Particle based Waterfall Simulation with Spray Cloud Emerging from
Basin
Nobuhiko Mukai, Yuto Hizono and Youngha Chang
Graduate School of Engineering, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya, Tokyo 158-8557, Japan
Keywords:
Physics based Simulation, Particle Method, Waterfall, Spray Cloud.
Abstract:
One of the most challenging issues is to simulate and visualize physical phenomena such as thunder, aurora,
avalanche and so forth. Among them, simulation of liquid behavior is difficult but most familiar to us. Then,
there are many researches related to water; however, there are not so many studies on waterfall. Some re-
searchers visualized waterfall behavior, which, however, was not based on physical simulations. Other papers
on physics based simulations demonstrated part of waterfall instead of the whole behavior. Then, we have
tried to visualize the whole behavior of waterfall with a physics based particle method. In order to simulate
the whole behavior of waterfall, huge amount of particles are needed so that our model divides the whole of
waterfall into three parts: water stream, splashing spray, and spray cloud. In the previous works, we have been
able to visualize the whole behavior of waterfall, where water stream was translated to splashing spray that
was also changed into spray cloud. Then, the previous method had splashing spray and spray cloud emerging
from water stream, while real waterfalls have spray cloud that appears from the basin instead of water stream.
Therefore, this paper proposes a model to generate spray cloud emerging from the basin instead of water
stream.
1 INTRODUCTION
In recent years, we can visualize almost everything
with computer graphics technology, and some studies
focus on the reality and others emphasize the accu-
racy. The reality based visualization is important for
games or movies; however, the accuracy is important
for the simulation. Unless there are physics based ex-
pressions, the reality is a little bit odd. Then, there are
many kinds of physics based simulations and the visu-
alizations. Among them, one of the most challenging
issues is to visualize liquid behavior, including ocean,
rivers, bubbles, spray and so on. However, there were
not so many studies related to waterfall. In addition,
some of them did not perform physics based simula-
tions, and others simulated part of waterfall instead of
the whole behavior.
Then, we have been researching how the whole
behavior of waterfall could be visualized since huge
amount of calculation resources are needed for
physics based simulations. In order to simulate the
whole behavior of waterfall with a normal PC, we
divide the model of waterfall into three parts: water
stream, splashing spray, and spray cloud. In addition,
water stream particles that flow from the lip of the
waterfall are translated into splashing spray particles
through isolated particles, and splashing spray parti-
cles are changed into spray cloud. Thus, spray cloud
emerges from water stream, while real waterfalls have
spray cloud appearing from the basin instead of water
stream.
Therefore, in this paper, we propose a waterfall
model that outputs spray cloud from the basin in-
stead of water stream. In the simulation, we employ
a hybrid method of particle and grid for the efficient
use of calculation resources, because the area of wa-
ter stream and splashing spray is limited, while spray
cloud disperses broadly. In addition, we employ ap-
propriate methods and the governing equations for
each part: particle based method with Navier-Stokes
equation, equation of motion for small particles, and
grid based method with Navier-Stokes equation.
2 RELATED WORKS
(Mould and Yang, 1997) surveyed some previous
studies related to water modeling, and indicated that
some were based on hydrodynamic theory, while oth-
ers presented the context based on the observation
Mukai, N., Hizono, Y. and Chang, Y.
Particle based Waterfall Simulation with Spray Cloud Emerging from Basin.
DOI: 10.5220/0006896500550061
In Proceedings of 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2018), pages 55-61
ISBN: 978-989-758-323-0
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
55

