## 5 Results and Discussion
### Foam Effects with Angular Momentum  
Three scenarios were devised to analyze the angular momentum-based foam advection technique proposed in this paper under various conditions. Based on these scenarios, the quality of foam effects using our method was compared with [PERSON] and [PERSON]'s method, which is a recent foam effects technique. First, as a simple test scenario to prove the superiority of the proposed technique, a scene in which water particles are sourced was produced. In this scene, the time-step was set to 0.006 and more than 200,000 water particles were used. Figure 10 is a foam simulation using the method of [PERSON] and [PERSON] [26]. When the sourced water particles interact with the still water surface, angular momentum is strongly generated and rotational motion appears in the foam particles as well. This phenomenon is similarly shown in Figure 7, and the motion in which the foam particles are rolled more strongly by angular momentum compared to the previous technique is well expressed (see **A** in Figure 10a). As shown in **B** to **D**, most of the angular motions were expressed well, and in the wave crest (the highest part of a wave, see Figure 11), remarkably lively foam particle motion was expressed. Figure 11: Wave concepts. Figure 10: Comparison of the rotating-emitter scene in water simulation. The related video has been submitted as Supplementary Material. (**a**) Our method. (**b**) [PERSON] and [PERSON] [26]. Figure 12 is a scene where two boxes stir water, and foam effects are naturally created according to the movement path of the box. Due to the interaction of the water particles and the box, the vortex motion was clearly conveyed to the foam effects (see **A** and **B** in Figure 12). Additionally, as shown in **C** to **E**, the wave crest created by the interaction of water particles clearly shows the motion as if the foam particles are being rolled in. As a result, compared to the previous method that expressed foam effects in an over-cloudy manner due to viscous foam particles, our method expressed the foam effects lively and showed clear foam patterns. Figure 13 is the foam effects that appear when an external force in the form of tornado is added. If the previous results were generated by the sourcing of water particles or the impact of the solid, Figure 13 is the foam effects that are expressed due to the external force that is gradually strengthened. In addition, we examined how angular momentum affects the foam effects by using an external force of an unusual form such as tornado. In the 38 th and 53 rd frames of Figure 13a showing the gradual rotation, our technique expressed the foam effects symmetrically, whereas the previous technique did not reflect the angular momentum, so the effect of the tornado force was not well shown. Figure 14 shows the foam effects created by the interaction of water particles inside the U-shaped corridor. In this figure, the blue region is where there is no angular momentum, and as the color goes from green to red, the angular momentum becomes stronger. It can be seen that the angular momentum of the foam particles is also strongly expressed as the water is bent into a curved shape by the interaction between the water particles and the wall (see **A** in Figure 14). As mentioned earlier, angular momentum was strong in the wave crest (see **B** and **C** in Figure 14), and foam particles that simply bounce into the air like splashes have relatively weak angular momentum, so it can be seen that they are expressed in blue (see **D** in Figure 14). Figure 15 shows the difference of foam effects with or without our method. The red particles are generated using the previous technique [26], and the white particles are generated by advection based on angular momentum. Although the results were made in the scenes of the same configuration, the difference in motion of the foam particles was clearly revealed. Figure 12: Rotating two boxes in water simulation. The related video has been submitted as Supplementary Material.