Vis Comput (2011) 27: 241–248
DOI 10.1007/s00371-010-0531-1
O R I G I N A L A RT I C L E
Realistic simulation of mixing fluids
Shiguang Liu · Qiguang Liu · Qunsheng Peng
Published online: 17 November 2010
© Springer-Verlag 2010
Abstract Recently, simulation of mixing fluids, for which
wide applications can be found in multimedia, computer
games, special effects, virtual reality, etc., is attracting more
and more attention. Most previous methods focus separately
on binary immiscible mixing fluids or binary miscible mixing fluids. Until now, little attention has been paid to realistic simulation of multiple mixing fluids. In this paper,
based on the solution principles in physics, we present a
unified framework for realistic simulation of liquid–liquid
mixing with different solubility, which is called LLSPH. In
our method, the mixing process of miscible fluids is modeled by a heat-conduction-based Smooth Particle Hydrodynamics method. A special self-diffusion coefficient is designed to simulate the interactions between miscible fluids. For immiscible fluids, marching-cube-based method is
adopted to trace the interfaces between different types of fluids efficiently. Then, an optimized spatial hashing method is
adopted for simulation of boundary-free mixing fluids such
as the marine oil spill. Finally, various realistic scenes of
mixing fluids are rendered using our method.
Electronic supplementary material The online version of this article
(doi:10.1007/s00371-010-0531-1) contains supplementary material,
which is available to authorized users.
S. Liu () · Q. Liu
School of Computer Science and Technology, Tianjin University,
Tianjin, China
e-mail: lsg@tju.edu.cn
S. Liu
State Key Lab of Virtual Reality Technology and System,
Beihang University, Beijing 100191, China
Q. Peng
State Key Lab of CAD&CG, Zhejiang University, Hangzhou
310058, China
Keywords LLSPH · Mixing fluids · Physically based
modeling · Miscible · Immiscible
1 Introduction
Fluid scenes are very common natural phenomena which
are often included in movie special effects and digital video
games. There are two categories of fluids, according to their
components: single fluids and mixing fluids. Single fluids
are such that there is only one fluid in them: such as smoke,
fire, water, etc. Mixing fluids consist of multiple interacting
fluids: such as floating oil in water, sugar dissolving in coffee, etc. Until now, most methods focused on realistic simulation of single fluids. In recent years, some researchers
proposed methods for simulation of mixing fluids. However,
these methods can only be used to simulate a special type
of mixing fluids, for example, immiscible or miscible fluids. Little attention has been paid to realistic simulation of
multiple mixing fluids. In this paper, based on the solution
principles in physics, we present a unified framework for realistic simulation of mixing fluids with different solubility.
We propose an LLSPH (Liquid–Liquid Smooth Particle Hydrodynamics) model to simulate the mixing process
of multiple fluids with different solubility. Differently from
previous works, LLSPH can express the interactions between different liquid–liquid mixings under a unified framework. For the simulation of miscible fluids, a special selfdiffusion coefficient is designed to calculate the interactions
among miscible fluids. For the simulation of immiscible fluids, marching-cube-based method is adopted to trace the interfaces among different types of fluids. To extend the simulation to boundary-free domain, we adopted an optimized
spatial hashing method for fast identification of the interior
domain and the boundary domain. The main contributions
of our paper can be summarized as follows:
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• LLSPH model is proposed to simulate the mixing process
of multiple fluids with different solubility under a unified
framework.
• By introducing solution principles in physics into LLSPH, different mixing fluids can be simulated by only designing a special self-diffusion coefficient and calculating
the interfaces which are simple and efficient.
• Our method can also be used to simulate mixing fluids in
a boundary-free domain such as the phenomenon of the
marine oil spill. As far as we know, it is the first attempt
to simulate such phenomenon in computer graphics.
The paper is organized as follows. In Sect. 2 we give a
brief survey of related work. Then we give an introduction
to SPH theory in Sect. 3. In Sect. 4 we propose the model of
LLSPH. Section 5 discusses the implementation of LLSPH
under different conditions and gives the rendering results.
