Proceedings of the 9th International and 49th National Conference on Fluid Mechanics and Fluid Power (FMFP)
December 14-16, 2022, IIT Roorkee, Roorkee-247667, Uttarakhand, India
FMFP2022–8305
Towards analysis of Corium hydraulics in liquid sodium
Ram Kumar Maity1, T. Sundararajan2, M. Rajendrakumar1, K. Natesan1
1
Thermal Hydraulics Division, Indira Gandhi Centre for Atomic Research, Kalpakkam-603102, India
2
Department of Mechanical Engineering, IIT Palakkad, Palakkad-678557, India
a mixture of reactor fuel, fission products and structural
material. An expected post-accident scenario with respect to
movement of molten corium and its relocation is depicted in
Fig. 1 [2, 3]. Moreover, during such an accident christened as a
severe accident in a fast reactor in addition to molten corium,
large volumes of fission gases are released into reactor pool as
well. Accident progression in the fast reactor is expected to be
distinct from that in the thermal reactor. Accidents, which can
lead to degradation and melting of the whole core, are classified
as severe accidents and specifically called a core disruptive
accident (CDA) in the context of fast reactors. Due to the
extreme low probability of occurrence of CDA, it is often
referred to as a hypothetical accident [4]. Despite such low
frequency, an accident like CDA needs to be analyzed in view
of its possible serious impact on the reactor systems and its
surroundings. It is to be demonstrated that any radioactivity
release to environment stays within specified limits [4].
ABSTRACT
During a hypothetical severe accident in a sodium cooled
fast reactor, core melt-down is expected to occur. As a result,
molten corium would fall through sodium as a jet. The
dispersion of the fissile material within the molten jet is
important to avoid re-criticality. This occurs in several stages
with different sequence of complex controlling physical
phenomena. The present work is in the direction of
understanding the initial jet behavior influenced primarily by
flow instabilities. An in-house force balanced two phase flow
solver namely THYC-MP has been used to study the behavior
of two dimensional jets of corium and simulant material namely
wood’s metal within liquid sodium. The expected physics and
jet breakup behavior is studied and described.
Keywords: Corium jet, two phase flow, Volume of Fluid,
severe accident, fast reactor
1. INTRODUCTION
Fast reactors are a type of nuclear reactors in which the
nuclear fission reaction that produces thermal energy is
sustained by fast (energetic) neutrons. In order to limit neutron
flux and fissile inventory, fast reactors work with compact cores
with enriched fuel. Because of low core volume, the power
density and linear power rating of fuel becomes high for such
reactors (typical power density is 400 to 500 kW/litre and linear
power rating is 400 to 500 W/cm of pin length). Such high
power densities warrant use of a highly efficient coolant
medium. Many operational fast reactors around the world
therefore use liquid metal in the form of liquid sodium as the
coolant. A pool type reactor is characterized by a large pool of
sodium which holds the primary heat transport system. A large
capacity fast reactor is preferentially designed as a pool type
reactor due to the large thermal inertia of reactor pool and
associated enhancements in reactor safety. Details of design of
Indian fast reactors can be found in ref. [1].
Accidents leading to core meltdown that can release
molten corium into reactor pool are rare events with very low
probability and is considered as beyond design basis event [2].
However, in the eventuality of an incident that leads to core
meltdown and release of molten corium into reactor pool the
transport of the same becomes important and needs to be
characterized. Corium is the material created in a nuclear
reactor during core meltdown accident and would primarily be
Figure 1: Schematic showing transport of molten corium
after a severe accident (top) [3].
It is necessary to develop tools that allow
characterization of transport of various species after a severe
accident in a fast reactor. Such characterization would allow to
improve and optimize accident management strategies. For
example, one of the major tasks after release of corium in
sodium pool is to avoid clumping (aggregation of fuel
fragments) that may result in re-criticality and further large
1
energy release. Development of such tools becomes even more
pertinent in view of the planned expansion of the Indian fast
reactor program [2]. The understanding of severe accident
related phenomena need to be improved substantially as a part
future advanced reactor designs. The focus of the present study
is on the movement of corium in liquid state within liquid
sodium with resolution of flow structures resulting from two
phase instabilities. Phase change is beyond the scope of the
present study. The studies are done using an in-house force
balanced two phase flow solver based on the volume of fluid
(VOF) method called THYC-MP.
movement of simulant material (wood’s metal) in water and
corium in sodium. Preliminary results from these studies are
presented.
