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Vortex Fluid for Gaseous Phenomena Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2005) K. Anjyo, P. Faloutsos (Editors) Vortex Fluid for Gaseous Phenomena Sang Il Park† and Myoung Jun Kim‡ † Carnegie Mellon University, Pittsburgh, PA – [email protected] ‡ Ewha Womans University, Korea – [email protected] Abstract In this paper, we present a method for visual simulation of gaseous phenomena based on the vortex method. This method uses a localized vortex flow as a basic building block and combines those blocks to describe a whole flow field. As a result, we achieve computational efficiency by concentrating only on a localized vorticity region while generating dynamic swirling fluid flows. Based on the Lagrangian framework, we resolve various boundary conditions. By exploiting the panel method, we satisfy the no-through boundary condition in a Lagrangian way. A simple and effective way of handling the no-slip boundary condition is also presented. In treating the no-slip boundary condition, we allow a user to control the roughness of the boundary surface, which further improves visual realism. Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Three-Dimensional Graphics and RealismAnimation 1. Introduction ibility condition, as does the superposition of multiple parti- cles. In this paper, we present a novel approach to generating gas animation based on the vortex formulation. Our scheme is Information about the strength of the circulation carried by different from previous methods in computer graphics be- the particles forms another field over space and time, called a cause we are concerned with a higher level field description. vorticity field. Vortex methods deal with the evolution of this Instead of simulating the individual velocity vector at every vorticity field and consequently make it possible to deter- fixed grid point, we simulate a set of localized basis veloc- mine the position and the circulation strength of each vortex ity fields. By placing and scaling each of the basis fields particle over time. Although these vortex methods have not based on underlying physical laws and user-given boundary yet received much attention in the graphics community, they conditions, we synthesize a plausible fluid animation. This have been regarded as distinct alternatives to the established method allows us to concentrate only on the region of impor- computational techniques such as finite difference methods tance that contributes to the whole flow area. Consequently, a in the computational fluid dynamics community. small number of computational elements are needed for gen- erating a fluid animation even though the whole flow area is Besides the efficiency from the localized computation as large or unbound. mentioned above, the vortex method is suitable for the sys- tem based on the Lagrangian description; we track the vortex We choose a purely circulating flow as a basis velocity field, particles convected along the velocity field freely without and represent it as a particle, vortex particle. This circulating grid. This Lagrangian description has several potential ad- fluid is an appropriate basic building block because the vivid vantages over grid-based Eulerian methods. First, it is free spinning movement and property of incompressibility often of numerical dissipation from which the grid-based methods make flow look fluid-like; a combination of even a few spin- generally suffer in advection process. Therefore, the gener- ning fields can generate a visually complex swirling move- ated flow is less damped and does not require any additional ment in the flow. Furthermore, a circulating flow field from reinforcement to keep the fluid moving. Second, the number each vortex particle automatically satisfies the incompress- of numerical elements can be easily adapted to the complex- c The Eurographics Association 2005. S. I. Park & M. J. Kim / Vortex Fluid for Gaseous Phenomena ity of the flow. Finally, it is suitable for either open or closed Smoothed Particle Hydrodynamics(SPH), Lagrangian meth- environments. ods have been gradually attracting more interest. The Lagrangian description also allows us to represent a Based on the Eulerian description, Kajiya and Von Herzen boundary condition easily and accurately. Grid-based meth- adapted fluid dynamics for the first time for generating fluid ods require that boundaries such as walls or obstacles should animation [KvH84]. Later, Foster and Metaxas presented a be discretized along with the grid or the grid should be realistic gas simulation by considering full 3D dynamics of well aligned with the boundaries in advance. However, our the fluid [FM97]. Although it could generate realistic move- method uses conventional geometry representations such as ment of the gas, it required a very fine timestep for a stable polygonal objects consisting of line segments in 2D or trian- simulation. By using a semi-Lagrangian formulation in the gles in 3D for describing the boundary condition. advection process, Stam guaranteed unconditional stability in simulation, which, thus, enables