CFD Analysis of the Mixing Patterns Produced by a
Turbulent Steam Jet in a Subcooled Water Pool
by
Charles Slayden
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Thesis Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2013
© Copyright 2013
by
Charles Slayden
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF SYMBOLS ........................................................................................................ ix
LIST OF ACRONYMS ................................................................................................... xii
ACKNOWLEDGMENT ................................................................................................ xiii
ABSTRACT ................................................................................................................... xiv
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Problem Description........................................................................................... 6
2. Methodology and Theory ............................................................................................ 7
2.1
2.2
CFD Models ....................................................................................................... 7
2.1.1
Turbulence.............................................................................................. 7
2.1.2
Level Tracking ..................................................................................... 13
Steam Condensation Region Model: Free Jet .................................................. 14
2.2.1
Penetration Length ............................................................................... 15
2.2.2
Jet Width/Velocity Profile.................................................................... 17
2.2.3
Jet Temperature Profile ........................................................................ 20
2.2.4
Fluid Conditions ................................................................................... 21
3. Model Development .................................................................................................. 23
3.1
Geometry .......................................................................................................... 23
3.2
Model Inputs and Boundary Conditions .......................................................... 28
3.2.1
Jet Parameters....................................................................................... 28
3.2.2
Buoyancy ............................................................................................. 39
3.2.3
Wall Parameters ................................................................................... 51
3.2.4
Volume of Fluid Parameters, Initialization, and Solver Options ......... 52
4. Results........................................................................................................................ 55
iii
4.1
4.2
Case 1 Results .................................................................................................. 55
4.1.1
Benchmark Results............................................................................... 56
4.1.2
Effects of Free Surface Level and Energy Models on Results ............ 58
4.1.3
Independence Study ............................................................................. 63
Case 2 Results .................................................................................................. 64
4.2.1
Benchmark Results............................................................................... 64
4.2.2
Effects of Free Surface Level and Energy Models on Results ............ 68
4.2.3
Independence Study ............................................................................. 71
5. Conclusions................................................................................................................ 73
5.1
Conclusions ...................................................................................................... 73
5.2
Suggestions for Future Work ........................................................................... 75
6. References.................................................................................................................. 77
Appendix A: SCRM Detailed Calculations ..................................................................... 79
Appendix B: FLUENT User Defined Functions ............................................................. 85
Appendix C: 600 kg/m2s Turbulent Parameter Sensitivity.............................................. 91
Appendix D: Closure Coefficient Study .......................................................................... 93
iv
LIST OF TABLES
Table 2-1: Closure Coefficients ....................................................................................... 10
Table 2-2: SST k-ω Closure Coefficients ........................................................................ 12
Table 2-3: “B” Definitions [7] ......................................................................................... 16
Table 2-4: Dimensionless Velocity Components [11]..................................................... 19
Table 2-5: Dimensionless Temperature Components [11] .............................................. 21
Table 3-1: JICO Test Conditions [8] ............................................................................... 23
Table 3-2: Fluid Properties Used in Calculations [16] .................................................... 25
Table 3-3: Geometry Dimensions Used .......................................................................... 26
Table 3-4: Boundary Conditions Used ............................................................................ 28
Table 3-5: Final Turbulent Intensity and Length Scales Selected ................................... 34
Table 3-6: Water Density................................................................................................. 50
Table 3-7: Solver Options ................................................................................................ 54
Table 3-8: Under-Relation Factors .................................................................................. 54
Table 4-1: Model Comparison Locations ........................................................................ 59
v
LIST OF FIGURES
Figure 1-1 - AP1000 RCS and Passive Core Cooling System [1] ..................................... 2
Figure 1-2 - JICO Experimental Facility [8] ..................................................................... 5
Figure 2-1: Flow Regimes Observed in Pool Mixing ........................................................ 8
Figure 2-2: Steam Jet Control Volume ............................................................................ 15
Figure 2-3: Submerged Axially Symmetric Jet [11]........................................................ 17
Figure 3-1: Tank and Nozzle Geometry [8] .................................................................... 24
Figure 3-2: Schematic of Point Source ............................................................................ 26
Figure 3-3: Design Modeler Image.................................................................................. 27
Figure 3-4: Boundary Conditions .................................................................................... 27
Figure 3-5: Velocity Profiles at Jet Exit .......................................................................... 29
Figure 3-6: Model Used to Determine Turbulent Parameters ......................................... 32
Figure 3-7: Case 1 Sensitivity to Turbulent Intensity (60mm axially) ............................ 33
Figure 3-8: Case 1 Sensitivity to Turbulent Intensity (80 mm axially) ........................... 34
Figure 3-9: Case 1 Sensitivity to Turbulent Intensity (100 mm axially) ......................... 35
Figure 3-10: Case 1 Sensitivity to Length Scale ............................................................. 36
Figure 3-11: Velocity Profile Isothermal v. Nonisothermal ............................................ 36
Figure 3-12: Temperature Profiles at Axial Location: 60 mm ........................................ 37
Figure 3-13: Temperature Profiles at Axial Location: 80 mm ........................................ 38
Figure 3-14: Temperature Profiles at Axial Location: 100 mm ...................................... 38
Figure 3-15: Schematic of Experiment in [17] ................................................................ 40
Figure 3-16: Grid and Initial Geometry of Buoyancy Benchmark .................................. 41
Figure 3-17: Non-Dimensional Velocity v. Non-Dimensional Position ......................... 43
Figure 3-18: Non-Dimensional Temperature (-) vs. Non-Dimensional Position (-) ....... 44
Figure 3-19: Non-Dimensional Velocity vs. Non-Dimensional Position in the Boundary
Layer ................................................................................................................................ 44
Figure 3-20: Non-Dimensional Temperature vs. Non-Dimensional Position in the
Boundary Layer ............................................................................................................... 45
Figure 3-21: SST+ Average Thot Y-Velocity vs. Time in Boundary Layer ..................... 46
Figure 3-22: Buoyancy Benchmark, Grid Independence ................................................ 47
vi
Figure 3-23: Velocity Grid Independence Study ............................................................. 47
Figure 3-24: Temperature Grid Independence Study ...................................................... 48
Figure 3-25: Velocity Timestep Independence Study ..................................................... 49
Figure 3-26: Temperature Grid Independence Study ...................................................... 50
Figure 3-27: Boundary Conditions Marked as "Wall" .................................................... 51
Figure 3-28: Illustration of Roughness Height [9] .......................................................... 52
Figure 3-29: Initial Model with Initialized Level ............................................................ 53
Figure 4-1: Local Flow Patterns Presented in [8] ............................................................ 55
Figure 4-2: Case 1 PIV [8] and CFD Results (at 30 seconds) ......................................... 56
Figure 4-3: Case 1 Location B Comparison .................................................................... 57
Figure 4-4: Case 1 Location A Comparison .................................................................... 58
Figure 4-5: Model Comparison Locations ....................................................................... 59
Figure 4-6: Case 1, Location 1 Model Effects ................................................................. 60
Figure 4-7: Case 1, Location 2 Model Effects ................................................................. 61
Figure 4-8: Case 1, Location 3 Model Effects ................................................................. 61
Figure 4-9: Case 1, Location 4 Model Effects ................................................................. 62
Figure 4-10: Case 1 Velocity Contours Mesh/TS Study ................................................. 63
Figure 4-11: Case 1 Velocity Vector Comparison .......................................................... 64
Figure 4-12: Case 2 PIV [8] and CFD Results (at 30 seconds) ....................................... 65
Figure 4-13: Case 2 Location B Comparison .................................................................. 66
Figure 4-14: Case 2 Location A Comparison .................................................................. 67
Figure 4-15: Case 2 Location C Comparison .................................................................. 67
Figure 4-16: Case 2, Location 1 Model Effects ............................................................... 68
Figure 4-17: Case 2, Location 2 Model Effects ............................................................... 69
Figure 4-18: Case 2, Location 3 Model Effects ............................................................... 69
Figure 4-19: Case 2, Location 4 Model Effects ............................................................... 70
Figure 4-20: NonIsothermal VOF/NonVOF Secondary Flow Comparison .................... 71
Figure 4-21: Case 2 Velocity Contours Mesh/TS Study ................................................. 71
Figure 4-22: Case 2 Velocity Vector Comparison .......................................................... 72
Figure 5-1: Improved Entrainment Assumption .............................................................. 76
Figure C-1: Case 2 Sensitivity to Turbulent Intensity (60mm) ....................................... 91
vii
Figure C-2: Case 2 Sensitivity to Turbulent Intensity (80mm) ....................................... 91
Figure C-3: Case 2 Sensitivity to Turbulent Intensity (100mm) ..................................... 92
Figure D-1: Turbulent Intensity Outer Parametric at 80 mm .......................................... 95
Figure D-2: Turbulent Intensity Inner Parametric at 80 mm ........................................... 95
Figure D-3: Velocity Contour Comparison for Case 1 @ 30 seconds ............................ 96
viii
LIST OF SYMBOLS
Symbol
Definition
Unit
a
Area
m2
ac
Condensed Area
m2
ae
Entrained Area
m2
as
Nozzle Area
m2
B
Jet Penetration Coefficient
-
C
Jet Penetration Coefficient
-
cp
Pressure Coefficient
-
D
Diameter
m
Dw
Cross Diffusion Term
-
E
Energy
J
F1
SST k-ω Blending Function
-
f
Subscript, fluid
-
fg
Subscript, at Saturation
-
g
Subscript, gas or gravity
- or m/s2
G
Mass Flux
kg/m2*s
Gk
Generation of k Due to Mean Velocity Gradient
kg/m*s3
Gm
Critical Mass Flux (275)
kg/m2*s
Gw
Generation of ω Due to Mean Velocity Gradient
kg/m*s3
H
Height
Meters
h
Enthalpy
k
Kinetic Energy
L
Length
Mp,g
kJ/kg*K
m2/s2
m
Mass Transfer Between Phases
kg/s
ṁ
Mass Flow Rate
kg/s
Pe
Entrained Pressure
psia
Pc
Condensed Pressure
psia
Po
Bulk Pressure
psia
Ps
Steam Pressure
psia
ix
Symbol
Pr
Sk,w
Definition
Unit
Turbulent Prandtl Number
-
User Defined Turbulence Source Terms
-
t
time
Seconds
T
Temperature
K
Tm
Midline Temperature
K
Tsat
Saturation Temperature
K
Tbulk
Bulk (or freestream) Temperature
K
u
Velocity
m/s
uc
Condensed Velocity
m/s
ue
Entrained Velocity
m/s
um
Mean Velocity
m/s
us
Steam Velocity
m/s
u’
Fluctuating Velocity
m/s
u*
Friction Velocity @ Wall
m/s
um
Midline Velocity
m/s
y
Distance
m
Yk
Dissipation Source
-
𝑢̅
Mean Velocity
m/s
β
Thermal Expansion Coefficient
1/K
β1, β2, β*
Turbulent Closure Coefficients
-
η
y/x Substitution
-
θ
Normalized Temperature
-
μ
Dynamic Viscosity
Pa*s
ν
Kinematic Viscosity
m2/s
ρ
Density
σk1, σk2, σω1,
σω1
kg/m3
Turbulent Closure Coefficients
-
τ
Reynolds Stress Tensor
lb/ft*s
φ
Dimensionless Location
-
ϕ1
Factor for Blending Function
x
Symbol
Definition
ψ
Stream Function
ω
Specific Dissipation Rate
Г
Effective Diffusivity
ΔT
Unit
1/s
m2/s
Temperature Difference
K
xi
LIST OF ACRONYMS
Acronym
Definition
BWR
Boiling Water Reactor
CFD
Computational Fluid Dynamics
CMT
Core Makeup Tank
DCC
Direct Contact Condensation
LDA
Laser Doppler Anemometer
PIV
Particle Image Velocimetry
PWR
Pressurized Water Reactor
RANS
Reynolds Averaged Navier-Stokes
RCS
Reactor Coolant System
SCRM
Steam Condensation Region Model
SST
Shear Stress Transport
UDF
User-Defined Function
VOF
Volume of Fluid
xii
ACKNOWLEDGMENT
I’d like to first thank my advisor, Dr. Ernesto Guitierrez-Miravete, for providing
guidance and direction when it was needed at any time during the day. I’d also like to
thank my parents, Gary and Diane Slayden, for nurturing an environment of learning and
discovery. Without them I would not be where I am today. Lastly, and definitely not
least, I would like to thank my girlfriend, Caitlin. Without her support and
understanding, I would have surely not been able to finish this work and probably would
have starved attempting so. Thank you for putting up with me and my endless nights of
babbling about case runs and meshes.
xiii
ABSTRACT
Over the past ten years, much research has been performed to predict the nature
of a steam jet injecting into a subcooled pool in the direct area of the jet (micro-scale). A
popular approach to characterizing the steam jet is through the use of the SCRM
developed by Kang and Song. Their model combines experimental correlations of
penetration length, mass, momentum and energy balance, along with turbulent jet theory
to come up with a boundary condition that can simulate a condensing steam jet. Much
study has been performed to determine the adequacy of this theory in the micro-scale but
no study has been performed to determine the adequacy of the model in the macro-scale;
or the interactions of the condensing steam jet in the overall tank.
This thesis investigated the mixing patterns produced by a turbulent steam jet
condensing into a subcooled pool by CFD with Fluent 14.0. The SCRM was used to
model the behavior of the steam jet in the subcooled pool. Variable mass flux, steam
temperature, and nozzle pressure for two cases (300 and 650 kg/(m2*s)) was
benchmarked to the experimental data by Choo [8].
