## 1 Introduction  
When two or more adjacent droplets merge on a superhydrophobic surface, spontaneous droplet removal can be induced. This droplet jumping phenomenon was reported by [PERSON] and [PERSON] in 1969 during experiments with mercury droplets and was examined by [PERSON] and [PERSON] in 2009, observing the jumping phenomenon in droplet coalescence [1]. Coalesced droplet jumping has garnered significant attention in various industrial fields, such as self-cleaning [2; 3], anti-icing/frosting [4; 5], and enhancement of condensation heat transfer [6; 7]. Many researchers have investigated the mechanism and applicability of droplet jumping, developing applied technologies by introducing various factors such as droplet-solid interactions and various surface structures [8]. Superhydrophobic surfaces can rapidly detach when colliding droplets, similar to lotus leaves [9; 10]. This surface was able to promote the spontaneous removal of droplets by maintaining the contact angle between droplet and solid at over \(150^{\circ}\) and keeping the contact angle hysteresis very small. Numerous experiments and theoretical studies have been conducted on the microstructure of superhydrophobic surfaces, and research on improving superhydrophobic surface structures continues. The performance of superhydrophobic surfaces is expressed by the very high contact angle or residence time of droplets, where the residence time can be approximated by \(\tau\sim(\rho D^{3}/8\sigma)^{1/2}\) under inviscid fluid conditions [11], where \(\rho\), \(D\), and \(\sigma\) correspond to density, equivalent diameter, and surface tension, respectively. For small droplets, behavior is dominated by inertia force and surface tension. Important dimensionless numbers to express this are the Weber number (\(We\)) and the Reynolds number (_Re_): the Weber number, _We_ = \(\rho DV^{2}/\sigma\), represents the inertia force relative to the surface tension, whereas the Reynolds number, \(Re=\rho DV/\mu\), represents the inertia force relative to the viscous force, where \(V\) is velocity and \(\mu\) is the viscosity of the liquid. A few research studies on droplet coalescence jumping have been investigated from various aspects, such as jumping speed and direction, as well as energy conversion. On a flat surface without introduced structures, the jumping speed when two droplets merge is roughly proportional to the inertia-capillary speed of \([\sigma/(\rho D)]^{1/2}\)[12]. Regarding energy conversion, when two droplets combine, surface energy is converted into kinetic energy as part of the motion, inducing a vertical component of velocity on the substrate. On flat surfaces, the jumping speed of droplets can only be limited to a certain maximum, so introducing structures on the surface to increase the energy conversion rate has been proposed [13; 14; 15; 16; 17; 18]. [PERSON] et al. [13] introduced triangular prisms to increase jumping speed and reported a high energy conversion rate. [PERSON] et al. [14] verified an increase in rebound speed and energy conversion rate by introducing an egg-tray structure. [PERSON] et al. [15] investigated the energy conversion mechanism by observing internal flow during droplet coalescence. [PERSON] et al. [16] and [PERSON] et al. [17] investigated jumping speed and energy conversion by introducing micro-grooves and U-shaped grooves, respectively, and discussed the horizontal component of jumping direction. [PERSON] et al. [18] proposed asymmetric ridge structures to investigate horizontal movement of droplets on superhydrophobic surfaces and validated the hypothesis through experiments and simulations. On a horizontal superhydrophobic surface, the dynamics of droplet coalescence-jumping exhibit variations in velocity and droplet morphology, depending on factors such as surface properties, structures, and temperature. Despite these differences, the fundamentals of coalesced droplet jumping remain similar, as follows. Droplets in close proximity form a small liquid bridge.