An icicle generation model based on the SPH method
Masaki Sato†
Jun Kobayashi††Tokyo Denki University
Adachi-Ku, Tokyo, Japan

Tomoaki Moriya†

Yuki Morimoto†
Tokiichiro Takahashi†‡‡
UEI Research{m-sato, j-kobayashi, moriya, yuki, toki}@vcl.jp

Fig. 1 Comparison between actual (Left) andcomputer generated water drop (Right)

1. Introduction
There has been a lot of research done on the water drop model.
However, for the icicle growth model, the one by [Kim] has beenreported. Although there has been a lot of research on melting iceand icicles, there is no model that can simulate the developing
state of an icicle as each water drop drips and freezes. Our model
can deal with the phase transition process from start to end, fromwater into ice, by using an extended model of the SPH method.

2. Previous Work
2.1 Interfacial tension method
In order to represent the shape of a real water drop, an interfacial
tension model, which is an extended SPH method, has beendevised by [Abe]. This model made it possible to represent theshape of a water drop with different contact angles, as well as theflat shape of a large volume water drop.
2.2 Ice dynamics as a thin-film Stefan problem
Kim et al. [Kim] proposed an icicle generation model that cangenerate complicated ice shapes, such as in the case of a limited
water supply due to a Stefan problem upon freezing, or in the caseof an icicle tip model based on an experiment. Their method
expresses realistic icicle modeling.

3. Proposed Method
3.1 Behavior of a water drop
An icicle grows due to freezing water drops that drip on its
surface. The shapes of icicles vary greatly according to how thewater drops adhere to them. However, as the interfacial tension
model devised by Abe et al. doesn't consider any kinetic contact
angles for water drops, it is difficult to precisely represent thestate of a water drop dripping onto an icicle.
We have extended the model of Abe et al. [Abe] and propose awater drop model that puts an emphasis on dynamic behavior. The
biggest difference between the resting state and the motion stateof a water drop shape is the contact angle. The contact angle of awater drop in the motion state is called the kinetic contact angle(See Supplement Figure 1). By using this model, and withconsideration of the kinetic contact angle, we can represent awater particle dripping on a solid surface with more realistic
movement (See Figure 1, Supplement Figure 2 and Video 1).
3.2 Heat Transfer and Freezing between Particles
In order to represent the state of freezing water, we install a heat

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Fig. 2 Comparison between actual icicles (Left) anda sequence of icicle growth simulation (Right)
transfer system between particles into the model mentioned in theprevious section. The temperature of a water particle i is regarded
as ti. This model doesn’t distinguish air, water particles, iceparticles or the water-ice interface, and regards anything thattouches particle i as particle j.

Qi as the amount of heat transferred to particle i can becalculated by the following Equation (1):.

(1)

Here, Ni is the total number of particles contacting particle i, isthe specific heat transfer rate for the contacting particles i and j,
is the contact area. After its temperature drops to 0˚C, thewater particle goes through a phase transition into ice and itslatent heat of condensation is lost.

4. Results
Figure 1, Supplement Figure 2 and Video 1 show the water drop
generated by our model. Due to the effect of the kinetic contact
angle, a water drop shape with tails extending both to thetraveling direction and the reverse direction was generated.
Figure 2 and Supplement Videos 2 and 3 show the result oficicles generated by our model. The icicles’ shape becomesthinner as it goes from its root to the tip. Furthermore, ripples aregenerated on the surface of the icicles.
Figure 3 shows the difference between Kim et al.’s results andours. Although our model is simple, we obtained a similar result.

Fig. 3 Comparison between Kim et al.’s result (Left) andour results (Right) of icicle generation.

5. Conclusions
We have proposed a model to simulate the developing state of anicicle as each water drop drips and freezes. By using this model,
we were able to generate a model that could generate an iciclewith ripples on its surface, the same as a real icicle, and thatgrows with each drip of water. We will consider high speed
processing for a larger simulation in the future.

References
ABE, T., IMURA, M., IKEDA, S., MANABE, Y., AND CHIHARA, K.,
2009. Particle-based liquid simulation for animation of waterdrops. Journal of Institute of Image Information andTelevision Engineers, Vol.63, No.7, 951-956.
THEODORE, K., DAVID, A., AND, MING, C, L., 2006. Modeling ice
dynamics as a thin-film Stefan problem. Proc. ACM
SIGGRAPH/Eurographics
SymposiumonComputer
Animation, 167-176.