Conclusions and future work conclude the paper.
2 Related work
Fluid simulation has been a hot research area in computer
graphics, with wide applications in movie special effects,
computer games, virtual reality, etc. Until now, there are
mainly two types of methods for realistic simulation of
fluids, which are the Eulerian based methods and the Lagrangian based methods. The former focus on 3D grids and
calculate the grid parameters at each time step while the latter represent fluids with a collection of particles and then
trace their motion. After Stam [1] proposed the method of
stable fluids allowing large time step calculation, Eulerian
based methods are widely used for fluid simulation [2–6].
However, to capture more details with Eulerian based methods, the division of the space should be fine, which would
require large amounts of calculations. On the contrary, Lagrangian based methods [7] are more flexible to model fluids with different resolutions. They are also suitable for implementation and user interaction. Among them, SPH [8] is
widely used which comes from computational astrophysics.
Stam et al. [9] were the first to apply SPH to simulate gas and
fire phenomena. Takeshita et al. [10] used particle method
for explosive flames. In addition to these compressible phenomena, SPH was also used for the simulation of liquids [7,
11]. Premože et al. [12] improved SPH by MPS (Moving
Particle Semi-implicit) method. MPS needs solving complex Poisson equations, and thus is time-consuming. Recently, Becker et al. [13] simulated free surface flows using weakly compressible SPH. This method can be used for
volume-preserving low-viscosity liquids while only appropriate for medium-scale and small-scale phenomena. The
above methods can deal with single fluids well, however,
they are not suitable for realistic simulation of mixing fluids.
S. Liu
Currently, there are a few methods for simulation of mixing fluids, which separately focus on miscible and immiscible fluids. For the simulation of miscible fluids, Mizuno et
al. considered volcanic clouds as consisting of two miscible fluids: one was magma and the other was the entrained
gas [14]. They simulated this phenomenon by employing a
single-phase non-viscosity Navier–Stokes equation, without
considering the interaction between the two components of
the mixture. Zhu et al. proposed a Two-Fluid Lattice Boltzmann Model [15]. With this model, they simulated miscible
binary mixtures like pouring honey into water, Coca Cola
into wine, etc. For the simulation of immiscible fluids, Hong
et al. presented novel techniques for simulation of bubbles, a
kind of mixing fluid of gas and water, considering the interaction between liquid and gas [16]. Volume-of-Fluid (VOF)
method was used to track the bubble surface in liquid. They
also upgraded the method of VOF to level set for simulation
of discontinuous fluids [17]. More recently, they simulated
bubbly water by incorporating a new bubble model based
on SPH into an Eulerian grid-based simulation [18]. By this
method, lively motion of bubbly water with small scale details can be animated efficiently. Long lasting bubbles simulated by level-set method can suffer from a small but steady
volume error that accumulates to a visible amount of volume change. Kim et al. [19] propose to address this problem
by using the volume control method. To realistically animate the pouring of a glass of beer, Cleary et al. [20] presented a discrete particle-based method allowing of creating
a very realistic animation of bubbles in fluids. The above
methods require a full 3D fluid solver with free surface and
surface tension. To simulate bubble and foam effect at high
frame rates, Thürey et al. [21] coupled SPH simulation with
a shallow-water-based particle model. Boling water is another kind of immiscible fluid. Mihalef et al. [22] simulated
the boiling phenomenon based on physics principles. Muller
et al. [23] proposed a method for modeling fluid–fluid interaction based on the SPH. This method made possible the
simulation such as boiling water, the dynamics of a lava
lamp, etc. For the simulation of multiple immiscible fluids, Losasso et al. [24, 25] simulated the interactions among
multiple liquids by extending the particle level-set method.
To handle both miscible fluids and immiscible fluid simulations, Park et al. [26] presented an LBM-based framework,
in which the low order advection procedure made their results diffusive. As LBM consumed a large amount of memory, this method is not suitable for high-resolution domains.
