4.1. Establishment of the self-similar solutions for the early stage of impact

The governing equations for liquid and vapour flows can be simplified under certain key approximations. (i) Negligible viscosity: the liquid’s viscous terms can be safely neglected, primarily due to the high Reynolds number ( $Re\sim O(10^3)$ ) characteristic of the cases under investigation. This assumption effectively reduces the liquid phase to a potential flow, aligning with Helmholtz’s theorem. (ii) Young–Laplace Relation: the pressure difference between the droplet ( $P_l$ ) and the thin vapour film ( $P_v$ ) adheres to the Young–Laplace relation. (iii) Kinetic boundary condition: the advancement of the liquid interface is governed by the kinetic boundary condition. (iv) Lubrication theory: the vapour flow within the thin film can be effectively described using lubrication theory. This approximation is valid since the lateral length scale ( $L = \sqrt {hR_0}$ ) significantly exceeds the film thickness ( $h$ ), while remaining notably smaller than the curvature radius ( $R_0$ ) of the droplet. Besides, in the case of Leidenfrost impact, where boiling occurs at the bottom of the droplet, it is also essential to account for heat transfer and vapour effects in the lubrication equation. We incorporate these approximations into the modelling approach and obtain the following equations:

(4.1) \begin{equation} \nabla \cdot \boldsymbol {u} = 0 ,\end{equation}

(4.2) \begin{equation} \frac {\partial \boldsymbol {u}}{\partial t} + \frac {1}{2}\nabla \left | \boldsymbol {u}\right |^2 = - \frac {1}{\rho _l}\nabla P_l, \end{equation}

(4.3) \begin{equation} P_g + \sigma \kappa = P_l ,\end{equation}

(4.4) \begin{equation} \frac {\partial h^{\pm }}{\partial t} + u_{l}^{\pm } \frac {\partial h^{\pm }}{\partial x} = v_{l}^{\pm } ,\end{equation}

(4.5) \begin{equation} \frac {\partial h}{\partial t}=\frac {1}{r}\frac {\partial }{\partial r}\left (\frac {rh^{3}}{12 \mu _v} \frac {\partial P_{g}}{\partial r}\right ) - \frac {1}{r}\frac {\partial }{\partial r} \left ( \frac {rh \left (u^+ + u^- \right )}{2}\right ) + \frac {\lambda _v \Delta T}{\rho _v \mathcal {L} h}. \end{equation}

The isothermal parts of these equations are identical with those provided by Mani et al. (Reference Mani, Mandre and Brenner2010) and Mirjalili & Mani (Reference Mirjalili and Mani2020), while (4.5) incorporates the evaporation effect on the right-hand side. In brief, (4.1) is the continuity equation of the liquid phase, (4.2) describes the potential motion within the liquid droplet, (4.3) represents the balance between the surface tension force and the pressure difference across the boundary of the ethanol droplet, (4.4) is the kinematic boundary condition for the drop and pool surfaces and (4.5) characterizes the flow field in the thin film using the lubrication theory, which stems from the continuity in the vapour. The last term on the right-hand side of (4.5) accounts for evaporation, which involves balancing the heat flux with the released latent heat, as proposed by Sobac et al. (Reference Sobac, Rednikov, Dorbolo and Colinet2014) and Maquet et al. (Reference Maquet, Sobac, Darbois-Texier, Duchesne, Brandenbourger, Rednikov, Colinet and Dorbolo2016).

Table 2.

The characteristic scales of thickness, lateral length, pressure and time for different superheat regimes.

To ensure a consistent and standardized analysis, the selection of appropriate non-dimensional scales is paramount. In the initial falling stage, depicted in figure 2(a) where the minimum vapour film thickness remains near the axis ( $t \lt 0.3\, \rm ms$ ), the vapour pressure is conventionally scaled as $P_g \sim \mu _v R_0 U_0/ h^2$ based on lubrication theory. This characteristic scale has been widely employed in previous studies for both isothermal cases (Mani et al. Reference Mani, Mandre and Brenner2010; Mirjalili & Mani Reference Mirjalili and Mani2020) and Leidenfrost cases (Chantelot & Lohse Reference Chantelot and Lohse2021). However, we advocate a more rigorous approach by assuming $P_g \sim \mu _v R_0 (U_0 + U_e)/h^2 \sim \mu _v R_0 U_0 (1 + U_e/ U_0)/h^2$ for Leidenfrost impact problems. Here, $U_e = \lambda _v \Delta T/\rho _v h \mathcal {L}$ accounts for the additional effect of evaporation-induced velocity. Consequently, we obtain $P_g \sim \mu _v R_0 U_0/ h^2$ at low superheats ( $U_0 \gg U_e$ ), while it transforms into $P_g \sim \mu _v R_0 U_e /h{^2}$ at high superheats ( $U_e \gg U_0$ ). We will substantiate the rationale behind this inference in subsequent discussions.

