With g'=oc,
however, 8=1, and the formation number has an upper limit of L/D=10.7. This implies the
interesting result that the ability of buoyancy to enhance circulation and delay pinch-off is
restricted by a maximum stroke length of 10.7D.
70
3.5 Summary and Conclusions
The present study examines the formation process of the buoyant buoyant jet over a wide range
from a pure jet to a thermal. To determine the formation number, a method to differentiate the
head vortex ring and the trailing stem is derived based on the evolution of the cross-sectional
circulation. Because the buoyancy flux continuously induces the circulation, a new strategy is
proposed to identify the formation number using the observed step-jump in the vortex ring
circulation. The strategy is validated by the results that it is able to reproduce the formation
number of a pure jet. In addition, the relationship between the presence of the trailing stem and
the formation number is also reconfirmed.
Numerical simulations are performed by adding buoyancy flux incrementally to simulations with
two turbulent source conditions of Re=2000 and 2500. Results show that the buoyant formation
number increases with the Richardson number following an error function relationship (Equation
3.12). In addition, since the momentum and buoyancy effects are uncoupled at the initial
development of the buoyant jet, the time for the step-jump occurrence is found to be invariant
through the range of added buoyancy flux. To bridge the gap between the time scaling of pure
jets and plumes, a buoyant time scale is formulated. Finally, with a slug model that incorporates
the buoyancy effect as well as the Kelvin-Benjamin variational principle, the buoyant formation
number is shown to follow the non-dimensional energy of 0.33, which is consistent with the
value for pure jets that was reported in the literature.
While this paper is in press, Marugan-Cruz et al. [31] reported that for negatively buoyant
starting jets, the formation number would decrease with increasing negative buoyancy. This
71
conclusion is consistent and mirrors our findings here that the number increases with increasing
positive buoyancy. It also implies a potential extension of our analysis to encompass the entire
range of density deficit of the injection fluid from positive to negative, which we are currently
pursuing.
72
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arrangement", Phys. Fluids 18, 033601-6 (2006).
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Fluids 37, 87-94 (2004).
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plume", J. Fluid Mech. 416, 173-185 (2000).
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[19] Y. Zang, "On the Development of Tools for the Simulation of Geophysical Flows." Ph.D
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(1997).
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[24] L. L. Yuan, R. L. Street, and J. H. Ferziger, "Large-eddy simulations of a round jet in
crossflow", J. Fluid Mech. 379, 71-104 (1999).
[25] R.
Q. Wang,
A. W. K. Law, E. E. Adams, and 0. B. Fringer, "Large Eddy Simulation of
Starting Buoyant jets", Environmental Fluid Mechanics, 11(6), 591-609.
[26] L. L. Yuan, "Large Eddy Simulations of a Jet in Crossflow", Ph.D. Thesis, Stanford
University (1997)
[27] H. B. Fischer, "Mixing in Inland and Coastal Waters", Academic Press, New York (1979).
[28] A. W. K. Law, J. Ai, S. C. M. Yu "Leading vortex of a starting forced buoyant plume."
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[29] B. R. Morton, "Forced Plumes", J. Fluid Mech. 5, 151-163 (1959).
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Mech. 239, 461-88 (1992).
[31] C. Marugan-Cruz, J. Rodriguez-Rodriguez, and C. Martinez-Bazan, "Negatively buoyant
starting jets", Phys. Fluids 21, 117101 (2009)
75
[32] Kaye, N. B., and Hunt, G. R. (2006). "Weak fountains." JournalofFluid Mechanics, 558,
319.
76
buoyant jet
free-slip
outflow
2
4
0
6
8
10
y
vr x
12
14
z
42
y/D
4
x/ D
(b)
(a)
Figure 3-1 The computational domain: (a) boundary conditions (not to scale); (b) a typical
grid mesh (only every
4 th
mesh point shown in each direction; the center of the jet outlet is
at x=3D, y=3D, Z=O)
77
t=3
(s)
t-8 (S)
t=-5 (s)
0
t=10 (S)
t=9 (s)
0
1
2
3
-5
4
5
6
7
-10
8
9
10
-5
0
1
x 10,
2
x/D
I)
-5
0
1
2
x/D
-5
0
2
1
*x
-5
x/D
0
2
x/D
1
X 0
-5
0
1
x* 1
-15
2
x/D
Vorticity
-
t=12 (s)
t=11 (s)
0
t=13 (s)
t=14 (s)
t=15 (s)
0
1
2
3
-5
4
L. . .
5
6
-10
7
8
9
10
-5
0* x 10
0
1
2
x/D
-5
0
x 10
1
2
x/D
-5
0
e x 104
1
2
xD
-5
0
M*x 10
1
2
xnD
-5
0
0)*X1 -
1
2
XJD
Figure 3-2 The formation process of vortex ring/trailing jet and the corresponding crosssectional circulation wo*; Re=2500, Ro=0.186, t=12
78
-15
0.025-
total
------vortex
0.02-
Ca)
0.015 0
e
0.01 -
0.005-
0
0
10
5
tU 0/D
Figure 3-3 Total and starting vortex circulation; Re=2500, RO=0.186, t=12
79
15
0.018
0.016
cE
.2
total circulaton
0.06- - tf=2
tf=3
0.014-
t;=4
tf=6
0.012-
t =12
0.01vortex ring
- 0.0080.0060.0040.002
0
5
10
tU0/D
15
20
Figure 3-4 Total and starting vortex circulation for pure jets with different formation
times; Re=2500, RO=O
80
5
UD=3
tU0/D=3
tUo/D=5
tU 0/D=8
tUO/D=15
4
1
3
2
2
3
1
4
0
5
2
-1
4
6
-2
8
-3
10
2
4
2
4
2
4
2
4
x/D
x/D
X/D
x/D
-5
L/D=5
tU0/D=5
tU0/D=3
5
tU0/D=15
tU 0/D=8
2
3
4
2
6
8
1
1110
0
2
-1
4
6
-2
8
-3
10
2
4
x/D
2
xD
4
2
x/D
4
2
x/D
4
_4
-5.
Vorticity
Figure 3-5 Comparison of the vorticity fields for different formation times at the center
plane; upper row: t=3 < formation number, and lower row: ty=5>formation number;
Re=2500; RO=0O
81
6x 10
--L/D=3
L/D=5-
54
-
I'
32-
0-10
0
~
2
6
4
8
10
Z/D
Figure 3-6 Comparison of the cross-sectional circulations of the forced plumes at tUOI/D=15;
Re=2500; RO=0
82
0.04
0.035-
c:R0=0
--
0.03-
---
R =0.186
R =0.264
0
Total
---- R =0.323
0.025C
e
R 0 =0.373
0.020.0150.01-
Head Ring
0.0050
0
2
4
6
8
10
12
14
t*
Figure 3-7 Total and head vortex ring circulations under different buoyancy fluxes;
Re=2500, t=12
83
6.5
+ 0
0
6-
0
0
L.
0
o3
0
C
40.50-
+
+0
0
cc
E4.5-
0
40-
+
0
0+
++
3.5
0
0.1
0.2
0.3
0.4
0.5
Re=2000
Re=2500
0.6
R0
Figure 3-8 Buoyant formation number as a function of Richardson number for two
Reynolds numbers
84
9F
-
8
jet
E)
7.
Tplume
- *-
6
5. -
321n
0
0.1
0.3
0.2
0.4
0.5
R0
Figure 3-9 Formation number as a function of Richardson number; Re=2500
85
0.65 r
Ro=0.0, Re=2500
0.6
0.55
R0=. 19,Re=2500
R =0.30,Re=2500
0
R=0.23,Re=2000
R=0.37,Re=2500
Ro=0.37,Re=2000
..-... R =0.47,Re=2000
0.50
.0.5
..
R=0.42,Re=2500
R=0.0, Re=2000
--
iUr-%e=
-
0
formation number
0.45
w0.4
0.35
0.30.250.23
4
5
6
7
8
L/D
Figure 3-10 Dimensionless energy for different buoyancy fluxes versus the formation time
L/D. The solid squares are the formation numbers determined from numerical simulations.
86
Chapter 4
Pinch-off and Formation Number of
Negatively Buoyant Jets
Abstract
Previous investigations of starting buoyant jets are extended towards negative buoyancy to
address key issues in the formation processes. A series of Large-Eddy Simulations (LES) is
performed to identify whether an optimal vortex can be formed with negative buoyancy, and if
so what the corresponding formation number would be. The numerical code was previously
validated for non-buoyant and positively buoyant jets, and is further validated here for negatively
buoyant jets using literature data on submerged fountains. Subsequently, jets with a range of
negative buoyancies are simulated using source Reynolds numbers of 2000<Re<3000.A revised,
directional form of Richardson Number (Rid) is proposed to accommodate the entire range of
buoyancy, with Rid > 0 for positively buoyant jets, Rid =0 for non-buoyant jets, and Rid < 0 for
negatively buoyant jets. Simulations identify two ranges of negative buoyancy.
For weakly
negatively buoyant starting jets (-0.05 ~< Rid <0), the pinch-off and formation of an optimal
starting vortex occur, and the corresponding formation number can be determined using a revised
algorithm, catering to negative buoyancy, based on the observed step jump in the vortex ring's
circulation. This algorithm suggests a continuation of the declining trend in formation number
*
Published as Wang, R.-Q., Law, A. W.-K., and Adams, E. E. (2011). "Pinch-off and formation
number of negatively buoyant jets." Physics ofFluids, 23(5), 052101.
87
with declining Rid identified previously for positively buoyant jets.
For strongly negatively
buoyant jets (Rid < -0.05), the starting vortex falls back onto the stem after the initial roll-up for ~
-2.0 <Rid < -0.05, and the pinch-off process does not exist for Rid
<
~ -2.0 with the starting
vortex and stem connected at all time. A plot of time-varying vortex circulation and penetration
for starting plumes with different Rid unifies our understanding of buoyant vortex dynamics.
4.1 Introduction
A buoyant jet (plume) is formed by pushing denser or lighter fluid out of an orifice. Such
phenomena are of inherent interest to environmental fluid dynamists, because they affect
concentrations produced by effluents from chimneys, wastewater outfalls, etc. Buoyant jets also
play important roles in other areas such as homeland security [1], building interior environments
[2], and volcanic processes [3].
As a buoyant fluid enters the environment, a shear layer is generated around the orifice; the shear
layer then rolls up and forms a ring-shaped concentrated vortex. This vortex ring leads the
penetration of the starting plume into the surrounding ambient, and is followed by a trailing stem
if the formation time is large enough. A seminal investigation of the pinch-off and formation
pattern of this "vortex ring-trailing stem" system was performed by Gharib et al. [4]. Through
experiments with a piston/cylinder arrangement, they identified two different flow patterns.
With a small piston aspect ratio (L/D, where L is the piston stroke length and D is the nozzle
diameter) of 2, an isolated vortex ring was formed without the development of a trailing stem
(i.e. the intrusion was totally absorbed in the starting vortex), whereas with a larger aspect ratio
of 14.5, a trailing stem was observed following the front vortex ring. The critical aspect ratio was
88
called the "formation number", above which a trailing stem is present. They found that the
formation number is approximately equal to 4.0, and remains relatively invariant through various
nozzle geometries and inflow configurations. The formation number was proposed as the
criterion for the "pinch-off' occurrence (i.e. detachment of the starting head vortex from trailing
stem), the reason being that the starting vortex could not absorb the additional intrusion from the
piston if L/D is higher, and would thus physically separate from the trailing stem. By inference,
when the formation number is reached, the vortex ring has achieved the upper limit of circulation
by the driving mechanism.
Following the pioneering study of Gharib et al. [4], a series of experimental and numerical
studies confirmed the existence of the formation number (e.g. Rosenfeld et al. [5]; Zhao et al.
[6]), and extended the concept towards a wide range of non-buoyant situations, including nonconservative generating forcing [7], time-varying nozzle diameters [8], gravity-driven starting
jets [9], and square nozzle starting jets [10]. (See Dabiri [11] for a review.) With these efforts,
the formation dynamics of non-buoyant starting jets is now relatively well studied. However,
there has been less study of buoyant starting jets.
A buoyant jet can be broadly classified as positively buoyant, if the buoyancy force and
momentum fluxes are aligned in the same direction, or negatively buoyant if the buoyant force
and momentum flux are opposed. The pinch-off and formation number of starting buoyant jets
were first studied by Shusser and Gharib [12] and Pottebaum and Gharib [13], for flows with
high buoyancy flux and negligible momentum flux. Subsequently, a study of starting positively
buoyant jets was presented by Ai et al. [14] whereby the concept of Period of Flow Development
89
(PFD) and Period of Developed Flow (PDF) were proposed to differentiate the transient from
steady state flow dynamics, and an analysis of the penetration dynamics during the PDF was
presented with experimental verification.
Numerically, a study of the formation number of
positively buoyant starting jets was performed by Wang et al. [15] using Large-Eddy Simulations
(LES). They extended Gharib et al.'s [4] method of determining the formation number in nonbuoyant jets, based on comparison of the circulation in the total domain and that in the vortex
ring, to positively buoyant jets, and identified a trend of increasing buoyant formation number
with increasing Richardson Number, Ri. In addition, the application of the Kelvin-Benjamin
variational-principle, proposed by Gharib et al. [4] to predict the formation number for nonbuoyant jets, was extended to starting buoyant jets and showed good agreement with numerical
results.
The study by Wang et al. [15] targeted specifically positively buoyant jets. A corresponding
comprehensive study on the formation process for starting jets with negative buoyancy has not
yet been presented. Only recently, the formation number under negatively buoyancy flux was
reported by Marugin-Cruz et al. [16] for a particular value of Ri. They reported a formation
number for this particular case that was smaller than that for a non-buoyant jet, which is
qualitatively consistent with the declining trend of formation number with decreasing Ri found
by Wang et al. [15]. Here, we aim to quantify the formation number for the negatively buoyant
range to provide a fuller understanding for starting buoyant jets. Our specific objectives include:
1) to further the physical understandings of the pinch-off and detachment of negatively buoyant
jets; 2) to examine whether the formation number exists for negatively buoyant jets, and if so to
90
identify the range Ri over which it occurs, and finally 3) to explore the necessary extension of the
buoyant formation model in Wang et al. [15] to negatively buoyant jets.
In the following, the numerical approach and the flow configuration are first introduced in
Section 2. In Section 3, the LES code is verified by comparing to the penetration height for
submerged fountains reported in the literature. In Section 4, the formation dynamics of
negatively buoyant jets are analyzed through numerical simulations, and the pinch-off for
negative buoyant plumes is further discussed with the model of Wang et al. [15].
Finally,
conclusions from the current study are summarized in Section 5.
4.2 Numerical Model
4.2.1 Numerical method
The numerical strategy includes discretizing and solving the filtered Navier-Stokes plus scalar
transport equations under the Boussinesq approximation, and modeling the sub-grid terms by a
Dynamic Mixed Model. The numerical method and code have been described in an earlier study
[15], so we will omit details here for brevity. The numerical model has been verified by
comparison with standard experiments, e.g., Cui and Street [17], and implemented in a number
of recent studies [15][18][19].
4.2.2 Flow configuration
The three-dimensional computational domain extends to 0.3 m x 0.3 mx 0.8 m in the Cartesian
coordinates, x, y, and z, respectively (Figure 4-la). A downward uniform velocity Uo issues from
91
a circular inlet with diameter of D=5 cm at the top of the domain. Thus, the domain occupies a
size of 6D x 6D x 16D, which has been verified to be sufficient for investigating near-field
formation processes (Wang et al. [20]). With a discharge density lesser than ambient, the
buoyancy flux and momentum flux are in opposite directions, producing a negatively buoyant
jet. The governing equations are solved in a stretched mesh which has denser grids at the center
axis through the nozzle (Figure 4-1b).
The computational domain is filled initially with stagnant fluid with uniform density, p0 . As
depicted in Figure 4-la, the top boundary is a free-slip surface, while the other boundaries are
"no-gradient" outflow boundaries following the recommendation of Yuan and Street [21]. A
constant volume flux, determined from the prescribed inflow velocity Uo, is imposed
instantaneously at the inlet at t=0 to initiate the simulation and terminated at t=12s., where the
non-dimensional stroke, L/D, is assumed to be long enough to exceed the formation number.