3. MATERIALS AND METHODS
Out of the different fixed grid two phase flow analysis
methods, the VOF method sees widespread use. This method is
capable of accurate conservation of mass for each phase.
Volume conservation is a very important aspect in two phase
flows and must be ensured in order to capture the right flow
physics. In two-phase problems, methods to track the physics
of an interface between the two phases present within the
domain need to be implemented in addition to the solving the
flow. As a results, two-phase flow problems are in general more
challenging than single phase flow problems. The VOF method
is a fixed grid ‘one-fluid’ approach that makes use of a marker
function to separate the two phases within the domain.
Integration of the marker function within a control volume or
computational cell gives the value of volume fraction ‘ ’. If the
presence of phase designated ‘k’ is specified by the marker
function ( , , , ), then the volume occupied by this phase
and subsequently the volume fraction ( ) within a cell can be
calculated using the following expressions:
2. LITERATURE REVIEW AND OBJECTIVE
A brief description of numerical studies on movement of
corium in coolant using two phase analysis methods with
capabilities similar to VOF method is presented here. The
motion of a jet of corium in coolant takes place primarily after
a severe accident has occurred. A majority of the studies till
now have been experimental in nature. This is due to the
complex physics involved and the lack of adequate
computational power along with suitable development on
numerical algorithms. The aim of these experiments includes
determination of jet instabilities and jet breakup length and
characterization of corium spread on structures like core
catcher. A comprehensive review of experimental works on jet
break-up and fragmentation phenomena can be found in ref. [5].
Oh et al. [6] described a study on the break-up behavior of a
molten metal jet in still gas when in the presence of sinusoidal
oscillations. The studies that included amplitude-modulated
waveform were done using a VOF based method. Thakre et al.
[7] presented a two dimensional VOF analysis of melt jet
fragmentation using commercial CFD code FLUENT. The
study included identification of jet fragmentation pattern,
influence of Weber number and physical properties. Lin et al.
[8] described studies using a VOF based method when a melt
droplet has a vapor film and is exposed to a pressure pulse. The
effect of surface tension is studied and found to be insignificant.
The results also indicated that the vapor film can be neglected
as well with material density and pressure pulse playing the
most important role in droplet deformation and surface wave
growth.
Zhou et al [9] presented a paper on metal jet breakup
that also analyzed cooling and solidification behavior.
Solidification was modelled using an enthalpy based approach.
The results were found to be in good agreement with
experimental observations at different Weber numbers. More
recently, using the color-gradient Lattice Boltzmann method, a
very detailed study on corium jet break-up and fragmentation
in sodium has been studied by Cheng et al. [10].
Several recent advances and application of two phase
flow algorithms to study the interactions of two immiscible
liquid metals or equivalent materials have been described in
literature. They also prove the viability of using two phase flow
analysis tools for characterization of corium behavior in liquid
sodium. Corium interaction with liquid sodium involves several
intricate interfacial and phase change phenomena. For the
present study a force balanced two phase solver based on the
VOF method is developed and described in more detail in the
next section. The code developed in applied to analyze
=
( , , , )
,
=
/
(1)
The value of volume fraction can take any value from 0 to 1.
The implications of the values taken by the variable are listed:
= 1 Cell completely filled with primary phase
0< <1
Mixed cell containing both phases
= 0 Cell completely filled with secondary phase
The primary fluid is also referred to as the dark fluid. The
first step of the overall method, involves reconstruction of the
interface between the primary and secondary fluids. This is
done using the volume fraction values of mixed cells. The
interface would pass through mixed cells and is a zone of
discontinuity across which a large change in fluid properties
occurs. Either with the help of the reconstructed interface or
directly operating upon the values of volume fractions, the next
step involves calculation of curvature. The calculated curvature
of interface is required for calculation of surface tension forces.
These forces arise due to the existence of the discontinuity in
medium at interface and need to be imbibed into the momentum
equations as part of the flow solver using special force balanced
schemes. After the flow field advances, the interface is advected
based on the prevalent flow velocities. Thus, a single time step
or iteration of calculations for two-phase flow is completed.