It was determined that the SCRM could be used to characterize the flow patterns in
the subcooled pool for the wall bounded and separated flow. It was able to predict the
behaviors of Choo [8] for these regions. However, the area of entrainment near the
nozzle, the jet width/penetration away from the nozzle, and the transition from a
turbulent jet to an impinging jet was overpredicted by the use of the SCRM. Therefore, it
can be concluded that the use of the SCRM should be further refined to better predict the
transition from a turbulent jet to an impinging jet. However, the SCRM can be used to
develop flow behaviors away from the jet, such as the wall bounded flow, areas of
circulations, and the introduction of the secondary flow, which was predicted by Fluent
and observed in Choo [8].
Keywords: CFD, Steam Jet, SCRM, Tank Mixing, Condensation, turbulent jet,
subcooled pool
xiv
1. Introduction
1.1 Background
Many advanced nuclear power plants recently developed have adopted a form of
steam discharge into a subcooled water pool as a safety related system. The Boiling
Water Reactors (BWRs) have used suppression pools for decades to help mitigate design
basis accidents where the reactor coolant leaking from the vessel flashes to steam and is
directed through vent pipes to the subcooled pool to condense; thereby limiting the
pressure increase within containment. Recently, Pressurized Water Reactors (PWRs),
such as the Westinghouse AP1000, have been using similar techniques to provide a way
to depressurize the plant in case reactor core cooling is needed through safety injection.
Figure 1-1 shows the general layout of the Westinghouse AP1000 Reactor Coolant
System (RCS) and Passive Core Cooling System. When passive safety injection through
the Core Makeup Tanks (CMTs) is required, the depressurization valves off the
pressurizer will actuate and begin to slowly depressurize the system. The high pressure
steam is then discharged through the sparger into the In-Containment Refueling Water
Storage Tank; which is designed for atmospheric pressure [1].
The Nuclear Energy Agency Committee on the Safety of Nuclear Installation
released a document for the extension of Computational Fluid Dynamics (CFD) codes
application to safety problems [2]. The document’s goal is to select a limited number
nuclear reactor safety issues that could benefit from two-phase CFD analysis and
provide data to the community that could be beneficial in solving the identified issues.
The phenomenon of the steam discharge into a subcooled pool is one of them.
According to [2], there are least two kinds of technical concerns during the actuation
of these steam discharging devices. The first is the concern on the thermo-hydraulically
induced mechanical loads on the structures of systems. These are induced by the direct
influence of the steam discharge or by the thermo-fluid processes of the
expansion/contraction of large air bubbles or pressure oscillation from the unstable
condensation of discharged steam in a water pool. The other concern is on the thermal
mixing in the subcooled water pool.
1
Figure 1-1 - AP1000 RCS and Passive Core Cooling System [1]
The initial steam mass is quickly condensed through direct contact condensation
(DCC) when it comes in contact with the pool water. When the local pool temperature
increases from continued steam discharge, pressure oscillations may cause large
mechanical loads on the pool wall from unstable condensation. In order to properly
predict the local pool temperature from the continued steam discharge, in general,
transient three dimensional analyses is required since the temperature is depended on the
turbulent mixing and/or possibly on the gravity-driven circulation with density
stratification [3].
Kang and Song [3-6] have done extensive research over the past few years on the
behavior of the steam jet and the thermal mixing of the subcooled pool induced by a
steam jet. Their work has culminated in a Steam Condensation Region Model (SCRM)
in which steam is perfectly condensed to water within the steam penetration length [4].
The results of the Kang and Song’s work show that the SCRM model appears to be
2
adequate to model the turbulent jet caused by the steam jet injection within an error rate
of 10% [6].
Kang and Song in [4] simulated the thermal mixing between a high steam mas
flux discharge and a subcooled pool to develop the methodology for numerical analysis
of the phenomena. Test conditions for the blowdown and condensation test facility,
which considered mass fluxes ranging from 250 to 1600 kg/m2s, was used to benchmark
the methodology. The test facility used a side discharge sparger consisting of 64 holes
and one bottom mounted hole with a tank measuring 3 meters in diameter and 4 meters
in height.
Kang and Song [4] developed a lumped SCRM to model the discharge holes into
a single boundary condition with uniform velocity and temperature conditions. Transient
cases were run using CFX and the results compared to the temperature readings of the
blowdown and condensation test facility. The comparison of the results show good
agreement to the test data within 7-8%. They also noticed a difference in the temperature
distribution at the upper and lower region where the condensed water jet comes back
after colliding with the tank wall. Kang and Song attributed these differences to the use
of the SCRM.
The SCRM was further developed in [5] when Kang and Song analyzed the
turbulent jet behavior induced by a steam jet discharge through a single side discharge.
The test at the GIRLS test facility was used to benchmark the results of the numerical
analysis. During the discharge tests, the temperature, pressure, and mass flow rates of the
steam were measured. The pool water temperature was estimated by averaging the
temperature signals from 6 thermocouples [5]. The tests considered 1,000 kg/m2s mass
flux injection.
To model the turbulent jet induced by the condensing steam, Kang and Song [5]
improved the SCRM from what they previously developed in [4]. Tollmien’s theory for
a submerged axisymmetric turbulent jet was utilized for the boundary conditions of the
SCRM. This allowed a detailed velocity and temperature profile to be declared along the
boundary condition rather than the previously assumed uniform profiles. They also
developed an entrainment model to better characterize the jet physics. Parametric studies
were performed to determine the appropriate coefficients needed to fully use Tollmein’s
3
theory. Additional parametrics were performed on the turbulent intensity at the jet
boundary to determine the sensitivity to the parameter.
Their results showed a dependence on the turbulent intensity at the inlet region of
the turbulent jet region and suggested that higher turbulent intensity would enlarge the
jet width by a momentum diffusion process along the radial direction [5]. They
concluded that the SCRM predicted the velocity differences in the radial direction within
10% when considering the entrainment model.
Kang and Song further developed the SCRM in [6] with GIRLS tests cases through
a vertical upward single hole in a subcooled water pool. The previous work in [5] using
Tollmein’s theory and the coefficients determined was extended under a vertical steam
discharge condition. The CFD experiments performed predicted the test results within
10% and reinforced the conclusion of the important of the turbulent intensity at the
boundary condition of the simulated turbulent condensing jet.
Moon [7] utilized a form of the SCRM to benchmark several experiments,
including local phenomena tests, cylindrical and annulus pool simulations. Moon
performed parametric studies on the penetration length correlations and constants from
several authors (Kerney, Weimer, Chun, Kim and Wu). It was concluded that the work
of Kim closely emulated the penetration lengths of a wide range of different steam mass
fluxes (150 to 750 kg/m2s) and pool temperatures (30 to 75°C). Moon [7] utilized a
lumped SCRM to simulate a multihole sparger and assumed a constant velocity and
temperature profile at the boundary of the penetration length.
The temperatures at several locations through the cylindrical and annulus tank
were compared to test data in [7]. He concluded that the SCRM overestimated the
temperature throughout the tank, especially around the sparger. However, the error
decreased farther away from the sparger. Moon attributed this difference to the
differences between the test and simulation conditions in the tank. Only a comparison of
the fluid temperature was performed and not of the fluid velocity patterns within the
tank.
In fact, no study has been performed to benchmark experimental data on mixing
against CFD predicted mixing profiles using the SCRM model within a subcooled pool.
One possible reason for this is the intrusive nature of many of the common measurement
4
techniques used to determine the mixing patterns. However, Choo [8] utilized a particle
image velocimetry (PIV) as a non-intrusive optical measurement technique to visualize
the mixing patterns in a subcooled water tank with steam jet discharge in the JICO
experimental facility (shown in Figure 1-2).
The JICO facility consists of a test section and an electric boiler to supply steam.
Two tests were performed in [8]; one turbulent jet experiment and a pool mixing
experiment. For the turbulent jet experiment the steam discharge nozzle was installed at
the bottom of the tank and oriented in the vertical direction. For the pool mixing
experiment, the nozzle was installed at the upper part of the pool with a downward
facing injection. For the pool mixing, downward injection is more advantageous when
compared with an upward one for inducing a strong internal circulation flow in the pool.
[8]. Choo performs experiments on pool mixing for three different steam mass fluxes
(300, 450, and 600 kg/m2s) and presents the results as a series of images depicting the
internal circulation patterns for different steam discharge flow rates.
Figure 1-2 - JICO Experimental Facility [8]
5
Choo concluded in [8] that the turbulent jet entrains the surrounding water and
creates an internal circulation pattern within a pool which then governs the content of the
pool mixing. The pool mixing behavior showed a strong internal driving flow with
increasing mass flux and the existence of a secondary flow with higher steam mass
fluxes. These results could be used as benchmarking data for the validation of CFD
simulations.
1.2 Problem Description
The topic of jet behavior of the steam discharge has been extensively researched
over the past few years. The SCRM [3 to 6] model has been developed and proven to
adequately model the behavior of the steam jet within a subcooled pool at the mesoscale. However, limited research has been performed that utilizes this model within CFD
to investigate the behavior of the flow pattern in the subcooled pool, or the macro-scale,
created by the steam discharge. Therefore, this research will investigate the flow patterns
created by the steam jet in a subcooled pool.
A CFD model will be developed using Ansys Fluent 14.0 [9] that will simulate the
experimental conditions used in [8]. Several Fluent internal models will be used to
characterize the turbulent flow and level tracking. To model the turbulent jet caused by
the condensing steam jet, the SCRM methodology developed by Kang and Song [3 to 6]
will be used. The improved SCRM methodology utilizes turbulent jet theory, such as
Tollmein’s Theory [11] to develop velocity and temperature profiles at the boundary
conditions.
The objective of the work presented in this thesis will be to benchmark the results
of [8]. These results will be a set of internal circulations based on steam mass flow rate
within the pool that can be compared to the results of [8] to farther validate the use of the
SCRM to predict conditions in a subcooled pool under the effects of a steam jet.
6
2. Methodology and Theory
The following sections will develop the theory and methodology used in the
development of the model of this paper. The sections will be subdivided into the
individual sub-models used within this analysis. ANSYS Fluent 14.0 [9] was selected as
the CFD program as it provides the models required to properly model the phenomena in
question (i.e., Volume-of-Fluid (VOF) Interface Tracking, k-ω SST, etc).
2.1 CFD Models
To properly characterize the flows that are experienced in the subcooled pool
during steam jet injection, several different models are required. To model the unsteady
nature of the flow through the tank leading to increased flow mixing, a turbulence model
is required that would capture the fluid to wall interactions as well as buoyancy and the
steam jet effects in the bulk fluid. The model chosen and the theory behind it will be
presented in Section 2.1.1.
Additionally, the level of water in the experiment [8] is only 60 mm from the
steam discharge nozzle. The interaction of the level of the water when the steam
discharge nozzle is engaged may play an important in the development of the circulation
within the pool. To characterize the interaction, a two-phase (water and air) immiscible
fluid level tracking model is required. The model chosen and the theory behind it will be
presented in Section 2.1.2.
Lastly, a model that will characterize the energy and velocities of the steam entering
the system through the steam nozzles is required. Research have shown that the SCRM
model of [3 through 6] have been popular and accurate to characterize these flows. With
this model, combined with turbulent jet theory of Abramovich [11], a boundary
condition can be developed to simulate the energy and momentum applied to the system
by the steam jet injection. The theory will be presented in Section 2.2.
2.1.1
Turbulence
The nuclear CFD best practice guidelines of [10] states the two-equation Reynolds
Averaged Navier-Stokes (RANS) model family behaves well for configurations at the
first level of complexity including impinging flows, flows dominated by buoyancy
7
leading to mixed or natural convection, and “canonical configurations” of geometries
such as plane walls. All of which characterize the phenomena in this analysis as seen in
Figure 2-1.
Line of Symmetry
Free Surface
Condensing Jet
Entrained Flow
Turbulent Jet
Wall Jet
Impinging Jet
Pool Wall
Figure 2-1: Flow Regimes Observed in Pool Mixing
The Shear-Stress Transport (SST) k-ω two-equation model in Fluent was selected
as the Turbulence model. The SST k-ω model is more accurate and reliable for a wider
class of flows than the standard k-ω model and uses a blending function to utilize the
standard k-ω model near the wall region and the k-ε away from the surface [10]. Many
of the special considerations of [10] utilized several different two-equation turbulence
models and concluded that SST k-ω predicts flows near the wall and in the free stream
well.
8
The RANS method decomposes the Navier-Stokes equations into mean or time
averaged and fluctuating components. The velocity terms yield:
𝑢 = 𝑢̅ + 𝑢′
where 𝑢̅ and 𝑢′ are the mean and fluctuating components, respectively.
Likewise, the other scalar quantities, such as pressure and energy, yield:
𝜙 = 𝜙̅ + 𝜙′
Applying these into instantaneous continuity and momentum equations and taking a time
average yields the ensemble-averaged momentum equations. Writing these equations in
Cartesian tensor form yields [9]:
𝜕𝜌 𝜕(𝜌 𝑢𝑖 )
+
=0
𝜕𝑡
𝜕𝑥𝑖
𝜕(𝜌 𝑢𝑖 ) 𝜕(𝜌 𝑢𝑖 𝑢𝑗 )
𝜕𝑝
𝜕
𝜕𝑢𝑖 𝜕𝑢𝑗 2 𝜕𝑢𝑙
𝜕
+
= −
+
[𝜇 (
+
− 𝛿𝑖𝑗
)] +
(−𝜌 ̅̅̅̅̅̅̅
𝑢′ 𝑖 𝑢′𝑗 )
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜕𝑥𝑗
𝜕𝑢𝑗 𝜕𝑥𝑖 3
𝜕𝑥𝑙
𝜕𝑥𝑗
These equations are known as the “Reynolds-averaged Navier Stokes (RANS)
equations” and are in the same general form as the instantaneous Navier-Stokes
equations with time-averaged values. The additional terms, (−𝜌 ̅̅̅̅̅̅̅
𝑢′ 𝑖 𝑢′𝑗 ) in the
momentum equations, are the Reynolds stresses that represent the effects of turbulence.
Additional information, as follows, is required to make the equation closed. [9]
The SST k-ω provides a two equation approach to compute these Reynolds
stresses. Developed by Menter [12], the model blends the popular standard k-ω
turbulence model near the wall region with the standard k-ε turbulence model, which is
converted to a k-ω formulation, in the bulk fluid. This blending function is designed to
activate the standard k-ω model near the wall region and activate the transformed k-ε
model away from the wall. It additionally incorporates a damped cross-diffusion
derivative term in the ω equation. The definition of the turbulent viscosity also accounts
for the transport of the turbulent shear stress. [12]
9
The k-ω family of models is empirical based on model transport equations for the
turbulence kinetic energy (k) and the specific turbulence energy dissipation rate (ω). The
transport equations for the SST k-ω are as follows [9];
Turbulence Kinetic Energy:
𝜕(𝜌𝑘) 𝜕(𝜌𝑘𝑈𝑗 )
𝜕𝑈𝑖
𝜕
𝜕𝑘
+
= 𝜏𝑖𝑗
− 𝛽 ∗ 𝜌𝑘𝜔 +
[(𝜇 + 𝜎𝑘1 𝜇 𝑇 )
]
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
Specific Turbulence Energy Dissipation Rate:
𝜕(𝜌𝜔) 𝜕(𝜌𝜔𝑈𝑗 )
𝜔 𝜕𝑈𝑖
𝜕
𝜕𝜔
+
= 𝛼 𝜏𝑖𝑗
− 𝛽1 𝜌𝜔2 +
[(𝜇 + 𝜎𝑤1 𝜇 𝑇 )
]
𝜕𝑡
𝜕𝑥𝑗
𝑘
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
Where,