To facilitate the analysis, the following parameters identified by Fischer et al. [22] are used:
, M _= -rD
= -D 2 U 09D0
QO=4
4
22,
Bo = -ng'D2Uo,Re=UOD
0D4
0U
V
(4.1)
where Qo, MO, and BO are the initial values of the volume flux, the kinematic momentum flux and
the kinematic buoyancy flux, respectively; g' is the reduced gravity (gAp /pa); Apo = Pa - po is
the difference between the ambient density pa and the discharge density po; Re is the jet
Reynolds number and the kinematic viscosity v = 10-6 m2 /s. Density differences are simulated
using concentration differences characterized by a Schmidt number of 725. Here g' and BO are
92
defined as positive for light jets and negative for dense jets. However, we define positively
buoyant and negatively buoyant jets by the direction of their buoyancy force relative to the
direction of their momentum flux (i.e., their trajectory). Accordingly, we proposed a revised
(directional) form of the Richardson Number (Rid):
Ri
-
6( 4
QOBosin
m512
g'D
U
4
2
sin
_sin_
_
F2 lp,
_
- P)
_(p
- POI
(4.2)
where 6 is the angle of the jet trajectory relative to horizontal (positive in the first quadrant).
Note that the magnitude of Rid is the square of the magnitude of the conventionally defined
Richardson number, Ri = Q0 BO 1 /M0' 4 , but has the advantage that it can be applied seamlessly
with negative values of BO.
Rid
> 0 for positively buoyant jets (e.g., dense fluid directed
vertically downward), Rid = 0 for non-buoyant jets, and Rid < 0 for negatively buoyant jets,
while by convention the densimetric Froude number F = UO / g p -po ID is positive in all
cases. Another advantage of (2) is that it is also potentially usable for inclined discharges, with
sin(6) accounting for the change in the gravitational acceleration along the discharge direction.
4.3 Penetration characteristics
Starting plumes resemble the beginning of fully developed plumes, and hence the extensive data
on the latter can be used to verify simulations of the former. In both cases, fronts display steep
gradients in concentration and velocity, making penetration characteristics relatively insensitive
93
to chosen concentration/velocity thresholds. Results for terminal penetration heights of
submerged fountains reported in the literature are summarized in Table 1. Note that the Froude
number, Fr, is often defined using the orifice radius, such that Fr = U
/
rg' = f2Fd.
Both
values are listed in Table 1 for reference.
The data on penetration can be used to identify three ranges: 1) "strong fountains" (Fd>2),
characterized by weak negative buoyancy, 2) "weak fountains" (0.7<Fd<2), characterized by
stronger negative buoyancy, and 3) "very weak fountains" (Fd<0.7), characterized by much
stronger negative buoyancy.
In the "strong fountain" range, the value of the penetration
coefficient varies significantly from 1.45 to 2.16. The mean value of 1.8 is consistent with Kaye
and Hunt [23], and is used here for later analysis. Fewer studies have been performed on "weak
and very weak fountains", though Lin and Armfield [24] and Armfield and Lin [25] performed
numerical studies in these ranges. Using the simulation results combined with experiments, Kaye
and Hunt [23] proposed the following fitting equations:
1.8FF
z/D= 0.9F2
0.6F 2/3
>~ 2)
(~0.7 < F<~ 2)
(F<~ 0.7)
(4.3)
4.3.1 Simulated Penetration height
We define the tip or front edge of the penetration as the location where the density difference is
10% of its initial density difference. For strong fountains, the penetration is significant and the
front edge can be clearly identified. However, for very weak fountains, the penetration height is
short and the smaller number of grids covering the plume height can lead to greater relative error
in the computed penetration height. And for very strong fountains, the penetration height can
94
exceed the length of the simulation domain. Thus, the present results shown in Figure 4-2 are
restricted to the range of 1 <Fd< 3.5.
For strong fountains (Fd>2), it is well known that the front of a starting plume first reaches a
maximum value, then retreats ~30%, and oscillates around a terminal penetration height
[26][23], z,, while weak and very weak fountains exhibit little or no oscillation [24][25]. Our
simulations show similar behavior.
From Figure 4-2, it can be observed that the simulated
terminal height increases with higher Froude number, Fd, and agrees well with experimental
data.
Note that Fd is used here to describe the penetration height in order to be consistent with other
numerical results.
For negatively buoyant jets, Rid
-Fd,
so the reverse trend should be
observed.
4.3.2 Penetration rate
The penetration rate of the starting plume is represented by the slope of the transient penetration
location over time. Figure 4-3 shows the decrease in penetration for several starting plumes with
negative buoyancy. The deceleration is caused by the entrainment of stationary ambient fluid, as
well as the opposing buoyancy flux. Based on simulations for positively buoyant starting jets,
Wang et al. [20] showed that the rate of penetration, defined by the location where the centerline
excess concentration (or relative density difference) was 10% of the initial difference, could be
expressed as the sum of separate contributions due to buoyancy and momentum. Generalizing to
starting jets of either positive or negative buoyancy,
95
zID=zB /D+zM /D=0.10
BO t 2 sin 6+0.47 m0
Q0D
Q2
t
(4.4)
where the first term on the right is positive for positively buoyant jets. Figure 4-3 shows that Eq
(4.4) is also applicable to weakly negatively buoyant jets (sin 6= -1), e.g., Rid =-0.035. This is
not surprising since Eq (4.4) is based on a decoupling of the penetration caused by the initial
buoyancy and momentum fluxes, and this is expected to be valid for positive or negative values
of B.
For somewhat stronger negative buoyancy (Rid =-0.09), theory and simulation begin to
diverge, while for even stronger negative buoyancy (Rid= -0.16) the divergence increases,
suggesting that the two mechanisms are no longer decoupled.
4.4 Formation number
Since first defined by Gharib et al. [4], the formation number has become a key parameter in the
understanding of jets and plumes. As discussed in the introduction, a number of previous studies
have analyzed the formation number under different flow configurations. However, the variation
of formation number in the range of negative buoyancy fluxes has not been presented. In the
following, LES results for negatively buoyant jets are analyzed and their formation numbers are
determined. Subsequently, the pinch-off and detachment of negatively buoyant jets are
discussed.
4.4.1 Determining formation number
96
To begin with, a cross-sectional circulation is defined to distinguish the vortex ring from the
trailing jet-head vortex system [15]:
(4.5)
w*(z)= fc,(x,z)dx
HCP
where HCP represents the half central plane, i.e. either left central plane (O<x<3D,y=3D), or
right central plane (3D<x<6D,y=3D), and w, represents the vorticity in the y direction. The
peak of the absolute value of the cross-sectional circulation, Iw *l, corresponds to the position of
the centre of the starting vortex ring. A minimum of Ico*1 can be found between the head vortex
and the trailing stem throughout the development process until the flow pattern disintegrates.
This minimum can be used to divide the flow field into: (1) the starting vortex region and (2) the
trailing stem region. Hereafter, the term "head vortex" refers to the starting vortex ring in order
to differentiate it from "leading vortex" which is the regenerated vortex ring formed after the
pinch-off [27].
Next, the formation number can be determined by comparing the circulation calculated over the
entire domain versus that calculated within the head vortex. During the formation process, when
the intrusion begins, the circulation of the head vortex increases as the head vortex is fed by the
trailing stem's leading vortices (TSLV). If and when pinch-off occurs, a portion of the TSLV
would be engulfed by the head vortex which would be recorded as a transient step-jump in the
head vortex circulation. As a result, the head vortex physically detaches from the TSLV. In
theory, the step-jump corresponds to the time when pinch-off occurs, and thus the formation
number can be determined by noting the time of the step-jump in the following manner. First, the
circulation value after the step-jump in head vortex circulation, Tj is obtained. The intersection
point on the total circulation line which has the same value of Fj is then noted. The
97
corresponding normalized L/D is finally determined as the formation number. This strategy is the
same as proposed by Wang et al. [15] and has been validated for pure jets as well as positively
buoyant jets.
4.4.2 Buoyant formation number
Using the above approach, formation numbers for negatively buoyant jets are determined as
shown in Figure 4-4. The formation number is found to decrease with increasing negative
buoyancy, i.e. the formation number decreases with decreasing Rid. The simulations are also
performed with Reynolds numbers of 2000, 2500, and 3000 which reflect varying injection
velocity. According to Figure 4-4, the formation numbers almost overlap, and thus it is
concluded that Reynolds number has insignificant effect on the buoyant formation number in this
range.
The formation number can be determined only down to Rid~-0.05. For lower Rid the present
method to determine the formation number breaks down, since no step jump in the vortex ring's
circulation is present. As illustration, in Fig 5 the circulation of the vortex ring has a step jump
for Rid=-0.035, but not for Rir=-0.052. The step jump entails the process whereby the vortex ring
engulfs some of the trailing stem, which is revealed in the vorticity distributions shown in the
upper two rows of Fig 6. Consistently, the engulfment is present for Rir=-0.035 and absent for
Rid=-0.052. Therefore, computed formation numbers are not shown for smaller Rid.
98
To offer a full spectrum of formation number variation versus Rid, the formation numbers of
positively buoyant jets based on Wang et al. [15] are also presented in Figure 4-4. Thanks to the
definition of the directional Richardson number Rid, if the extreme cases of Rid-0.23 and 0.26 are
excluded, the formation numbers can be fit to a straight line with a slope of 18. The experimental
data by Marugan-Cruz et al. [16] are also shown on Fig. 4. Their data closely follow the present
trend, although the exact values for both neutral and negatively buoyant cases are lower by ~0.5.
The systematic difference may be attributed to the different methods used to determine the
formation number.
Interesting phenomana are observed at the extremes of the fitting line in Fig 4. At the high
extreme Rid>0.15, the formation number has reached an asymptote, i.e. the formation number
remains a constant value of about 6.5. If the fitting line is extrapolated in the opposite direction,
it intersect with the "zero" formation number at Rid=-0.2. Possible dynamical interpretations are
discussed in the following.
4.4.3 A physical map of the vortex ring dynamics
To summarize the effects of positive and negative buoyancy on vortex ring dynamics, the
circulation and the center of the vortex ring core are extracted from the vorticity distributions at
the central cross-section and shown in Fig 7. The magnitude of the circulation is normalized by
the slug model's prediction, i.e. FVR
0.5fM Uo D. (fM is the formation number of 4 for non-
buoyant vortex ring.) The time interval between neighboring points is one second. All curves
have Reynolds number = 2500 and the directional Richardson number varies from -0.35 to 0.17.
99
Plotting in such manner, Fig 7 provides a sort of unifying map of vortex ring dynamics, from
which several observations can be made.
Firstly, the time varying circulation, position of the vortex ring, and certain transitional behaviors
can be determined for a given Rid . For example, for Rid=O. 17 between 9 to 10 s, the circulation
of the vortex ring engulfs a portion of the trailing stem at pinch-off, and thus experiences a "step
jump" as reported before. After 13s, the circulation starts to decay due to the drag from the
surrounding ambient, and the vortex ring begins to lose its coherent structure.
Secondly, the status of the vortex ring can be predicted from the map. The green line on the map
connects the engulfing time when the vortex ring engulfs and pinchs-off, whereas the blue line
connects the decaying points that differentiate the stable and unstable phases of the vortex ring.
Therefore, given a time and a Rid, whether the vortex ring is stable or not and whether the "step
jump" has happened can be predicted by the map.
Finally, the physical map helps explain the buoyant vortex ring's complex behavior at different
Rid.
In particular, the map provides possible explanations for the upper limit and the "zero" value
of formation number.
By extending Gharib et al's Kelvin-Benjamin variational principle model, Wang et al. [15] found
an upper limit of the formation number at L/D = 10.7 as Ri ->oo.
Fig 7 offers a possible
explanation for this limit. As Ri increases, the "engulfing" time is nearly constant while the
engulfing location is further away from the source. Therefore, the lines of stability creteria and
100
engulfing time are expected to intersect, at which point the vortex ring would become unstable
before reaching the maximum circulation limit. As a result, the maximum circulation that the
vortex ring could reach would not be limited by Rid but by instability, after which the formation
number would depart from its original trend.
At the other end of the spectrum, extrapolating the linear fit in Fig. 4 suggests a formation
number of zero at Rir=-0.2. Recall that the formation number was defined as the critical value of
a piston's aspect ratio, L/D, above which the trailing stem would be present along with the
leading vortex ring.
Equivalently, it is the aspect ratio above which "pinch-off' can occur.
Therefore, if the formation number is equal to zero, the trailing stem would be present no matter
how short the piston stroke is. In another words, pinch-off would always occur.
Overall, Fig 7 provides a conceptual model that could explain the zero formation number and the
"pinch-off' criteria. With negative buoyancy, the trailing stem has a finite length because the
buoyancy causes the stem fluid to decelerate until it reaches a terminal height.
Moreover,
buoyancy would cause the vortex ring to decay and eventually disappear when its circulation
approaches zero. If Rid>-0.2 , buoyancy is relatively weak and the vortex ring can eject from the
end of the trailing stem, which forms the "pinch-off'. Conversely, for Rid<-0.2 , the vortex ring
would decay strongly once formed, and be unable to detach from the trailing stem. The key to
the conceptual model is thus the finite length of the trailing stem. One interpretation is to assume
that the trailing stem length is fully reflected by the terminal penetration height as determined in
Fig 2. Comparing the terminal penetration height with the final position that the vortex ring can
reach in Fig 7, the leading vortex ring is found to disappear around the end of the penetration
101
height of xdisappea/D~3 at Rijz -0.2 (corresponding to z/D=3 at Ffz 2 in Fig 2). The vortex ring
decays beyond the terminal penetration height at Rid>-0.2 (e.g.
Xdisppear/D=7 >
z/D=4 for Ria=-
0.09) and before the terminal penetration height at Rid<-0.2 (e.g. Ri=-O.35). This model might
also explain observations in classical experiments investigating terminal penetration heights. As
reported by Turner [26] and Kaye and Hunt [23], negatively buoyant starting plumes rise to an
intial height of zm and then retreat to the terminal penetration height of z,, with zm ~1.43zt.
However, as observed in Fig 5 of Lin and Armfield [24], zm is not much higher than z, for "weak
fountains" or Rid<-0.2. Using the conceptual model above, zm might simply be the highest
position of a decaying vortex ring front. Once the vortex ring disappears , the penetration height
would retreat to the front of the trailing stem zt. Since the vortex ring can not depart from the
trailing stem for weak fountains, an obvious retreat is not observed, and zm is not much greater
than z,.
To support the conceptual model, the vorticity distribution of the starting negatively buoyant
plumes are shown on the second and third rows in Fig 6 for Rid=-0.09 and -0.35. Clearly, the
vortex ring pinch-offs at Ri=-0.09, i.e. formation number is positive, and always connects to the
trailing stem for Rid=-0.35, where the formation number is not physically meaninful . More
numerical and experimental investiagations are encouraged to further the understanding.
To further examine the flow patterns around the point of neutral buoyancy, the distributions of
relative density difference (p - p, )/p, for a negatively buoyant jet (Rid=-0.035), a positively
buoyant jet (Rid=0.035), and a neutrally buoyant jet (Rid=0) are extracted at t-14s and shown in
Figure 4-8. Compared to the pure jet, the head vortex is stretched in positively buoyant jets, and
102
compressed in negatively buoyant jets. As a result, the positively buoyant head vortex has a
larger volume than the pure jet, while the negatively buoyant one has a smaller volume. Linden
and Turner [28] have attributed the formation number to the limitation of a vortex ring's volume.
They matched the volume of the head vortex to that in Hill's spherical vortex [29] and concluded
that a trailing stem would occur if the total volume occupied by the jet exceeds the Hill's
spherical vortex. Following their conclusion, a larger volume of positively buoyant head vortex
should have a higher formation number and vice verse, which is consistent with our observation.
