# Numerical Investigation of Air Entrapment Dynamics for High-Speed Thermal Spraying
## 4 Results and Discussion  
This model was applied to simulate a molten nickel droplet (\(d_{0}\) = 54 \(\mu\)m, \(h_{0}\) = 3 \(\mu\)m, \(T_{\text{d0}}\) = 2100 K) impacting on a stainless-steel substrate (\(L\times H_{2}\) = 160 \(\mu\)m \(\times\) 25 \(\mu\)m, \(T_{\text{s0}}\) = 293.15 K) with an initial velocity v and air surroundings (\(L\times H_{1}\) = 160 \(\mu\)m \(\times\) 90 \(\mu\)m, \(T_{0}\) = 293.15 K). Their physical properties are listed in Table 1.  
Triangular and mapped mesh were used, and the domains of the droplet and the contact area with the substrate were refined, as shown in Figure 3a. A mesh independence test was performed by checking the pressure at the point (1,1) at \(t\) = 0.1 \(\mu\)s. As shown in Figure 3b, when the mesh number is larger than 221,651, the mesh independence could be achieved, and this mesh is used for the following simulation. In general, the entrapped air bubbles during the whole process could be divided into impacting air bubbles (IM bubbles) and spreading air bubbles (SP bubbles) (Figure 4). The former results from the instant impact and the latter is entrapped due to the inhomogeneous solidification of the molten droplet. IM bubbles universally exist for impacting droplets either with [25] or without phase transition [5]. As shown in Figure 4A1, an air film with ripples is entrapped just after the droplet impacts the substrate (\(t\) = 0.12 \(\mu\)s) due to the capillary waves generated by the energy of the impaction [5]. The propagation speed of the wave could be estimated by 1.5 (2\(\pi\sigma/\rho\lambda\))\({}^{0.5}\) (\(\lambda\) is the capillary wave length, \(\lambda\) << \(d_{0}/2\)) [26], which is much greater than the retraction speed \(-(\sigma/\rho\delta)^{0.5}\) (\(\delta\) is the air film thickness) [10]. Consequently, capillary waves converge when the crests touch the substrate, dividing the air film into individual toroidal bubbles. Meanwhile, to minimize the surface tension, the IM bubbles retract towards the center (\(t\) = 0.16-2.8 \(\mu\)s) [5] and finally merge into a \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Melt Point** & **Density** & **Thermal Conductivity** & **Specific Heat** & **Latent** & **Viscosity** & **Surface Tension** \\  & **(K)** & **(kg/m\({}^{3}\))** & **(W/m-K)** & **(J/kg\({}^{3}\)K)** & **(kJ/kg)** & **(P\({}^{3}\)-s)** & **(N/m)** \\ \hline \hline \multirow{2}{*}{Nickel} & \multirow{2}{*}{1728} & 7850 (L) & 69.2 (L) & 735 (L) & \multirow{2}{*}{292} & \multirow{2}{*}{0.006 (L)} & \multirow{2}{*}{\(1.6\) (L)} \\  & & 8450 (S) & 80 (S) & 595 (S) & & & \\ \hline Stainless- & \multirow{2}{*}{1723} & \multirow{2}{*}{7854} & \multirow{2}{*}{15} & \multirow{2}{*}{480} & \multirow{2}{*}{—} & \multirow{2}{*}{—} & \multirow{2}{*}{—} \\ Steel & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties of Ni and stainless steel (L: Liquid and S: Solid). Figure 2: Comparison with experimental results of [PERSON] [25]: (**a**) Evolution of the impacting process (Reprinted with permission from Ref. [25]. Copyright 2000 Elsevier) and (**b**) spreading factor. sizeable toroidal bubble (\(t=3.0\) us). The formed toroidal bubble is finally detached from the substrate (\(t=3.10\) us), leaving a void in the center of the splat after complete solidification; this was observed in both experiments [27] and previous numerical simulations [28]. In addition to the IM bubbles, as shown in Figure 4, more bubbles are entrapped during the spreading process. Solidification is gradually completed with the spreading propagating from the bottom, which directly contacts the substrate upwards and from the edge to the center. The splat edge is unstable during this process, and fingers could be formed, as shown in Figure 4, at \(t=0.16\) us, \(0.26\) us, \(0.40\) us, \(1.20\) us. The fingers will either fall back to the substrate, entrapping air, or be cut off as secondary droplets (\(t=1.0\) us, \(2.4\) us).