𝜏𝑖𝑗 is the reynolds stress tensor

𝜏𝑖𝑗 𝜕𝑥 𝑖 is the production of turbulence kinetic energy

𝛽 ∗ 𝜌𝑘𝜔 is the dissipation of turbulence kinetic energy

𝛼 𝑘 𝜏𝑖𝑗 𝜕𝑥 𝑖 is the production of specific turbulence energy dissipation rate

𝛽𝜌𝜔2 is the dissipation of the specific turbulence energy dissipation rate

𝛼, 𝛽1 , 𝛽 ∗ , 𝜎𝑘1 , 𝜎𝑤1 are closure coefficients
𝜕𝑈
𝑗
𝜔
𝜕𝑈
𝑗
The closure coefficients are experimental constants determined from benchmarking
experimental tests to the models. The closure coefficients for both Wilcox [13] and
documented in Fluent [9] are in Table 2-1.
Table 2-1: Closure Coefficients
Coefficient
Wilcox [13]
Fluent [9]
α
0.09
0.09
β
0.5
0.5
β*
0.56
0.52
σk1
0.075
0.072
σω1
0.5
0.5
10
To obtain the SST k-ω, the original k-ω model is blended with a transformed k-ε
turbulence model. Menter in [12] transforms the k-ε model into a k-ω formulation in the
form of:
Turbulence Kinetic Energy:
𝜕(𝜌𝑘) 𝜕(𝜌𝑘𝑈𝑗 )
𝜕𝑈𝑖
𝜕
𝜕𝑘
+
= 𝜏𝑖𝑗
− 𝛽 ∗ 𝜌𝑘𝜔 +
[(𝜇 + 𝜎𝑘2 𝜇 𝑇 )
]
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
Specific Turbulence Energy Dissipation Rate:
𝜕(𝜌𝜔) 𝜕(𝜌𝜔𝑈𝑗 )
𝜔 𝜕𝑈𝑖
𝜕
𝜕𝜔
1 𝜕𝑘 𝜕𝜔
+
= 𝛼 𝜏𝑖𝑗
− 𝛽2 𝜌𝜔2 +
[(𝜇 + 𝜎𝑤2 𝜇 𝑇 )
] + 2𝜌𝜎𝑤2
𝜕𝑡
𝜕𝑥𝑗
𝑘
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜔 𝜕𝑥𝑗 𝜕𝑥𝑗
Where,