of real phenomena. (Iglesias, 2004) also researched
papers on realistic modeling, rendering, and anima-
tion of water, which were published during 1980s and
1990s. In addition, (Darles et al., 2011) presented
a survey on ocean simulation and rendering in com-
puter graphics field, and described that there were
two types of works: physically based methods us-
ing Navier-Stokes equations, and oceanographic re-
searches based on empirical laws.
Related to the presentation of ocean, (Hinsinger
et al., 2002) proposed interactive animation methods
of ocean waves that were located far from the coast,
and (Cui et al., 2004) presented another real-time sim-
ulation method for irregular long crest waves. In ad-
dition, (Dupuy and Bruneton, 2012) demonstrated a
scalable method of vast ocean scene. These methods
rendered ocean waves that were basically continuous
surfaces, and mesh models were used for many re-
searches. However, mesh models require re-meshing
if the topology change such as river stream breaking
occurs.
Then, (M¨uller et al., 2003) used a particle method
called SPH (Smoothed Particle Hydrodynamics),
which solved Navier-Stokes equations with surface
tension. They visualized water pouring into a glass
with point splatting and surfaces generated by march-
ing cubes. (Kipfer and Westermann, 2006) also used
SPH for realistic rendering of rivers that flowed from
a rock and filled the lake below.
For the simulation of liquid behavior, there are ba-
sically two types of methods: Eulerian (grid based)
and Lagrangian (particle based) methods. (Chentanez
and M¨uller, 2011) proposed an Eulerian simulation
method and optimized the grid for GPU (Graphics
Processing Unit) implementation, and (Nishino et al.,
2012) used a grid based method to present freezing
ice that had air bubbles.
On the other hand, (Foster and Fedkiw, 2001)
employed semi-Lagrangian method to handle viscous
liquids that interacts with 3D objects, and (Busaryev
et al., 2012) proposed a particle based algorithm
to simulate bubble behavior, where bubble particles
were generated with Voronoi diagram.
In addition, some researchers used hybrid meth-
ods of Eulerian and Lagrangian approaches. For in-
stance, (Hong et al., 2008) combined a particle based
method for bubbles with a grid based one for the back-
ground that was composed of large amount of water
and air, and (Chentanez and M¨uller, 2010) presented
a hybrid water simulation method to visualize spray,
splash and foam. (Miller, 1989) also introduced a
method for animating viscous fluids by consideering
collision between particles and obstacles, and (Sims,
1990) proposed a parallel particle rendering system
allowing to treat different shapes, sizes, colors and
transparencies. (Greenwood and House, 2004) pre-
sented a particle level-set fluid simulation algorithm
to generate various kinds of bubble shapes. Moreover,
(Geiger et al., 2006) used a particle level set algo-
rithm to present the main body of fluid and fine splash
particles, and (Kim et al., 2007) employed a level set
method to produce bubbles in liquid and gas interac-
tion. (Losasso et al., 2008) proposed a two-way cou-
pled simulation framework using both particle level
set for dense liquid volume and particle method for
diffuse regions.
In terms of waterfalls, (Mallinder, 1995) proposed
an idea of string texture that was continuous stream
since huge number of particles were needed if each
particle was independent. On the other side, (Foster
and Metaxas, 1997) developed a system for animators
to be able to control fluid animations without knowl-
edge on governing equations of fluid. (Howes and
Forrest, 1997) also developed a simple yet flexible
simulation system that generated visually convincing
rendering. In addition, (Sakaguchi et al., 2007) cre-
ated tools for generating a variety of animations from
underwater to waterfall, and the tools was used for
a movie of “Pirates of the Caribbean 3”. (Hardie,
2007) generated a waterfall with a particle method,
which works was accepted for ACM SIGGRAPH Art
Gallery. The aim of these methods was for animators
to generate realistic water related behavior.
On the other hand, (Takahashi et al., 2003) pro-
posed a new method called CIP (Cubic Interpolated
Propagation) to visualize dynamic behavior of flu-
ids with splashes and foam. (Hoetzlein and H¨ollerer,
2009) presented a method to extract surfaces from
particles. (Miyashita and Funahashi, 2012) also de-
veloped VR (Virtual Reality) learning system, where
the envelope surface was rendered around the key par-
ticles in real time. In addition, (Nielsen and Østerby,
2013) used an Eulerian two-continua simulation of air
and water to visualize waterfall behavior.
In fact, waterfall are composed of three parts: wa-
ter stream, splashing spray, and spray cloud; however,
the works mentioned above did not propose methods
to present the whole behavior of waterfall by physics
based simulation. Then, (Mukai et al., 2014) pro-
posed a method to visualize the whole behavior of a
waterfall including water stream, splashing spray, and
spray cloud, and (Nishibe et al., 2015) also visualized
spray cloud that was soaring from the basin. In ad-
dition, (Mukai et al., 2016) presented various kinds
of spray cloud behavior that changes according to the
environments. These results, however, showed spray
cloud emerging from water stream, while spray cloud
appears from the basin instead of water stream in real
SIMULTECH 2018 - 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
56