The present implementation is on a fully staggered grid that
stores quantities like pressure, volume fraction, temperature,
fluid properties and curvature at center of continuity cells. Two
separate cells viz. u-momentum and v-momentum cells are used
for integration of u, v momentum equations and for storage of
u and v velocities. Specifically, u-momentum cells are centered
around centers of left and right faces and v-momentum cells are
centered around centers of top and bottom faces of a continuity
cell. Further details of the main steps of the VOF method along
with their implementation as part of the present work are
described in the succeeding sections.
2
These relations are relevant for right face (marked as
R) of the computational cell and are used to determine
coordinates of points 3 ( 3, 3) and 4 ( 4, 4) with respect to
coordinates of points 1 ( 1, 1) and 2 ( 2, 2). Flux polygons
are constructed for all other sides of the cell sussing similar
relations. The method is an unsplit advection method that is
mass conserving and uses edge matched flux polygons with no
overlap or uncovered region. Geometric integration requires the
construction of flux polygons from the prevailing velocity field
and subsequently, the area of intersection between the flux
polygon and the dark polygon (region covered by primary/dark
fluid) derived from the interface reconstruction step is
calculated that gives the flux of primary fluid through
respective face. This is depicted in a schematic as given in Fig.
4. The area of flux polygon represents the total fluid flux
passing through each edge of cell. The sum of these areas over
must be zero (to satisfy overall continuity).
1
(6)
+
+
+ )
, = (
4
3.1 Interface Reconstruction
The first step is reconstruction of interface using the
prevailing volume fraction field. An interface will lie within
mixed cells alone and is defined using piecewise linear
segments. Such methods are called Piecewise Linear Interface
Construction (PLIC) methods. The equation of a line segment
can be defined in terms of components of normal and a constant
as:
+
+ = 0. Thus, defining a linear segment
representative of the interface within a mixed cell requires
values of interface normals ( , ) and a line constant ( ). The
method used for the present work for determination of interface
normals is the E-PLIC method. This has been discussed in detail
in ref. [11]. The method uses a c0 correction template (shown
in Fig.3) that is imposed over a primary PLIC interface.
Averaged
interface
Averaged
interface
A2
Original
interface
A1
C2
B1
C1
Cell 1
Cell 0
B2
Cell 2
B0
Primary Fluid
A0
C3
Original
interface
A3
Cell 3
B3
Figure 3: c0 correction template for PLIC interface
For the present study this primary PLIC interface is the
Linear least squares (LLS) method [12]. The LLS method is
known to be one of the most accurate PLIC methods reported
in literature. The application of the c0 correction template has
been shown to further improve the accuracy of PLIC
reconstruction [11].
3.2 Interface Advection
A major step in two phase flow simulation is the
computation of two phase interface kinematics. This requires
obtaining a solution for the equation of conservation of volume
fraction shown below:
(4)
+ ⃗⋅∇ =0
Where, ⃗ is the velocity vector,
is the volume
fraction. The equation is integrated and discretised
geometrically to allow a diffusion free advection of the
interface. This maintains sharpness of interface through a two
phase flow transient being studied. For the present work the
edge matched flux polygon algorithm (EMFPA) proposed by
Lopez et. al. [13] is implemented. The method requires the
following constraints are respected during construction of the
flux polygons (referring to Fig. 4):
/
=
=
=
/
(
(
−
−
−
−
Figure 4: Flux polygons constructed using EMFPA
(top) and method used to calculate geometric
advection using flux polygon and dark polygon.
/
,
=
)−(
/
)−(
/
Similar relations are used to derive
, ,
, ,
, with the
following condition to be satisfied (L-left, T-top, B-Bottom):
(7)
, +
, +
, +
, =0
In the present work a set of analytical relations are
developed to derive the flux polygons [14]. This makes the
implementation of the method highly efficient. The integration
/
−
⁄
)
−
⁄
)
(5)
3
of Eq. 4 is done over a continuity cell. Using Gauss Divergence
theorem and integrating Eq. 4, we get:
1
(8)
=
+
,
The value of capillary pressure ( ) is determined such that the
surface tension force is equal to the derivative of . The
divergence of the difference between surface tension force and
gradient of capillary pressure is made zero as shown in the
following equation:
(12)
∇⋅ ⃗ −
=0⇒ . ⃗ = .
The above equation is integrated and discretised over
the continuity cell. The equation is discretised as an implicit (in
time) equation and is stored by using a matrix solver. The
velocity and pressure fields are coupled using a projection
method described in detail in ref. [17].