Considering the necessity for deformation, where $\partial \boldsymbol {u}_l/\partial t \sim \nabla P_l/\rho _l \sim \nabla P_g/\rho _l$ , we can establish relevant scales for thickness, lateral length, pressure and time (see Appendix C for more details about the derivation process). It is noteworthy that these scales differ between impacts at low and high superheat regimes due to the distinct contributions of $P_g$ . Initially, for impacts in the low superheat regime ( $U_0 \gg U_e$ ), the scales converge with those from isothermal scenarios, as extensively documented in previous studies (Mani et al. Reference Mani, Mandre and Brenner2010; Hicks & Purvis Reference Hicks and Purvis2011; Bouwhuis et al. Reference Bouwhuis, van der Veen, Tran, Keij, Winkels, Peters, van der Meer, Sun, Snoeijer and Lohse2012), and are outlined in the left column of table 2. Conversely, in the high superheat regime ( $U_e \gg U_0$ ), where evaporation velocity dominates during the initial falling stage, we derive new scaling correlations, as presented in the right column of table 2. Notably, these characteristic scales deviate significantly from their isothermal counterparts. In order to confirm the presence of the high superheat regime and differentiate it from the conventionally established isothermal (or low superheat) regime, we calculate the evaporation-induced velocity $U_e$ with the thickness $h$ obtained from the numerical results. This velocity is then compared with the falling velocity $U_0$ for various impact conditions, as illustrated in figure 6. It is evident that, as the superheat increases above 200 K, marking the onset of the high superheat regime as depicted in figure 5(b), the ratio of $U_e/U_0$ is larger than 1, which means that the evaporation-induced velocity $U_e$ surpasses the falling velocity $U_0$ and the characteristic scale of thickness transforms to $h \sim R_{0}We^{-2/5}\mathcal {E}^{2/5}$ . Therefore, we can utilize the value of $U_e/U_0 = 1$ as a threshold to distinguish the two regimes. In addition, we will show in § 4.3 that, below this threshold, different characteristic scales of thickness should be adopted.

Figure 6.

Ratio of the evaporation-induced velocity $U_e$ to the initial falling velocity $U_0$ under different impact conditions, and the threshold is $U_e/U_0 = 1$ to distinguish those two superheat regimes. Note that the vapour thickness $h$ is obtained from numerical results to determine the evaporation-induced velocity, given by $U_e = \lambda _v \Delta T/\rho _v h \mathcal {L}$ .

Figure 7.

Numerical results depicting the minimum height of the central dimple ( $h_d$ ) at various impact conditions within the high superheat regime. (a) Illustrates the relationship between $h_d$ and $U_0$ for different $\Delta T$ values, while (b) shows the variation of $h_d$ with $\Delta T$ for different $U_0$ settings. The observed power-law behaviour closely aligns with the theoretical prediction in (4.6b), providing strong evidence that, in the high superheat regime, the evaporation term indeed becomes dominant during the initial falling stage.

Furthermore, according to (4.6b), if $U_e$ prevails over $U_0$ during the falling stage, we anticipate that the minimum central dimple height at the axis, $h_d$ , scales as $h_d \propto U_0^{-4/5}\Delta T^{2/5}$ . To validate this adjusted scaling hypothesis, we depict the numerical results of $h_d$ against $U_0$ and $\Delta T$ while varying the impact parameters in figure 7. Evidently, figure 7(a) discloses a power exponent governing $h_d \propto U_0^{-4/5}$ , while figure 7(b) demonstrates a discernible correlation of $h_d \propto \Delta T^{2/5}$ in the high superheat regime for $\Delta T \gt 200 \,\mathrm {K}$ . Consequently, these established correlations, substantiated by numerical data, affirm the validity of (4.6b), thereby validating the pressure scale of $P_g \sim \mu _v R_0 U_e/h^2$ in this regime.