𝛽2 , 𝜎𝑘2 , 𝜎𝑤2 are closure coefficients
Wilcox’s k-ω model is then multiplied by a blending function, F1, and the transformed
k-ε model is multiplied by (1-F1). The corresponding kinetic energy and dissipation rates
for each model are then added together to form the SST k-ω [12]:
Turbulence Kinetic Energy:
𝜕(𝜌𝑘) 𝜕(𝜌𝑘𝑈𝑗 )
𝜕𝑈𝑖
𝜕
𝜕𝑘
+
= 𝜏𝑖𝑗
− 𝛽 ∗ 𝜌𝑘𝜔 +
[(𝜇 + 𝜎𝑘 𝜇 𝑇 )
]
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑗
Specific Turbulence Energy Dissipation Rate:
𝜕(𝜌𝜔)
𝜕𝑡
+
𝜕(𝜌𝜔𝑈𝑗 )
𝜕𝑥𝑗
𝜔
𝜕𝑈
𝜕
𝜕𝜔
= 𝛼 𝑘 𝜏𝑖𝑗 𝜕𝑥 𝑖 − 𝛽2 𝜌𝜔2 + 𝜕𝑥 [(𝜇 + 𝜎𝑤 𝜇 𝑇 ) 𝜕𝑥 ] + 2𝜌(1 −
𝑗
𝑗
𝑗
1 𝜕𝑘 𝜕𝜔
𝐹1 )𝜎𝑤2 𝜔 𝜕𝑥
𝑗
𝜕𝑥𝑗
Menter [12] uses the blending function 𝐹1 = tanh(𝜙14 ), where ϕ1 is:
500𝜇 4𝜌𝜎𝑤2 𝑘
√𝑘
𝜙1 = min[max (
; 2 ), 2
]
0.09𝜔𝑦 𝜌𝑦 𝜔 𝑦 𝐶𝐷𝑘𝑤
𝐶𝐷𝑘𝑤 = max(2𝜌𝜎𝜔2
11
1 𝜕𝑘 𝜕𝜔
, 10−20 )
𝜔 𝜕𝑥𝑗 𝜕𝑥𝑗
In the previous equation, y is the distance from the nearest wall. Table 2-2 lists the
closure coefficients suggested by Menter [12] and those in Fluent [9]. Neither [12] or [9]
document the origin of the closure coefficients; therefore, the Fluent values will initially
be used in the model. Appendix D reruns the case with the Menter coefficients to
determine the difference.
Table 2-2: SST k-ω Closure Coefficients
Coefficient
Menter [12]
Fluent [9]
σk1
0.85
1.176
σk2
1.0
1.0
σω1
0.5
2.0
σω2
0.856
1.168
α
0.31
0.31
β
0.075
0.075
β*
0.09
0.0828
Inspection of the SST k-ω model in Fluent shows that a term associated with
buoyancy is not present. As a hot fluid enters the domain through the steam jet,
buoyancy is expected to play a role in the development of the flow. Therefore, an
investigation into the adequacy of the SST k-ω turbulence equation in predicting flow in
buoyancy induced scenarios will be performed in Section 3.2.2.
The effects of buoyancy on turbulence for the k-ε model are the generation of
turbulence given by [9]:
𝐺𝑏 = 𝛽 ∗ 𝑔 ∗
𝜇 𝜕𝑇
∗
𝑃𝑟𝑡 𝜕𝑥
Where,

𝑃𝑟𝑡 is the turbulent Prandtl number

𝛽 is the coefficient of thermal expansion

𝑔 is a gravity term

𝜇𝑡 is the dynamic viscosity

𝜕𝑇
𝜕𝑥
is the temperature gradient in the “x” direction
12
2.1.2
Level Tracking
The level of water to air transition above the steam jet nozzle is only 60 mm.
Therefore, the interaction of the sloshing water in the tank may interfere with the
development of the mixing within the tank. To properly model this interaction, a model
is selected that will track the interface of two immiscible fluids. In this case, water and
air. ANSYS Fluent uses the Volume of Fluid (VOF) model to model two or more
immiscible fluids.
The VOF model solves a single set of momentum equations and tracking of the
volume of fraction of each of the fluids throughout the domain. The resulting velocity
field is shared among the phases. The momentum equation is dependent on the volume
fractions of all phases through the properties ρ and μ. [9]
𝜕(𝜌𝑣⃑)
+ ∇(𝜌𝑣⃑𝑣⃑) = −∇𝑝 + ∇[𝜇(∇𝑣⃑ + ∇𝑣⃑ 𝑇 )] + 𝜌𝑔⃑ + 𝐹⃑
𝜕𝑡
To track the interface between the two phases, the continuity equation is solved
for the volume fraction of one or more of the phases. The equation of the q-th phase has
the following form: [9]
𝑛
1 𝜕(𝛼𝑞 𝜌𝑞
[
+ ∇(𝛼𝑞 𝜌𝑞 𝑣⃑𝑞) = ∑(𝑚𝑝𝑞 − 𝑚𝑞𝑝 )]
𝜌𝑞
𝜕𝑡
𝑝=1
Where,

𝑚𝑝𝑞 is the mass transfer from phase p to q.

𝑚𝑞𝑝 is the mass transfer from phase q to p.
Fluent additionally has the ability to add a source term to the continuity equation.
However, this term is neglected in the formulation in this paper as no source term is
required to track the interface of this system.
The model also has an energy equation that is shared among the phases: [9]
𝜕(𝜌𝐸)
+ ∇(𝑣⃑(𝜌𝐸 + 𝑝)) = ∇(𝑘𝑒𝑓𝑓 ∇𝑇)
𝜕𝑡
The model treats E and T, the energy and temperature, respectively, as mass-averaged
variables of the form:
13
∑𝑛𝑞=1 𝛼𝑞 𝜌𝑞 𝐸𝑞
𝐸=
∑𝑛𝑞=1 𝛼𝑞 𝜌𝑞
Where,