4.4.4 Application of analytical model
A model based on the Kelvin-Benjamin variational principle was first used for pure jets [4][30],
and later extended by Wang et al. [15] for positively buoyant jets. With their modified version,
Wang et al. [15] were able to predict the observed trend of increasing formation number with
increasing buoyancy (increasing Rid), as well as the occurrence of the asymptotic behavior for
large Rid. In the following, we show that for negatively buoyant jets the model predicts that the
formation number decreases with increasing (negative) buoyancy (decreasing Rid), which is
consistent with the above simulations as well as the recent study by Marugain-Cruz et al. [16].
The modified slug model consists of the following set of equations:
E=E +E 2
F
,= +72 ]
1= 11 + 12
1 ZpD2L
8
= UOL +
2
U2++1
4
4
2
Lg't sin 0
=-fpD 2 UOL+ IpD
4
rpD2Lg'(UOt+ Igt2 sin 6)
4
(4.6)
(4.7)
2 Lg'tsino
(4.8)
103
where E is the kinetic energy, F the circulation, and I the impulse, with the subscripts "1" and
"2" referring to the contributions of the initial momentum and buoyancy fluxes, respectively.
A dimensionless energy can be derived by the Kelvin-Benjamin variational principle as
E,nd
-
1-2
E/2i]/2
7D
32(4.9)
2 L
Pl/f/J/
where
g't sin 0
g't sin 0+ 2U0
(4.10)
The buoyancy coefficient J represents the increasing effect of buoyancy on the non-dimensional
energy as a function of time. If we replace t by L/UO, i.e.
=
g'Lsin
g'Lsin0+2U
2
(4.11)
End represents the non-dimensional energy at the end of a discharge, and hence the limiting value
when L/D equals the formation number. Note that J is negative for negatively buoyant jets.
Gharib et al. [4] proposed that the formation number for pure jets corresponds to End=0.33. For
positively buoyant jets, Wang et al. [15] derived a similar general value of End, and reported that
all of their simulated formation numbers fell within a narrow range of 0.31 < End < 0.35, with a
median value of 0.33. Their results are repeated in Fig 9 along with the formation numbers found
for negatively buoyant jets in this study. While End increases gradually with increasing negative
buoyancy, the criteria of 0.3 1<End<0.35 is satisfied in the range -0.017 < Rid, while the criteria
has to be relaxed to 0.3 1<End<0.43 in the range -0.05 < Rid, suggesting the model is limited for
more negatively buoyant jets.
104
4.5 Summary and Conclusions
The present study investigates numerically the formation processes of negatively buoyant
starting jets using LES. As validation of the numerical approach, the simulated penetration
heights for submerged fountains are shown to compare favorably with existing experimental
data.
Wang et al.'s [20] model for the penetration of positively buoyant starting jets (Eq. 4.4) was
found to also apply to weakly negatively buoyant starting jets. The range of agreement between
theory and simulation corresponds approximately with the range over which pinch-off is
possible. With stronger negative buoyancy (Rid<~-0.2 ), pinch-off does not occur, and the link
between the head vortex and the stem remains at all time. For the range of weak negatively
buoyant jets through positively buoyant jets, the buoyant formation number increases
monotonically with Rid.
A unified physical map of vortex circulation and penetration is
proposed, allowing several interpretations of vortex dynamics. Finally, Wang et al.'s modified
slug model [15] based on the Kevin-Benjamin variation principle, is found to be applicable for
weakly negatively buoyant jets (Rid>-0.05), but to break down for strongly negatively buoyant
jets(Rid<-0.05).
105
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109
Table 1 Summary of maximum penetration distance at steady state in literature
Literature
Formula
Based on Fr
z/r=2.46Fr
Based on Fd
Abraham [32]
z/r =2.74Fr
z/D=1 93Fd
Seban et al. [33]
z1/r =2.52Fr
z/D=1. 78Fd
List [34]
zt/r =2.05-2.74Fr
z/D=1.45-1.94Fd
Mizushina et al.
z/r =2.34Fr
z/D= 1. 65Fd
zJ3.06Fr (Fr >7)
r
1.7Fr (Fr <7)
Z,
Turner [17]
Baines et al.[31]
z/D=1. 74Fd
Fr
Re
<250
-
6.6-52.7
7701923
5-100
17405420
0.37-36.2
170025500
[35]
Zhang &
Baddour [36]
D
2.16F
(F > 4.95)
12.67F, .(Fr <4.95)
Friedman &
Katz [37]
z/r =2.4OFr
z/D=1. 69Fd
0.01-100
100030000
Bloomfield &
Kerr [38]
z/r =2.28Fr
z/D=1.6Fd
<65
-
Lin & Armfield
[24]
Z, =0.198+1.165Fr
Zt =0.198+1.65F
0.1-1
200
Lin & Armfield
[25]
zt=1.26Fr2 3 (Re=200)
zj=1.59Fj'3 (Re=200)
213
zI=0. 1615+0.3803Re-
2.5 xI0^3
-0.2
Kaye & Hunt
[23]
Papanicolaou &
Kokkalis [39]
Marugin-Cruz
et al. [16]
2 13
zt=0.1615+0.3803Re-
(Fr=0.05)
2.46Fr
_ = 0.90Fr2
r
0.94Fr2/3
zi/r =2.66Fr
(Fr >3)
(1< Fr <3)
(0 < Fr <1)
(F=0.035)
1. 8 F
D
(F
z-~0.9F,2
2
0.6Fd2 /3
z/D 1.88Fd
Consistent with Kaye & Hunt [23]
110
>-
2)
0.7 < Fd<(Fd <~ 0.7])
1-100
>0.2
770-5
840
848-2
578
buoyant jet
free-slip
outflow
2
4
6
0
8
10
12
y
14
z
4
x/D
y/D
(a)
(b)
Figure 4-1 The computational domain: (a) boundary conditions (not to scale); (b) a typical
grid mesh (only every
4 th
mesh point shown in each direction; the center of the jet outlet is
at x=3D, y=3D, Z=O)
111
7
present simulations
fitting curve of Kaye & Hunt (2006)
data of Kaye & Hunt (2006)
o
6.
+
+
0
5+
+
+
3
0
21
00
'
0.5
1
1.5
2
2.5
3
3.5
4
Fd
Figure 4-2 Comparison of penetration height with experiment fitting line proposed by Kaye
and Hunt [23].
112
+
o
x
6
o
5
-
Rid=0
Rid=-0.035
Rid=-0.087
Rid=-0.174
Rid=0
Rid=-0.035
Rid=-0.087
Rid=-0. 174
++
x
xx
x
xxx
4-
xxx
x
3
xX
-
00x000
0
00000
yxxx
xx0000..........
12
0
0
3-
00
2
4
0
x
6
tU 0/D
0
8
0
---------
10
12
Figure 4-3 Penetration height versus non-dimensional time.
(Marker: simulations; Lines: model equations from Wang et al. [20] adapted for negatively
buoyant jets.)
113
87-
/
neutral
buoyancy
x
+
/
6x
E
5
C
C
0
4-
EU
0
3-
7xx
21
0
+
/
x
fitting /'
line A
-0.2
-0.1
o
A
Re=2500
Re=2000
Re=3000
Marugan-Cruz et al. (2009)
0.1
0
0.2
I
0.3
Rid
Figure 4-4 Determined buoyant formation number versus directional Richardson number
114
0.015
-
e
*
0.01 -
Rid=-0.052 Total
Rid=-0.052 Head Vortex
Rid=-0.035 Total
Rid=-0.035 Head Vortex
E
0
ip
0.005-
0
2
4
6
time t*
8
10
12
Figure 4-5 Circulation within total domain and within head vortex rings for Rid =-0.035 and 0.052
115
t=6s
t=7s
t=8s
x/D
x/D
V/D
t=9s
t=lOs
x/D
x/D
t=lls
t=12s
V/D
Figure 4-6 Vorticity distributions for Rid=-O.04 (1st row), -0.09 (2nd row), and -0.35 (3rd
row).
116
VfD
3.5 r
Rid=0. 17
3 -
Unstable
Engulfing
2.5-
Rid=0.05
1-O
d
Engulfment
0.5-
i
criteria
Rid=-O.0
Ri d=-417
'd=-
Rid=-0.03
R~d=d=O
2I
2
4
Zero formation number
6
x/D
8
10
12
Fig 4-7 Physical map of vortex ring dynamics, where 'VR = O.5fM Uo D predicted by slug model,
wherefm is the formation number of 4 for non-buoyant jets.
117
1=4
x 0
1S)
1
-0.2
1
9
1
18
2
-0A4
2
8
2
16
3
-0.6
3
7
3
14
4
-0.8
4
6
4
12
-1
5
5
5
10
-1.2
6
4
6
8
7
3
7
6
-1.6
8
2
8
4
-1.8
9
1
9
2
0
10
S5
6
7-1.4
8
9
10
1
2
3
4
5
-2
10
=4()
1
2
x/D
3
x/D
x~g-
4
5
W=4 (s)
12
34
x 10'
5
x/D
Figure 4-8 Comparison of flow patterns around neutral point. Left panel: Rit=-0.035;
middle panel: Rit=O; right panel: Rid=0.035
118
0
0.8
0.7
Neutral buoyant line Rid=O
0.6
\%
%%l
0.5
*0
C
w
--
0.4
Rid=-0.050
Rid=-0.035
Rid=-0.017
Rid=-0.009
Rid=-0.002
Rid=0
Rid=0.035
Rid=0.090
Rid=0.139
Rid=0.174
*%*
0.3
0.2
0.12
3
4
5
6
L/D
Figure 4-9 Modified slug model applied to buoyant formation numbers
119
7
8
120
Chapter 5
LES Study of Settling Particle Cloud
Dynamics*
Abstract
A series of Euler-Lagrangian four-way coupling Large Eddy Simulations (LES) are performed to study the dynamics of settling particle clouds. The numerical method is first
validated by comparing with existing experimental results. Then, a parametric study is
performed with various initial release shapes, particle sizes, and cloud buoyancies. Three
issues are examined in detail, including the initial release aspect ratio, two-phase interactions and polydispersion. Incorporating these factors, empirical relationships are developed
to predict the phase separation time and height, the position of the cloud front and the edge
radius based on numerical results. The entrainment rate and deposition patterns are also
analyzed. These relationships should be useful for engineering applications
*This chapter has been submitted to International Journal of Multiphase Flows as a paper.
121
5.1
Introduction
A particle cloud settling through a water column is inherently a two-phase interactive process that is closely coupled. Once released, the particle cloud first accelerates due to the
density difference (so called "acceleration stage"), and then forms a circulating system
with the solid particles and a rotating fluid vortex ring (so called "thermal stage"). Finally,
individual particles separate from the vortex ring and descend with their terminal settling
velocity, while the fluid vortex ring is left behind due to the removal of driving force by the
heavier particles (so called "dispersive stage") [1]. In this study, we focus on the thermal
and dispersive stages of the particle cloud with the initial release of particles in the middle
of the water column. The "acceleration stage" is very short with this release configuration,
and can therefore be ignored.
The key in understanding the coupled system is the two-phase interactions that govern
the settling dynamics. The route of momentum transfer differs between the thermal and
dispersive stages. In the thermal stage, momentum is transferred from the particles to the
fluid generating the vortex flow, which then redistributes the solid phase and feeds the
momentum back to the particles. Contrary to this closed loop, the dispersive stage has
an open route - the momentum is primarily transferred from the solid to the fluid phase
through the settling process, which is then dissipated by viscous diffusion. In other words,
momentum is lost after the transfer.
Beyond the importance of basic understanding, including the fluid phase in the engineering analysis has practical implications. Sediment particles released in the water body
may possess dissolvable or volatile chemical substance, e.g. organic pollutants in disposed
dredged sediment. The fluid-phase pollution can pose a secondary impact on the surrounding environment alongside the solid phase. Detailed knowledge of the solid-fluid phase
interactions can help control and minimize the consequences. Until recently, the tracking
of both phases has received little attention. [2] presented the first study that addresses the
dynamics of both phases using the simplified approach of the Hill's vortex for the fluid
122
phase. We aim to address this issue in fuller detail using the Large Eddy Simulation (LES)
numerical approach as the first objective of the present study.
From a physical perspective, the process of particle settlement is an initial value problem where the boundaries are relatively far away compared to the cloud sizes. Thus, the
computational domain is relatively simple in geometry, and the initial release conditions set
the tone for the settlement process. In previous investigations, the initial conditions often
covered a selected range of several basic parameters, including particle sizes, materials,
densities, and total mass ([1], [3], and [4]). The typical particle cloud characteristics that
were measured included the penetration depth of the particle cloud front (or centroid) and
the cloud radius.
[5] was among the first to systematically vary all the basic parameters mentioned and
generalize his results into an integral model. In addition, he also explored the initial condition of moisture content and quantified the mass in the trailing stems. Subsequently,
additional ambient and initial conditions were studied, including ambient stratification [6],
currents [7], waves [8], and release height [9].
The previous studies have largely ignored the release shape, which is highly relevant,
e.g., in the design of particle cloud generators, buckets used for open water sediment disposal, or injection chambers used in plant seed dispersions. In addition, due to facility
constraints, past researchers were mostly preoccupied by a particular release mechanism,
e.g. a bowl shape container in [1], a cylindrical pipe in [5], and two sizes of funnels in [6].
This restricted the research to certain release aspect ratios, precluding a systematic parametric study. As the second objective of the present study, we aim to complete a parametric
study of the initial release conditions numerically as described in the following.
Finally, the third objective of the present study concerns polydispersion: the particles
in the release may be non-uniform and possess different sizes. Again, there have been very
few systematic studies involving multi-size particles. [3, 10] observed qualitatively that the
particle distribution has a significant effect on the growth and settlement of particle clouds.
123
Here, we aim to clarify the quantitative details of the polydispersion dynamics through the
parametric study.
As discussed above, we adopt LES for the present investigation. LES has great flexibility in terms of simulating the particle cloud behavior with respect to the initial release
shape and particle size distributions. In contrast, most classical integral models originally
developed by [11] and extended by others [5, 6] assumed a specific cloud shape (e.g. a
sphere or ellipse). The novel semi-integral method recently developed by [2] approximated
the fluid phase motion as an expanding Hill's spherical vortex, and tracked the motion of
individual particle groups. The simplified approach is flexible for various applications, but
is also limited by the approximate nature of the assumed fluid flow field. More advanced
CFD methods can potentially provide more accurate predictions with various initial release
conditions, if they can be validated. Previously, [12] performed a two-fluid simulation with
a mixing length model as turbulence closure. [13] simulated the two-phase flows directly.
The fluid phase motion was solved by Reynolds Averaged Navier-Stokes method with k - E
turbulence model and the solid phase motion was computed with Lagrangian tracking. [14]
replaced the fluid phase computational approach with LES, and obtained more detailed turbulence structures and better agreement with experimental data. Generally speaking, a
relatively small number of particles were used in these previous studies to reduce computational resources, and particle-particle collision was typically ignored which might affect
the computational results in the high particle concentration region near the point of release.
In this study, we develop a new higher order numerical scheme that can circumvent the
limitations and accommodate higher particle concentrations.
In the following, section 2 introduces the numerical method and computational domain;
section 3 validates the code by comparing with existing experiments; sections 4 and 5 analyze the monodispersion results to address the issues of phase separation and the effect of
initial release condition; section 6 focuses on the entrainment rate and deposition patterns;
the particle size distribution effect of polydispersion is shown in section 7; and conclusions
124
are drawn in section 8.
5.2
Numerical methods
A series of 3-D numerical simulations was performed with Euler-Lagrangian LES using
the software CFDEM, which couples the open source CFD toolbox (OpenFOAM) and the
Discrete Element Method (DEM) package (LIGGGHTS) [15]. For the fluid phase, the void
fractioned Navier-Stokes equations,
2L+
V -(aGuf) = 0(.)
D(afuf)
+ V- (afu=ur)
at
Pf
- Rs1 + vV - Vuf +- 'T,
P
are solved numerically, where af is the volume fraction of the fluid, uf is the velocity of
the fluid phase, t is time, p is pressure, pf is the density of the fluid, v is the viscosity, and
Ri is the inter-phase momentum transfer term. A finite volume method is adopted and
all variables are filtered except the sub-grid stress tensor r, which is resolved by the local
dynamic one equation eddy viscosity sub-grid model (locDynOneEqEddy) [16].