Kang et al. [27] improved it by using a method of volume
fractions. Lenaerts [28] simulated the mixing phenomena
between fluids and granular materials such as sand particles.
This method focused on the interactions between liquid and
granular materials which cannot be used to model liquid–
liquid mixing with different solubility. Recently, Bao et al.
Realistic simulation of mixing fluids
[29] proposed a volume-fraction-based method for simulation of mixing fluids. It is a grid-based method which can
generate good mixing results of multiple fluids.
Although some methods are proposed for simulation of
miscible fluids and some for immiscible fluids, little attempt
was reported on handling the mixing fluids in a unified way.
How to simulate multiple mixing fluids with different solubility under a unified framework is really a challenging
work.
3 SPH model
SPH is an interpolation method for particle systems [8]. The
field quantities that are defined only at discrete particle locations can be evaluated anywhere in space by a continuous
function A(r) at a position r. The contributing particles are
determined by a kernel function W of finite support associated with each particle. The smoothing kernel is important
for the accuracy and validity of the simulation result. A suitable kernel must have the following properties:
W (r, h) dr = 1,
lim W (r, h) = δ(r),
(1)
Ω
h→0
where r is any point in Ω, W is a smoothing kernel with h
as the width, δ is Dirac’s delta function. The kernel in (1)
must be normalized. In addition, it must also be positive.
The numerical equivalent of the kernel function can be approximated by a summation interpolant,
A(r) =
Aj Vj W (r − rj , h),
(2)
where j is the iterator over all particles, Vj is the volume
attributed to particle j, rj the position, and Aj is the value
of any quantity A at rj . If we use m and ρ to represent the
mass and density of the fluid, respectively, then (2) can be
expressed as
mj
W (r − rj , h).
(3)
A(r) =
Aj
ρj
Equation (3) is the basic formulation of the SPH method,
which can be used to approximate any continuous quantity
field.
4 LLSPH model for mixing fluids
To simulate mixing fluids realistically, we must first clear
the solubility relationships and the properties of the liquid
solvent. Solvents are usually divided into two categories: polar solvents and non-polar solvents. Polar solvents can dissolve ionic compounds, as well as the dissociated covalent
compounds. However, the polar solvents can only dissolve
non-polar covalent compounds. For example, the salt, as an
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ionic compound, can be dissolved in water, but cannot be
dissolved in ethanol [30]. We will take binary mixing fluids
as an example to introduce LLSPH model for the simulation
of mixing fluids.
4.1 Simulation of binary miscible and immiscible fluids
According to the above theories, we divide mixing fluids
into polar fluids and non-polar fluids. Polar fluids and nonpolar fluids are both miscible, but a polar fluid and a nonpolar fluid are immiscible. In this paper, we mark the particles belong to polar and non-polar fluids as 1 and −1,
respectively. Suppose fluids a and b are miscible, and the
mass ratios of fluids a and b at position r are respectively
Ca and Cb . Since the sum of Ca and Cb is 1, if Ca is calculated, Cb can be obtained simultaneously. Denote Ca as C,
then we can express its changes during the mixing process
as ρ dC/dt = D∇ 2 C, where ρ is the density of fluid a and
D the self-diffusion coefficient. Based on the SPH model,
the above equation can be calculated as
dCi
D mj 2
Cj
=
∇ W (ri − rj , h).
dti
ρi
ρj
(4)
Unfortunately, (4) does not obey the law of conservation of
mass, and its solution is unstable if the particles are in random arrangement. Take two particles i and j as an example:
the exchanged energy calculated from particle i and particle
j using (4) is not the same. To address this problem, we introduce the heat conduction theory [31] to SPH model and
calculate C as the following:
dCi mj 4Di Dj
(Ci − Cj )F |ri − rj | , (5)
=
ρi ·
dt
ρ j Di + Dj
(6)
(ri − rj )F |ri − rj | = ∇W (ri − rj , h),
where Di and Dj are the self-diffusion coefficients of particle i and particle j, Ci and Cj are the ratios of fluid a in
particle i and particle j . From (5) and (6) we can see that
the mass ratio of each fluid in each particle is determined by
Ci − Cj . If Ci is larger than Cj , the concentration of fluid a
is higher than of fluid b, and then fluid a will diffuse to fluid
b; and vice versa. The special design of the self-diffusion
coefficient makes both fluids obey the law of conversation
of mass.