4.2. Reconstruction of the self-similar solutions at high superheat regime

Recognizing that the impact problem in the early stage is featured by different sets of characteristic scales depending on the degree of superheats, we initially focus on the high superheat regime where evaporation significantly influence the behaviour of the thin vapour film. The governing equations (4.1)–(4.5) for liquid and vapour flows can be non-dimensionalized with the scaling relations of (4.6b)–(4.9b) as follows:

(4.10) \begin{equation} \frac {\partial \tilde {\boldsymbol {u}}}{\partial \tilde {t}} + We^{-3/5}\mathcal {E}^{1/5} \tilde {\nabla } \left | \tilde {\boldsymbol {u}}\right |^2 = -\tilde {\nabla } \tilde {P_l}, \end{equation}

(4.11) \begin{equation} \tilde {P_g} + We^{-6/5}\mathcal {E}^{1/5} \tilde {\kappa }= \tilde {P_l}, \end{equation}

(4.12) \begin{equation} \frac {\partial \tilde {h}^{\pm }}{\partial \tilde {t}} + We^{-3/5}\mathcal {E}^{1/5} \tilde {u}_{l}^{\pm } \frac {\partial \tilde {h}^{\pm }}{\partial \tilde {r}} = \tilde {v}_{l}^{\pm }, \end{equation}

(4.13) \begin{align} \frac {\partial \tilde {h}}{\partial \tilde {t}} &= \frac {St We^{-3/5}\mathcal {E}^{3/5}}{\tilde {r}}\frac {\partial }{\partial \tilde {r}}\left (\frac {\tilde {r}\tilde {h}^{3}}{12 } \frac {\partial \tilde {P}_{g}}{\partial \tilde {r}}\right )-\frac {We^{-3/5}\mathcal {E}^{1/5}}{\tilde {r}}\frac {\partial }{\partial \tilde {r}} \left ( \frac {\tilde {r}\tilde {h} \left (\tilde {u}^+ + \tilde {u}^- \right )}{2}\right )\nonumber\\[5pt] &\qquad + \frac {St We^{-3/5} \mathcal {E}^{3/5}}{\tilde {h}}, \end{align}

where the tilde symbol ‘ $\sim$ ’ is used to denote dimensionless variables.

Given that the impact problems under consideration generally exhibit high values of the Weber number ( $We\sim O(10)$ ) and small values of the evaporation number ( $\mathcal {E}\sim O(10^{-6})$ ), the second terms on the left sides of (4.10)–(4.12) can be safely disregarded. Additionally, the Stokes number is also very high ( $St\sim O(10^5)$ ), which results in $We^{-3/5} \mathcal {E}^{1/5} \ll St We^{-3/5} \mathcal {E}^{3/5} \sim O(1)$ . Consequently, the second term on the right-hand side of (4.13) is significantly smaller in comparison with the other terms. This allows us to derive the leading-order equations from the aforementioned equations

(4.14) \begin{equation} \frac {\partial \tilde {\phi }}{\partial \tilde {t}} = -\tilde {P}_l ,\end{equation}

(4.15) \begin{equation} \tilde {P}_g = \tilde {P}_l ,\end{equation}

(4.16) \begin{equation} \frac {\partial \tilde {h}}{\partial \tilde {t}} = c\frac {\partial \tilde {\phi } }{\partial \tilde {y}} ,\end{equation}

(4.17) \begin{equation} \frac {\partial \tilde {h}}{\partial \tilde {t}}=\frac {St We^{-3/5}\mathcal {E}^{3/5}}{\tilde {r}}\frac {\partial }{\partial \tilde {r}}\left (\frac {\tilde {r}\tilde {h}^{3}}{12 } \frac {\partial \tilde {P}_{g}}{\partial \tilde {r}}\right ) + \frac {St We^{-3/5}\mathcal {E}^{3/5}}{ \tilde {h}}, \end{equation}

where $\phi$ denotes the velocity potential such that $\boldsymbol {u} = \nabla \phi$ . The pre-factor $c$ in (4.16) depends on the material of the impacted substrate. Specifically, it takes a value of ‘1’ when a droplet impacts a solid substrate (Mani et al. Reference Mani, Mandre and Brenner2010; Duchemin & Josserand Reference Duchemin and Josserand2011) and ‘2’ when a droplet impacts the same liquid pool with low viscosity (Hendrix et al. Reference Hendrix, Bouwhuis, van der Meer, Lohse and Snoeijer2016; Mirjalili & Mani Reference Mirjalili and Mani2020). Given that the silicon oil pool has higher viscosity than the ethanol droplet, it is reasonable to assume that $c$ remains constant within the range of $1 \lt c \lt 2$ in our study. Importantly, the self-similar solutions established later are independent of the specific value of $c$ except for the prefactors, thus the particular value of $c$ is unimportant.