The discretisation scheme for volume fraction (f) and
capillary pressure (P ) follows the same exact discretisation in
order to maintain perfect force balance. The time integration of
all conservation equations including the energy equation is
done using fully implicit schemes. After discretisation these are
solved using matrix solvers. For the present study, a BiConjugate Gradient stabilised (Bi-CGSTAB) linear solver is
used. The flow solver has been validated extensively against
complex two phase flow problems not described as part of the
present study. In the next section results from application to the
problem of movement of corium in sodium and simulant
wood’s metal in water is presented. The standard energy
equation is solved along with flow equations. The buoyancy
forces due to temperature differences are not turned on due to
the large buoyancy forces imparted by the density differences
between the primary and secondary fluids. The code is named
as Thermal Hydraulics Code – Multi-Phase viz. THYC-MP.
3.5. Calculation domain and properties
, , ,
Here,
represents area of cell (i,j). The second term
of Eq. 5 represents the area of intersection of flux and dark
polygons as shown in Fig. 4.
3.3. Calculation of curvature
Calculation of curvature required for determination of
surface tension force is done using the Height Functions
method. The height functions used here is the variable template
version of Hernandez et. al. [15]. The basic principle of the
method involves the following three steps:
1. The dominant component of interface normal is found in
order to orient the direction of height functions. For the present
study components of normal from Young’s PLIC are used.
2. Next the volume fractions are summed up along the dominant
direction of the interface normal from step 1. This gives the
local values of height functions X(y) or Y(x).
3. Differentiation of height functions gives the value of
Curvature. This step is valid only when the height function lies
within the cell dimensions.
Thus, if
< , Three height functions viz. Yi-1,j , Yi1,j , Yi+1,j
can be defined:
,
=∑
,
(
−
)
(9)
Curvature can be calculated from height functions as:
=
(
)
1+
(10)
Inlet
=−
,
500 mm
The original height functions [16] uses a stencil of size
7×3 cells. The version used in this work is a more accurate
version that uses a variable stencil size depending on the
volume fraction field. The steps for these can be found in ref.
[15]. The surface tension force for u-momentum equation (for
u-momentum cell ( , ) on the left face of continuity cell
( , )) is given by:
(11)
,
`
Local curvatures are calculated and stored for
continuity cells and are to be interpolated to u-momentum cells
using and v-momentum cells. The surface tension forces are
calculated using these curvature values.
3.4. Flow solver
The inclusion of surface tension forces need to be
carefully implemented using a ‘well balanced’ or a ‘force
balanced’ method. This means the discretised form of the
momentum equation(s) should be able to describe the Laplace
equilibrium correctly. In the present work the scheme (PROST)
proposed by Renardy and Renardy [17] is implemented for a
staggered grid system. Components of velocity viz. u-velocity
is stored on the left and right faces of a continuity cell with the
v-velocity being stored at the top and bottom faces of the same.
In this method, the pressure is divided into two components
( = + ) where component
represents the capillary
pressure component arising due to presence of surface tension.
Outlet
Inlet
Outlet
75 mm
Walls
(Not to scale)
Figure 5: Details of domain and boundary conditions
Two systems of fluids viz. Wood’s metal-water and
Corium-sodium are studied. The former system is a popular
choice as simulant system to study hydraulics of the later. It
must be noted that no phase change is considered as part of this
work. Phase change is otherwise expected to occur and is the
subject of future extension. The flow domain, boundary
conditions and fluid properties are similar to those used in ref.
[7]. The domain selected is a 75 mm width ×500 mm depth
column of fluid (secondary fluid – water and sodium in study 1
4
and 2 respectively) as shown in Fig. 5 kept at an initial
temperature of
. A slot jet of width 5 mm is injected at
0.5 m/s and temperature of
. The top boundary otherwise
(excluding faces through which jet is injected) is an outlet
boundary. All the remaining walls (side and bottom) are
adiabatic walls. Flow and thermal properties for the present
study are listed in Table 1. and are sourced from ref. [18]. A
uniform grid of 150×1000 cells is used along with a time step
of 1e-4 s for the numerical studies. The Woods metal-water
system corresponds to an Ambient Weber number (
) of
1.25 while that for Corium Sodium study is 2.2.