The solid phase motion is resolved using the ODE by LIGGGHTS, i.e.
du
Sdt
MP dt = EF
(5.2)
where mp is the particle mass, up is the particle velocity, and EF is the total force on the
particle, which can be specified as
EF=FG+Fs+FD+FL + FA
(5.3)
where FG is the gravity force, Fs is the fluid stress, FD is the drag force, FL is the lift
125
force, and FA is the added mass force. These forces can be modeled by
FG = mp(1 - 7)g
(5.4)
Fs = -Vp
(5.5)
3
3CDMpduP
4
FD
-
U (up
3
3CLmnpy
Up
U (Up
-
(5.6)
U)
du)
dt)
DC
FL
-
-
(5.7)
U) x W/1WI
(5.8)
where the density ratio y = pf/pp. The drag coefficient is
24
(1 + 0.15Re 6. 87 ), Rep < 800
p
CD={Re
(5.9)
0.44,
Rep, > 800.
The added mass coefficient is CA= 0.5, and the lift coefficient is
12.92
C*n= 1 -
0.3
CL =J*
Re
{0.825 + 0.15tanh [0.28
(Qq
1 + tanh 2.5
(10go
ReC~
-
+ 0.191
p ,eq CQ
(5.10)
2)]} tanh (0.18Rep/2 )
(5.11)
+ tanh 6
-1.92
(5.12)
*,eq =2
(1 - 0.0075Re,) (1 - 0.062Re1 / 2 - 0.001Rep)
Up
-
Uf I
(5.13)
(5.14)
v is the kinetic viscosity, and dp is the diameter of the particle [17]. All of the source code
to implement these force models can be found in the latest version of CFD-DEM.
The inter-phase momentum term Rs, needs to be adjusted every time step to conserve
126
S
the total momentum by the interface of CFDEM, i.e. R,1 =
n±1
,n
At
, where s is the
number of particles within a single grid cell, Vj is the volume of the i-th particle and V is
the volume of the cell. In the present study, the four-way inter-phase coupling is realized
by LIGGGHTS, i.e. particle-particle interaction is incorporated. We assume the particles
can overlap by a small distance 6 when they collide. Then, the normal force and tangential
force can be computed by
Fn= -kn6 + CAVn,
(5.15)
and
t
Ft = kt jAV
(T) dT
(5.16)
t + CtAVt,
tc,O
where AVn is the normal relative velocity at the contact point, AV7 is the relative tangential
velocity, t is the tangential vector at the contact point, te,o is the time of contact start, and
Ct and C,, are model constants. As a result, the particles' motion after collision can be
modeled by
mrnRi
(5.17)
= F+ Ft + mpg
After every simulation, a passive tracer is applied by solving the scalar transport equation,
a (afT) + V - (afufT) = 0,
at
(5.18)
where T is the the scalar concentration. The scalar transport reveals the fluid phase motion.
The computational domain was discretized with an unstructured mesh in a cylindrical
shape as shown in figure 6-1. Finer grid cells were concentrated along the axis of symmetry to better capture the shear layer. The domain extended to
Hdepth =
1.2m in depth
and 0.567m in diameter (two exceptions are BL4 and BL16, whose length scales were increased by factors of 4 and 16). Zero-gradient open boundary conditions were applied to
all the surfaces. The particles were released in the middle of the quiescent domain with
127
the same upper bound at 4.33% of Hdepth from the top. The initial release was contained
in a predefined cylinder with radius RO and height H, the geometry of which was varied
to obtain different aspect ratios of A
=
H/Ro. The particles in the initial release were
compacted with a volume fraction a,= 1 - af = 7r/6 = 0.524. The non-buoyant passive
tracer was released at a concentration of 1 g/cm 3 in the fluid inside the particles volume.
The boundary of the fluid phase was outlined by 1 % of the maximum scalar concentration.
All the simulations were executed with 32 processors in the High Performance Cluster in
the Nanyang Technological University, Singapore.
5.3
Validation of numerical method
A non-trivial issue in multiphase simulation meshing is the relationship between the particles and grid cells. In principle, LES should pass the grid convergence test, which guarantees that the grid length scale is within the turbulence cascade range and thus that the
final results are independent of the grid resolution. This requires a relatively small grid
length. At the same time, in the present numerical strategy, the grid cells have to be bigger
than the particles to properly apply the point model of particles in the governing equations.
If the particles are relatively small or the Reynolds number is fairly low, there is no conflict. However, this is not the typical situation and thus the conflict needs to be reconciled.
Here, we propose a new numerical scheme: instead of using small grid cells, large grids
are adopted but with the spatial discretization enhanced to high order accuracy, which also
leads to a quicker grid convergence with the reduced number of grid cells. Specifically,
the convection terms are discretized using the Total Variation Diminishing (TVD) scheme
with a cubic limiter, and the cubic discretization schemes are applied to second-order differential terms. The success of this new numerical scheme for two-phase simulations is
demonstrated by the following grid convergence test.
The grid convergence test was performed by systematically increasing the resolution
128
of the grid. The conditions are summarized in table 5.1. Five different grids with 6098,
13186, 17695 and 26931 cells were used. From figure 5-2, we can observe that the different
grids yielded almost the same penetration rate of the particle cloud, and only the grid with
6098 cells was different from the others in cloud growth. Therefore, for all the following
studies, we used the grid with 26931 cells, for which results can be considered to be gridindependent. It should be noted that the smallest grid cell still had a volume that was 1.5
times that of the particles. Note that the penetration rate is defined as the time derivative
of the cloud front position. The solid phase front is defined by the lowest particle within a
cloud, and the fluid phase is the lowest point of the 1 %tracer concentration.
Three experiments from [2] were used to validate the present numerical method. The
flow cases are specified in the first three rows of table 5.2, the only difference among them
being the particle size.
The comparison for Case A is shown in figure 5-3. The green dots are the particle group
(solid phase) and the bright yellow region indicates the tracer distribution (fluid phase). At
t = 0.2s, the fluid phase still remained interwined with the solid phase, with the particle
cloud starting to accelerate and a vortex ring being forming in the fluid phase. The numerical simulation results generally matched the experimental measurements - overlap can be
observed in both the left and right columns in figure 5-3. At t = 0.4s, the particles were
half inside the tracer cloud and half outside, which indicated that the particle cloud and the
fluid were separating. Again, the numerical results well captured the transient separation
process shown in the experiments. At t = 0.8s, the phase separation was nearly complete,
and by t = 1.2s, the solid particles began to settle individually. The numerical simulations
were able to predict the position and bowl shape of the particle cloud. Similar observations
can also be made in cases B and C (omitted here for brevity). Note that there is a small difference in size comparison. This is due to the particle size distribution in the experiments.
In the experiments, the particles have a distribution in sizes, but the particles in numerical simulations are strictly uniform. In the dispersive stage, particles with non-uniform
129
size would settle with different velocities. As a result, the experimental particle cloud is
stretched vertically with smaller particles settling slowly and bigger particles settle faster.
This is consistent with observation in the following section of poly-dispersion.
Quantitative comparison further validated the present numerical approach. We focused
on four variables - the frontal depth and cloud radius of both the solid and fluid phases
which are shown in figure 6-4. The solid lines denote the numerical results, and the error
bars bound the lower and upper limits of five realizations of the same experiment. As shown
in figure 6-4a, good agreement was found in the frontal position of the solid phase (z,), despite a small overestimation in cases A and C and a slight underestimation in case B. A
better comparison was noted in the fluid phase front (zf), where the numerical predictions
were generally within the ranges of experimental uncertainties (figure 6-4b). Furthermore,
the cloud radius of the solid phase (r,) shown in figure 6-4c was well predicted, although a
slight discrepancy can be identified at earlier times in case C. The most challenging comparison was in the fluid phase radius (rf) shown in figure 6-4d. The numerical predictions
for cases A and B generally followed the experimental measurements, while case C had a
significant discrepancy at the beginning, which was removed at later times. In summary, the
numerical results compared well to the experimental data which verified the applicability
of the numerical approach.
5.4
Phase separation
After the code validation, a parametric study with 14 simulations was implemented covering three sizes of particles, with a range of total mass and aspect ratios. The flow cases
are summarized in table 5.2, which also lists the corresponding Rayleigh number, Ra, and
the characteristic Reynolds number, Re = (Bg2/ps)1/ 6 Ro/v, for each case. The non130
dimentional Rayleigh number is defined as
Ra =
B
pows2R0'
(5.19)
S
and characterizes the ratio of buoyancy force to viscous drag force, where B = g (1 - 7)
is the total buoyancy, S is the total number of particles within a particle cloud, and w8 is
the settling velocity of an individual particle. Ra can also be used to scale up the results
from lab-scale experiments to field scale [5].
The most pronounced phenomenon in such two-phase flows is phase separation. While
it is inherently a continuous process, for analytical convenience we define a phase separation time Tsp as the time that the centroid of the particle cloud reaches the front of the fluid
phase. Because the front of the fluid phase is usually easy to observe in the experiments
and the particle centroid is normally available after image processing, this criterion can be
easily applied to the experimental measurements.
Figure 5-5a sketches the separation time as discussed above. Figure 5-5b further illustrates this time for case B through plotting the positions of the particle cloud centroid and
fluid front. In the figure, it can be observed that the particle centroid started slightly behind
the fluid front, but eventually accelerated ahead of the fluid front (a red dash line marks the
time of crossing over). Also shown in the figure is the portion of particles still held inside
the fluid phase cloud. As the phase separation developed, the portion reduced from I (full
overlap) to around zero.
Using this definition, the phase separation time for different cases was determined and
shown in figure 5-6 normalized by a time scale of
TS=
R2
0
.
IsBipo
(5.20)
Including the aspect ratios A in the normalization, all the separation times collapse to a
131
m
straight line, as follow:
TsA = 0.85
/p
A
.
(5.21)
For comparison, the phase separation time determined alternatively by the instant that the
particle front velocity approaches the settling velocity is also shown in figure 5-6. Here,
1.4w, is defined to be the threshold, below which the particle cloud is considered in the
dispersive phase [10]. The figure shows that these two methods are roughly the same and
can be fitted by the same equation 5.21.
In a similar fashion, the ratio of the phase separation height over the separation time
collapses to a straight line in figure 5-7. Using equations 5.21 and 5.22, the phase separation
time and heights can be predicted according to the flow condition. Furthermore, the fact
that the aspect ratio A is raised to a small power (1/4) in the equation of Tsp indicates that
the separation has only a weak dependence on the initial aspect ratio. The fit in figure 5-7
is given by
H8 , = 2.4wsTsp.
(5.22)
Combining equations 5.21 and 5.22, we can draw a comparison to [6] as shown in figure
5-8. [6] gave a lower and an upper limit for their empirical equation for the separation
height (solid lines in figure 5-8). The initial release aspect ratio was not reported in their
experiments, hence we assumed a range of I to 3 based on their experimental description.
Furthermore, they defined the phase separation time differently - as the instant that the
cloud centroid reached the individual particles' settling velocity - which should give a
slightly larger phase separation height compared to ours. However, generally speaking,
figure 5-8 shows that equation 5.22 is able to predict the trend and range of phase separation
heights observed in their experiments.
132
5.5
Penetration and growth
The phase separation results derived above can help understand the general settling characteristics including the penetration and growth, which are represented by the time evolution
of the front position and the radius of the solid and fluid phases of the settling particle
cloud. They were extracted from the numerical predictions in figures 5-9, 5-10, and 5-13.
As shown below, the proper scaling is able to simplify the present numerical results and
collapse all the data.
Defining the length scale of L, = V B/po/wS, the penetration distance can be normalized and generalized into simple relationships. For dimensional scaling, the penetration
distance before and after phase separation has to be treated separately. The difference between the two normalizations is the presence of the aspect ratio A. Equations 5.23 and 5.24
are best fit equations of the penetration distance shown in figures 5-9 and 5-10
2.IAO-"17
Lm
1.89AO
34
t
./
(t < TSP)
S2/(5.23)
t-TSP
(t > TSP)
+ 1.1
VB/po/w2S
and
0.3
t
Z0.5
.)3/5
/
-r
0.45
(t<
4/5
S
1.36A 04 5 + 0.75
t-T 8 /
(t > TSP)
--
(5.24)
The fitting results reveal that the settling dynamics before and after the phase separation
(i.e. in the thermal and dispersive stages) are different, as anticipated. Before phase separation, particle clouds develop as "thermals", whereas after phase separation, particles settle
individually. This was confirmed in the fitting exponents of the non-dimensional time in the
solid phase, i.e. in equation 5.23. For t < Tsp, the exponent was 2/3, which was consistent
with the penetration rate of starting plume [18]; for t > T, the penetration was linear in
time, and close to the settling velocity of the particles. Although theoretically the coeffi133
cient of the second term should be equal to one, a value of 1.1 was obtained in the present
observation. This may be attributed to our definition of the dispersive stage, which covers a
section where some particles still remain in the fluid phase and contribute buoyancy to the
penetration. The difference in the settling dynamics in the two stages can also be observed
in the fluid phase as demonstrated in equation 5.24, but no direct analogy can be drawn to
known phenomenon (e.g. thermals/puffs or starting plumes/jets [19]). Finally, we address
the effect of the aspect ratio A, whose influence is only seen before phase separation. In
other words, particle clouds only "remember" the initial shape in the thermal stage, and the
dispersive stage is not directly affected. Equations 5.23 and 5.24 also clarify the difference
between the solid and fluid phases. First, as expected, the fluid phase penetrates slower
than the solid phase, which results in the phase separation. Second, the behavior of the
fluid phase is more sensitive to the initial release shape than the solid phase, as indicated
by the exponents in these equations. Similarly, the growth of the particle cloud can also be
extracted in figure 5-13 and fitted by
L__
Lm
Lm
0.7( IB/ p/w2)
Lm
Bp/,
)0.3
(5.25)
Different from the penetration results, the growth cannot clearly be separated into two
stages. The growth of the solid and fluid phases followed the same temporal rate. The
effect of the initial aspect ratio A was also not observed, which suggests that the growth
was independent of the initial release shape. To summarize, equations 5.23, 5.24 and 5.25
can be used to predict the position and size of the particle cloud released in the water
column, which could be useful in engineering applications.
5.6
Entrainment and deposition
One of the key ingredients in integral modeling is the entrainment rate, which determines
the general behavior of the model in terms of the particle cloud settlement and growth. Un134
der the self-similar assumption, the entrainment rate is often represented by the entrainment
coefficient, which is the ratio of the velocities of entrainment over the cloud settling. In this
section, the entrainment coefficient is extracted by post-processing the present two-phase
Large Eddy Simulation results aiming to further reveal the basic settling dynamics.
The entrainment flux can be derived by integrating the velocity normal to the sphere
that contains the whole particle cloud. Specifically, we find the particle cloud center and
the distance from the center to the farthest particle (figure 5-12). Then, the containing
sphere is built with the center at the determined particle cloud center and the radius equal
to the determined distance. At last, the normal flux to the containing sphere is numerically
integrated to yield the entrainment flux.
The entrainment coefficients with different particle sizes and initial aspect ratios are
shown in figures 5-13a and 5-13b, respectively. The entrainment coefficient has a sharp
drop in the beginning; then, it quickly climbs up to the peak; and it slowly decays to zero.
As observed from all the simulations, the entrainment coefficient always converges to a
curve, which can be fitted by a, = 0.45 exp (-t/(16
B/po/w )).
As shown in figure 5-13a, the cloud with smaller particles has a lower maximum entrainment coefficient and a smaller overshoot. In figure 5-13b, the cloud with a higher
initial aspect ratio has a longer increase in entrainment coefficient. The overshoot is more
apparent in low aspect ratios and case B 15 has no overshoot.
The deposition pattern is relevant in certain engineering applications; however, it's not
the main topic here. As a brief discussion, we focus on a particular case - case B. Figure
5-14 shows the radial distribution of particle projections onto the bottom at different time.