a large timestep and inter- Our contribution in adapting the vortex method to visual ap- active controls over the fluid [Sta99]. However, this method plications is as follows: First, for efficient simulation, we ex- suffered from the rapidly vanishing movement due to the nu- ploit the Lagrangian nature of the vortex method and present merical dissipation. Fedkiw et al. alleviated this problem by an efficient method for evolving the vorticity fields without adding the artificial vorticity confinement term [FSJ01]. introducing a system of matrix equations. Second, we re- solve the various boundary conditions in a Lagrangian way, Most methods based on the Lagrangian description are based in which the boundary geometry is treated accurately. Fi- on the SPH. Stam and Fiume were the first in the graphics nally, we address the importance of the no-slip boundary community to exploit it to simulate fire and gases [SF95]. condition for visual appearance of the fluid and present a Desbrun and Cani-Gascuel used it both for modeling a de- simple and effective scheme for it. We also suggest a con- formable surfaces and solving the governing equation of trollable parameter for the roughness of the boundary sur- substances [DCG98, DCG99]. SPH and its extension Mov- face. ing Particle Semi-implicit (MPS) were also used to simulate water [MCG03,PTB∗03]. The remainder of this paper is organized as follows: We first review related work in Section 2. The basic concept and the Differently from SPH, the vortex method has not got much formulation of the vortex method are given in Section 3.In attention since it was first introduced by Gamito et al. to Section 4, we present our numerical method in detail. We the graphics community [GLG95]. Although they showed explain how to satisfy the boundary condition in Section 5. impressive demonstrations of a turbulent jet stream, their Experimental results are provided in Section 6. In section 7, method had several limitations: they limited their simulation we discuss limitations and further issues and conclude this to simple 2D cases without any boundary walls or obstacles. paper. Furthermore, their method relied on a fixed grid for the vor- ticity evolution and did not take advantage of the Lagrangian nature of the vortex method. In this paper, we fully exploit 2. Related Work the potential of the vortex method for visual simulation of In the computer graphics community, the essence of fluid gasses including full 3D extension, imposition of bound- animation is laid upon visual appearance but not upon strict ary conditions and control of the boundary surface prop- physical accuracy. Thus, early methods concentrated on de- erty. Most recently, Salle et al. presented a method, in which veloping ad-hocs which can imitate a visual behavior of a they exploited vortex particles for the vorticity confinement fluid based on intuition of the authors [SF92, SF93, Sta97]. on top of the conventional Eulerian method [SRF05]. Our However, due to the many control parameters provided, the method is different from theirs because our method is en- quality of the resulting animations was heavily dependent tirely based on the Lagragian description. upon the skill of an animator. Another problem is the lack of interactivity: it was very hard to make the fluid react on an external force or satisfy the user given boundary conditions. 3. Vortex Method To overcome those problems, the computer graphics com- In this section we will briefly introduce the basic concept munity has adopted physics-based schemes from the Com- and formulations of the vortex method. First, we introduce putational Fluid Dynamics. There have been two different the definition of the vorticity and its transportation equation points of view in dealing with the fluid dynamics. The first describing the fluid dynamics (Section 3.1). Second, the dis- view is to describe the flow field at the fixed computational cretization of the vorticity field is presented for numerical points, called the Eulerian method. The other method, called computation (Section 3.2). Finally, we deal with calculat- the Lagrangian method, concentrates on the motion of an in- ing the velocity field numerically back from the descritized dividual particle which roughly imitates a molecule of the vorticity field (Section 3.3). For further mathematical de- fluid. Eulerian methods have been important in simulating tails and applications of vortex methods, we refer the read- fluid dynamics. However, due to the recent development in ers to the book by Cottet and Koumoutsakos [CK98], and c The Eurographics Association 2005. S. I. Park & M. J. Kim / Vortex Fluid for Gaseous Phenomena the survey articles by Leonard [Leo80] and Anderson and 3.2. Vorticity Discretization Graham [AG91]. We represent the vorticity field as a set of numerical ele- ments in order to facilitate the numerical computation. In 3.1. Vorticity Equation grid-based methods, this is done by using regularly distrib- uted nodal points, where the spatial differentiation of a given Let us start with the definition of vorticity.
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