Table 1: Properties for the present study
Woods Metal
Property
– water
9700
Density
(kg/m3)
998.16
Dynamic
1.94e-3
Viscosity
1.002e-3
(Ns/m2)
168
Specific Heat
(J/kgK)
4180
Thermal
18.8
Conductivity
0.59953
(W/mK)
200°C
50°C
Surface
Tension
1
Coefficient
significantly higher than that for the wood metal studies. It can
also be seen that jet breakup is seen later and at longer jet
lengths in the case of corium studies. Moreover, there is
similarity between the behaviour of the two primary fluids.
Corium in Sodium
CoriumSodium
8756
857
6.49e-3
0.281e-3
460
1278.5
2.5
0.01 s
0.1 s
0.2 s
0.3 s
0.4 s
0.5 s
0.01 s
0.1 s
0.3 s
Evolution of
Temperature
Evolution of Volume Fraction
Woods metal in water
71.2
3000°C
460°C
0.484
4. RESULTS AND DISCUSSION
The evolution of volume fractions of corium (in
sodium) and woods metal (in water) as predicted by THYC-MP
code is shown in Fig. 6. The computational elements used for
the present study are of size 0.5 mm. The total number of
elements used is 1.5 lakhs. The studies reported here are
preliminary in nature designed to be more like demonstration
studies. From the contours of volume fraction in Fig. 6 it can be
seen that both large scale and smaller scale instabilities are
present. Due to the deceleration of the jet as it enters further into
the secondary fluid, the whole jet tends to bend and eventually
breakup. Smaller structures continuously break and separate
from the main jet. These structure mainly arise from the trailing
edge of the mushroom cloud like structure that arises in both
the cases studied. The mushroom like structure at the leading
edge of the jet of primary fluid can be ascribed to the presence
of Rayleigh Taylor instability modes. As the jet travels further
blobs of primary fluid break out from the trailing edge as can
be seen clearly from the plots of Fig. 6. It must be remembered
that the smallest structures predicted as part of this study is
limited by the mesh size used. The mesh size is limited by the
computational resources available and the efficiency of the
code. At present steps are being taken to improve the efficiency
of the code in addition to parallelisation that would allow
studies on finer grids. The jets of corium and woods metal are
injected into respective secondary fluids at the same velocity.
This makes the Weber number for the corium studies
0.01 s
0.1 s
0.2 s
0.3 s
0.4 s
0.5 s
0.01 s
0.1 s
0.3 s
Evolution of Volume Fraction
Evolution of
Temperature
Figure 6: Evolution of volume fraction and temperature for
the two fluid systems studied.
In addition, the shedding of blobs of primary fluid from
the trailing edge of the mushroom like structure is much more
pronounced in the case of corium-sodium combination. This
can be explained by the significantly higher Weber number for
corium flow in sodium. Even though the velocity of the jet
injection is same, the movement of corium is significantly faster
into the column of secondary fluid. From the contours of
temperature, the effect of high thermal conductivity of sodium
with respect to water can be clearly seen. This can be concluded
from the significantly cooler corium jet w.r.t the woods metal
jet after 0.3 s. The range for the two plots is decided by the
extreme domain temperatures (
to
). Both the jet
5
and ambient temperatures in the case of corium-sodium study
is significantly higher. In addition, phase change is not
modelled. Thus, these results are indicative in nature. Thus, as
seen in both the case studies the jet of corium (or woods metal)
becomes completely fragmented after penetrating into the
column of secondary fluid for a distance of about half of the
column depth. This is desirable as it proves that corium jet
would be completely fragmented before reaching tank bottom
and would be well distributed without any clumping.
[5]
[6]
[7]
[8]
[9]
[10]
Figure 7: Jet Break-up length with respect to time for the
two fluid systems studied.
The jet break-up length extracted from plots of jet
interface are presented in Fig. 7. It is seen that jet break-up
length increases i.e. the jet grows in time, eventually breaking
at an intermediate length. This leads to sudden shortening of the
jet. The corium and woods metal jets grow at similar rates even
though the length of corium jet is slightly greater.
[11]
[12]
5. CONCLUSION
The development of a VOF based code for simulation
of two phase flow and heat transfer (THYC-MP) is described.
The code developed uses advanced algorithms for interface
reconstruction, advection and curvature calculations. The flow
solver is force balanced and is capable of reproducing Laplace
equilibrium conditions exactly. Application on the important
problem of corium movement in sodium arising as a result of a
severe accident in fast reactor is shown. The results show the
capability of the developed code towards capturing important
jet instability modes. Further extension in the form of including
phase change physics and speeding up through parallelisation
is in progress.