waterfalls. Therefore, this paper proposes a waterfall
model how spray cloud emerges from the basin.
3 WATERFALL MODEL
Figs.1 and 2 show the previous waterfall model,
which is divided into three parts: water stream,
splashing spray and spray cloud. The waterfall is
composed of many particles, and water particles flow
from a river into the lip and fall down to the basin
with a stream, which is called “water stream” in the
method. Particles in water stream are also divided
into three kinds: main stream, free surface, and iso-
lated particles, respectively as shown in Fig. 2. These
three particles are defined according to the density as
follows.
ρ
free
= α
free
ρ
main
(1)
ρ
iso
= α
iso
ρ
main
(2)
0 < α
iso
< α
free
< 1 (3)
where, ρ
main
, ρ
free
and ρ
iso
are the densities of main
stream, free surface and isolated particles, respec-
tively. ρ
main
keeps a constant value because water is
incompressible fluid.
Figure 1: Previous model of waterfall.
In the previous model, isolated particles gener-
ate splashing spray particles, which change into spray
cloud after a constant time. Then, there are many
isolated particles are generated in water stream be-
cause it spreads out gradually during falling down to
the basin, and the density reduces. As a result, spray
cloud emerges around water stream.
In real waterfalls, however, spray cloud emerges
from the basin instead of water stream. Then, we
consider that some water particles are broken and
changed into small splashing spray particles when
they collide with the water surface of the basin, and
splashing spray particles change into spray cloud that
emerges from the basin. Fig. 3 shows this concept.
Figure 2: Previous particle model of waterfall.
Figure 3: New particle model for splasing spray.
The amount of splashing spray particles depends
on the amount and the velocities of main stream par-
ticles that collide with the water surface, and the ve-
locity of air. The equation that decides the amount of
splashing spray particles is defined as the following.
Q
s
=
C
a
∗ Q
m
|V
m
|
|V
a
|
(4)
where, Q
s
and Q
m
are the amounts of splashing spray
and main stream particles, respectively. V
m
is the ve-
locity of a main stream particle that collides with the
water surface, and V
a
is the velocity of air. In addi-
tion, C
a
is an attenuation coefficient. The above equa-
tion is based on conservation of momentum, and it is
supposed that splashing spray particle has the same
velocity as that of air. Moreover, wall boundary con-
dition is applied to the velocities of main stream parti-
cles that collide with the water surface, which means
that the vertical component of the velocity is reversed
and the horizontal component is preserved consider-
ing the attenuation.
Particle based Waterfall Simulation with Spray Cloud Emerging from Basin
57