A clear ring shape pattern can be observed, which is enlarging with few particles in the
center and a peak at the boundary. Beyond the peak, the particle number quickly decreases
to zero. Note that this projection pattern can reflect the final deposition pattern if the cloud
reaches the bottom in the dispersive stage. However, the final deposition pattern might be
quite different from the projection pattern mentioned above, if the particle cloud impacts a
135
soft bottom and cause the sediment to surge. More experimental studies on the deposition
can be found in [7].
5.7
Polydispersion
In real applications, the released particles are not always uniform in size or "monodispersed". A spectrum of particle size is much more likely, yet there are limited studies on
this issue. Questions remain such as how the "polydispersion" changes the dynamics; how
significant is the effect; and whether there is a simple way to predict the cloud size and
position.
To address these questions, several "polydispersion" releases were simulated numerically. Their configurations can be found in table 5.3. The set of simulations covered two
types of particle distribution by number, i.e. uniform and Gaussian. The median particle
diameter by number was around 0.51 mm; the specific range was listed in the table and
their distributions can be found in figure 5-15. For engineering applications, the particle
diameter d50 , defined by the size that 50% of the mass is finer, is also listed. Corresponding to d5o and the maximum particle diameter respectively, the median and the maximum
settling velocities were calculated using the formula offered by [20] and are shown in the
same table. All the other flow parameters including initial particle cloud radius, total mass
and initial shape are the same as case B in table 5.2.
As shown in figure 5-16, the cloud patterns were similar before phase separation but
different afterwards. In the thermal stage (figure 5-16a), the particles were well mixed and
distributed similarly to the monodispersion cases. In contrast, the particles settled according to their sizes or individual terminal settling velocities in the dispersive stage (figure
5-16b). As a result, the particles were sorted by their sizes, with bigger particles settling
faster. The vertical distribution of size was also consistent with the particle distribution by
number at release.
136
This observation is easy to understand. In the thermal stage, the two phases are closely
coupled and the solid phase is continuously re-distributed by the fluid phase. This offers
a good stirring mechanism keeping the particles well-mixed. Conversely, in the dispersive stage, particles are more separated and able to settle individually. This analysis thus
suggests that the thermal stage dynamics for polydispersed particles is similar to that for
monodispersed particles, and that the different particle sizes can be modeled using the
same formulae developed for monodispersion. In addition, the phase separation results of
monodispersion can be applied in a similar manner to the polydispersion.
To test the above hypothesis, we develop an empirical model to compare with the polydispersion results. First, we define L, 1 = VB/po/w5 o and Lp2 =
IB/po/Wmax. Dic-
tating that LP = L,1 at t < T and Lp = Lp2 at t > T,, the polydispersion data are
normalized and the results callapse as shown in figures 5-17 and 5-18 using Lp in penetration and L
and L
2
2
in growth. Correspondingly, replacing Lm by Lp in equations 5.23 and 5.24
in 5.25, the collapsed data can be fitted well by the current empirical model. The
most significant discrepancy is observed in the radius of the fluid phase, where the present
empirical model overestimates the numerical results by around 20%.
5.8
Summary
In the present paper, a new numerical scheme with LES is developed for two-phase simulations to address three specific issues related to the settlement of particle clouds in the
water column: the two-phase separation, the effect of the initial cloud shape and polydispersion dynamics. Three series of simulations were performed. The first series validated
the current numerical approach. The second series provided quantitative results on the
monodispersion dynamics including phase separation, the penetration and growth of the
particle cloud, and the entrainment rate and deposition patterns. Finally, the last series
clarified the polydispersion dynamics.
137
With proper scaling, we obtain best-fit equations of the present numerical results. They
show that the initial aspect ratio plays a weak role, being more important in penetration
than in growth, and it only affects the behavior before phase separation. These equations
can be easily applied in engineering analysis.
138
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139
[8] Bing Zhao, Adrian W. K. Law, Zhenhua Huang, E. Eric Adams, and Adrian C. H.
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141
Figure 5-1: Mesh of the computational domain
10
9-
,
40
8-
35.
7-
30
0
6-
-grid
25
sizeof69
of 6098
-grid size of
o 13186
38
grid size of 17695
grid size of 26931
20
15
'5 5--o 4
-
3-
-
10
-
grid size of 6098
grid size of 13186
grid size of 17695
grid size of 26931
2
1
5
'
0
0.5
1
1.5
2
time (s)
2.5
3
'
3.5
4
0
0.5
1
Figure 5-2: Grid convergence test
142
1.5
2
time (s)
2.5
3
3.5
4
Figure 5-3: Visual comparison between numerical simulations (left) and experiments
(right).
143
0.9
0.45
0.8
5
0.4
0.7-
0.35
0.6
0.3
-A-
0.5
0.25
Nr
0.4
0.2
0.3
0.15
0.2
0.1
0.1
0.05
--
--
l-
0
6
4
2
10
8
2
0
6
4
8
10
12
time (s)
time (s)
(b) Fluid phase front
(a) Solid phase front
0.12,
0.12
1
0.1
0.1
0.08
0.08
-A
0.06
0.06
F-A
-0.04
B
0.04-
B
--
-c0.02
0.02
-c
0
2
6
4
8
8
6 m2
time (s)
4
10
time (s)
10
12
14
(d) Fluid phase radius
(c) Solid phase radius
Figure 5-4: Penetration and growth of particle clouds compared to experiments (lines:
numerical results)
-
1
Particle portion in fluid phase
Particle cloud center depth (m)
Fluid cloud front depth (m)
-
-0.8
0.6
-
0.4
-~.
0.2
I
0
6
2
8
time (s)
(a) A sketch of phase separation
(b) Definition of phase separation
Figure 5-5: Illustration of phase separation
144
10
1200
0.
1000
800
0/
600
0
--
0
/y
C',
in
00
400
0
200
o by center-fron t cross
0/
0
1000
500
1500
Ra
Figure 5-6: Phase separation time determined by two methods (symbols: determined separation time; dash line: equation 5.21)
1600
140120100-
0
0
.1e
C,
.00
80-
.0o
6040-
.01
20-
401
0
10
0
20
30
T p/(R /ws)
40
50
60
Figure 5-7: Phase separation height (symbols: determined separation height; dash line:
equation 5.22)
145
3500-
3000
-
-
-
-
Bush et
Bush et
Present
Present
al. upper limit
al. lower limit
formula A=1
formula A=3
2500
-0
C,
2000
1500
1
-0
-0
000
500
0
200
400
600
Ra
800
1000
1200
Figure 5-8: Comparison of the phase separation height between [6] (solid lines) and the
present equation 5.22 (dash lines)
-A
-B
-c
5
C9
-
4
-A
- B
A3
B3
-C3
33
I-
I
2
0
-A15
- -
-B15
--- C15
- - - BL4
- -- BL16
-0-formula
I-
1
- -
1*.
U,
+
2(Re/(Bp
0
)1I2) 3
4
-B3
-C3
15
-
10
- -
U,
-BA15
--- C15
-- - BL4
--- BL16
--- formula
5
5
t/((B/p )12/w 215i
50
(b) After phase separation (t > Tp)
Figure 5-9: Normalized penetration of solid phase front
146
B5
--010
- --015
C15
0
11)0
(a) Before phase separation (t < Tp)
A3
-
co
- B5
--- C5
010
0
-c
20
A.
-A
-B
-C
-A3
-- 3
-C3
-5
--- C5
C10
-- -A15
--- B15
--- C15
3.5
3
2.5[
2
C~)
.
1.5
22A'2
1
N
10
8
6
0
4
2
- - -1BL4
0.5
- -
--
0
-A
-- B
-C
-A3
-- 3
-C3
-5
--- C5
C10
--- A15
--- B15
--- C15
-- -1BL4
- -- BL16
-+-formula
-BL16
formula
C
t/(R 0 /(B/p0 ) )
5
10
i
t/((B3/p )
)21V
20
(b) After phase separation (t > Tp)
(a) Before phase separation (t < Top)
Figure 5-10: Normalized penetration of fluid phase front
-A
-B
-C
-A3
-B3
-C3
-B5
--- C5
C10
--- A15
--- B15
--- C15
- - - BL4
--- BL16
-+-formula
1.5
C,
C" 0-
1
II
0.5
0
102
t(R 0/(B/p 0 )
2.5
2
C,
M
1
102
0.5
-
)
102
1/220
t/(R0 /(B/p 0 ) )
(b) Fluid phase
(a) Solid phase
Figure 5-11: Normalized growth of particle clouds
147
-- BL16
-+-formula
C
1/20
-.22
-A
-B
-C
-A3
-63
-C3
-635
--- C5
C10
-- -A15
--- B15
--- C15
- - - BL4
Figure 5-12: The entrainment flow of the particle cloud in case B (the dash line marks the
surface of the containing sphere).
0.15
0.15
-A
-- B
0.1
C
C
0.05
0.05
0
C
0.1
0
0
-c
--
'-0.05
-0.05
-0.1
-0.1
B15
wi
Ci
I
-0.15
-0.2
C
5
10B/p)1/21)
20
25
(a)
B
B3
-ft15
2 2 .5
.0
5 1/2 2
t/((B/p)
//ws)
10
(b)
Figure 5-13: Entrainment coefficients with (a) different particle sizes and (b) initial aspect
ratios (Dash lines are given by ae = 0.45 exp (-t/(16/B/po/w )) ).
148
Q. X 107
a)
-0.02
s
0.06 s
-0.1
s
-0.3
s
-0.5
s
-1
s
1.5s
-2
s
q6-
0-
o4
a)
-0
E
C2
0
0.02
0.04
0.1
0.08
0.06
Radius (m)
0.12
Figure 5-14: Radial distributions of particle projections for case B
3000-
2500-
2000
Bu2
-
E
|
1500-
Bu3
1000-
500
0
0 .2
0.3
0.4
0.5
0.6
Particle diameter (mm)
0.7
Figure 5-15: Particle size distribution
149
0.8
0.9
(a) Thermal stage (t =0.Is)
dp (mm)
$
0.60
~1
0. 56
0.48
*2
0.40
(b) Dispersive stage (t = 6.0s)
Figure 5-16: Comparison of polydispersion (from left to right: cases Bul, Bu2, Bu3, Bgl,
Bg2 as in Table 3)
30
25
20
Bu1
-
Bu2
-
Bg1
12
Bu3
10
-Bg2
-4-empirical model
8
15
6
Bu1
10
5
-6
4 4
---
Bu2
Bu3
22
----
Bg1
Bg2
--
empirical model
2-11/
t/(R 0/(B/p)
20
5
25
(a) Solid phase
10
2/(
/( p
1/2)
ti(R 0/(B/pa0)s
20
(b) Fluid phase
Figure 5-17: Normalized penetration of particle clouds in polydispersion
150
25
30
1.8
1.6
1.41.2
1
1
0.8
-
0.5
-+-
0
2
4
2 6
1/2
t/(R;/(B/po)
2)
Bul
Bu2
Bu3
Bgl
Bg2
empirical model
8
10
-
0.6
-
0.4
2-1-
0.2
12
--
5
210
t/(R 0/(B/p 0 )
1/
Bu2Bu3
Bgl-
Bg2
B/2
empirical model
15
(b) Fluid phase
(a) Solid phase
Figure 5-18: Normalized growth of particle clouds in polydispersion
151
20
Table 5.1: Flow configuration for grid convergence
Case
dp (mm)
Ro (mm)
Total mass (g)
H (mm)
GC
0.513
8.4
8.4
50.4
Table 5.2: Flow configuration for monodispersion studies
Case
dp (mm)
Ro (mm)
Total mass (g)
H (mm)
Aspect ratio
Ra
Re
A
B
C
A3
B3
C3
B5
C5
CIO
A15
B15
C15
BL4
BL16
0.73
0.51
0.26
0.73
0.51
0.26
0.51
0.26
0.26
0.73
0.51
0.26
0.99
2.20
9
9
9
6.7
6.7
6.7
5.6
5.6
3.9
3.9
3.9
3.9
36
144
3
3
3
3
3
3
3
3
2
3
3
3
192
12288
11
11
11
20
20
20
28
28
39
58
58
58
43
171
1.2
1.2
1.2
3
3
3
5
5
10
15
15
15
1.2
1.2
19
41
214
36
76
398
106
557
772
104
220
1158
41
41
2667
2667
2667
1956
1956
1956
1654
1654
1072
1147
1147
1147
21339
170709
Table 5.3: Flow configuration for polydispersion studies
Case
dp(mm)
0.26-0.73
0.26-0.76
0.43-0.60
0.45-0.57
d50 (mm)
Bul
Bu2
Bu3
Bgl
Distr. by num.
uniform
uniform
uniform
Gaussian
Bg2
Gaussian
0.22-0.81
152
Wmax(Cm/s)
0.61
0.64
0.53
0.52
w50 (cm/s)
8.8
9.3
7.6
7.4
0.57
8.2
12.0
10.6
11.2
8.7
8.3
Chapter 6
Scaling Particle Cloud Dynamics - From
Lab to Field*
Abstract
Open-water disposal of sediment is an important component in many coastal engineering
projects. Numerous studies have focused on small scale dynamics and claimed their results
can be scaled up by Cloud number scaling. However, this scaling method is largely empirical and unexamined. The present paper confirms that the Cloud number scaling provides
a rational way to extrapolate small scale lab experiments or numerical simulations to the
field operations.
6.1
Introduction
Large quantities of sediment are released in open fresh and coastal water in support of land
reclamation, coastal protection, navigational dredging, and contaminant sediment isolation
projects. Studies to support these projects, mostly experimental but some numerical, are
mainly small scale due to the limited volume of laboratory tanks and the fact that compu*This chapter has been submitted to Journal of Hydraulic Engineering as a technical note.
153
tational methods, which depend on simulating individual particles, get expensive when the
number of particles increases. Small scale results need to be extrapolated to field scale.
This is normally done by so-called Cloud number scaling, but this approach has seen limited validation.
Sediment cloud dynamics involves a balance of inertia and buoyancy, so dynamic similarity scaling requires that a densimetric Froude number be maintained before and after
scaling, i.e. Fr = w-(g'L)1/ = constant, where w, is the particle slip velocity or the
terminal settling velocity, L is the length scale, and g' is the reduced gravity. If we pick L ~
cloud radius, or the cube root of the initial cloud volume V, then Fr = w,/(g'1/ 2 V1/6).
If the subscriptions before and after scaling are 1 and 2, the length scale ratio is then
r = L 2 /LI.
Geometric scaling would require that all lengths be proportional (i.e. obey the same
ratio r), so the default scaling (Approach 1) would have the particle diameter d2 /di =
L2/Li
= r. To keep Fr constant after scaling up, i.e. Fr2 = Fri, the particle slip
velocity has to be w, 2 = (L 2 /LI)1/ 2 w,,. This would happen automatically with the default
scaling if w, ~ L1 /2 , which occurs if settling is based on a balance of buoyancy and drag
(consistent with Froude scaling). But for small particles, this is not the case, so the adopted
approach is to guarantee Ws2 = (L 2 /L1 ) 1/ 2 WS1 . This is tantamount to preserving the Cloud
number, Nc, which is related to the Rayleigh number, Ra, as shown below.
Originally formulated by [1], the "Cloud number" is defined by
Nc =,
, IBO/po / R
(6.1)
where B 0 = mA12g
is the initial buoyancy of the particle cloud, and m is total released
PO
mass, ps, po, and Ap = ps - po are respectively the densities of particles, fluid, and their
difference, g is gravitational acceleration, and Ro is the initial radius of particle cloud. The
"Cloud number" is a variation of the Froude number (Fr)or the Rayleigh number (Ra),
i.e.
154
Ra -
B0
o
p0W2R2
-
1
Nc1
_
NC2
1
3
2)
(6.2)
02Fr2
where / is a coefficient related to the aspect ratio at release.
In the following, we test the Cloud number scaling by simulating different scaled releases. Not surprisingly, it works and is much better than the default. We introduce the
numerical method in the next section, show the numerical results in the following section,
and draw conclusion in the final section.