Eq for each phase is based on the specific heat and the shared temperature

The density and thermal conductivity (keff) are shared by the phases
Like the continuity equation, a source term is omitted that contains the contributions
from radiation and any user defined volumetric heat sources.
2.2 Steam Condensation Region Model: Free Jet
Characterizing the flow of condensing steam jet into a subcooled pool is a complex
problem that requires sophisticated computational analysis. Analyzing this complex twophase flow directly using the state-of-the-art CFD techniques is still very immature to
provide any accurate or reasonable results. Kang and Song [3-6] developed a technique
that assumes that the steam jet was condensed within the steam jet penetration length.
The steam discharge condenses over a short distance within the subcooled pool which
allows the liquid leaving the condensation region to be analyzed for its effects on mixing
within the tank.
To utilize this model several parameters are required to be calculated. The steam
jet penetration length will be determined through empirical correlations while the width
will be determined through turbulent jet theory. The condensation region will be
modeled as a cylindrical control volume with these lengths. Condensed liquid is
discharged at the exit of the cylinder from the injected steam flow. Water from the pool
is entrained in the cylindrical boundaries of the control volume. To calculate the
conditions of the injecting water, mass, energy, and momentum balances will be applied
to the control volume in Figure 2-2.
The original SCRM from Song and Kang assumed a uniform velocity profile
across the entering condensed water. This velocity profile uses the velocity calculated
through the mass and momentum balance to calculate the fluid conditions. However, it is
believed that to properly model the mixing in the subcooled pool, a more realistic
14
entering profile is required. This will be obtained through Tollmein’s Axially Symmetric
Turbulent Source Theory in Abramovich [11]; which will also be used to determine the
width and velocity profile of the jet at the penetration length.
as
P s, u s
Po
ac-as
x
Steam Inlet Plane
y
Vapor
Core
ae
Pe
ue
Entrained Water Surface
Condensed Water Plane
ac
uc
Pc
Figure 2-2: Steam Jet Control Volume
2.2.1
Penetration Length
According to Song in [3], several author’s (Kerney et al., Weimer et al, Chun et
al., Kim et al, Wu et al,) studies developed empirical correlations into jet penetration
length at varying mass flux and temperatures. A generalized form was developed for the
steam jet penetration length correlation as such:
𝐿
𝐺
= 𝐶𝐵 −1 ( )0.5
𝑑
𝐺𝑚
The definition of “B” ranges from author to author as shown in Table 2-3.
15
Table 2-3: “B” Definitions [7]
Kerney
Weimer
Chun
Kim
Wu
𝐵=
𝑐𝑝 (𝑇𝑠𝑎𝑡 − 𝑇∞ )
ℎ𝑓𝑔
𝐵=
𝐵=
ℎ𝑓 − ℎ𝑔
ℎ𝑠 − ℎ𝑓
𝑐𝑝 (𝑇𝑠𝑎𝑡 − 𝑇∞ )
ℎ𝑓𝑔
𝐵 = 𝑐𝑝
𝐵=
𝑇𝑠 − 𝑇∞
ℎ𝑠 − ℎ∞
𝑐𝑝 (𝑇𝑠𝑎𝑡 − 𝑇∞ )
ℎ𝑓𝑔
The constant “C” varies from 0.2588 (Kerney) to 17.75 (Weimer). Moon [7]
compared the results of the steam penetration length correlations of each author to test
data and found that Kerney, Chun, Kim and Wu underestimated the steam jet penetration
length while Weimer overestimated the penetration length. Moon states that the “B”
definition of Kim physically represents condensation better and performs sensitivities on
“C” to obtain a value that correlates to test data well. It was determined for the test
conditions in Moon, 30 to 70°C and a steam mass flux between 150 to 750 kg/m2s, that a
“C” value of 0.8 fit test data appropriately. The test performed by Choo [8] had a pool
temperature of 45°C and ranged in steam mass flux of 300 to 650 kg/m2s. These
conditions are well within the ranged of the study performed by Moon, therefore the
combination of Kim’s “B” definition and Moon’s “C” coefficient of 0.80 yields:
𝐿
𝑇𝑠 − 𝑇∞ −1 𝐺 0.5
= 0.80(𝑐𝑝
) ( )
𝑑
ℎ𝑠 − ℎ∞
𝐺𝑚
This equation will be used to determine the steam jet penetration length for the
condensation zone and the boundary distance from the nozzle to apply the velocity and
temperature boundary conditions.
16
2.2.2
Jet Width/Velocity Profile
The jet width and velocity profile following the condensation zone is characterized
by use of Tollmein’s Axially Symmetric Turbulent Source Theory from Abramovich
[11]. The theory analyzes the flow in the downstream part of a turbulent submerged
axially symmetric jet by examining the flow from a turbulent point source as shown in
Figure 2-3.
Figure 2-3: Submerged Axially Symmetric Jet [11]
The theory positions the coordinate system at the source of the jet and notes that
the dimensionless flow remains constant along any radial line from the source and lying
within the downstream region of the jet. Therefore;
𝑢
𝑦
= 𝑓( )
𝑢𝑚
𝑥
Assuming a round cross section in the downstream portions of the jet, we can obtain a
general form of the law of velocity and it’s profile of the downstream portions of the jet.
The velocity in the center section of an axially symmetric submerged jet is inversely
proportional to the distance from the source.
𝑢𝑚 =
17
𝑚
𝑥
With this, it is possible to obtain a formula for the velocity profile. The components of
velocity in an axially symmetric flow can be decomposed with a stream function. For
simplicity,
𝑦
𝑥
is substituted with η.
𝑢 = 𝑢𝑚 𝑓(𝜂) =
1 𝜕𝜓
1 𝜕𝜓
𝑢 = 𝑦 𝜕𝑦
𝑣 = − 𝑦 𝜕𝑥
𝑚
𝑓(𝜂)
𝑥
𝜓 = ∫ 𝑢𝑦𝑑𝑦 = 𝑚𝑥 ∫ 𝑓(𝜂)𝜂𝑑𝜂
Introducing new notation and rewriting the expressions for the stream function,
longitudinal and transverse velocity yields:
𝐹(𝜂) = ∫ 𝑓(𝜂)𝜂𝑑𝜂
𝑢=
𝑚 𝐹′(𝜂)
𝑥
𝜂
𝑣=
𝑚
𝑥
1
[𝐹 ′ (𝜂) − 𝜂 𝐹(𝜂)]
𝜓 = 𝑚𝑥𝐹(𝜂)
The problem is now to determine the function (F(η)) and it’s derivatives. The solution to
the problem is determined by placing a control surface around Figure 2-1, symmetrical
with respect to the axis and doing a momentum balance.
The momentum equation becomes:
1 𝜕 𝑦 2
𝜕𝑢
𝑢𝑣 +
∫ 𝑢 𝑦𝑑𝑦 + 𝑐 2 𝑥 2 ( )2 = 0
𝑦 𝜕𝑥 ∞
𝜕𝑦
Abramovich [11] calls the third term on the left hand side the “turbulent shearing stress,”
derived from Prandtl’s Theory of Free Turbulence. The experimental constant c is
𝑦
removed from the momentum balance by changing the coordinate system to x, 𝜑 = 𝑎𝑥,
3
and setting 𝑎 = √𝑐 2 . Shifting the velocity components to this new system and
substituting into the momentum balance yields a fundamental equation of the form:
18
[𝐹 ′′ (𝜑) −
1 ′
𝐹 (𝜑)]2 = 𝐹(𝜑)𝐹′(𝜑)
𝜑
The following boundary conditions are applied to solve the differential equation.
1) The transverse component of velocity must vanish on the jet axis. Therefore; 𝑣 =
𝑢
0 𝑤ℎ𝑒𝑛 𝜑 = 0. In addition, 𝑢 = 1 𝑤ℎ𝑒𝑛 𝜑 = 0.
𝑚
2) The longitudinal component of velocity vanishes on the jet boundary.
The differential equation is reduced in order and solved by successive approximations to
determine the coefficients. Abramovich [11] continues by creating the dimensionless
parameters for the longitudinal and transverse velocity components from the solution of
this problem. To determine the jet width and velocity profile at a given jet penetration,
the concern is only for the transverse component. The values for the transverse
component are in Table 2-4.
Table 2-4: Dimensionless Velocity Components [11]
𝝋=
𝒚
𝒂𝒙
𝒖
𝑭′(𝝋)
=
𝒖𝒎 𝑭(𝝋)
𝝋=
𝒚
𝒂𝒙
𝒖
𝑭′(𝝋)
=
𝒖𝒎 𝑭(𝝋)
𝝋=
𝒚
𝒂𝒙
𝒖
𝑭′(𝝋)
=
𝒖𝒎 𝑭(𝝋)
0
1.000
1.2
0.510
2.4
0.094
0.1
0.984
1.3
0.470
2.5
0.075
0.2
0.958
1.4
0.425
2.6
0.059
0.3
0.922
1.5
0.378
2.7
0.046
0.4
0.884
1.6
0.340
2.8
0.034
0.5
0.843
1.7
0300
2.9
0.024
0.6
0.795
1.8
0.265
3.0
0.017
0.7
0.748
1.9
0.230
3.1
0.011
0.8
0.700
2.0
0.198
3.2
0.007
0.9
0.653
2.1
0.169
3.3
0.003
1.0
0.606
2.2
0.140
3.4
0.000
1.1
0.555
2.3
0.117
19
The experimental constant “a” will be defined in the model development section of this
paper from research papers on this subject. This constant with the calculated penetration
length and the mean velocity will approximately characterize the velocity profile and jet
width entering the bulk fluid from the condensed steam jet.
2.2.3
Jet Temperature Profile
The same type of analysis performed for the velocity profile can be extended to
the temperature profile with minor differences. Abramovich [11] states that the general
expression for the temperature difference of a turbulent axisymmetric jet at an arbitrary
point is in the form:
∆𝑇 = ∆𝑇𝑚 ∗ 𝜃(𝜂)
From Abramovich [11], a heat balance yields:
∆𝑇𝑣 +
1 𝜕 𝑦
𝜕𝑢 𝜕𝑇
∫ ∆𝑇𝑢𝑦𝑑𝑦 + 2𝑐 2 𝑥 2
=0
𝑦 𝜕𝑥 ∞
𝜕𝑦 𝜕𝑦
The same stream functions and coordinate shift as used in the velocity profile can be
used here lead to a similar equation as before. After manipulation and integration with
the boundaries that 𝑢 = 𝑢𝑚 (i.e., F’(η)/η = 1) and that ΔT = ΔTm (i.e., θ(η)=1), the
dimensionless temperature difference at an arbitrary point of the cross section of an
axially symmetric submerged jet equals the square root of the dimensionless velocity at
the same point [11]:
∆𝑇
𝑢
=√
∆𝑇𝑚
𝑢𝑚
The values are in Table 2-5.
20
Table 2-5: Dimensionless Temperature Components [11]
𝝋=
2.2.4
𝒚
𝒂𝒙
𝜽=
∆𝑻
∆𝑻𝒎
𝝋=
𝒚
𝒂𝒙
𝜽=
∆𝑻
∆𝑻𝒎
𝝋=
𝒚
𝒂𝒙
𝜽=
∆𝑻
∆𝑻𝒎
0
1.000
1.2
0.714
2.4
0.307
0.1
0.992
1.3
0.686
2.5
0.274
0.2
0.979
1.4
0.652
2.6
0.243
0.3
0.960
1.5
0.615
2.7
0.215
0.4
0.941
1.6
0.583
2.8
0.185
0.5
0.918
1.7
0.548
2.9
0.155
0.6
0.892
1.8
0.515
3.0
0.131
0.7
0.865
1.9
0.480
3.1
0.105
0.8
0.837
2.0
0.445
3.2
0.085
0.9
0.808
2.1
0.411
3.3
0.055
1.0
0.779
2.2
0.374
3.4
0.000
1.1
0.745
2.3
0.342
Fluid Conditions
The experiments give the conditions of fluid as it leaves the nozzle. As the
SCRM assumes that the steam is fully condensed within the penetration length the
conditions of the fluid as it leaves penetration length is required. Additionally, water is
entrained into the jet as it injects into the bulk fluid. The entrained water is expected to
be small compared to the steam and condensed water flow. However, the entrained water
density and flow area are relatively large and cannot be neglected. To determine the
mass flow rate and fluid conditions of the condensed water and entrained water, a
control volume, shown in Figure 2-2, is constructed and the conservation of mass,
momentum, and energy is applied to it.
Conservation of Mass:
𝑚̇ 𝑠 + 𝑚̇ 𝑒 = 𝑚̇ 𝑐
21
Conservation of Energy:
𝑚̇𝑠 ℎ𝑠 + 𝑚̇𝑒 ℎ𝑒 = 𝑚̇𝑐 ℎ𝑐
To derive the momentum balance, several assumptions are to be made.
1) At the steam inlet plane, a uniform pressure, density, and velocity enters the
control volume.
2) At the condensed water plane, only water with uniform pressure and density exit
the control volume.
3) At the entrained water plane, water enters the control volume in the normal
direction with a uniform velocity, density, and pressure.
4) The drag force is neglected.
5) The pressure on the washer inlet area of the control volume is the static pressure
of the bulk fluid at that depth.
With these assumptions, the momentum balance becomes;
Conservation of Momentum:
𝜌𝑠 𝑢𝑠2 𝑎𝑠 + 𝑃𝑠 𝑎𝑠 + 𝑃∞ (𝑎𝑐 − 𝑎𝑠 ) = 𝑃𝑐 𝑎𝑐 + 𝜌𝑐 𝑢𝑐2 𝑎𝑐
These conservation equations, along with the measured experiment data, will be used to
determine the fluid conditions at the previously calculated jet boundaries (penetration
length, jet width).
The jet penetration length (Section 2.2.1), jet width and velocity profile
(Section 2.2.2), the jet temperature (Section 2.2.3) and the fluid conditions
(Section 2.2.4) together form the basis for modeling the steam jet region in the
subcooled pool. They will be used to form the condensation region of the model and to
determine the boundary conditions that will be applied to this region to simulate the
turbulent jet from the condensing steam jet. These parameters will be calculated in the
appropriate subsection in the model development section of this paper.
22
3. Model Development
The following sections describe the ANSYS Fluent model developed to analyze
the mixing caused by a turbulent steam jet condensing into a subcooled pool. The
sections will calculate all constants and parameters required to develop the models
previously described. Choo [8] presented the measured mixing patterns of two cases in
the JICO test facility. Therefore, these sections will develop the inputs required for two
cases; one at a steam mass flux of 300.81
𝑘𝑔
𝑚2 𝑠
and another at 650.77
𝑘𝑔
𝑚2 𝑠
. The test
conditions are summarized in Table 3-1.
Table 3-1: JICO Test Conditions [8]
Mass Flux
Steam Temp.
Nozzle Pres.
Pool Temp.
( 𝒎𝟐 𝒔)
𝒌𝒈
(°C)
(kPa)
(°C)
Case 1
300.81
135.95
204.12
45.10
Case 2
650.77
165.12
571.86
44.96
3.1 Geometry
Developing the geometry is the first part of constructing a CFD model. The model
simulates the experiment performed by Choo in [8]. The JICO test facility consists of
two open tanks, an inner cylinder as a mixing pool and an outer square tank to reduce
visual distortions, and an injection nozzle. Figure 3-1 shows the dimensions of the inner
tank and placement of the nozzle.
The tank is cylindrical with the nozzle centered, injecting in the downwards
direction at a level of 790 mm. A 2-Dimensional vertical slice through the tank would be
symmetric around all planes. Therefore, to reduce computational time and required
nodes, a 2-Dimensional Axisymmetric model will be the basis of the geometry. The
radius of the modeled tank will be 390 mm. The actual height of the JICO test facility
tank is 2000 mm [8]. Preliminary runs showed that the interaction of the level in the tank
can play an important role in developing the mixing within the tank. A tank height of
1200 mm is modeled which will allow roughly a 50% increase in water level before
additional modeling space is required.
23
Figure 3-1: Tank and Nozzle Geometry [8]
The SCRM is used to model the condensing turbulent jet. The model assumes that
the steam is fully condensed in the penetration length of the jet and allows the user to
characterize the flow as a set of boundary conditions. As such, incorporated into the
geometry is a region that represents the flow in the jet. The penetration length and width
of the jet are determined through the theory developed in Section 2.2.1 and 2.2.2,
respectively. The detailed mathematics of the problem is presented in Appendix A for
the each of the test conditions provided in Table 3-1. Table 3-2 presents the various fluid
properties used in determining the steam jet penetration length.
24
Table 3-2: Fluid Properties Used in Calculations [16]
Property
Case 1
Case 2
cp (Steam)
2.2504 𝐾∗𝑘𝑔
cp (Liquid)
4.17876 𝐾∗𝑘𝑔
Enthalpy (Steam)
2729.5 𝑘𝑔
Enthalpy (Liquid)
188.306 𝑘𝑔
𝑘𝐽
𝑘𝐽
2.2370 𝐾∗𝑘𝑔
𝑘𝐽
𝑘𝐽
4.17875𝐾∗𝑘𝑔
𝑘𝐽
𝑘𝐽
2782.2 𝑘𝑔
𝑘𝐽
𝑘𝐽
187.720 𝑘𝑔
𝑘𝐽
Density (Steam)
1.671
Density (Liquid)
988.34 𝑚3
2.330
𝑚3
𝑘𝐽
𝑘𝐽
𝑚3
𝑘𝐽
988.34 𝑚3
Using these properties with the penetration length equation from Section 2.2.1 yields:
𝐿 = 𝐷 ∗ 0.80 (𝑐𝑝
𝑇𝑠 −𝑇∞ −1
ℎ𝑠 −ℎ∞
)
𝐺
0.5
(𝐺 )
𝑚
36 𝑚𝑚 (𝐶𝑎𝑠𝑒 1)
={
41 𝑚𝑚 (𝐶𝑎𝑠𝑒 2)
The constant “a” for the spreading of a single phase jet is required to determine the
velocities. Song in [3] recommends a value of 0.082 for nonuniform velocity
distributions. Two velocity profiles are required to represent each case.
The turbulent source from Tollmein’s theory has the axisymmetric turbulent jet
originating from a point source. This point source would be recessed some distance from
the nozzle outlet. To determine this distance, the jet penetration length is iterated upon
until the jet width equals the diameter of the nozzle at the exit (5 mm). This distance was
determined to be 18 mm. A schematic of this is shown in Figure 3-2. All jet width and
velocity profile calculations will originate from this point source.
Using the theory from Section 2.2.2, a coefficient of 0.082 for nonuniform velocity
distributions and the location of this point source yields a jet width at the boundary of:
15.055 𝑚𝑚
𝑤𝑖𝑑𝑡ℎ = 0.082 ∗ (𝐿 + 18𝑚𝑚) ∗ 3.4 = {
16.566 𝑚𝑚
25
Nozzle Wall
Point Source
18mm
5 mm
Figure 3-2: Schematic of Point Source
The geometry and jet dimensions are summarized in Table 3-3 with the design
modeler image presented in Figure 3-3. Note that Fluent requires that the axis of
symmetry for an axisymmetric problem be the x-axis. Therefore, the domain is rotated
90° counter-clockwise.
Table 3-3: Geometry Dimensions Used
Description
Parameter Label
Value
Tank Height
H2
1200 mm
Tank Width
V1
390 mm
Nozzle Depth
H3
410 mm
Nozzle Outer Radius
V4
11 mm
Steam Jet Penetration Length
H5
Steam Jet Radius
V6
26
36.0 mm (Case 1)
41.0 mm (Case 2)
15.055 mm (Case 1)
16.566 mm (Case 2)
(2)
(1)
Injecting Nozzle
H3
Tank Side
V4
(1)
Steam Condensing Region
H5
(5)
H2
V6
Tank Bottom
(1)
Entrained Boundary Condition
Turbulent Jet Exit
V1
(3)
(4)
Figure 3-4: Boundary Conditions
Figure 3-3: Design Modeler Image
27
3.2 Model Inputs and Boundary Conditions
The following subsections will detail the development of the required inputs and
boundary conditions used in the model. Figure 3-4 and Table 3-4 detail the boundary
conditions used in the model. The numbers on the figure correspond to a boundary
condition in Fluent and are correlated in Table 3-4.
Table 3-4: Boundary Conditions Used
Number
Boundary Condition
(1)
Wall
(2)
Pressure Outlet
(3)
Velocity Inlet (set to negative)
(4)
Velocity Inlet
(5)
Axis
In general, the list below summarizes the areas where input is required:

k-ω SST Turbulence Model Inputs. The inputs required for this model are
localized to the required intensity and length scale. A benchmark to
experimental data for buoyancy was performed to ensure the model is able to
predict the effects of buoyancy.

Wall Parameters. The roughness constant and roughness height are required
inputs for the wall boundary condition.