Differently from miscible fluids, no diffusion occurs between immiscible fluids, so the mass ratio of each fluid
at each particle will be unchanged. In our experiments,
the mass ratio of each fluid at each particle is kept constant. To render the immiscible fluids realistically, the interfaces between them should be calculated. Here, we adopted
marching-cube principle to extract the interfaces [32, 33].
Figure 1 shows a cell for marching-cube calculation. The
numbers from 0 to 7 identify each vertex, among which the
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S. Liu
Fig. 1 A cell in the
marching-cube calculation
value of vertex 6 is more than the threshold of the interface
in the cell, and it is shown in red. The points of intersection
of the interface can be calculated using the method of linear
interpolation. After the triangles in each cell are obtained,
we connect them to generate the final interfaces. We call the
above method to model the interactions among the liquid–
liquid mixing fluids as LLSPH (Liquid–Liquid Smooth Particle Hydrodynamics) in this paper.
4.2 Simulation of multiple mixing fluids
LLSPH can be easily extended for simulation of multiple
mixing fluids, some of which may be miscible and others
immiscible. So for each particle i, there will be several kinds
of fluids mixing in it. To model the interactions between
these fluids, we should calculate the mass ratio of each kind
of fluids. Suppose there are n kinds of fluids 1, 2, . . . , n in
particle i, among which 1, 2, . . . , n − r are non-polar fluids
and the other are polar fluids. The mass ratio Cit of fluid t in
particle i can be expressed as
ρit ·
dCit mit 4Dit Dj
(Cit − Cj )F |rit − rj | , (7)
=
dt
ρj Dit + Dj
where 1 ≤ t < n, 1 ≤ j ≤ n, j = i and particle j has the
same polarity with particle i; the meaning of the parameters
in (7) is the same as in (5). With (7), the mass ratio of n − 1
kinds of fluids can be obtained. The mass ratio of the last
kind of fluid can be calculated as
Cin = 1.0 −
n−1
Cit .
(8)
t=1
If there are polar fluid and non-polar fluid together, the final
interfaces to be rendered are collections of each interface
generated by a polar fluid and a non-polar fluid.
4.3 Simulation of boundary-free mixing fluids
Previous works on simulation of mixing fluids assumed a
boundary restriction which simplified the solution of partial differential equations. In this paper, LLSPH can also be
extended to boundary-free domain. The key difficulty is to
Fig. 2 Hashing values to be calculated for certain cells
cancel the restrictions of the interior and the boundary of
the calculation domain and allocate the memory to the cells
containing fluid particles at high rates. We adopted the optimized spatial hashing which was used for collision of deformable objects to address this problem [34].
We first divide the coordinates of the particle position r
(x, y, z) by the given grid cell of size h and rounded down
to the next integer. Then, using the hash function hash, the
above discretized 3D position (i, j, k) will be mapped to a
1D index id and the coordinate information will be stored
in a hash table (see Fig. 2) at this index id which can be
expressed as id = hash(i, j, k). In our implementation, we
adopt the following hash function:
hash(x, y, z) = (xp1 xor yp2 xor zp3) mod n,
(9)
where n is the size of the hash table, p1, p2 and p3 are large
prime numbers. To improve the efficiency of the algorithm,
we select the particles in a spherical neighborhood of radius
h which is just the size of a grid cell for calculation.