6.2
Numerical method
The present paper implements 3-D Euler-Lagrangian LES using the software CFDEM,
which couples the open source CFD toolbox (OpenFOAM) and the Discrete Element Method
(DEM) package (LIGGGHTS) [2]. For the fluid phase, the void fractioned Navier-Stokes
equations,
aa + V - (azguf) at
= 0(.3
D(afuf) + V- (aciufur) =-aV
at
(6.3)
PO
-
R + V -T,
are resolved by OpenFOAM, where af is the volume fraction of the fluid, uf is the velocity
of the fluid phase, t is time, p is pressure, pf is the density of the fluid,
T
is the shear
stress tensor, and R,1 is the inter-phase momentum transfer term. OpenFOAM uses the
finite volume method with LES and the local dynamic one equation eddy viscosity subgrid model (locDynOneEqEddy).
The solid phase is obtained by solving the ODE by LIGGGHTS, i.e.
du
= EF
m d
P dt
(6.4)
where mp is the particle mass, up is the particle velocity, and EF is the force applied on the
particle, which can be specified as
155
EF = FG + Fs + FD + FL + FA
(6.5)
where FG is the gravity force, Fs is the fluid stress, FD is the drag force, FL is the lift
force, and FA is the added mass force. These forces can be modeled by [3]. In addition,
the particle-particle collision effect is incorporated. After every simulation, a passive tracer
is released from the void space and tracked to show the fluid phase motion. More details
can be found in [4] and [5].
The flow configuration is the same as in [5]. Specifically, the computational domain
is an unstructured mesh within a cylindrical shape (Fig. 6-1). More grid cells are placed
around the symmetry axis to better capture the shear layer. The domain extends to 1.2m
in depth and 0.567m in diameter for the "Original" case and is scaled up accordingly for
other cases (i.e. magnified by 4 times and 16 times). Zero-gradient open boundaries are
applied to all the surfaces. All the particle clouds are released in the middle of the quiescent
domain. The initial release is contained in a predefined cylinder with radius Ro and height
H, which dictates the aspect ratio of H/RO = 1.2 for all the cases. The particles in the
initial release are compact with a volume fraction oz,
1 - of = -F/6 = 0.524. All the
simulations are executed with 32 processors in the High Performance Cluster of Nanyang
Technological University. The code has been validated and gird convergence has been
achieved with 26931 cells, as is further detailed in [5].
6.3
Results and discussion
A series of simulations is performed including four cases tabulated in Table 6.1. The "Original" case is set up as a control. The geometrically scaled up case is denoted "Lr4", and
has all lengths, including the grain diameter, magnified by 4 times. In comparison, BL4
and BL16 are cases where the source dimension Ro is scaled up by 4 and 16 times, while
keeping the Cloud number constant. Note that since the relative length ratio r is higher in
156
BL4 and BL16, these cases contain more particles than the first two as shown in Table 6.1,
which results in increased simulation time. In the following, we compare the results from
these four cases.
A typical flow pattern is shown in Fig. 6-2. The particles are solid dots and the contour
represents the tracer concentration that is generated by releasing the tracer along with the
particles. Several geometric parameters are defined: z, and z, are distance penetrated by
the solid and fluid phase fronts; r, and rf are the radii of the solid and fluid phases. The
boundaries of the fluid phase are defined by locations where the scaler concentration is 1%
of its maximum concentration. The rest of the paper focuses on these major characteristics
of the particle cloud.
The normalized solid phase penetration distance, ze, is shown in Fig. 6-3 for the "Original" and Lr4 cases. These two cases share the same length scale ratio of r. It is clear
that, after normalization, the two phenomenon are not consistent. Lr4 is penetrating more
slowly than the original. This suggests that pure geometric scaling cannot produce the
needed similarity.
More comparisons are drawn among the "Original", BL4, and BL16 cases in Fig. 6-4.
These cases share the same Nc, and the results are found to collapse to a single curve after
normalization. Specifically, all the penetration distances (z, and z,) overlap each other.
In addition, the solid phase growth of the radius also exhibits good overlap. The worst
agreement comes from the fluid phase radius, where all the results follow the same trend
but the later time shows some discrepancy, which might be due to the low resolution close
to the boundaries and accumulated numerical error. But the major particle cloud dynamics
follow the same behavior if their Nc is the same.
157
6.4
Conclusion
The present paper overviews two possible methods to scale the dynamics of settling particle
clouds. By numerical simulations, the Cloud number scaling is found to be appropriate,
confirming conventional methodology.
158
Bibliography
[1] Hamid Rahimipour and David Wilkinson. Dynamic Behavior of Particle Clouds. In
11th AustralasianFluid Mechanics Conference, Hobart, Australia, 1992.
[2] Christoph Kloss, Christoph Goniva, Alice Hager, Stefan Amberger, and Stefan Pirker.
Models, algorithms and validation for opensource DEM and CFD-DEM. Progress in
ComputationalFluid Dynamics, an InternationalJournal, 12(2):140-152, 2012.
[3] E. Loth and A. J. Dorgan. An equation of motion for particles of finite Reynolds
number and size. EnvironmentalFluid Mechanics, 9(2):187-206, March 2009.
[4] Ruo-Qian Wang, Adrian Wing-Keung Law, and E. Eric Adams. Large Eddy Simulation
of Starting and Developed Particle-Laden Jets. In Proceedingsof the 8th International
Conference on Multiphase Flow, Jeju, Korea, 2013.
[5] Ruo-Qian Wang, Adrian Wing-Keung Law, and E. Eric Adams. LES study of settling
particle cloud dynamics, submitted.
159
Figure 6-1: Grid mesh of the computational domain
Table 6.1: Simulation configurations
Case
Ro(cm)
M(g)
H(m)
d(mm)
Original
0.9
3
0.011
Lr4
3.6
192
BL4
3.6
BL16
14.4
r
Nc
0.51
17.5
0.192
0.043
2.04
17.5
0.364
192
0.043
0.99
36.4
0.192
12288
0.172
2.20
65.5
0.192
160
Figure 6-2: Sketch of a two-phase particle cloud
300
250200150 N
100-
-Original
-
50-
0
200
100
e -)
Lr4
300
400
1/2t
Figure 6-3: Normalized penetration history of case Lr4
161
120
50-
100
40-
80
30-
CD
-Original
60
N
N
-Original
-BL4
-BL16
20-
40
10
20
C
b
200
100
500
400
300
6
6 )0
100
200
B)1/2
Po
(B)1/
I
I
I
8
0D
4
0
600
1 01
6
2
500
(b) Fluid phase front
8
C)
400
R
(a) Solid phase front
1o
3002
-
100
200
300
-Original
4
-BL1
2
6
400
500
W0
600
1/2
P0
6
-Original
-BL4
-BL1 6
100
200
300
400
500
600
B)1/2
Ra
s
(c) Solid phase radius
(d) Fluid phase radius
Figure 6-4: Penetration and growth of particle clouds for cases Original, BL4, and BL16
162
Chapter 7
An Axisymmetric Steady State Vortex
Ring Model*
Abstract
Based on the solution of [I], a theoretical model for axisymmetric vortex flows is derived
in the present study by solving the vorticity transport equation for an inviscid, incompressible fluid in cylindrical coordinates. The model can describe a variety of axisymmetric
flows with particular boundary conditions at a moderately high Reynolds number. This
paper shows one example: a high Reynolds number laminar vortex ring. The model can
represent a family of vortex rings by specifying the modulus function using a Rayleigh
distribution function. The characteristics of this vortex ring family are illustrated by numerical methods. For verification, the model results compare well with the recent direct
numerical simulations (DNS) in terms of the vorticity distribution and streamline patterns,
cross-sectional areas of the vortex core and bubble, and radial vorticity distribution through
the vortex center. Most importantly, the asymmetry and elliptical outline of the vorticity
profile are well captured.
*This chapter has been submitted to Applied Mathematical Modeling as a paper.
163
7.1
Introduction
The studies of vortex rings can be traced back more than one and half centuries, when
William Barton Rogers, founder of MIT, conducted the first systematic vortex ring experiments [2]. Inspired by his and other pioneers' work, a series of theoretical endeavors have
been made for the mathematical description of vortex rings. An early example is the famous Hill's vortex, which assumes that vorticity is linearly proportion to radius within a
spherical volume, with potential flow outside [3]. The assumption is relatively simple, yet
it is able to generate realistic-looking streamlines, and is arguably the most popular vortex
ring model in applied science and engineering, e.g. Lai et al. [4]. From a different starting
point, Fraenkel [5] analyzed the vortex ring by extending the theoretical solution of a vortex filament to allow for a small finite thickness. To bridge these two models, Norbury [6]
treated the Hill's spherical vortex and Fraenkel's thin ring as two asymptotic members of a
series of generalized vortex rings. He then numerically determined a range of intermediate
rings, now referred to as the Norbury-Fraenkel (NF) vortex ring family. The NF family delivers more accurate streamlines and more precise vortex ring outlines. However, its linear
distribution of vorticity is still unrealistic [7].
Recently, a solution for viscous vortex rings at low Reynolds numbers (Re) was obtained by [8]. They derived a generalized solution to the diffusing viscous vortex ring, by
directly solving the axisymmetric vorticity-stream function equations without the nonlinear convection terms. Their solution delineates a donut shape outline and a more realistic
Gaussian distribution of vorticity in the radial direction. Kaplanski et al. [9] extended
the solution to turbulence by adopting an effective turbulent viscosity. However, recently
Danaila and Helie[7] conducted Direct Numerical Simulations (DNS) and reported elliptical cross-sections and radial asymmetry for the vorticity distribution, which both NorburyFraenkel and Kaplanski-Rudi models are unable to capture. Realizing the issue, Kaplanski
et al. [10] added two adjustable parameters to allow an elliptical cross-section, but the radial asymmetry of the vorticity distribution is still amiss due to their symmetric Gaussian
164
distribution.
Laboratory experiments and numerical simulations have also shed light on vortex ring
dynamics as reviewed by [11] and [12]. In particular, Gharib et al. [13] performed experiments in which a piston generated a non-buoyant vortex puff. They observed that if the
aspect ratio of piston stroke length L to diameter D was less than about four, the generated
vorticity could be incorporated into the head vortex, whereas for larger aspect ratios a trailing stem occurs. The phenomenon is now referred to as the "pinch-off", and the critical
aspect ratio is called the "formation number". Because the trailing stem stops supplying
vorticity to the head vortex after the pinch-off, the "saturated" head vortex ring in the postformation stage is relatively stable, and should resemble the steady state situation in Hill's
vortex. The experimental results can advance the state-of-the-art by enabling a meaningful
comparison with idealized theoretical models and numerical simulations or experiments
with a saturated head vortex ring. An example can be found in [7], and the present study is
also initiated in the same spirit.
The best theoretical model would be the analytical solution to the Navier-Stokes equations, which accurately depicts the dynamics of the flow but is difficult to derive. As a compromise, a theoretical model could be built on the solution to the Euler equations, which
ignores the viscous effect but captures the more important non-linear dynamics of the flow.
In the present case, we focus on the steady state axisymmetric Euler equation, the target
of which is to solve an elliptical second order partial differential equation. This equation
is a particular form of the Grad-Shafranov equation [14, 15], which is related to the magnetostatic equilibrium in a perfectly conducting fluid and is well known in magnetohydrodynamics (MHD). Specifying different forms of the involved arbitrary functions, plasma
physicists have derived a series of analytical solutions to this equation, e.g. the simplest
solution of the Solov'ev equilibrium [16], the Herrnegger-Maschke solution [17, 18], and
the recent more general solution by [1] (hereafter referred to as AGLM). It can be shown
that the Hill's spherical vortex corresponds to the Solov'ev equilibrium, yet no counterparts
165
of the more advanced solutions have been explored for vortex dynamics. This encourages
us to take advantage of them, especially the AGLM, to reach a more accurate model of
vortex rings. Beyond that, a higher accuracy model can also meet the industrial demand to
improve the description of vortex structures to replace the aged Hill's vortex, e.g. [4]. In
particular, a more realistic vorticity distribution may lay a better foundation to address the
particle-vorticity interactions, which motivates us to make the following study.
As mentioned earlier, existing models have improved progressively to describe more
detailed features of real vortex rings. However, an outstanding issue is the asymmetry of
the vorticity distribution. The present paper provides an alternative analysis that is capable
of incorporating the asymmetry, by deriving a theoretical model for high Re laminar vortex
rings based on the solution to the axisymmetric inviscid vortex flow. The governing equations are first presented in section 7.2 and shown to relate to the Grad-Shafranov equation.
The alternative vortex ring solution is shown in section 7.3, and its properties are derived
in section 7.4. Then, the model is compared to existing simulation results in section 7.5.
Summary and conclusions are given in section 7.6.
7.2
A particular solution
We adopt the cylindrical coordinate system (r,#, x), where x is the local longitudinal coordinate. The incompressible axisymmetric vorticity transport equations can be stated as
[19]
-(U0) = p
9t + 0r (VO) + ax
where t is time,
9X2
+
Or2
+
ror
r2
(7.1)
is the vorticity, u and v are velocity components in x and r directions
respectively, p is the dynamic viscosity, and p is the density.
Introducing the Stokes stream function 0, the velocity components can then be expressed as
U=
-
r oBr
+ U, I
166
=
r 8x'
,
(7.2)
where U., is the translational velocity of the vortex centroid. The vorticity can be defined
by the stream function as
92
02V
__+
2
Ox
Or
100
2
r
Or
(7.3)
_
= -_r_
'
which provides the closure to solve (7.1). Equations (7.1)-(7.3) can therefore be applied
to any axisymmetric flows in theory. Substituting (7.2) into (7.1) and normalizing the
variables by a length scale 1 and a velocity scale U, i.e.
U= Uu*,
v
=
Uv*,
x
r = lr*,
-
Uxt = lx*,
= l 2 UO*,
V
= Ul--*,
(7.4)
a non-dimensionalized form of (7.1) can be derived as
UX0*+
aDCr*)
a
r*
(*
( *
(*
__*
__)
0
( U
(
+
(
Re
*
1x*2
)
(r*2
+
r* Dr*
r *2
(7.5)
where Re = pUl/u. U and 1 can be specified for a particular application. Note that if
Re < 1, the nonlinear terms on the LHS of (7.5) can be neglected, leading to the theoretical
solution obtained earlier by [8].
For Re > 1 but before transition into turbulence, the viscous effect (RHS of equation
(7.5)) can be ignored in the high Re laminar flows. Assuming a steady translation such that
U, is constant, and dropping the * sign for brevity, then
a
ar
Using (
=
)
r (9x
ax
( -Da )
r ar
rf (V) [12] and substituting into (7.3), we get
Dr2 ± Dx 2
r Dr
167
=0.
(7.6)
As mentioned earlier, equation (7.7) is a particular form of the Grad-Shafranov equation:
2
ar 2
'
02 b
I
X2
r
_r 2 f (<) + g (b),
=
Dr
(7.8)
with the arbitrary function g(o) = 0.
The simplest particular solution to (7.8) is the Solov'ev equilibrium solution, which
assumes
f(0) = A,
where A and B are constants. If B
=
g()) = B
(7.9)
0, this reduces to the Hill's spherical vortex solution
[3], which can be solved in spherical coordinates by separation of variables, while confining
the vorticity within the spherical vortex boundary. Although Hill's streamlines are similar
to experimental and numerical observations, the vorticity distribution is too simplified and,
as will be further discussed, differs significantly from observations [7].
In MHD, more solutions are available and have the potential to improve the accuracy over the simplest Hill's spherical vortex. One example is the particular solution with
f (4)
=
A0 and g (0) = B0, which is known as the Herrnegger-Maschke solution [17, 18].
In addition, a more general solution of AGLM [1] covers both situations above, assuming
a series expansion of f (0) and g(o), i.e.
f (x) = A0 + B,
g(x) = Co + D,
(7.10)
where A, B, C and D are constants.
The AGLM solution inspires us to generate a counterpart model of vortex rings as a
meaningful extension to the Hill's spherical vortex. If C = D = 0, equation (7.7) becomes
9r 2
+
a
-
r2 O9r
-
Ar 2o - Br 2.