4 GOVERNING EQUATIONS
We employ a particle method of SPH (Smoothed par-
ticle Hydrodynamics) for main stream, free surface
and isolated particles since the simulation area is lim-
ited, and these particles obey Navier-Stokes equa-
tions. On the other hand, splashing spray particles
are very small so that equation of motion for small
particle is used as the governing equation. Spray
cloud also obeys Navier-Stokes equation, however,
it spreads broadly in the simulation space. Then, a
grid based method is applied to the simulation. Each
method is described briefly as follows.
4.1 Water Stream
Navier-Stokes equation for a particle method is writ-
ten as the following (S. Koshizuka, 2008).
∂u
u
u
∂t
= −
1
ρ
∇p+ ν∇
2
u
u
u+ f
f
f (5)
where, u
u
u is velocity, t is time, ρ is density, p is pres-
sure, ν is viscosity, and f
f
f is external acceleration.
A physical quantity φ(x
x
x
i
), its gradient ∇φ(x
x
x
i
) and its
Laplacian ∇
2
φ(x
x
x
i
) at a position x
x
x
i
are defined respec-
tively as follows.
φ(x
x
x
i
) = Σ
j
m
j
φ
j
ρ
j
W(x
x
x
i
− x
x
x
j
) (6)
∇φ(x
x
x
i
) = Σ
j
m
j
φ
j
ρ
j
∇W(x
x
x
i
− x
x
x
j
) (7)
∇
2
φ(x
x
x
i
) = Σ
j
m
j
φ
j
− φ
i
ρ
j
x
x
x
j
− x
x
x
i
|x
x
x
j
− x
x
x
i
|
∇W(x
x
x
i
− x
x
x
j
)
(8)
where, m
j
, ρ
j
and x
x
x
j
is the mass, the density and the
position of a particle j, respectively. W is called ker-
nel function, and W
d
, W
p
, and W
v
are the kernel func-
tions of density, pressure, and viscosity, respectively
(M¨uller et al., 2003; Fujisawa, 2013).
In addition, the gravity and the air resistance are
considered as the external acceleration f
f
f, which is de-
fined as follows.
f
f
f = g
g
g+
1
ρ
p
V
p
F
F
F
air
(9)
F
F
F
air
=
1
2
π

D
p
2

2
ρ
a
C
r
|u
u
u
p
|u
u
u
p
(10)
where, g
g
g is gravity, and ρ
p
, V
p
, D
p
, and u
u
u
p
are den-
sity, volume, diameter and velocity of a particle, re-
spectively. ρ
a
and C
r
are density of air and coefficient
of air resistance, respectively, and the air resistance is
calculated by approximating the particle as a sphere.
4.2 Splashing Spray
Equation of motion for small particle, which is used
as the governing equation of splashing spray, is de-
scribed as the following (Ushijima, 2004).
F
F
F
r
+ F
F
F
b
= ρ
p
π
6
D
3
p
du
u
u
p
dt
(11)
F
F
F
r
=
1
2
π

D
p
2

2
ρ
a
C
r
|u
u
u
a
− u
u
u
p
|(u
u
u
a
− u
u
u
p
)
(12)
F
F
F
b
= (ρ
p
− ρ
a
)
πD
3
p
6
g
g
g (13)
where, ρ
a
and u
u
u
a
are density and velocity of air, re-
spectively and other parameters are the same as those
in Eqs.(9) and (10). Splashing spray particles come
out of isolated particles, and the initial velocity of
splashing spray particle is the same value as that of
isolated particle. On the other hand, the velocity of
air is the same value as that of spray cloud, which is
calculated in the next section.
4.3 Spray Cloud
We employ Stable Fluid (Stam, 1999) as the grid
based method that guarantees stable behavior of fluid.
Navier-stokes equation for spray cloud is written as
follows, which is different from Eq.(5) because ad-
vection term (u
u
u· ∇)u
u
u should be considered in a grid
based method.
∂u
u
u
∂t
= −(u
u
u· ∇)u
u
u−
1
ρ
∇p + ν∇
2
u
u
u+ f
f
f (14)
∇· u
u
u = 0 (15)
In addition, we consider vapor density for visual-
ization of spray cloud. Vapor density (ρ
v
) is calcu-
lated with the following equation (Kondo, 1994).
ρ
v
= 1.293
273.15
273.15 + T
p
1013.25
(1− 0.378
e
p
) (16)
where, 1.293[kg/m
3
] is the density of dried
air, 273.15[degree] is a coefficient of translation
from Celsius temperature to absolute temperature,
T[degree] is Celsius temperature, p[hPa] is pressure,
1013.2[mb] is air pressure, 0.378 is a coefficient of
translation from the gravity of dried air to the gravity
of wet air, and e[hPa] is vapor pressure.
The density difference generates the force that
makes spray cloud move up. Then, the following ac-
celeration is added to f
f
f in Eq.(14) as an external ac-
celeration of the lower grid.
SIMULTECH 2018 - 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
58