VOF Initialization. After the initial conditions of the model are set, the
region will need to be patched through Fluent’s patch initialization routine to
create a level of water in the model.
3.2.1
Jet Parameters
The boundary conditions representing the condensing steam jet are set to a turbulent
velocity inlet. The velocity profile at the jet exit is required to characterize the
downstream flows of the jet. Appendix A presents the detailed SCRM calculations for
28
the values of the velocity profiles. The mean velocity at the penetration length is
calculated through the momentum balance formed in Section 2.2.4. This yields a mean
condensed velocity (uc) of 2.6 and 3.4 m/s for Case 1 and 2, respectively.
To determine the velocity profile, the normalized velocity profile, determined from
the values in Table 2-4 at the penetration length, is integrated over the width of the jet to
determine the appropriate scaling factor. This scaling factor is then divided into the
mean velocity to determine the centerline velocity of 6.867 and 9.204 m/s for Case 1 and
2, respectively. The resulting velocity profiles are plotted in Figure 3-5.
10
9
8
Jet Velocity (m/s)
7
6
650 kg/(m^2*s)
300 kg/(m^2*s)
5
4
3
2
1
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Jet Width (m)
Figure 3-5: Velocity Profiles at Jet Exit
To input these profiles into Fluent, a User-Defined Function (UDF) is created that
iterates on the faces of the jet exit and apply the velocity to the given face. A UDF is a
Fluent coding language that can be compiled and attached to models in Fluent. They can
be used to control almost any parameter set in Fluent. The profiles in Figure 3-5 are
29
curve fit with a fourth-order polynomial. With an R2 value of 1.0, the following curves
are used to represent the velocity profiles (“y” represents the radial distance from the
nozzle centerline):
Case 1:
𝑚
𝑣( ) = −270,637,760.51 ∗ 𝑦 4 + 10,469,848.09 ∗ 𝑦 3 − 104,292.20 ∗ 𝑦 2 − 339.41 ∗ 𝑦 + 6.867
𝑠
Case 2:
𝑚
𝑣( ) = −254,545,220.23 ∗ 𝑦 4 + 10,759,081.25 ∗ 𝑦 3 − 117,096.77 ∗ 𝑦 2 − 416.37 ∗ 𝑦 + 9.204
𝑠
The resulting UDFs for each velocity profile are presented in Appendix B. Once
compiled into Fluent, they are used as the velocity function in the boundary condition
menu.
Additional turbulence parameters are required for the jet boundary. Fluent allows
several combinations of turbulence parameters to be entered to characterize the flow;
including the turbulent intensity and length scale. The turbulent intensity (I) is defined
as the ratio of the root-mean-square of the velocity fluctuations to the mean flow
velocity [9]. This value has the relationship to the turbulent kinetic energy term as:
𝑘=
3
(𝑢 𝐼)2
2 𝑎𝑣𝑔
The turbulence length scale (l) is a physical quantity related to the size of the large
eddies that contain the energy in turbulent flows [9]. This value has the relationship to
the specific dissipation rate as:
𝜔=
𝑘 0.5
𝑐𝜇0.25 𝑙
The Fluent manual defines how to estimate these parameters for classical problems.
Unfortunately, the correlations presented are not applicable to the problem analyzed
30
here. Typical experiments will give the measure of turbulent intensity as a parameter.
However, Choo did not for the PIV measurements performed in [8]. Therefore, an
alternative approach must be done to obtain these needed parameters.
Aloysis [14] performed CFD analysis of buoyant and non-buoyant jets of several
classical jet experiments. The work performed in [14] was isolated to a horizontal plane
jet of relatively high velocity (when compared to the JICO experiments). What Aloysis
concluded was that the higher the turbulence intensity, the more symmetrical the
predicted behavior of the jet profile. Additionally, it was observed that there was a
strong correlation between the turbulence intensity and the distance for the flow to fully
develop [14]. This is also supported in Abramovich [11] where it was stated that in tests
that artificial increased production of turbulent flow creates a faster decay of the jet. For
the experiments that Aloysis was attempting to simulate, a turbulent intensity of 5.0%
reproduced the results well. He also concluded that there was no influence from the
turbulent length scale for a range of values between 0.025 m to 1.0 m.
However, Song points out in [3] that the turbulent intensity for a condensing
steam jet near the beginning of the turbulent jet region can behave with a turbulent
intensity of up to 25-35%. Much larger than a single phase jet. Therefore, the use of the
turbulence intensity of 5.0%, as used in the experiments in [14], for this model may not
be accurate. To determine the proper values for the turbulent intensity and length scale, a
parametric study is performed on a much smaller and simpler Fluent model. A single
phase (water) 2D axisymmetric 150 mm x 50 mm model is created for each jet
dimension of Table 3-3 and the velocity profiles of Figure 3-5. Several distances away
from the jet exit are chosen and the velocity profile is calculated with the dimensionless
parameters in Section 2.2.
Fluent requires that a 2D axisymmetric be symmetric about the x-axis and that no
cells extend below the x-axis or into the negative y-direction. The SST k-ω model is
applied to the domain of the problem. Gravity and buoyancy are neglected for this
problem as the derivations in [11] have no gravity or buoyancy terms. The outer edges of
the domain are declared as pressure outlets with a constant pressure across the domain.
Two cases, an isothermal and non-isothermal case, are run to determine the impact of
31
temperature on the development of the turbulent jet. The nonisothermal case injects the
fluid at the condensed temperature found in the SCRM calculations in Appendix A.
The User Defined Functions (UDFs) from Appendix B for the velocity profiles are
declared for the jet exit and sides. An unstructured triangle grid, as shown in Figure 3-6
is applied for ease of refinement in the mesh studies. The mesh and time steps were
refined until the model showed mesh and time step independence for the starting
assumed turbulent intensity and length scale (for these cases 5.0% and 0.1 m).
Figure 3-6: Model Used to Determine Turbulent Parameters
The turbulent intensity and length scale were then iterated upon until the Fluent
calculated normalized velocity profile at the chosen distance away from the jet exit
matched closely to the theoretical solution from Section 2.2.2. The mesh and time steps
were then once again checked for independence. Figure 3-7 through Figure 3-10 each
show the normalized velocity (to the centerline velocity) profiles for several turbulent
intensity and length scales compared to the theoretical for the non-isothermal case run.
The figures show that the behaviors concluded in [14] for the turbulent intensity
and length scale persist for this modeling. As the turbulent intensity is increased, the
velocity profile becomes flatter. This is seen in Figure 3-7 through Figure 3-9. The
increasing turbulence causes the centerline velocity to decrease faster as it moves down
the jet length. The decreasing centerline velocity causes the normalized velocity profile
32
to not decay as quickly. Hence, as the turbulent intensity is increased, the normalized
velocity profiles would move up on the plot.
As the turbulent jet moves down the model, the different turbulent intensities
predict the normalized velocity profiles differently when compared to the theoretical. At
60 mm, the results show that only the 10% intensity is fully over predicted when
compared to the theoretical. However, at 100 mm, all three intensities are fully over
predicted by Fluent. This is caused by Fluent decaying the centerline velocity faster than
the theoretical profile by [11].
1.2
Normalized Axial Velocity (-)
1
Theoretical
2%
5%
10%
0.8
0.6
0.4
0.2
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Radial Position (m)
Figure 3-7: Case 1 Sensitivity to Turbulent Intensity (60mm axially)
However, the full behavior of the velocity profile at the edges of the theoretical
profile doesn’t fully decay away but asymptotes roughly to the same value for each case.
This is consistent with the figures in [5] and [6] where the velocity profiles reach a nonzero bulk velocity. Kang and Song in [6] attribute this to the turbulent intensity at the
inlet region of the turbulent jet region enlarging the jet width by a momentum diffusion
33
process along the radial direction. This enlargement of the jet width may be decaying the
centerline velocity faster when compared to the theoretical answer.
1.2
Normalized Axial Velocity (-)
1
Theoretical
2%
5%
10%
0.8
0.6
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Radial Position (m)
Figure 3-8: Case 1 Sensitivity to Turbulent Intensity (80 mm axially)
Figure 3-11 plots the velocity profiles for Case 1 under an isothermal and
nonisothermal condition at the 60 mm penetration. The figure shows that there is no
difference between the two cases. At this point in the jet, the velocity is dominated by
momentum effects. The sensitivity to the turbulent intensity will be valid for the final
cases where a nonisothermal condition is considered.
Table 3-5: Final Turbulent Intensity and Length Scales Selected
Case 1
Case 2
Turbulent Intensity (%)
5.0
10.0
Length Scale (m)
0.1
0.1
34
1.2
Normalized Axial Velocity (-)
1
Theoretical
2%
5%
10%
0.8
0.6
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Radial Position (m)
Figure 3-9: Case 1 Sensitivity to Turbulent Intensity (100 mm axially)
Turbulent intensity levels of 5.0% and 10.0% were chosen for Case 1 and 2,
respectively. These intensity levels closely matched the behavior predicted by
Tollmein’s Theory in Abramovich at the jet core. The length scales showed very little
sensitivity to the profile and therefore, an arbitrary value of 0.1 m was selected. These
turbulent intensities and length scales that closely matched the theoretical profiles, as in
Table 3-5, are applied at the jet boundary conditions of for the main models.
35
1.2
Normalized Axial Velocity (-)
1
0.8
Theoretical
0.01 m
0.1 m
1.0 m
0.6
0.4
0.2
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Radial Position (m)
Figure 3-10: Case 1 Sensitivity to Length Scale
1.2
Normalized Axial Velocity (-)
1
0.8
Isothermal
Nonisothermal
0.6
0.4
0.2
0
0
0.005
0.01
0.015
Position (m)
Figure 3-11: Velocity Profile Isothermal v. Nonisothermal
36
0.02
The same sensitivity is repeated for Case 2. The figures are located in Appendix D.
The temperature distribution of the injecting jet was also checked to ensure the
expected behavior was occurring. The temperature was non-dimensionalized to the
following form:
𝑇 − 𝑇𝑏
𝑇𝑚 − 𝑇𝑏
Where,
T = Temperature, °F
Tb = Bulk Temperature, °F
Tm = Maximum, Mid-Line, Temperature, °F
The non-dimensionalized profile is plotted against the theoretical temperature profile of
Tollmien’s Theory from Abramovich [11] for three different location in Figure 3-12
through Figure 3-14.
1.2
Dimensionless Temperature (-)
1
0.8
Theoretical
Model
0.6
0.4
0.2
0
0.000
0.005
0.010
0.015
0.020
Radial Position (m)
Figure 3-12: Temperature Profiles at Axial Location: 60 mm
37
0.025
1.2
Dimensionless Temperature (-)
1
0.8
0.6
Theoretical
Fluent
0.4
0.2
0
0.000
0.005
0.010
0.015
0.020
0.025
0.030
Radial Position (m)
Figure 3-13: Temperature Profiles at Axial Location: 80 mm
1.2
Dimensionless Temperature (-)
1
0.8
0.6
Theoretical
Fluent
0.4
0.2
0
0.000
0.005
0.010
0.015
0.020
0.025
Radial Position (m)
Figure 3-14: Temperature Profiles at Axial Location: 100 mm
38
0.030
The temperature profiles follow the same trend; however, do not decay to the
same value. This can be attributed to the non-decay of the velocity profile to the
theoretical value in the velocity sensitivity. With the velocity not fully decayed at the
edges of the turbulent jet, additional entrained liquid would enter the fringes of the
turbulent jet. As the bulk fluid is cooler than the turbulent jet, this would decrease the
temperature of the fluid at the edges of the turbulent jet.
3.2.2
Buoyancy
As noted in Section 2.2.1, the SST k-ω model in Fluent does not have a
buoyancy term in the formulation. A temperature gradient is expected to occur between
the condensing steam and the bulk fluid of the subcooled pool. This causes a density
difference as the condensing steam mixes in the tank. The hotter, lower density fluid
would rise in the subcooled pool and create a buoyancy driven flow away from the jet
that would help develop the mixing in the tank.
As this phenomena is expected to be important, an investigation is performed to
determine the adequacy of the SST k-ω in predicting buoyancy driven flow. To test the
adequacy of buoyancy in the SST k-ω Turbulence Model, a series of benchmark cases
were performed to compare the results of a modified SST k-ω, the standard SST k-ω,
and the well test k-ε turbulence models to the results of the test performed by Ampofos
and Karayiannis in [17].
Section 2.1.1 details the formulation for the buoyancy term used in the kinetic
energy formula of the k-ε turbulence model in Fluent that will be added to the SST k-ω
turbulence equation. This equation will be programmed in a Fluent UDF. The gravity
term of the source term equation is set by the user in the settings of the model. The
turbulent Prandtl, viscosity, and temperature gradient terms are stored in the Fluent
memory and can be used directly. However, the thermal expansion coefficient is not
calculated in Fluent. Therefore, the UDF will have to calculate this parameter. As Air
can be assumed to be behave as an ideal gas, the thermal expansion coefficient is simply
the inverse of the temperature.
39
The test in [17] studies the details of the natural convection of an air filled cavity as
shown in Figure 3-15. A single wall on each side is heated and cooled to 50°C and 10°C,
respectively while the air in the cavity is at an initial temperature of 30°C. The
temperature and velocities were simultaneously measured at different locations in the
cavity using laser Doppler anemometer (LDA) and a micro-diameter thermocouple. The
results for many measured parameters, including temperature and directional velocities,
are presented in Table 2 of [17].
Figure 3-15: Schematic of Experiment in [17]
The experiment is reproduced in Fluent 14 as a 2D slice of the domain. The
boundary conditions keep the Thot and Tcold at the experimental conditions. The top and
bottom walls are adiabatic to ensure no heat transfer occurs. The initial geometry used an
unstructured grid with 10,849 amount of cells. The geometry and initial grid is shown in
Figure 3-16.
A set of three cases are run to determine the differences of the modified k-ω SST
(with a buoyancy source term added to the turbulent kinetic energy equation – this
formulation will be known at SST+ from now on), the standard k-ω, and the well tested
40
k-ε turbulence models. The cases are run with a second order implicit transient
formulation with a time step of 0.25 seconds. The Y-Velocity in the boundary layer of
the Thot (estimated from preliminary cases) is monitored to determine steady state. When
the Y-Velocity becomes sufficiently flat, the case run is terminated and the results postprocessed.
Figure 3-16: Grid and Initial Geometry of Buoyancy Benchmark
The case is initialized to zero flow in the cavity, the density differences, which is set
to change as an ideal gas for air, will begin natural convection in the cavity as the air is
heated/cooled. As the conditions in the cavity are continually changing prior to steady
state, the mesh is adapted to maintain a y+ near 1.0. (the suggested value when the
boundary layer is important in [9]). y+ is non-dimensional wall distance for a wall
bounded flow and is defined as:
𝑦+ =
𝜇∗ 𝑦
𝜐
Where,
μ* = velocity at the wall
y = distance to the wall
υ = kinematic viscosity of the fluid
41
Fluent calculates and can adapt the mesh according to this parameter under the
“Adapt” menu of Fluent. Therefore, to keep the y+ in the region suggested, the following
procedure is developed:
1) The transient is run for 1.0 second.
2) The Y+ value is checked, if the Fluent calculated Y+ value is not between 0.95 <
Y+ < 1.05 then the mesh is refined by the Fluent adapt menu. Typical values for
the initial mesh in Figure 3-16 is ~2.8.
3) The transient is run for 1.0 second intervals and step 2 is repeated for an
additional 9 seconds (Flow Time = 10 seconds). This is where the changes in the
Y+ values are negligible for a one second change in time.
4) Each Time Step should be fully converged (all Fluent calculated scaled residuals,
with exception of Energy is < 10-3, while Energy < 10-6). The solution typically
converges within 15 attempts.
5) Run the transient for 10 second intervals (40 timesteps) and repeat Step 2 while
ensuring Step 4. Run till 100 seconds. Save Case Files at Flow Time = 50 and
100 seconds.
6) At 100 seconds, the flow will not change significantly in the 10 second time
interval. The intervals are increased to 50 seconds. Repeat Step 2 while ensuring
Step 4. Save Case Files at every 50 second interval.
7) If after the 50 seconds, the monitored Y-Velocity in the Thot boundary layer is
sufficiently flat. Begin Post-Processing.
This process is performed for each of the turbulence models. The SST+ uses the
source term UDF documented in Appendix B.
In [17], the values are given in the mid-line of the cavity in a non-dimensional
form. All the velocities are non-dimensionalized by the following factor:
𝑉𝑜 = √𝑔𝛽𝐻Δ𝑇 = 0.974571
Where,
g = acceleration of gravity = 9.81 m/s2
42
β = Thermal Expansion Coefficient = 3.36 * 10-3 1/K
H = heigh of the cavity = 0.75 m
ΔT = Temperature Difference = 40 K
The temperatures are normalized as follows:
𝑇 − 𝑇𝑐𝑜𝑙𝑑
Δ𝑇
Where,
T = Temperature from Fluent
Tcold = Cold Temperature = 283.15 K
ΔT = 40 K
The non-dimensional parameters Y-Velocities and Temperatures for the midline
are plotted in Figure 3-17 and Figure 3-18 at the steady state condition. The nondimensional parameters in the boundary layers are plotted in Figure 3-19 and Figure
3-20. A representative monitor of the Thot boundary layer Y-Velocity versus time to
presented in Figure 3-21.
0.3
SST+
"benchmark" [16]
SST
KE
Non-Dimensional Y-Velocity (-)
0.2
0.1
0
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
-0.1
-0.2
-0.3
Non-Dimensional Distance (-)
Figure 3-17: Non-Dimensional Velocity v. Non-Dimensional Position
43
1.00
Non-Dimensional Temperature (-)
1.2
SST+
"benchmark [16]"
SST
KE
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Non-Dimensional Position (-)
Figure 3-18: Non-Dimensional Temperature (-) vs. Non-Dimensional Position (-)
0.3
SST+
"benchmark" [16]
SST
KE
Non-Dimensional Y-Velocity (-)
0.2
0.1
0
0.00
0.02
0.04
0.06
0.08
0.10
-0.1
-0.2
-0.3
Non-Dimensional Distance (-)
Figure 3-19: Non-Dimensional Velocity vs. Non-Dimensional Position in the Boundary Layer
44
The buoyancy parametric shows that all three turbulence models have
comparable results and models the benchmark closely. The added buoyancy source term
provides an insignificant effect on the results of the natural convection within the cavity.
These results also show that the standard SST k-ω is adequate to predict the effects of
buoyancy induced flow.
It is understood that the medians between experiments [14] and [8] are different.
This study was performed to understand the capabilities of the SST k-ω in calculating
buoyancy induced flow. It was an additional goal to determine the difference between
using the standard SST k-ω and adding a production term that the standard k-ε
turbulence model uses for buoyancy in the SST k-ω. As the added production term
added an insignificant different, it is concluded that the added source term would
provide an insignificant effect to the model. Therefore, the added source term to emulate
the buoyancy source term of the k-ε turbulence model will not be added to the standard
SST k- ω for the final case runs.
1.2
SST+
"benchmark [16]"
SST
KE
Non-Dimensional Temperature (-)
1
0.8
0.6
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Non-Dimensional Position (-)
Figure 3-20: Non-Dimensional Temperature vs. Non-Dimensional Position in the Boundary Layer
45
0.14
0.12
Velocity (m/s)
0.1
0.08
0.06
0.04
0.02
0
0
50
100
150
200
250
Time (seconds)
Figure 3-21: SST+ Average Thot Y-Velocity vs. Time in Boundary Layer
To confirm grid independence the SST+ case was performed again using an
initial grid size of 12,301 cells (~20% increase in cells). The same procedure was used to
run the case until a final steady state value was achieved. The initial grid is displayed in
Figure 3-22. The boundary layer temperature and Y-Velocities are shown in Figure 3-23
and Figure 3-24. The figures show the solutions yield comparable results. The
differences in the Y-Velocity can be explained by the placement of the cells in the
boundary layer and the way that Excel linearly interpolates the plots. Comparison of the
values at the actual points show that there is an insignificant difference between the
values. Therefore, the solution is grid independent.
46
Figure 3-22: Buoyancy Benchmark, Grid Independence
0.3
SST+ 12k
Non-Dimensional Velocity (-)
0.2
Benchmark
SST+ 10k
0.1
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
-0.1
-0.2
-0.3
Non-Dimensional Position (-)
Figure 3-23: Velocity Grid Independence Study
47
0.08
0.09
0.1
1.2
Non-Dimensional Temperature (-)
1
SST+ 12k
benchmark
SST+ 10k
0.8
0.6
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Non-Dimensional Position (-)
Figure 3-24: Temperature Grid Independence Study
An additional study was performed that reran the SST+ case with a reduced
timestep from 0.25 seconds to 0.1 seconds. The results are presented in Figure 3-25 and
Figure 3-26. The study shows that there is no difference between the solutions.
Therefore, the solution is timestep independent.
48
0.3
Non-Dimensional Velocity (-)
0.2
0.1
0
0
0.005
0.01
0.015
0.02
-0.1
0.025
0.03
SST+ 0.10 sec
Benchmark
SST+ 0.25 sec
-0.2
-0.3
Non-Dimensional Position (-)
Figure 3-25: Velocity Timestep Independence Study
With the standard SST k- ω turbulence model benchmarked, the pool mixing
model can be developed with the proper buoyancy inputs. To model buoyancy in Fluent,
the gravity term and material density calculations needs to be set. Since Fluent requires
that a 2-D axisymmetric case be symmetric around the x-axis, the gravity term is set to
9.81 m/s2 in the positive “x”-direction. The fluid density is calculated through a
piecewise linear temperature versus density curve fit. The temperature ranges chosen to
cover the range of expected values and the densities are extracted from [16] and
presented in Table 3-6.
With the standard SST k- ω turbulence model benchmarked to an experiment that
undergoes buoyancy induced flow, the pool mixing model can be developed with the
proper buoyancy inputs. To model buoyancy in Fluent, the gravity term and material
density calculations needs to be set. Since Fluent requires that a 2-D axisymmetric case
be symmetric around the x-axis, the gravity term is set to 9.81 m/s2 in the positive “x”direction. The fluid density is calculated through a piecewise linear temperature versus
density curve fit. The temperature ranges chosen to cover the range of expected values
and the densities are extracted from [16] and presented in Table 3-6.
49
1.2
Non-Dimensional Temperature (-)
1
SST+ Fine
0.8
benchmark
SST+
0.6
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Non-Dimensional Position (-)
Figure 3-26: Temperature Grid Independence Study
Table 3-6: Water Density
Temperature (K)
Density (kg/m3)
Temperature (K)
Density (kg/m3)
295
997.771
320
989.395
300
996.510
325
987.170
305
995.032
330
987.192
310
993.339
335
982.272
315
991.456
50
3.2.3
Wall Parameters
The boundary conditions marked in Figure 3-27 are set to “Wall.” The
experimental setup in [8] says that the outer tank walls are made of “acrylic material”
and that the nozzle to the steam jet that is submerged is covered in a thermal insulation
to minimize heat loss.
Figure 3-27: Boundary Conditions Marked as "Wall"
In Fluent [9], two parameters are used to determine the roughness at the wall
boundary condition; the roughness height (Ks) and the roughness constant (Cs). These
are used to modify the Law-of-Wall formulation and the physical location of the wall.
Both modifications are well-correlated to the non-dimensional roughness height Ks+.
𝐾𝑠+ =
𝜌𝑢∗ 𝐾𝑠
𝜇
Where,
u* = friction velocity = Cμ1/4k1/2
According to [9], there are three distinct regimes in the surface of the non-dimensional
roughness height:

Hydrodynamically smooth (Ks+ < 2.25)

Transitional (2.25 < Ks+ < 90)

Fully rough (Ks+ > 90)
51
Roughness effects are negligible in the hydrodynamically smooth regime (the shifts for
the Law-of-the-Wall and physical wall location is set to zero), but become increasingly
important in the transitional regime, and take full effect in the fully rough regime. Cs is
used in the terms to shift the Law-of-Wall in the transitional and fully rough regimes. Ks
represents the height of the actual height of the sand grain roughness near the wall as
seen in Figure 3-28.
Figure 3-28: Illustration of Roughness Height [9]
It is reasonably assumed that the acrylic material at the tank walls are
hydrodynamically smooth and will therefore neglect the effects of surface roughness. In
Fluent, Ks is set to zero. Cs is left at the default value as it’s term is neglected with the
zero Ks.
3.2.4
Volume of Fluid Parameters, Initialization, and Solver Options
The model is initialized with the water at the experimental temperature
(45°C/318K). This initially sets the model up with the entire domain with the primary
phase (water). To initialize the VOF with a level within the domain, a “patch” is
performed on the initialization. The region that will consist of air is highlighted through
Fluent’s “Adapt – Region” menu. The patch is performed on this region declaring the
volume fraction of the secondary phase (air) to 1.0 in this region. Figure 3-29 is a
contour plot of the initialized model with the level. This figure represents the domain
that Fluent will solve the jet regimes of Figure 2-1.
52
Figure 3-29: Initial Model with Initialized Level
Additionally, the pressure outlet representing the top of the tank is set to only
allow air to cross the boundary. This is important when reverse flow on the boundary
occurs for model stability. Otherwise, divergence would immediately occur when
reverse flow through the boundary happens.
The solver options used are found in Table 3-7. Table 3-8 presents the underrelaxation factors used.
53
Table 3-7: Solver Options
Description
Option
Pressure-Velocity Coupling
SIMPLE
Gradient
Least Squares Cell Based
Pressure
PRESTO!
Momentum
Second Order Upwind
Volume Fraction
Geo-Reconstruct
Turbulent Kinetic Energy
First Order Upwind
Specific Dissipation Rate
First Order Upwind
Energy
Second Order Upwind
Transient Formulation
First Order Upwind
Table 3-8: Under-Relation Factors
Description
Value
Pressure
0.3
Density
1
Body Forces
1
Momentum
0.7
Turbulent Kinetic Energy
0.8
Specific Dissipation Rate
0.8
Turbulent Viscosity
1
Energy
1
54
4. Results
4.1 Case 1 Results
The following sections document the results of the cases run for the Case 1 (see
Table 3-1) conditions (as defined in Table 3-1). Section 4.1.1 represents the results of a
benchmark to the results in Choo [8] using the turbulence, VOF, and energy models.
Section 4.1.2 determines the impact on the separate models of the overall development
of the flow patterns. These cases separately turned off the energy and volume of fluid
models to determine how buoyancy and the free surface level impact the development of
mixing in the tank. Section 4.1.3 reruns the cases with an decreased mesh size and
decreased time step to ensure that mesh and time step independence is achieved.
Each case run utilizes a time step of 0.001 seconds. This time step was
determined by iteration until the model, scaled residuals, and flow behaviors behaved
well. Choo [8] did not specify the time in which the PIV measurement were taken.
Therefore, screen shots at many intervals of the velocity contours and vectors were taken
and compared to the data in Choo [8]. This was done up 30 seconds of transient time.
Figure 4-1: Local Flow Patterns Presented in [8]
Choo [8] presents the overall flow patterns and local flow patterns in Figure 11
of [8]. The locations of the images are presented in Figure 4-1 as “A”, “B”, and “C”. A
comparison of the CFD predictions to these locations will be performed for each of case
conditions.
55
4.1.1
Benchmark Results
Figure 4-2 shows the comparison between the PIV measurements and the
velocity contour of the Fluent run at 30 seconds into the transient. Comparison of the
images show similar behaviors of the impinging and wall bounded flows. The jet
impinges on the bottom of the tank; redirects towards the tank wall and then up the tank
wall. The fluid momentum decay and entraining jet flow eventually separate the flow
from the wall at the approximately the same location (as marked on Figure 4-2) in the
PIV and Fluent prediction. A circulation of flow occurs in the bottom right hand corner
of the tank from the entrained and wall bounded flow. As shown in Figure 4-3 (which
shows the same geographic location), Fluent predicts the center of this circulation at
approximately the same location and magnitude. This location, as marked on Figure 4-3,
was pinpointed by expanding the velocity vectors until the center was distinct enough to
select. This center of circulation is slightly higher due to the expanded jet \width pushing
up the center.
Figure 4-2: Case 1 PIV [8] and CFD Results (at 30 seconds)
56
-45
-50
-55
Center of Circulation
X
-60
-65
(28.1, -61.3)
-70
-75
20
25
30
35
Figure 4-3: Case 1 Location B Comparison
Comparison of Location A in Figure 4-4 shows differences in the jet
impingement location. Figure 4-2 shows that the jet width and penetration is
overpredicted by Fluent (note the differences in color scale). This translates to the jet
impingement velocity being too high when compared to the PIV. Additionally, the PIV
measurements shows a circulating current that moves inward. However, due to the
overpredicted penetration length of the jet, the flow moves outward from the jet in the
computed results.
Choo’s [8] image for Case 1’s Location C appears to have been doubled with
Case 2’s Location C. The velocity vectors show in Figure 11 of [8] for Location C show
the same behaviors and magnitudes. Therefore, Figure 10 of [8] is zoomed in on in the
location of Location C and compared to the prediction of Fluent. Fluent shows that the
no secondary flow is developed in the area of the level which is consistent with the
results of Choo [8].
57
-60
-65
-70
-75
5
0
10
15
20
Figure 4-4: Case 1 Location A Comparison
Despite the overpredicted jet penetration and width, the comparison of Location
B and C show similar behaviors to that observed in the PIV measurements. Namely,
Fluent was able to predict the center of circulation caused by the fluid entrainment and
wall bounded flow well and showed no creation of a secondary circulation created near
the free surface of the tank.
4.1.2
Effects of Free Surface Level and Energy Models on Results
This section will compare the velocity magnitudes at several locations to
determine the effects of each of the models on the development of the velocity patterns
in the tank. Positions are chosen that are thought to be impacted by the different models
in the flow as shown in Figure 4-5 and tabulated in Table 4-1.
58
(1)
(2)
(3)
(4)
Figure 4-5: Model Comparison Locations
Table 4-1: Model Comparison Locations
Location
X-Location (cm) Y-Location (cm)
(1)
10 to 30
0
(2)
29-39
+5
(3)
29-39
-20
(4)
29-39
-60
59
Figure 4-6 through Figure 4-9 show the case results at Location 1 through 4 of
Table 4-1 at 30 seconds into the transient. The results show that the velocities at all
locations are momentum dominated and decrease as they moved up the tank wall. At
lower locations, the highly momentum driven flow is the same for each model. As the
flow moves up the tank wall and begins to lose its momentum, the effects of each model
begins to become apparent.
0.02
0.018
Velocity (m/s)
0.016
0.014
Isothermal/VOF
0.012
Nonisothermal/NonVOF
Nonisothermal/VOF
0.01
0.008
0.006
0.004
0.002
0
0.1
0.15
0.2
0.25
0.3
Position (m)
Figure 4-6: Case 1, Location 1 Model Effects
The temperature effects had no impact on the velocity development. This can
possibly be explained by the mass proportions between the injecting steam and the bulk
fluid. While the entering steam has a larger energy content compared to the bulk fluid, it
would require a longer time of injecting before the bulk fluid is heated enough for
buoyancy effects to show.
60
0.004
0.0035
Isothermal/VOF
Velocity (m/s)
0.003
Nonisothermal/NonVOF
Nonisothermal/VOF
0.0025
0.002
0.0015
0.001
0.0005
0
0.29
0.31
0.33
0.35
0.37
0.39
Position (m)
Figure 4-7: Case 1, Location 2 Model Effects
0.03
0.025
Isothermal/VOF
Velocity (m/s)
0.02
Nonisothermal/NonVOF
Nonisothermal/VOF
0.015
0.01
0.005
0
0.29
0.31
0.33
0.35
Position (m)
Figure 4-8: Case 1, Location 3 Model Effects
61
0.37
0.39
The cases that utilized level tracking showed a small effect on the development
of the flows closer to the level of the tank. This is attributed to the free surface acting as
a pseudo-wall and causing the flow to develop differently. The flow would change
direction due to the presence of the free surface and lose momentum; therefore, reducing
the velocity. This effect is more prominent in Case 2 where the secondary flow develops
near the level interface. This will be discussed in more detail in Section 4.2.1.
0.25
0.2
Isothermal/VOF
Velocity (m/s)
Nonisothermal/NonVOF
0.15
Nonisothermal/VOF
0.1
0.05
0
0.29
0.31
0.33
0.35
Position (m)
Figure 4-9: Case 1, Location 4 Model Effects
62
0.37
0.39
4.1.3
Independence Study
Two studies are performed to determine the independence of the model; a time
step and mesh study. A different time step and mesh is constructed and the solutions are
compared to the original results. If the final solutions vary by a small amount than the
original results are proven to be mesh and time step independent.
Figure 4-10: Case 1 Velocity Contours Mesh/TS Study
The original mesh was increased from 18,316 cells to 22,579 cells (~23% increase
in cells) and rerun for 30 seconds with a 0.001 second time step. A second case was run
that kept the same mesh as the original run and reduced the time step to 0.0005 seconds
(doubling the amount of time steps) for a 30 seconds transient run. Figure 4-10 compares
the velocity contours of each of the runs. The image shows the same behaviors between
each of the runs. The velocity vectors of the case runs were also compared to ensure that
no additional mixing or velocity patterns developed between the three cases. Figure 4-11
is an example of the velocity vectors for the three case runs.
63
Figure 4-11: Case 1 Velocity Vector Comparison
Based on these comparisons, it is determined that the original Case 1 using the 18,316
Cells and a time step of 0.001 seconds is mesh and time step independent.
4.2 Case 2 Results
Similar to the Case 1 results, the results of the Case 2 run are compared to
Figures 10 and 11 of Choo [8]. The same locations are used to compare the development
of the mixing patterns through the tank. The same time step was used (0.001 seconds)
for a 30 second case run.
4.2.1
Benchmark Results
Figure 4-12 compares the velocity contours of the Fluent run to that of the PIV
measurements. This case shows higher velocity magnitudes through the tank from the
higher mass flux entering the domain. The separation from the right hand wall occurs
approximately at the same location for the PIV and the Fluent case runs. Additionally,
the strong circulation at the bottom right hand corner develops from the faster entraining
fluid and wall bounded flow.
64
Figure 4-12: Case 2 PIV [8] and CFD Results (at 30 seconds)
Figure 4-13 through Figure 4-15 compare the velocity vectors at Locations A, B,
and C of Choo [8]. The strong circulation shown in Location B of Choo develops in the
Fluent model at approximately the same location. The center of circulation is found by
expanding the velocity vectors of the Fluent run until a center is distinct enough to
select. The wall bounded flow creates this circulation as it separates from the wall and
becomes entrained in the injecting jet. Location A shows the strong main jet current
from the jet. However, much like Case 1, Fluent overpredicts the jet width and
penetration length.
65
-45
-50
-55
Center of Circulation
X
-60
(27.8, -57.8)
-65
-70
-75
20
25
30
35
Figure 4-13: Case 2 Location B Comparison
Location C shows the development of the secondary flow. This behavior was not
observed in the Case 1 Fluent runs or the associated PIV measurements. Even with the
overpredicted jet width and penetration length, the CFD calculations observed the proper
behavior for this region in terms of initial development of the secondary flow. However,
the location of Fluent’s center of circulation for the secondary flow, extracted by
expanding the velocity vectors of the Fluent run until a center is distinct enough to
select, showed to be lower and more towards the center of the tank. This could be
attributed to the overprediction of the jet width and penetration length which may carry
more momentum through the wall bounded flow to this region. This additional
momentum can cause the secondary flow to be larger, both in physical space and in
velocity, when compared to the PIV which would push the center of circulation farther
away from the wall and the free surface.
66
-60
-65
-70
-75
0
5
10
15
20
Figure 4-14: Case 2 Location A Comparison
5
0
-5
-10
Center of Circulation
X
(22.3, -11.9)
20
25
-15
30
35
Figure 4-15: Case 2 Location C Comparison
67
4.2.2
Effects of Free Surface Level and Energy Models on Results
As was done in Section 4.1.2, the velocity magnitudes were observed at several different
locations throughout the model. Figure 4-5 and Table 4-1 details the locations. Figure
4-16 through Figure 4-19 plots these velocity magnitudes at these locations.
0.045
0.04
Velocity (m/s)
0.035
Nonisothermal/NonVOF
NonIsothermal/VOF
0.03
Isothermal/VOF
0.025
0.02
0.015
0.01
0.005
0
0.1
0.15
0.2
0.25
0.3
Position (m)
Figure 4-16: Case 2, Location 1 Model Effects
As expected, the velocity magnitudes are higher than that of Case 1 due the
higher mass flux entering the tank. Similar to Case 1, the buoyancy effects played an
insignificant role in the development of the flow patterns as the case runs assuming
isothermal and nonisothermal VOF predicted almost the same magnitude. Very slight
differences are seen towards the higher ends of the tank as the momentum decays away
and a small amount of buoyancy induced flow starts.
68
0.02
0.018
0.016
Nonisothermal/NonVOF
Velocity (m/s)
0.014
NonIsothermal/VOF
0.012
Isothermal/VOF
0.01
0.008
0.006
0.004
0.002
0
0.29
0.31
0.33
0.35
0.37
0.39
Position (m)
Figure 4-17: Case 2, Location 2 Model Effects
0.017
0.016
Nonisothermal/NonVOF
NonIsothermal/VOF
Velocity (m/s)
0.015
Isothermal/VOF
0.014
0.013
0.012
0.011
0.01
0.29
0.31
0.33
0.35
Position (m)
Figure 4-18: Case 2, Location 3 Model Effects
69
0.37
0.39
The interesting development is the impact of the level on the development of the
flow for this high of a mass flux. Figure 4-16 and Figure 4-17 imply that towards the
higher end of the tank (near the level), the secondary flow that is seen in Figure 4-15
develops differently with the interaction of the level. In this case, the level acts as a
pseudo-wall, carrying the fluid momentum through the circulation and preventing the
circulation from expanding. This is observed in Figure 4-20 with a comparison vector
plot of the secondary circulation at the same location. Notice that velocity around the
circulation for the VOF models are higher (more green) than those of the non-VOF
model.
0.36
0.31
Velocity (m/s)
0.26
0.21
0.16
Nonisothermal/NonVOF
0.11
NonIsothermal/VOF
0.06
Isothermal/VOF
0.01
0.29
0.31
0.33
0.35
Position (m)
Figure 4-19: Case 2, Location 4 Model Effects
70
0.37
0.39
Figure 4-20: NonIsothermal VOF/NonVOF Secondary Flow Comparison
4.2.3
Independence Study
Just like Section 4.1.3 for Case 1, two studies are performed to determine the
independence of the model; a time step and mesh study. A different time step and mesh
is constructed and the solutions are compared to the original results. If the final solutions
vary by a small amount than the original results are proven to be mesh and time step
independent.
Figure 4-21: Case 2 Velocity Contours Mesh/TS Study
71
The original mesh was increased from 25,692 cells to 30,423 cells (~20% increase
in cells) and rerun for 30 seconds with a 0.001 second time step. A second case was run
that kept the same mesh as the original run and reduced the time step to 0.0005 seconds
(doubling the amount of time steps) for a 30 seconds transient run. Figure 4-21 compares
the velocity contours of each of the runs. The image shows the same behaviors between
each of the runs. The velocity vectors of the case runs were also compared to ensure that
no additional mixing or velocity patterns developed between the three cases. Figure 4-22
is an example of the velocity vectors for the three case runs.
Figure 4-22: Case 2 Velocity Vector Comparison
Based on these comparisons, it is determined that the original Case 2 using the 25,692
Cells and a time step of 0.001 seconds is mesh and time step independent.
72
5. Conclusions
5.1 Conclusions
Much research has been performed to predict the nature of a steam jet injecting
into a subcooled pool in the direct area of the jet (micro-scale). A popular approach to
characterizing the steam jet is through the use of the SCRM developed by Kang and
Song [3 through 6]. Their model combines experimental correlations of penetration
length, mass, momentum and energy balance, along with turbulent jet theory to come up
with a boundary condition that can simulate a condensing steam jet. Much study has
been performed to determine the adequacy of this theory in the micro-scale but not much
has been performed to determine the adequacy of the model in the macro-scale; or the
interactions of the condensing steam jet in the overall tank.
A CFD model of the experiment in [8] was created using the SST k-ω turbulence
model, the VOF level tracking model, and the SCRM model developed by Kang and
Song to simulate the flow patterns developed in subcooled pool under the influence of an
injecting steam jet. Two different mass injections of [8] were benchmarked in Section 4;
300 and 650 kg/m2*s. Additional cases were run without Fluent’s energy formulation
and level tracking to determine the effects of the level and density effects on the mixing
pattern development.
The results of the analysis suggest that the use of the SCRM to predict mixing in a
subcooled pool with steam jet injection is possible. Many of the patterns observed in the
PIV measurements of [8] were predicted within the CFD model.
The centers of
circulation, fluid entrainment, and wall bounded flows were all predicted well by Fluent.
However, there are two noticeable areas of deficiency. The SCRM combined with
Tollmein’s Theory has proven to be effective in predicting the velocities close to the
injecting nozzle in the research performed in [4]. However, as one goes farther down the
penetration length, it is noticed that the jet width is over predicted. This causes the large
fluid momentum of the injecting jet to not decay as expected and create a large overall
jet injection length. The impinging jet to wall jet transition created by this appears to
decay the momentum enough to allow the behaviors from the experiment [8] to
propagate through the model. Kang and Song observed similar behavior, overprediction
73
of jet width, in their micro-scale studies in [6] and attributed this to the turbulent
intensity at the inlet region of the turbulent jet region enlarging the jet width by a
momentum diffusion process along the radial direction.
Another possibility is Fluent’s capabilities in predicting the transition from a
turbulent jet to an impinging jet. Several modeling features were changed to determine
if they would improve the results of these case runs, including:

Finer/Coarser Meshes

Reduced/Greater Timesteps

Use of Different Turbulence Models (k-ε Turbulence Model)

Higher/Lower Turbulent Intensities (1.0% to 40.0%)