(7.11)
Thus, the Hill's vortex solution can be viewed as the zeroth order approximation, A
168
=
0, which linearizes the governing equations for solvability. Note that higher order series
expansions can be used in the source term of (7.7), which involves the nonlinear term and
no explicit solution is available so far.
Recalling that
the vorticity
= rf(V)), the improvement of (7.10) over (7.9) is the manner in which
correlates with the stream function 0. As discussed earlier, Hill's model
delivers realistic-looking stream lines, and the linear vorticity distribution is also advantageous for the handling of the viscous terms in (7.5) and thus becomes the only explicit
solution to the Beltrami flow [20]. However, the vorticity distribution is unrealistic. The
present solution of (7.10) assumes that the vorticity distribution is the first radial moment
of the stream function, and by doing so directly correlates the vortex and flow structures. It
will be shown later that an improved matching of the stream function and vorticity distribution can be obtained in this manner. Furthermore, the correlation better reflects the physics
that the vorticity is transported by the flow structure.
In the following, we will show how the explicit exact solution can be sought with this
first order approximation, which is similar to [1] but with improvements in details. The
inhomogeneous nature of (7.11) leads to a solution consisting of two parts, i.e. ) = ' +
V' no,
where 00 and 4no are the homogeneous and inhomogeneous parts, respectively. We
shall obtain a general solution for the homogeneous problem first, and then complete the
solution with a particular inhomogeneous solution.
Taking the Fourier transform of the homogeneous problem with respect to x, we have
,92
Or2
1-
r Or
+ (Ar 2 - A 2 )
= 0,
(7.12)
where the Fourier transform is defined by
j
4'
4'e\dA.
169
(7.13)
Equation (7.12) can be simplified by introducing j = ar 2
192V)
8i12
1
+
A2
4
)
4agj
(7.14)
0,
=
which has a solution of
= CIM__2 1 (ar
2
4a '2
) + CA2 W2 1(ar 2)
4o
(7.15)
'2
where Mk,m(x) and Wk,m(x) are the Whittaker M and W functions, i is the imaginary
unit, and A, CAI, CA2 are constants to be determined, a
=
\/7i if A
> 0 and a =
-A
if A < 0 [21]. The solution in AGLM is also consistent with the present result but has a
much lengthy form. In other words, the present solution is more compact and succinct.
When A > 0, the solution can also be expressed as:
,A
=CA1Fo
A4v/A
2
)
±CA2Go
,;
(4A-X
Ar
2
(7.16)
where FL(1, x) and GL(q, x) are the Coulomb wave functions of the first and the second
kind. This solution is consistent with the Herrnegger-Maschke solution.
Subsequently, the stream function and vorticity can be recovered by taking the inverse
Fourier transform as
0
:
_OC
[CAM
A2 1(ar2) + C
4ca
12
2
W
2',
4o '2
(ar2)] [cos(Ax) - i sin(Ax)]dA.
(7.17)
The inhomogeneous solution 0,, should satisfy the equation of
a2 no
9r2
+ 0"
a22
1 Or"- = -Ar
r D
2 Vno -
Br2 .
(7.18)
The simplest particular solution is '/no =-B/A. To summarize, the solution to the reduced
170
Grad-Shafranov equation is
CxM
A2
1(Cr
2
) + CA2W
A2
1(r2)
[cos(Ax) - i sin(Ax)]dA - B/A, (7.19)
and
j
=
r[Cu1M
2
(ar2 ) + CA2W
-)r.
[cos(Ax) - i sin(Ax)]dA + (A -
21(r2)]
(7.20)
In theory, (7.19) and (7.20) apply to any axisymmetric inviscid flow in steady state (i.e.
with constant translation Ux) so long as the first order approximation in (7.10) is adequate.
We now develop a theoretical model of the high Re laminar vortex ring using these solutions
together with the necessary boundary conditions.
7.3
Vortex ring model
We seek an axisymmetric vortex ring solution with a finite boundary that is also symmetric
w.r.t. the plane x = 0, i.e.
=
const.
(r = 0),
(7.21 a)
=0
(x = 0),
(7.21b)
(at Q),
(7.2 1c)
Ox
V= const.
where the boundary of Q is toroidal in the 3-dimensional domain. Q cannot be determined
a priori, and is abstracted from the observation [7].
Here, we only consider the case of A > 0, because A < 0 gives rise to a monotonic
increase of the integral in the r direction and thus an infinite boundary extent. In this case,
boundary condition (7.21 a) requires that the axis of symmetry is a streamline, and without
loss of generality, B/A is chosen to be the streamfunction value. Because W,
171
(0)
#
0,
we have to set A2= 0 to prevent the stream function from varying at r = 0.
The boundary condition (7.21 b) comes from the observation that the vorticity distribution typically peaks at the center, and the vortex shape is nearly symmetrical with respect to
the plane of x = 0. Note also that the real part of the Whittaker M function M_
(ar2 )
is not integrable, and thus CA has to be a pure imaginary number. Hence, the vortex ring
solution can then be expressed as
iC(A)MX2 i
=
r
4V
2 i)
cos(Ax)dA + B/A
('i/Zr 2i) cos(Ax)dA + (A - B/A)r
iC(A)MXA
JO
1(Vr
12
(7.22)
(7.23)
where C(A) is a real function that can be viewed as the modulus function. Because the
integral is symmetric with respect to A = 0, the integral limits are constrained to the positive
range.
C(A) needs to be resolved in order to obtain the solution. Mathematically, if C(0) 4 0,
the equation is reduced to a parallel shearing flow, which has no finite boundary in the x
direction; and if C(oo)
#
0, the integral is not finite. Therefore, for a closed boundary and
for integrability, C(A) has to vanish as A -+ 0 or oc. Physically, C(A) can be seen as a
spectrum of vortex length scales, which has reached a self-balanced state after the vortex
ring formation. Hence, it is reasonable to assume that C(A) is continuous and smooth.
Although many functions are able to satisfy the conditions mentioned above, the simplest
candidate would be the Rayleigh distribution function, i.e.
C(A)
A
2
>,2
2,
(7.24)
where o- is a constant. Taking A = 1 and B = 0, we reach a normalized form of equation
(7.22), i.e.
=
[0 A
0
i-e
(
A2
(12 r 2 i) cos(Alx)dA.
20-2 IMV
42
172
(7.25)
Accordingly, the vorticity distribution becomes
0
-e-a
20,2
(7.26)
( 2 r2 i) cos(Alx)dA.
2
For illustration, two typical vortex ring patterns are shown in figures 7-1 and 7-2. Note
that the pattern has been further normalized by the length scale I so that the x extent of
the vortex ring is within the range from -1 to 1. Comparing figures 7-1 and 7-2, the aspect
ratio, 43
rmax/Zmax, is greater with a greater a, and the boundary is more rectangular.
Generally, equations (7.25) and (7.26) depict a family of new vortex ring models. In
the following, we obtain the properties of the present model by numerical methods.
Properties of the vortex ring family
7.4
The current model of equation (7.25) has only one free parameter, or, by which a family of
vortex rings can be modeled. The basic properties of this model can be described by a series
of integrals of the vortex ring dynamics including the circulation (F), the hydrodynamic
impulse (I), and the energy (E), which are defined as below:
F=
j
dxdr,
J
0 -o0
I= F
j
r2
dxdr,
E=
-o0
j JO
dxdr.
(7.27)
By varying o, we can derive the variation of 3, F, I, and E, which are shown in figure
7-3. The aspect ratio and all the integrals generally increase with larger a. Before a = 0.7,
the aspect ratio 3 is increasing almost linearly. Afterwards it increases dramatically and
approaches infinity at a =~ 0.78. The integrals increase much more quickly than 3 and
also approaches infinity at the same place. Note that when a
=
0, the r extent approaches
zero (or 1 -+ oc), and thus the vortex ring is reduced to an infinite line. As a result, all the
integrals also vanish.
A useful parameter in vortex ring dynamics is the propagation velocity, Ux, which is
173
defined earlier in this paper. Theoretically, it can be estimated by the following general
equation [22, 23],
() - 6x k)dxdr
f =0 11 2 dxdr
Sf'2
(7.28)
It is difficult to derive an explicit expression for Ux. Hence, we use a numerical method
to compute U2, which is shown in figure 7-4. From this figure, the propagation velocity
Ux decreases with larger or, but the slope of the decrease is much reduced when o is near
~ 0.78. At or
-3
0, Ux
-±
oc. Note that the results in figure 7-4 are normalized. The
absolute values depend on the scaling.
Also note that the effect of A
ratio 0 and the length scale 1. If B
1 can be incorporated by a correction to the aspect
#
0, the aspect ratio of the streamlines remains, but the
vorticity and other integrals will be increased accordingly.
7.5
Comparison to numerical simulations
The present solution takes advantages that at high Re laminar flow, the viscous effects are
present but negligible. This enables us to make a meaningful comparison with corresponding direct numerical simulations in this range.
To examine the properties of vortex rings, Danaila and Helie [7] performed axisymmetric Direct Numerical Simulations of a piston generated vortex ring and compared the
results with the Norbury-Fraenkel family and Kaplanski and Rudi's analytical solutions.
Their simulations are based on the formation number of L/D = 4 which is close to that of
a saturated vortex ring. The comparison included the streamlines and vorticity distribution
at tUo/D = 30, which are shown in figure 7-6(a) - (f).
To quantify the characteristics of the vortex ring using the present approach, the parameter, a, has to be specified. By close examinination, the aspect ratio of DH is 3 = 1.23,
which corresponds to o = 0.585 in the present model from figure 7-3. Furthermore, the
normalized standard form should be adjusted in its magnitude to compare with the simula174
tion results, with C(A) scaled by a factor of 0.65 and 1 = 0.94. Using this configuration,
we can visualize the three dimensional structure of the vortex ring in figure 7-5. The vortex ring has a torus shape with a deformed cross-section. The outside of the vortex ring
is narrower than the inner side. The toroid is also sliced to show the cross-section, where
the vorticity distribution can be seen. In the following, we draw a series of comparisons to
show that the present results agree well with the numerical simulations.
Overall, figure 7-6 shows that the present solution provides a better agreement than the
other peer models. A close match of streamlines and vorticity distributions is obtained as
shown in figure 7-6 (g) and (h). The streamlines of figure 7-6 (b), (d), (f) and (h) illustrate that all models essentially share similar characteristics, but the present solution in (h)
achieves better matching. More notably, the vorticity distribution comparison demonstrates
a significant improvement. As discussed before, the Norbury-Fraenkel's family has a major shortcoming due to the unrealistic linear vorticity distribution. The Kaplanski-Rodi's
model improves with a continuous vorticity distribution at the boundaries, but the vorticity contours dictate a circular shape which is also unrealistic. The present solution better
depicts the asymmetry of the vorticity distribution by accommodating multiple harmonics.
The small discrepancy at the boundary might be attributed to the absence of viscous effects.
With the present solution, the boundaries of the vortex cores and vortex bubbles are
also found to better match the numerical simulations by [7]. The isocontour of 5% of the
maximum vorticity is taken as the vortex core boundary shown in figure 7-7(a). The present
solution shares similar characteristics of a sharper top and flatter bottom with the numerical
results. In addition, the outline of the vortex bubble where the stream function decays to
zero is shown in figure 7-7(b). With the appropriate a-, the present solution has the same
aspect ratio as DH, but underestimates slightly the r and the x extents, which may again
be due to the fact that the viscous effect is neglected. Although the cross-sectional area is
underestimated by 18%, this is a considerable improvement over the peer models in which
~ 40% or more underestimation is common.
175
The radial distribution of vorticity through the vortex center is shown in figure 7-8. The
numerical results show a smooth peak profile, with a long tail expanding to the axis of symmetry and a steep slope at the outside. Quantitative comparison reveals that the NorburyFraenkel family has an unrealistic linear profile; while the Kaplanski-Rudi model features
a symmetrical Gaussian distribution which by definition cannot reproduce the asymmetry
observed in the numerical results.
Besides the good comparison with [7], similar qualitative agreement in vorticity distribution contours is also obtained with observations reported in the literature, e.g. [24].
However, for quantitative comparison, to the authors' knowledge, only [25] (MRC) and
[26] (ATC) offered high quality data that can be compared alongside [7] (DH). Both MRC
and ATC also adopted Direct Numerical Simulations to reproduce the vortex ring flows
similar to DH. Specifically, MRC focused on the pinch-off process. The radial vorticity
distributions through the vortex ring center at different times are shown as MRC 1 -MRC4
in figure 7-9. In contrast, ATC simulated the transition of vortex rings from laminar to
turbulent, and only a snap shot of the radial vorticity distribution in the laminar range for
both thin and thick cores were reported. They are shown as ATC 1 and ATC2 respectively
in figure 7-9.
The maximum value, (max, and the peak radius,
1
R,
can be extracted directly from
MRC and ATC. They are used as the vorticity and length scales for normalization of their
reported data. Meanwhile, we show the present solution in figure 7-8 using the best fitting
coefficient -= 0.585 with DH, but with the output of the results normalized by (max and lR.
In figure 7-9, clearly the present solution compares well with the numerical results by MRC
and ATC (the abbreviations and their corresponding flow conditions can be found in table
7.1). Specifically, MRC reported their vorticity distributions as a time series of a specific
simulation. In their case, the distribution was narrower in earlier time when the vortex was
still developing, and became wider later on when U, approached constant. The latter is
closer to our steady state assumption in the derivation of (7.6) (see the constant slope of the
176
vortex center position in figure 1 of MRC). Strictly speaking the present solution is only
appropriate to be compared with the steady state MRC4, where an excellent comparison is
found. Note that in MRC2, a secondary peak can be identified near the axis of symmetry,
reflecting the merging trailing stem.
ATC observed in the late laminar phase before turbulent onset that "the vorticity profile
across the core region relaxes towards a new equilibrium state, when the axisymmetric inviscid ideal is solely a function of the stream function V); the distribution becomes skewed,
decreasing faster toward the bubble edge than the ring centre." This observation is consistent with the present model development, and validates our comparison exercise. Of the
two cases in ATC, one was with a relatively saturated state of development shown as ATC 1,
and the other a thin core vortex ring with a much narrower peak that had not yet reached the
saturated phase shown as ATC2. Therefore, it's more appropriate to compare with ATC 1
instead of ATC2. As expected, the present model, which is derived with the steady state assumption, matches well the former. Finally, DH is also normalized and shown in figure 7-9
for comparison. To summarize, the present solution is able to match closely the DNS results of laminar vortex rings with high Re reported in the literature when the post-formation
stage is approached, which covers a wide range of Re = 1400 - 5500.
Recently, [27] derived a family of vortex rings with varying levels of saturation. By
examining the elliptical boundaries, they proved the existence of the elliptical vortex rings.
Using numerical methods, they offered data on the stream function distribution in the radial
direction. Corresponding to their most saturated situation, a comparison is drawn in figure
7-10 after normalization with the radius and the value of the maximum of 4', i.e. 1Ri and
. Although the present solution is relatively narrower, the shapes of these profiles
agree well with the results from [27]. The slight discrepancy may be due to the fact that
there is no superimposed potential flow in the present solution.
177
7.6
Summary and conclusions
The present study proposes a theoretical model of a family of steady state vortex rings.
First, a solution to the Grad-Shafranov equation from magnetohydrodynamics is derived,
which is more succinct and compact compared with the existing solution by [1]. Using
the analogy between the equations of hydrodynamics and magnetohydrodynamics, a theoretical model to high Re laminar vortex rings is developed based on this solution, which
can satisfy the D-shaped boundary conditions that are commonly used in magnetohydrodynamics and suitable for vortex rings. A new family of theoretical vortex rings is derived
based on this model with a free parameter o-. A numerical method is then used to calculate
the properties of this family, including aspect ratio, circulation, impulse, energy, and propagation velocity. By choosing o- = 0.585, a good match is obtained comparing the results
from the present model and direct numerical simulations. The present model is found to be
superior to the Norbury-Fraenkel family by virtue of its improved accuracy in the vorticity
distribution and streamline pattern. Most importantly, the asymmetry and elliptical outline
of the vorticity profile are well captured. Further comparison is performed with the normalized vorticity and stream function distributions reported in the literature and the present
solution. Again, close agreement is obtained. The improved matching with the present
solution can yield better assessment and models for engineering applications.