[13]
[14]
[15]
REFERENCES
[1]
S. C. Chetal et al., “The design of the Prototype Fast
Breeder Reactor,” Nucl. Eng. Des., vol. 236, no. 7–8,
pp. 852–860, 2006,.
[2]
B. Raj, P. Chellapandi, and P. R. V. Rao, Sodium Fast
Reactors with closed fuel cycle. Boca Raton: CRC
Press Taylor and Francis Group, LLC, 2015.
[3]
A. J. Sudha et al., “Progress in Nuclear Energy
Experimental simulation of downward molten material
relocation by jet ablation of structures and fuel coolant
interaction in a fast reactor,” Prog. Nucl. Energy, vol.
105, no. December 2017, pp. 194–201, 2018.
[4]
J. Rouault et al., “Severe Accidents,” in Handbook of
[16]
[17]
[18]
6
Nuclear Engineering, D. G. Cacuci, Ed. New York:
Springer Science+BusinessMedia LLC, 2010, pp.
2651–2661.
Y. Iwasawa and Y. Abe, “Progress in Nuclear Energy
Melt jet-breakup and fragmentation phenomena in
nuclear reactors : A review of experimental works and
solidi fi cation e ff ects,” vol. 108, no. March, pp. 188–
203, 2018.
Y. S. Oh et al., “Breakup behavior of a molten metal
jet,” Int. J. Heat Fluid Flow, vol. 50, pp. 27–37, 2014.
S. Thakre, L. Manickam, and W. Ma, “A numerical
simulation of jet breakup in melt coolant interactions,”
Ann. Nucl. Energy, vol. 80, pp. 467–475, 2015.
M. Lin, M. Zhong, Y. Li, M. Yuan, and Y. Yang,
“Numerical analysis on molten droplet hydrodynamic
deformation and surface waves under high pressure
pulse,” Ann. Nucl. Energy, vol. 77, pp. 133–141, 2015.
Y. Zhou, J. Chen, M. Zhong, J. Wang, and M. Lv,
“Numerical simulation of metal jet breakup, cooling
and solidification in water,” Int. J. Heat Mass Transf.,
vol. 109, pp. 1100–1109, 2017.
H. Cheng, S. Cheng, and J. Zhao, “Study on corium jet
breakup and fragmentation in sodium with a GPUaccelerated color-gradient lattice Boltzmann solver,”
Int. J. Multiph. Flow, vol. 126, p. 103264, 2020.
R. K. Maity, T. Sundararajan, and K. Velusamy, “An
enhanced piecewise linear interface construction
template based on c0 correction,” in Proceedings of
48th National Conference on Fluid Mechanics and
Fluid Power (FMFP), 2021.
R. Scardovelli and S. Zaleski, “Interface reconstruction
with least-square fit and split Eulerian-Lagrangian
advection,” Int. J. Numer. Methods Fluids, vol. 41, no.
3, pp. 251–274, 2003.
J. López, J. Hernández, P. Gómez, and F. Faura, “A
volume of fluid method based on multidimensional
advection and spline interface reconstruction,” J.
Comput. Phys., vol. 195, no. 2, pp. 718–742, 2004.
R. K. Maity, T. Sundararajan, and K. Velusamy, “An
accurate interface reconstruction method using
piecewise circular arcs,” Int. J. Numer. Methods Fluids,
vol. 93, no. 1, pp. 93–126, 2020.
J. Hernández, J. López, P. Gómez, C. Zanzi, and F.
Faura, “A new volume of fluid method in three
dimensions—Part I: Multidimensional advection
method with face-matched flux polyhedra,” Int. J.
Numer. Methods Fluids, vol. 58, no. 8, pp. 897–921,
Nov. 2008.
S. J. Cummins, M. M. Francois, and D. B. Kothe,
“Estimating curvature from volume fractions,”
Comput. Struct., vol. 83, no. 6–7, pp. 425–434, 2005.
Y. Renardy and M. Renardy, “PROST: A parabolic
reconstruction of surface tension for the volume-offluid method,” J. Comput. Phys., vol. 183, no. 2, pp.
400–421, 2002.
L. Manickam, S. Bechta, and W. Ma, “On the
fragmentation characteristics of melt jets quenched in
water,” Int. J. Multiph. Flow, vol. 91, pp. 262–275,
2017.