(
ρ
l
− ρ
u
ρ
l
)g
g
g (17)
where, ρ
l
and ρ
u
are the densities of lower and upper
grids, respectively, and g
g
g is gravity.
5 SIMULATIONS
We have performed the waterfall simulation based on
the method described above. Waterfall is composed
of three parts: water stream, splashing spray, and
spray cloud, and three kinds of methods are applied
to each part: Navier-Stokes equation based on parti-
cle model, equation of motion for small particle, and
Navier-Stokes equation based on grid model.
In the simulation, huge amount of particles are
needed. Then, the simulation is divided into two
stages: algorithm for water stream, and algorithm
for splashing spray and spray cloud. Here, the water
stream simulation can be performed independently
because it is not affected by the result of splashing
spray and spray cloud. However, splashing spray and
spray cloud simulation is affected by the result of
water stream. Then, it should be performed after the
water stream simulation has finished. The simulation
algorithm is as follows.
<A: For water stream>
A1 Initialize and set parameters.
A2 Add main stream particles to the lip and remove
others from the basin.
A3 Detect free surface and isolated particles.
A4 Analyze the behavior of each particle with SPH.
A5 Calculate position, velocity and density of each
particle.
A6 Continue from A2 to A5.
< B: For splashing spray and spray cloud>
B1 Input positions, velocities and densities of main
stream, free surface, and isolated particles.
B2 Generate splashing spray particles and remove
them after a constant time.
B3 Analyze the behavior of splashing spray particles
with equation of motion for small particles.
B4 Generate spray cloud and remove them after a
constant time.
B5 Analyze the behavior of spray cloud with Stable
Fluid.
B6 Update air velocity, which affects splashing spray
particles, with the velocity of spray cloud for the
next time step.
B7 Continue from B2 to B6.
6 SIMULATION RESULTS
We have performed the waterfall simulation with a
normal PC that has Intel Core i7 2.8GHz CPU, 4GB
main memory, and GeForce GTX 670 with 4GB
memory. The maximum number of particles was
262,144, and the grid size for spray cloud was 80
3
.
Simulation results are shown in Fig. 4. In the figure,
spray cloud is colored in green for easy recognition.
In the previous method, splashing spray particles
are generated from isolated particles, which exit in
water stream and the basin. Then, spray cloud, which
is translated from splashing spray particles, appears
along water stream and from the basin as shown in
Fig. 4 (a). On the other hand, in the proposed
method, splashing spray particles only come out of
main stream particles that collide with the water sur-
face of the basin. Thus, spray cloud emerges only
from the basin, and it does not appear along water
stream and emerges only from the basin as in Fig.
4(b).
In addition, Fig. 5 shows the comparison of the
simulation result with a real waterfall called “Kegon
no Taki”, which is very famous waterfall in Japan.
Water falls down from the lip to the basin along wa-
ter stream, which spreads out gradually and has some
water masses, which are generated by considering air
resistance as one of the external forces. From the fig-
ure, the simulation result by the proposed method is
very similar to the real waterfall.
7 CONCLUSIONS
In this paper, we have proposed a model that gen-
erates splashing spray particles from the basin, and
performed the waterfall simulation. According to the
proposed model, splashing spray particles are gen-
erated from main stream particles that collide with
the water surface in the basin, and the amount of
the particles is defined according to the theory based
on conservation of momentum. As a result, spray
cloud emerged only from the basin and did not ap-
pear along water stream. On the comparison of the
simulation result with a real waterfall, the simulation
result was very similar to the real one with spreading
water stream and some water masses in them. How-
ever, spray cloud did not soar up because this simula-
tion did not consider the terrain surrounding the wa-
terfall, while the previous simulation used some ter-
rains, which helped spray cloud to soar up. In fact,
Particle based Waterfall Simulation with Spray Cloud Emerging from Basin
59

Figure 4: Comparison of the simulation result with the previous method.
Figure 5: Comparison of the generated waterfall with a real one.
some real waterfalls have spray cloud that soars up
with the help of the surrounding terrains, while oth-
ers have spray cloud soaring up from the basin even
in vast space. Therefore, in the future, we have to
consider the method for spray cloud to soar up with-
out the surrounding terrains by considering the force
generated from the density and the temperature differ-
ence, and also wind around the waterfall.
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