178
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[3] Micaiah John Muller Hill. On a spherical vortex. Proceedings of the Royal Society of
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[4] Adrian C. H. Lai, Bing Zhao, Adrian Wing-Keung Law, and E. Eric Adams. Twophase modeling of sediment clouds. EnvironmentalFluid Mechanics, February 2013.
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March 1973.
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laminar vortex ring. Physics of Fluids, 20:073602, 2008.
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evolution in a viscous fluid. Physics of Fluids, 24(3):033101, 2012.
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[19] Y. Fukumoto and F. Kaplanski. Global time evolution of an axisymmetric vortex ring
at low Reynolds numbers. Physics of Fluids, 20(5):053103, 2008.
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ring formation. Phys. Fluids, 12:589-596, 2000.
[25] Kamran Mohseni, Hongyu Ran, and Tim Colonius. Numerical experiments on vortex
ring formation. Journalof FluidMechanics, 430:267-282, March 2001.
[26] P. J. Archer, T. G. Thomas, and G. N. Coleman. Direct numerical simulation of
vortex ring evolution from the laminar to the early turbulent regime. Journal of Fluid
Mechanics, 598:201-226, 2008.
[27] Yan Zhang and Ionut Danaila. Existence and numerical modelling of vortex rings
with elliptic boundaries. Applied MathematicalModelling, 37(7):4809-4824, 2013.
181
1
1
0.5
0.5F
S 0
0
-0.5[
-0.5F
1
-1
-0.5
0
r/I
0.5
-1
-0.5
0
0.5
1
/i
Figure 7-1: Streamlines for - = 0.2
and # = 0.39
Figure 7-2: Streamlines for -= 0.6 and 3 = 1.32
300
5
4-
-
- -
250
.....
-- E
200
3
-150F
100
2
1
50
.
'0
-
n
0
0.2
0.4
a
0.6
0.8
0.2
0.4
(F
0.6
0.8
Figure 7-4: The variation of the propagation velocity, U, against the free parameter,
a.
Figure 7-3: The aspect ratio, /, the circulation, F, the impulse I and the energy E
against the free parameter, a-
182
0.5,
01
-0.5,
-1.5
0.5
-0.5
0
0
0.5
-0.5
1.5
Figure 7-5: Three-dimensional vortex ring structure of vorticity.
1.
Norbury-Fraenkel
Simulation()
(c)
()
Kaplanski-Rudi
P resent
Solution
(g)
1.4
1.2
0.6
0.05
02.0
1.6
00.
Simulation
Norbury-Fraenkel
(b)
12
-0.6
0.4
02
00.1
(d)
Kaplanski-Rudi
(f)
V#VyAx
1.4.11U
Present Solution
# Vmx
(h)
-0.3
r
L i0)
T
5
10.5
11
X11.6
12
12.
10.5
11
11.5
12
12.510.S
11 X
11.5
12
Figure 7-6: Comparison of the ideal models of Norbury-Fraenkel, Kaplanski-Rudi, and the
present solution with the direct numerical simulations by[7]. The upper row is the vorticity
distribution and the lower is the streamline. (a)-(f) modified from [7].
183
1.5
(b)
1.5
(a)
I
1
---simulation by DH'
-Present
model
---simulation by DH
model
-present
0.5[
0.5[
0
10.5
11
x
11.5
12.5
12
0
10.5
12.5
12
11.5
11
x
Figure 7-7: Comparison of boundary contours with DH: (a) the vortex core and (b) vortex
bubble.
-Simulations
of DH
4
MRC1
1
5 - -- Norbury-Fraenkel
OMRC2
MRC3
0
- Kaplanski-Rudi
-I Present model
0. 8-
*MRC4
.ATC1
,p 3F
-!
4kI
0. 6,
6
0. 4-
.model
0. 2
2
0
.
*ATC2
DH
Present
.
U000.:
-
1
0
0.2
0.4
0.6
0.8
1
1.2
1
Figure 7-8: Radial distribution of vorticity through the vortex center: conmparison to different vortex ring models
184
0.5
1
r/l,
1.5
2
ovrtcity through the vortex center: comparison to different numerical simulations in
literature. Legend entries can be found
in table 7.1.
Figuire 7-9: Radial ditrbio
1.4
1.2[
-present
-Zhang
model
and Danaila (2013)
1
E
0.8
0.6
0.4
0.2
0
C
0.5
r/IR1
1
1.5
Figure 7-10: Radial distribution of the stream function through the vortex center compared
to [27].
185
Table 7.1: Selected numerical results in the literature
Abbreviation
MRC1
MRC2
MRC3
MRC4
ATC 1
ATC2
DH
Literature
Mohseni et al.(2001)
Re
3800
Archer et al.(2008)
5500
Danaila and Helie (2008)
1400
186
x/D
3
5.8
12.4
19.9
11.35
Chapter 8
Summary and Example for
Sediment-Disposal Field Operations
8.1
Summary
In chapters 2 to 6, single- and multiphase LES are applied to study starting buoyant jets
with and without particles. These methods are implemented in two different computational
codes. Their performance is validated comparing to lab experiments. From numerical
results, the physics of starting jets are illustrated, and corresponding theoretical models are
developed to explain the observation.
8.1.1
Large-Eddy Simulation of Starting Buoyant Jets
A numerical study using the LES approach has been conducted to investigate the penetration behavior of starting buoyant jets during the Period of Flow Development, which is the
beginning stage of the jet penetration. The behaviors of the two asymptotic cases of a pure
jet and a lazy plume are first reproduced to validate the numerical code. The corresponding
transient simulations are also consistent with the experiments reported previously. After the
validation, the model is used to simulate starting turbulent buoyant jets with three differ187
ent Reynolds numbers from 2000 to 3000 and a wide range of buoyancy effects from pure
jets to lazy plumes. The penetration front generally advances faster with higher buoyancy.
More importantly, the penetrative distances induced by the initial buoyancy fluxes and by
the initial momentum fluxes are found to be independent; therefore, the total penetration
distance can be treated as a linear combination of these two parts.
8.1.2
Buoyant Formation Number of A Starting Buoyant Jet
The same code is used to study the formation process of buoyant jets over a wide range
from a pure jet to a plume. To determine the formation number, a method to differentiate the head vortex ring and the trailing stem is derived based on the evolution of the
cross-sectional circulation. Numerical simulations are performed by adding buoyancy flux
incrementally to simulations with two turbulent source conditions of Re=2000 and 2500.
Results show that the buoyant formation number increases with the Richardson number
following an error function relationship. In addition, since the momentum and buoyancy
effects are uncoupled at the initial development of the buoyant jet, there is a simple and
reliable method to determine the formation number. Finally, with a slug model that incorporates the buoyancy effect as well as the Kelvin-Benjamin variational principle, the
buoyant formation number is shown to have a non-dimensional energy of 0.33, which is
consistent with the value for pure jets that was reported in the literature.
8.1.3
Pinch-off and Formation Number of Negatively Buoyant Jets
The same numerical method is also extended to explore the formation processes of negatively buoyant starting jets using LES. As validation of the numerical approach, the simulated penetration heights for submerged fountains are shown to compare favorably with
existing experimental data. The analytical model for the penetration of positively buoyant
starting jets was found to also apply to weakly negatively buoyant starting jets. The range
of agreement between theory and simulation corresponds approximately with the range
188
over which pinch-off is possible.
From numerical results, we observe that with stronger negative buoyancy, pinch-off
does not occur, and the link between the head vortex and the stem remains at all time.
For the range of weak negatively buoyant jets through positively buoyant jets, the buoyant
formation number increases monotonically with Rid. A unified physical map of vortex
circulation and penetration is proposed, allowing several interpretations of vortex dynamics. Finally, the modified slug model based on the Kevin-Benjamin variation principle, is
found to be applicable for weakly negatively buoyant jets, but to break down for strongly
negatively buoyant jets.
8.1.4
LES Study of Settling Particle Cloud Dynamics
In multiphase simulations, a new numerical scheme with LES is developed for two-phase
simulations to address three specific issues related to the settlement of particle clouds in
the water column: the two-phase separation, the effect of the initial cloud shape and polydispersion dynamics. Three series of simulations are performed. The first series validates
the multiphase numerical approach. The second series provides quantitative results on the
monodispersion dynamics including phase separation, the penetration and growth of the
particle cloud, and the entrainment rate and deposition patterns. Finally, the last series clarifies the polydispersion dynamics. With proper scaling, we obtain best-fit equations of the
present numerical results. They show that the initial aspect ratio plays a weak role, being
more important in penetration than in growth, and it only affects the behavior before phase
separation.
8.1.5
Scaling Particle Cloud Dynamics - From Lab to Field
Traditionally, the Cloud number scaling method has been used to extrapolate lab-scale
experimental results to field scale for engineering purposes, but this method has not been
validated. We compared two possible methods to scale the dynamics of settling particle
189
clouds. By numerical simulations, the Cloud number scaling is found to be appropriate,
confirming the conventional methodology.
8.1.6
An Axisymmetric Steady State Vortex Ring Model
Finally, a family of steady state vortex rings are derived analytically to explore the driving mechanism behind the starting jet. This family of models is inspired by a solution we
obtained to the Grad-Shafranov equation from magnetohydrodynamics. Using the analogy
between the equations of hydrodynamics and magnetohydrodynamics, a theoretical model
for high Re laminar vortex rings is developed based on this solution, and can satisfy the
specified boundary conditions that are commonly used in magnetohydrodynamics and suitable for vortex rings. A new family of theoretical vortex rings is derived based on this
model using a free parameter or. A numerical method is then used to calculate the properties of this family, including aspect ratio, circulation, impulse, energy, and propagation
velocity. By choosing a = 0.585, a good match is obtained comparing the results from the
present model and direct numerical simulations. The present model is found to be superior
to the Norbury-Fraenkel family by virtue of its improved accuracy in the vorticity distribution and streamline pattern. Most importantly, the asymmetry and elliptical outline of the
vorticity profile are well captured.
8.2
Suggestions for sediment disposal in field operations
The present study is aiming to reduce the quantity of sediment lost in land reclamation and
dredged material disposal. Through our studies, we found two methods that can help us
achieve this goal: 1) delaying the phase separation, and 2) getting rid of the trailing stem.
These two methods have traditionally been neglected, because the dynamics behind the
phenomenon were not clear.
From an engineering perspective, a set of guidelines can be developed following these
190
two principles and the benefits are multiple. First, if the self-circulating pattern of the settling particle cloud could be sustained for a longer time, the fluid phase would encapsulate
and accelerate the particles for a longer distance. As a result, a faster settlement leaves
less chance that the sediments would be washed away by ambient turbulence, currents, or
waves, and thus less sediment dispersion; the strong coupling flow in the thermal stage
(see the definition in chapter 5) could help particle clouds to resist the ambient disturbance;
and the quicker settling reduces the interval time between each individual releases allowing
more disposal dumps within a given window of time. Second, if the trailing stem striped
from the main cloud can be avoided, the sediment would not disperse into the environment
as a result of the loose structure of the trailing stem (see chapter 3 and 4 for more details).
According to the conclusion from chapter 5, there are at least two ways to delay the
phase separation: increasing the Rayleigh number Ra =
BR2
or the aspect ratio A
of the initial release. According to its definition, to enhance Ra requires a greater total
buoyancy B, or a smaller ambient density Pa, settling velocity w, or initial release radius
Ro. For a particular task, the ambient water density and the sediment size are fixed, which
dictates the values of pa and w. So the only adjustable parameters are B and R 0 , which
together ask for a heavier load to the release and a smaller opening at release. For dredging
machine providers, it is recommended to design dredge buckets and barges with smaller
openings and slender shapes to optimize Ro and A.
However, the enhancement might be limited by various conditions. First, a fast sinking
turbulent cloud hitting the bottom may cause a non-trivial surge, scouring bottom sediment
or jeopardizing the integrity of bottom capping. Second, if Ra is too high (e.g. Ra >
1000), particles "clump" are produced to form, destroying the particle cloud coherence and
causing greater sediment dispersion into the environment [1]. To summerize, it might be
optimal that the thermal phase ends at the bottom of the water to fully make use of the
thermal phase while Ra is still lower than 1000. When both situations cannot be satisfied,
a compromise between the operation efficiency and environmental impact should be drawn
191
depending on the specific disposal plan.
Using these principles, an optimization procedure can be outlined: 1) Given the average
settling velocity we, which can be derived from the average particle grain size dp, and letting
the phase separation height equals to the water depth, i.e. Hp = H, we can obtain the phase
separation time Tp = Hp /(2.4w,) according to equation (5.22). 2) The equivalent radius
of the release mechanism opening, RO, can be extracted from the device design. 3) The
volume of each release can be calculated from the equation
1.38 poTp2 W4 V
=4w
[13V 0
And 4) check that Ra
B
8
g (pp - PO)
< 1000 and A
1
2/3
(8.1)
0
< 6.5 so that no extra sediment loss
=
mechanisms are possible.
An example involving representative parameters is provided following the above procedures. We assume a barge of sediment is released by a dredge bucket with an opening
of R = 0.8m in radius. The settling velocity ws is 7.1 cm/s corresponding to the particle density of p, = 2.5g/cm3 and the average particle diameter dp
that the dumping site has a water depth of H
=
=
0.5 mm. Assume
31m and the phase separation height
H, = H. Therefore, we can obtain the phase separation time Tsp = 182s according to
equation (5.22), which is also the time that the particle cloud reaches the bottom. To derive
the optimal amount of sediment to be released by the bucket, we can use the equation (8.1),
so the total volume in each release is 0.22 M 3 . To avoid cloud disintegration loss due to
instability, we have to check the Rayleigh number Ra = 982 < 1000, so the present design
satisfies the basic requirement.
At last, the aspect ratio is checked to be lower than the formation number according to
the conclusions from chapter 3 and 4. In the field, the aspect ratio is not always available
and might be difficult to estimate by experience. As a rough guess, the aspect ratio for a
slip hull barge might be the ratio of the total release volume over the cubic of the opening
192
length scale, while a grab bucket's initial aspect ratio might be A = UentTent/RL, where
Ugrit is the velocity that the particles enter the water, Tent is the period that all the particles
enter the water, and RL is the opening length scale. In the above example, assuming the
initial velocity is negligible, so the Richardson number being infinitely large, we have to
use the asymptotic formation number of ~ 6.5 (see chapter 3 for more details). The aspect
ratio in the above example is 0.13, which is smaller than this formation number, so the
present aspect ratio meets the requirement.
In field operations, the ambient flows, including currents, waves, and turbulence, may
further complicate the particle cloud dynamics and the applications of the derived principles. Some previous studies have focused on these different effects. For example, the
current effect has been studied by [2], who found that a strong current beyond a certain
threshold could destroy the particle cloud coherence, whereas the wave effect, investigated
by [3], cannot significantly change the dynamics within a practical range. In other words,
the particle cloud solution derived in this thesis is still valid to a weak current and in practical wave conditions if corrections of passive current and wave transport are superposed.
But the solution would be invalid if a strong current was present. In addition, a series of
studies focusing on the turbulence effect is under investigation in our group. We are measuring the particle cloud dynamics within a homogeneous and isotropic turbulent flow. We
expect that the ratio of turbulence intensity over the particle cloud settling velocity may be
the key controlling parameter. More results will be available in the near future.
193
194
Bibliography
[1] B. Zhao, A.W.K. Law, E.E. Adams, and J.W. Er. Formation of particle clouds. Journal
of Fluid Mechanics, 746:193-213, March 2014.
[2] RJ Gensheimer, EE Adams, and AWK Law. Dynamics of particle clouds in ambient currents with application to open-water sediment disposal. Journal of Hydraulic
Engineering, 139(2):114-123, February 2012.
[3] Bing Zhao, Adrian W. K. Law, Zhenhua Huang, E. Eric Adams, and Adrian C. H.
Lai. Behavior of Sediment Clouds in Waves. Journalof Waterway, Port, Coastal,and
Ocean Engineering, 139(1):24-33, January 2013.
195