5 Implementation and results
Based on the above model, we have generated various
scenes of mixing fluids using Pov-ray. All simulations were
run on a Pentium IV with 2.0 G RAM, nVidia 7800 display
card. The resolution of the rendering image is 1024 × 768
and the rendering rate is about several minutes.
The kernel function is important for the rendering result.
For the calculation of the density, C field and the surface
tension, we used the following kernel function: W (r, h) =
315/64πh9 · (h2 − |r|2 )3 (0 ≤ |r| ≤ h) [7]. However, using
this kernel function to calculate gradient and Laplace operator will generate negative values. So, Spiky kernel function [7] and viscous kernel function are adopted to calculate the pressure and the viscosity force, respectively. The
gravity
forces of each particle include the gravity fi
, the prespressure
pressure
sure fi
, the viscous force fi
and the surface tension fisurface (see the Appendix).
Figure 3 shows the simulation result of miscible mixing
fluids, in which the blue ink is flowing into a glass of water.
Realistic simulation of mixing fluids
245
Fig. 3 Simulation of miscible mixing fluids
Fig. 4 Simulation of binary immiscible mixing fluids
Table 1 Parameters for the simulation of miscible mixing fluids
Fluids m (kg)
ρ0 (kg/m3 ) k
Ink
1 × 10−3 1000.0
Water
1 × 10−3
1000.0
μ
σ
5.0 10.0 0.0728
2.0 5.0
D
dt (s)
1.5 1 × 10−3
0.00728 1.5
1 × 10−3
One can see from Fig. 3(a) through (c) that the ink and water
are diffusing to each other. Figure 3(d) through (f) shows
the enlarged area of interest. The parameters used in this
experiment are listed in Table 1.
An obvious visual feature of the immiscible fluids is the
interface between them. Figure 4 is the simulation result of
binary immiscible mixing fluids. The interface between two
fluids during their interaction is clearly generated using our
method. The parameters for these two fluids used in our implementation are listed in Table 2.
Our method can be extended to easily simulate multiple
mixing fluids (see Fig. 5). Among these three kinds of fluids, the white one is miscible with the red one, while the
Table 2 Parameters for the simulation of immiscible mixing fluids
Fluids m (kg)
ρ0 (kg/m3 ) k
μ
σ
D
dt (s)
Blue
1 × 10−3 1000.0
10.0 25.0 0.0728 1.5 1 × 10−3
Green
1 × 10−3
10.0 25.0 0.0728 1.5 1 × 10−3
1000.0
green one is immiscible with the above two. According to
their properties, our algorithm can simulate the accurate result of their interaction. The numbers of the white fluid, the
red fluid and the green fluid are 50,000, 4000 and 4000, respectively. The parameters for these three fluids used in our
implementation are listed in Table 3. Figure 6 shows the simulation result of mixing fluids in an open area. Figure 6(a)
through (c) shows the animation of oil in water. The parameters for Fig. 6 are listed in Table 4.
Park et al.’s [26], Kang et al.’s [27] and Bao et al.’s [29]
methods can achieve good rendering results for different
mixing fluids. They all are Eulerian grid-based methods. It
is not easy to deal with arbitrary boundary conditions using
246
S. Liu
Fig. 5 Simulation of multiple mixing fluids
Fig. 6 Simulation of boundary-free mixing fluids
Table 3 Parameters for the simulation of multiple mixing fluids
Fluids m (kg)
ρ0 (kg/m3 ) k
White 1 × 10−3 1000.0
Red
1 × 10−3
Green
1 × 10−3
1000.0
1000.0
μ
2.0 3.5
σ
D
dt (s)
0.00728 1.5 1 × 10−3
5.0 12.5 0.00728 1.5
1 × 10−3
5.0 12.5 0.00728 1.5
1 × 10−3
them. On the other hand, our new algorithm is a particlebased method. By combining with the specially designed
Table 4 Parameters for the simulation of boundary-free mixing fluids
Fluids m (kg)
ρ0 (kg/m3 ) k
μ
σ
D
dt (s)
Water
1 × 10−3
1000.0
2.0 5.0 0.00728 1.5 1 × 10−3
Oil
1 × 10−3
850.0
4.0 8.0 0.00728 1.5 1 × 10−3
spatial hashing method, it can easily tackle with arbitrary
boundaries.