Modifying the SST k-ω’s turbulence coefficients (Appendix D)
All changes exhibited the same behaviors as what was seen in Section 4. The outer
diffusivity turbulent closure coefficient, which modified the momentum diffusion in the
SST k-ω turbulence model, was reduced to determine the effects of the closure
coefficients to the overall results. Additionally, the original closure coefficients
suggested by Menter in [12] was used. Altering Fluent’s standard turbulence coefficients
showed some improvement in the jet penetration length but no improvement to the jet
width as shown in Appendix D.
The second deficiency appears in the entrainment boundary condition of the
SCRM. The assumption states that the entrained velocity leaves the domain
perpendicular to the injecting jet. Moon [7] confirms this assumption and states that test
data supports the assumption. On the micro-scale, it can visualized that this assumption
would have little impact on the development of the overall flows. However, put into the
context of the macro-scale, the interaction of the wall separating flow from the tank wall
appears to create a flow away from the entraining jet flow farther down the penetration
length. This is observed in Figure 4-22 where the flow separating from the wall is
rotating around the main circulation. It was expected that the entraining fluid from the jet
injection would fully continue this rotation. However, the negative velocity boundary
condition disturbed this rotation and diverted some of the flow up towards the boundary.
This deficiency can also partially explain some of the overpredicted jet width.
Additional entrained fluid may cause additional turbulence at the boundary of the
74
turbulent jet. This additional turbulence can work to decay the effects of the jet more
quickly and possible align more with the PIV measurements of the jet portion.
Therefore, with further refinements in the area of transition from a turbulent jet
to an impinging jet, the SCRM can be used to characterize the flow patterns in the
subcooled pool. It was able to the predict the overall behaviors of Choo [8] somewhat
well. However, great care should be taken in using the results to predict the jet boundary
and penetration length far away from injection.
It was additionally observed that the buoyancy effects of the experiment played
very little into the development of the flow. This can be attributed to temperature quickly
dispersing into the tank due to the high turbulence of the jet and not causing as much as
a density difference as originally expected in the time frame of interest. Over time, the
temperature increase in the tank from injecting steam could be large enough to begin a
buoyancy induced flow away from the entrained fluid. This suggests that for this
experiment, the momentum of the jet is the driving force. Further experimentation would
have to be performed to determine the size of tank, temperature of injecting steam, and
time scale where buoyancy driven flow would begin to play a significant role.
The level of the tank played a significant role in developing the fluid flow. The
level acts as a pseudo-wall to create a strong secondary flow from higher mass fluxes.
Without this level, the secondary flow would just expand through the upper reaches of
the tank and in diameter; causing slower circulation. This could cause a significant force
on the wall at this location at higher injections – which would cause greater secondary
circulation flow rates from the wall separated flow.
5.2 Suggestions for Future Work
There are many areas of future work that can be explored with this topic. The
detailed macro-scale prediction of the effects of steam jet injection into a subcooled pool
is fairly undeveloped. The simplification of the SCRM is convenient in computational
space and time; however, it has been shown to have some deficiencies.
An investigation on being able to more accurately predict the jet injection
characteristics at locations farther away from the nozzle can uncover a more accurate
model to fully characterize the flow in the tank. Perhaps by determining a method to
75
characterize an impinging jet as a boundary condition, the SCRM could be extended past
the turbulent jet region and into the impinging jet region which could further simplify
the modeling of this transition.
Additionally, a better assumption can be employed at the entrainment boundary
condition to better predict the flow around the nozzle. As noted in Section 5.1, the
SCRM assumes that the velocity leaves the domain perpendicular to the fluid injection.
A more appropriate assumption can be devised that could assume the boundary to be at
some angle to the injection as shown in Figure 5-1. This could develop the flow near the
entrainment boundary in a more downward direction, pulling from the upper reaches of
the tank. This could possible reduce the velocities leaving the entraining rotation and
moving towards this boundary condition.
Vent
Θ
Figure 5-1: Improved Entrainment Assumption
Not related to CFD analyses, but additional PIV measurements for this
phenomenon would increase the pool of available experiments to benchmark to. Right
now, very little information has been gathered on the mixing behaviors in a tank due to
downward steam jet injection. Experiments with the placement of the level and size of
the tank may give insight into how to better design the tanks to minimize the hydraulic
forces caused by this phenomenon.
76
6. References
1. Cummins, W. et al, “Westinghouse AP1000 Advanced Passive Plant,”
Proceedings of ICAPP ’03, Paper 3235, May 2003.
2. Bestion, D. et al., “Extension of CFD Codes Application to Two-Phase Flow
Safety Problems Phase 2,” Nuclear Energy Agency Committee on the Safety of
Nuclear Installations, July 2010.
3. Song, C. et al, “Steam Jet Condensation in a Pool: From Fundamental
Understanding to Engineering Scale Analysis,” Journal of Heat Transfer, 134:
2012.
4. Kang, H. and Song, C., “CFD Analysis of Thermal Mixing in a Subcooled Water
Tank under a High Steam Mass Flux Discharge Condition,” Nuclear
Engineering and Design, 238: 292-501, 2008.
5. Kang, H. and Song, C., “CFD Analysis of Turbulent Jet behavior Induced by a
Steam Jet Discharge Through a Single Hole in a Subcooled Water Tank,”
Nuclear Engineering and Design, 240: 2160-2168, 2010.
6. Kang, H. and Song, C., “CFD Analysis of Turbulent Jet Behavior Induced by a
Steam Jet Discharged Through a Vertical Upward Single Hole in a Subcooled
Water Pool,” Nuclear Engineering and Design, 420, 2160-2168, 2010.
7. Moon, Y. et al, “CFD Simulation of Steam Jet-Induced Thermal Mixing in a
Subcooled Water Pool,” Nuclear Engineering and Design, 239: 2849-2863, 2009
8. Choo, Y. et al, “PIV Measurement of Turbulent Jet and Pool Mixing Produced
by a Steam Jet in a Subcooled Water Pool,” Nuclear Engineering and Design,
240: 2215-2224, 2010.
9. ANSYS, Inc., “ANSYS Fluent 14.0 Theory Guide,” 2012.
10. Mahaffy, J. et al, “Best Practice Guidelines for the use of CFD in nuclear
Reactor Safety Application,” Nuclear Energy Agency Committee on the Safety
of Nuclear Installations, May 2007.
11. Abramovich, G. N., “The Theory of Turbulent Jets,” Massachusetts, The M.I.T.
Press, 1963.
77
12. Menter, F. R., “Two-Equation Eddy-Viscosity Turbulence Models for
Engineering Applications,”AIAA Journal, 32: 1598-1605, 1994.
13. Wilcox, D. C., “Turbulence Modeling for CFD,” California, DCW Industries
1993.
14. Aloysis, A. and Wrobel, L., “ALAQS CFD Comparison of a Buoyant and Non
Buoyant Jet,” EEC/SEE/2007/002, May 2007.
15. Patek, J. et al, “Reference Correlations for Thermophysical Properties of Liquid
Water at 0.1 MPa,” Journal of Physical and Chemical Reference Data, Volume
38, Number 1, 2009.
16. NIST/ASME Steam Properties, Database 10, Version 2.11, 1996.
17. Ampofo, F. and Karayiannis, T. G., “Experimental Benchmark Data for
Turbulent Natural Convection in an Air Filled Square Cavity,” International
Journal of Heat and Mass Transfer, 46: 3551-3572, 2003.
18. Ljubola, M. and Rodi W., “Predication of Horizontal and Vertical Turbulent
Buoyant Wall Jets,” Journal of Heat Transfer, 103: 343-349, 1981
19. Townsend, A., “The Structure of Turbulent Shear Flow,” New York, Cambridge
University Press, 1976.
78
Appendix A: SCRM Detailed Calculations
𝟑𝟎𝟎
𝒌𝒈
𝑪𝒂𝒔𝒆:
𝒎𝟐 𝒔
79
80
81
𝟔𝟓𝟎
𝒌𝒈
𝑪𝒂𝒔𝒆:
𝒎𝟐 𝒔
82
83
84
Appendix B: FLUENT User Defined Functions
𝑽𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝑷𝒓𝒐𝒇𝒊𝒍𝒆 𝑼𝑫𝑭, 𝟑𝟎𝟎
𝒌𝒈
𝑪𝒂𝒔𝒆:
𝒎𝟐 𝒔
/************************************************************************
UDF to Define Boundary Conditions of the Jet
************************************************************************/
#include "udf.h"
/* jet_exit velocity*/
DEFINE_PROFILE(jet_exit,thread,index)
{
real x[ND_ND];
/* this will hold the position vector*/
real velocity, loc;
face_t face;
real quart, cubic, square, single, um;
begin_f_loop(face,thread)
{
F_CENTROID(x,face,thread);
loc = x[1];
um = 6.867;
quart = -270637760.51*loc*loc*loc*loc;
cubic = 10469848.09*loc*loc*loc;
square = -104292.20*loc*loc;
single = -339.41*loc;
F_PROFILE(face,thread,index) =
quart+cubic+square+single+um;
/* units of m per s */
}
end_f_loop(face,thread)
}
DEFINE_PROFILE(jet_temp,thread,index)
{
real x[ND_ND];
/* this will hold the position vector*/
real velocity, loc;
face_t face;
real quart, cubic, square, single, tm, tb;
begin_f_loop(face,thread)
{
F_CENTROID(x,face,thread);
loc = x[1];
tm = 4.717;
tb = 318.0;
cubic = 917848.60*loc*loc*loc;
square = -24029.87*loc*loc;
single = -145.88*loc;
F_PROFILE(face,thread,index) =
units of K */
}
end_f_loop(face,thread)
}
/* jet_entrained velocity */
85
cubic+square+single+tm+tb;
/*
DEFINE_PROFILE(entrained_flow,thread,index)
{
real x[ND_ND];
/* this will hold the position vector*/
real velocity;
face_t face;
begin_f_loop(face,thread)
{
F_CENTROID(x,face,thread);
velocity = x[0];
F_PROFILE(face,thread,index) = -0.446;
}
end_f_loop(face,thread)
}
86
/* units of m per s */
𝑽𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝑷𝒓𝒐𝒇𝒊𝒍𝒆 𝑼𝑫𝑭, 𝟔𝟓𝟎
𝒌𝒈
𝑪𝒂𝒔𝒆:
𝒎𝟐 𝒔
/************************************************************************
UDF to Define Boundary Conditions of the Jet
************************************************************************/
#include "udf.h"
/* jet_exit velocity*/
DEFINE_PROFILE(jet_exit,thread,index)
{
real x[ND_ND];
/* this will hold the position vector*/
real velocity, loc;
face_t face;
real quart, cubic, square, single, um;
begin_f_loop(face,thread)
{
F_CENTROID(x,face,thread);
loc = x[1];
um = 9.204;
quart = -254545220.23*loc*loc*loc*loc;
cubic = 10759081.25*loc*loc*loc;
square = -117096.77*loc*loc;
single = -416.37*loc;
F_PROFILE(face,thread,index) =
quart+cubic+square+single+um;
/* units of m per s */
}
end_f_loop(face,thread)
}
DEFINE_PROFILE(jet_temp,thread,index)
{
real x[ND_ND];
/* this will hold the position vector*/
real velocity, loc;
face_t face;
real quart, cubic, square, single, tm, tb;
begin_f_loop(face,thread)
{
F_CENTROID(x,face,thread);
loc = x[1];
tm = 5.209;
tb = 318.0;
cubic = 866698*loc*loc*loc;
square = -23906*loc*loc;
single = -152.91*loc;
F_PROFILE(face,thread,index) =
units of K */
}
end_f_loop(face,thread)
}
cubic+square+single+tm+tb;
/* jet_entrained velocity */
DEFINE_PROFILE(entrained_flow,thread,index)
{
real x[ND_ND];
/* this will hold the position vector*/
87
/*
real velocity;
face_t face;
begin_f_loop(face,thread)
{
F_CENTROID(x,face,thread);
velocity = x[0];
F_PROFILE(face,thread,index) = -0.559;
}
end_f_loop(face,thread)
}
88
/* units of m per s */
UDF for Incorporating Buoyancy into SST k-ω:
/************************************************************************
UDF to Define Buoyancy Source Term for K-Omega
************************************************************************/
#include "udf.h"
#include "mem.h"
real R = 461.51805; /* J/kg*K - Gas Constant*/
real Tr = 10; /* K - Reference Temperature [14]*/
real Ta = 593; /* K - Reference Temperature [14]*/
real Tb = 232; /* K - Reference Temperature [14]*/
real Po = 0.1; /* MPa - Atmospheric Pressure */
real alpha, beta, vto, vt, vtoAsum, vtoBsum, vtAsum, vtBsum; /* calculated values
*/
real thermcoeff, Gb, source;
int i = 0;
/* all matrix values from [14]*/
real a[6] = {0.0193763157, 6744.58446, -222521.604, 1002312470.0, -1635521180.0,
8322996580.0};
real b[6] = {0.00578545292, -0.0153195665, 0.0311337859, -0.0423546241,
0.0338713507, -0.0119946761};
real n[6] = {0, 4, 5, 7, 8, 9};
real m[6] = {1, 2, 3, 4, 5, 6};
DEFINE_SOURCE(buoyancy, c, t, dS, eqn)
{
alpha = Tr/(Ta-C_T(c,t));
beta = Tr/(C_T(c,t)-Tb);
dS[eqn] = 0.0;
vtoAsum = 0;
vtoBsum = 0;
vtAsum = 0;
vtBsum = 0;
while(i <= 5) /* calculating summations for thermal coefficient */
{
vtoAsum = n[i]*a[i]*pow(alpha,n[i]+1)+vtoAsum;
vtoBsum = m[i]*b[i]*pow(beta,m[i]+1)+vtoBsum;
vtAsum = b[i]*pow(beta,m[i])+vtAsum;
vtBsum = a[i+1]*pow(alpha,n[i+1])+vtBsum;
i = i+1;
}
if(NNULLP(T_STORAGE_R_NV(t, SV_T_G))) /* error check */
{
vto = (R/Po)*(vtoAsum+vtoBsum);
vt = ((R*Tr)/Po)*(a[0]+vtAsum+vtBsum);
thermcoeff = vto/vt;
Gb = thermcoeff*C_MU_T(c,t)/M_keprt*NV_DOT(M_gravity,C_T_G(c,t));
}
else Message0("Temperature Gradient Not Stored in Memory%n");
source = Gb;
89
return source;
}
90
Appendix C: 600 kg/m2s Turbulent Parameter Sensitivity
1.2
Velocity Magnitude (-)
1
Theoretical
2%
5%
10%
0.8
0.6
0.4
0.2
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
Position (m)
Figure C-1: Case 2 Sensitivity to Turbulent Intensity (60mm)
1.2
Velocity Magnitude (-)
1
Theoretical
0.8
2%
0.6
5%
10%
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
Position (m)
Figure C-2: Case 2 Sensitivity to Turbulent Intensity (80mm)
91
0.03
1.2
Velocity Magnitude (-)
1
0.8
0.6
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Position (m)
Figure C-3: Case 2 Sensitivity to Turbulent Intensity (100mm)
92
0.035
Appendix D: Closure Coefficient Study
Several techniques were attempted to align the results of Fluent more towards the
PIV measurements of Choo [8]. One of those techniques altered the closure coefficients
of the turbulence model. This appendix documents the results of that study and
determined its effects to the full model.
The turbulent closure coefficients are selected to mimic a wide range of
experimental data. For example, Ljuboja and Rodi [18] provides insight in the
adjustment of the k-ε turbulence model for horizontal and vertical turbulent buoyant wall
jets. The standard empirical constants typically used in the k-ε model were replaced with
functions which are derived by reducing model forms of the Reynolds-stress and heatflux transport equation to algebraic expressions.
When originally formulating the standard k-ω turbulence model, Wilcox in [13]
discusses the choice of coefficients. The coefficients relating to the dissipating terms and
the Reynolds shear stress were chosen by applying the model to decaying homogeneous,
isotropic turbulence that matches the experimental observation of Townsend [19]. The
viscosity coeffficients were related to data that examined the log layer, or the boundary
layer sufficiently distant from the surface that molecular viscosity is negligible relative
to eddy viscosity [13]. Wilcox [13] later states that the specific arguments selected for
determination of the closure coefficients are a free choice of the developer and should be
selected to yield an optimized model restricted to the flows of interest.
Inspection of the turbulence equations for the SST k-ω turbulence equation in [12]
yields a set of closure coefficients that applies to different portions of the equations. σk1,
σk2, σω1, σω1, are the turbulent Prandtl numbers for k and ω, respectively. They are
connected to the diffusivity terms of the turbulence models. α affects the calculated rate
of turbulent viscosity while βi1 and βi2 affects the calculation for the dissipation of k and
ω. The subscripts 1 and 2 represent the “inner” and “outer” portions of the blending
functions of the SST k-ω.
The results in Section 4 for both cases show that the mixing in the tank is highly
momentum driven. Adjustment of the coefficients relating to the diffusivity of
turbulence may provide some insight into closing the gap between the Fluent results and
93
the PIV measurements. It is expected that by decreasing the diffusivity, the momentum
transfer would decrease and improve the jet characteristics (length and width) during the
transition to an impinging jet. If the effect of impinging jet portion of the flow were to be
decreased, then it is possible that the predicted flows can converge on the PIV
measurements of Choo [8]. Therefore, the σ coefficients will be decreased in the Case 1
model to see if improved results occur.
The diffusivity terms are broken to “outer” and “inner.” The “outer” diffusivity
terms, σk2 and σω2, are related to the modified k-ε which is blended with the k-ω and
becomes more dominate farther from the nearest wall. The “inner” diffusivity terms, σk1
and σω1, are used with the standard k-ω that is used near the wall. As the jet penetration
and width is being overpredicted and is not directly wall bounded, the “outer” diffusivity
terms are altered. Lowering the diffusivity terms would reduce the momentum transfer
across the jet. Two different reductions are attempted to determine the impact of
reducing this paramter. The initial Fluent [9] SST k-ω closure coefficients for σk2 and
σω2 are reduced from 1.0 and 1.168 to 0.75 and 0.75 and then reduced again to 0.50 and
0.50. The jet studies in Section 3.2.1 were repeated to ensure a proper turbulent intensity
is declared at the boundary. The results are shown in Figure D-1.
Figure D-1 plots the theoretical, the profile used in the original model, and the
normalized velocity profile for the new closure coefficients. The results show that the by
using the new closure coefficients, the turbulent intensity had to be increased to 9% for
the 0.75 diffusivity and 19% for the 0.5 diffusivity from the original 5% to properly
mimic the theoretical velocity profile.
To determine the impact of increasing the “inner” diffusivity terms, the reduction
to 0.75 case was rerun reducing σk1 and σω1 from 1.176 and 2.0 to 0.75 and 0.75,
respectively. Figure D-2 shows the results of this run. Decreasing the inner diffusivity
terms showed a very small impact to the normalized velocity profiles. Inspection of the
peak velocities showed an insignificant difference between the two centerline velocities;
3.12 and 3.46 m/s for the “outer” and “inner” parametrics, respectively. Therefore, only
the “outer” diffusivity closure coefficients have an impact on the development of a
turbulent jet.
94
1.2
Normalized Axial Velocity (-)
1
0.8
Theoretical
5%
Closure Coefficients = 0.50
Closure Coefficients = 0.75
0.6
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Radial Position (m)
Figure D-1: Turbulent Intensity Outer Parametric at 80 mm
1.2
Normalized Axial Velocity (-)
1
0.8
Theoretical
9% - 0.75 Outer
9% - 0.75 Inner
0.6
0.4
0.2
0
0
0.005
0.01
0.015
0.02
0.025
Radial Position (m)
Figure D-2: Turbulent Intensity Inner Parametric at 80 mm
95
0.03
0.035
Additionally, Section 2.1.1 notes a difference in the closure coefficients
suggested by Menter [12] and those used in Fluent [9]. To test the difference between
the closure coefficients, the Case 1 model was rerun using the Menter closure
coefficients in Table 2-2.
Original Case 1
“Outer” Diffusivity = 0.75
Menter
“Outer” Diffusivity = 0.50
Figure D-3: Velocity Contour Comparison for Case 1 @ 30 seconds
Figure D-3 shows the original velocity contour and the rerun for the new
turbulent closure coefficients at 30 seconds. The results show the by decreasing the outer
diffusivity terms does reduce the magnitude of the penetration length of the jet.
However, it increases the jet width as the closure coefficient is reduced. Reducing the
closure coefficients had little to no effect on the impinging jet portion of the flow. The
turbulent jet transition to an impinging jet still widened the jet width such that the
behaviors for position “A” (the bottom left of the image) of Choo [8], namely the inward
flow from the entrained jet, were not observed. Additionally, the coefficients in Menter
[12] showed little improvement to the jet behaviors.
It is concluded that altering the closure coefficients of the SST k-ω turbulence
model provides no benefit to converging the results of Fluent to Choo [8].
96