## 5 Results and Discussion  
### Validation Test for Angular Momentum  
We conduct experiments in four scenarios to test the validation of the angular momentum method proposed in this paper. In the first experiment, the angular momentum proposed in this paper was applied to rotational motion conversion (see Figure 5). The particles were placed in a normal distribution in 2D space, and initial velocity and angular velocity were applied. In order for the particle to rotate, the initial velocity of the particle, \(\mathbf{v}_{p}^{0}\), was set as follows according to the relative position based on \(\mathbf{x}_{0}\), the center of mass of the area where the particles are distributed: \(\mathbf{v}_{p}^{0}=(\mathbf{x}_{0}-\mathbf{x}_{p})\times\mathbf{\omega}_{0}\) where \(\mathbf{\omega}_{0}\) is the initial angular velocity and was set to \((0,0,10)\) in the above experiment. The experiment in Figure 5a shows the result when all particles move with angular velocity \(\mathbf{\omega}_{p}=\mathbf{\omega}_{0}\), and the experiment in Figure 5b shows the general particle-based fluids. In Figure 5a, as the particles continue to rotate while maintaining their initial rotation momentum, the region is separated by centrifugal force,and it is confirmed that the rotation momentum of the separated region is also maintained. In Figure 5b, it can be seen that the rotation momentum gradually weakens. Second, we tested the rotational motion of particles in a particle-based water simulation (see Figure 6). In the falling water scene, rotational motion expressed when water particles interact with each other was observed. In the figure, the particle color is set to easily recognize the rotational motion, and when the falling water particles interact, it can be seen that the rotational motion is strongly expressed in the swaying area. Figure 7 shows a close-up of the part where the particles interact. In Figure 7b, the rotational motion of particles is hardly expressed, whereas in Figure 7a, our method well expresses the particle rotation by angular momentum that appears in the process of the particles interacting with each other. This rotational motion also affects the foam particles, improving the detail of secondary effects. In the third scenario, we tested the rotational motion of particles in a particle-based smoke simulation with buoyancy force (see Figure 8). It was designed to source smoke particles after filling the simulation space with ghost particles. The smoke particle color is expressed by diffusing the color instead of the smoke density in a way that diffuses to nearby smoke and ghost particles, and the SPH gradient kernel (\(\
abla\)W) is used. Position-based fluids [11] were used as the basic algorithm. Figure 5: Angular momentum in the process of rotation. The related video has been submitted as Supplementary Material. (**a**) With angular momentum (our method). (**b**) Without angular momentum. Figure 6: Two-dimensional water particle simulation with our method. The related video has been submitted as Supplementary Material. Figure 7: Comparison of close-up views in Figure 6. The related video has been submitted as Supplementary Material. (**a**) Our method. (**b**) Without angular momentum. In the fourth scenario, a two-dimensional experiment was conducted to check whether the intrinsic patterns of foam particles were well expressed by the angular momentum we proposed (see Figure 9). The movement of the water particles was controlled by the wave function, and the experiment was conducted based on the same distribution of foam particles. The wave function to move the water particles is as Equation (17)). \[(x_{i},y_{i})\leftarrow\left(x_{i0}+v_{0}^{x}t,y_{0}sin\left(2\pi\frac{x}{ \lambda_{0}}\right)cos(2\pi f_{0}t)\right) \tag{17}\]  
where \(v_{0}^{x}\) is the velocity of the wave, and \(x_{i0}\) is the initial \(X\)-axis coordinate of the water particle. \(\lambda_{0}\) is the wave length and \(f_{0}\) is the wave frequency.