Realistic simulation of mixing fluids
6 Conclusions and future work
This paper presents a novel method for realistic simulation
of liquid–liquid mixing under a unified framework, which
is called LLSPH. This method can deal with both multiple
miscible mixing fluids and immiscible mixing fluids. Our
algorithm can also be used to simulate mixing fluids in a
boundary-free domain such as the marine oil spill, making
realistic simulation of mixing phenomena in a large-scale
open area possible.
LLSPH can simulate the details of the mixing process
of multiple fluids very well. This method can also be used
to model the fluid–solid coupling by sampling particles on
the surface of the solid. However, our method suffers from
the high computing cost for simulation of large-scale mixing
fluids. LLSPH combined with grid-based method may be a
solution, in which grid-based method can be adopted to simulate the areas with little mixing process between different
fluids. This is one of the focuses for our future research. Tai
equation [35] and the Predictive-Corrective Incompressible
SPH described in [36, 37] may further optimize the incompressibility of the fluids in our simulation. Our future work
also anticipates the use of the advanced GPUs to further accelerate the algorithms.
Acknowledgements This work was supported by Natural Science
Foundation of China under Grant No. 60803047, the Specialized Research Fund for the Doctoral Program of Higher Education of China
under Grant No. 200800561045, the Open Project Program of the State
Key Laboratory of Virtual Reality Technology and System, Beihang
University under Grant No. BUAA-VR-10KF-1, the Open Project Program of the State Key Lab of CAD&CG, Zhejiang University under
Grant No. A0907. The authors would also like to thank the reviewers for their insightful comments which greatly helped improving the
manuscript.
Appendix
Each fluid particle will be exerted pressure force by other
particles surrounding it, which can be expressed by LLSPH
as
(pi + pj ) mj pressure
fi
∇W (ri − rj , h),
=−
2
ρj
where pi = k(ρi − ρ0 ), pj = k(ρj − ρ0 ), k is a coefficient
and ρ0 is the density of the stationary fluid. The friction between fluid molecules reduces the kinetic energy, converting
to heat. Internally generated resistance in the fluid is called
the viscous force, which is:
mj
viscosity
∇ 2 W (ri − rj , h),
fi
=μ
(ui − uj )
ρj
where μ is the viscous coefficient. The above two forces belong to internal forces. Gravity is a kind of external force
247
gravity
= ρi g, g is the accelerawhich can be expressed as fi
tion of gravity. The other important external force is surface
tension which can be calculated as follows:
ni
surface
· ni ,
= σ · −∇
fi
|ni |
where σ is the surface tension coefficient, ni is the outward
normal of the fluid surface, which can be obtained as
mj W (ri − rj , h).
ni = ∇
ρj
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Shiguang Liu is an Associate Professor at School of Computer Science and Technology, Tianjin University, P.R. China. He graduated
from Zhejiang University and received a PhD from State Key Lab
of CAD & CG in 2007. His research interests include realistic image synthesis, computer animation
and virtual reality, etc.
Qiguang Liu is a graduate student
at School of Computer Software,
Tianjin University, P.R. China. His
research interest is computer graphics and computer vision.
Qunsheng Peng is a Professor in
the State Key Lab of CAD&CG at
Zhejiang University. His research
interests include realistic image synthesis, virtual reality, infrared image synthesis, Scientific visualization and biological calculation, etc.
He graduated from Beijing Mechanical College in 1970 and received a
Ph.D. from the Department of Computing Studies, University of East
Anglia, in 1983. He is currently
the Vice Chairman of the Academic Committee, State Key Lab of
CAD&CG, Zhejiang University and
is serving as a member of the editorial boards of several international
and domestic journals.