Since we are more concerned with the dynamics near the spreading front, the lubrication equation can be simplified further. It should be noticed that the squeezing term always poses $ {\partial h}/{\partial t} \lt 0$ due to the decreasing interface height, while the evaporation term $ {\lambda _v \Delta T}/{\rho _v \mathcal {L} h} \gt 0$ is set up due to continuous evaporation. Therefore, these two terms cannot be equivalent in (4.17). Moreover, from a physical standpoint, the increase in pressure within the vapour film results from the compression of film and evaporation production. Hence the pressure term should inherently balance either the compression or evaporation term in a simplified model; however, the latter two cannot be equated by neglecting the pressure term. As a consequence, a plausible conjecture emerges, suggesting that the evaporation-induced flow flux should be balanced by the pressure term in the high superheat regime. To support this hypothesis, we calculate the squeezing and evaporation terms appeared in (4.17) at $t = 0.36\, \rm ms$ for a case with $U_0 = 0.394\, \rm m\, s^-{^1}$ and $\Delta T = 400$ K ( $We = 8$ and $\mathcal {E} = 3.55 \times 10^{-6}$ for dimensionless values), as shown in figure 8(a). The dashed line corresponds to the position of the spreading front. It is evident that the squeezing term is negligible at the neck region, while the evaporation term is significantly larger. Besides, our numerical data verify that the two terms have opposite signs around the location of spreading front. Furthermore, in figure 8(b), the temporal evolution of the absolute value of the ratio between the evaporation term and the squeezing term is depicted for various $\Delta T$ values at the spreading front. This representation illustrates that the evaporation term consistently dominates and becomes more significant at higher superheat conditions. Based on our numerical observations, it is reasonable to simplify (4.17) as follows:

(4.18) \begin{equation} -\frac {1}{\tilde {h}}=\frac {1}{\tilde {r}}\frac {\partial }{\partial \tilde {r}}\left (\frac {\tilde {r}\tilde {h}^{3}}{12 } \frac {\partial \tilde {P}_{g}}{\partial \tilde {r}}\right ), \end{equation}

which indicates the evaporation term balances the pressure term at the high superheat regime actually.

Figure 8.

Comparison of the squeezing term ${\partial h}/{\partial t}$ (dash dotted line) and the evaporation term $({\lambda _v \Delta T})/({\rho _v \mathcal {L} h})$ (solid line) appearing in the lubrication equation (4.5). (a) Distribution of the two terms along the radial axis, corresponding to the particular case of $U_0 = 0.394\,\rm m\,s^{-1}$ , $\Delta T = 40\, \rm K$ at $t = 0.358\, \rm ms$ and $\Delta T = 400\, \rm K$ at $t = 0.36\, \rm ms$ , respectively, while the dashed line is the spreading front; (b) time evolution of the absolute value of the ratio between the two terms for various superheats of $\Delta T = 120, 400$ and $600$ K. It is clear that, near the spreading front, the evaporation term always dominates over the squeezing term while the two have opposite signs.

To find a self-similar solution at the spreading front, self-similar ansatze are widely adopted (Mani et al. Reference Mani, Mandre and Brenner2010; Duchemin & Josserand Reference Duchemin and Josserand2011; Mirjalili & Mani Reference Mirjalili and Mani2020; Chantelot & Lohse Reference Chantelot and Lohse2021). Such an ansatz provides self-similar expressions for the film thickness, vapour pressure and velocity potential, as denoted by the following equations:

(4.19a,b,c) \begin{align} \tilde {h} \left (\tilde {r},\tilde {t} \right ) = \tilde {h}_{min }(\tilde {t}) H(\xi ), \quad \tilde {P}_g \left (\tilde {r},\tilde {t} \right ) = \tilde {P}_{max }(\tilde {t}) \Pi (\xi ), \quad \tilde {\phi } \left (\tilde {r},\tilde {y},\tilde {t} \right ) = \tilde {\phi }_{0}(\tilde {t}) \Phi (\xi , \eta ) .\end{align}

In these expressions, $\xi = (\tilde {r}-\tilde {r}_{0}(\tilde {t}))/\tilde {l}(\tilde {t})$ and $\eta = \tilde {y}/\tilde {l}(\tilde {t})$ represent self-similar variables in the neck region. Here, $\tilde {r}_0(\tilde {t})$ corresponds to the $r$ -coordinate of the spreading front, while $\tilde {l}(\tilde {t})$ signifies the horizontal length scale. It is important to note that the solution in the neck region exhibits wave-like characteristics to the leading order. For instance, the time derivatives of variable $\tilde {h} (\tilde {r},\tilde {t} )$ have three contributions as follows:

(4.20) \begin{equation} \frac {\partial \tilde {h}}{\partial \tilde {t}} = H(\xi )\frac {{\rm d} \tilde {h}_{min }(\tilde {t})}{{\rm d} \tilde {t}} - \frac {\xi \tilde {h}_{min }(\tilde {t})}{\tilde {l}(\tilde {t})} \frac {{\rm d} \tilde {l}(\tilde {t})}{{\rm d} \tilde {t}}\frac {{\rm d} H(\xi )}{{\rm d} \xi } - \frac {\tilde {h}_{min }(\tilde {t})}{\tilde {l}(\tilde {t})} \frac {{\rm d} \tilde {r}_0(\tilde {t})}{d \tilde {t}}\frac {{\rm d} H(\xi )}{{\rm d} \xi },\end{equation}

which correspond to the decrease in the minimum height, change of the horizontal length scale and radial movement of the spreading front, respectively. Mani et al. (Reference Mani, Mandre and Brenner2010) evaluated the relative size of these terms and find that the last advection term is dominant. Thus it can be hypothesized that the neck region is wave-like in character as $\partial \tilde {h}/ \partial \tilde {t} \approx -(\dot r_0 \tilde {h}_{min }/\tilde {l} )\partial H / \partial \xi \approx -\dot r_0 \partial \tilde {h} / \partial \tilde {r}$ , where $\dot r_0(t)$ represents the front velocity, and a similar proposal can also be used for the pressure and velocity potential fields. Consequently, temporal derivatives can be approximated using spatial derivatives, represented as $\partial / \partial \tilde {t} \approx -\dot r_0(\tilde {t})\partial / \partial \tilde {r}$ . Substituting these self-similar expressions into (4.14)–(4.16) and (4.18), we obtain the following scaling relationships:

(4.21) \begin{equation} \frac {\dot r_{0} \tilde {\phi }_0 (\tilde {t})}{\tilde {l}(\tilde {t})}\frac {\partial \Phi }{\partial \xi } = -\tilde {P}_{max }(\tilde {t}) \Pi,\end{equation}

(4.22) \begin{equation} \frac {\dot r_0 \tilde {h}_{min }(\tilde {t})}{\tilde {l}(\tilde {t})}\frac {\partial H}{\partial \xi } = c \frac {\tilde {\phi }_0(\tilde {t})}{\tilde {l}(\tilde {t})} \frac {\partial \Phi }{\partial \eta }, \end{equation}

(4.23) \begin{equation} - \frac {1}{H \tilde {h}_{min }(\tilde {t})} = \frac {\tilde {h}^3_{min }(\tilde {t}) \tilde {P}_{max }(\tilde {t})}{\tilde {l}^2(\tilde {t})}\frac {\partial }{\partial \xi }\left (\frac {H^{3}}{12 } \frac {\partial \Pi }{\partial \xi }\right ) .\end{equation}

By combining equations (4.21) to (4.23) and utilizing the method of separating variables, a set of scaling laws for the dimensionless maximum vapour pressure and the horizontal length scale can be derived

(4.24) \begin{equation} \tilde {P}_{max } \sim \tilde {h}_{min }^{-2/3} ,\end{equation}

(4.25) \begin{equation} \tilde {l} \sim \tilde {h}_{min }^{5/3} ,\end{equation}

(4.26) \begin{equation} \tilde {\kappa }_{max } \sim \tilde {h}_{min }/\tilde {l}^2 \sim \tilde {h}_{min }^{-7/3} .\end{equation}

Upon obtaining these self-similar solutions, figure 9 presents the values of $\tilde {P}_{max }$ and $\tilde {\kappa }_{max }$ for various impact conditions, varying $U_0$ and $\Delta T$ , in the high superheat regime. These dimensionless values, re-scaled according to the newly established scaling correlations (4.24) and (4.26), collapse onto single curves. Furthermore, an additional noteworthy observation emerges from the descending rate of the curve featured in figure 9, revealing a conspicuous correlation between $\tilde {P}_{max }$ , $\tilde {\kappa }_{max }$ and $\tilde {h}_{min }$ . In order to give a more quantitative comparison between the numerical results and the derived self-similar solutions, we also conduct a linear fit on the data presented in figure 9. In figure 9(a), the fit gives the exponent of $-0.7$ , aligning closely with the predicted value of $-2/3$ as outlined in (4.24). In figure 9(b), in contrast to the single exponent of $-7/3$ predicted by (4.26), we have to acknowledge that there is a change in the slope from $-2.1$ to $-2.55$ for the numerical results. Considering the complexity of the problem and the comprehensive inclusion of the neglected effects in our numerical simulations, such as inertia, viscosity and surface tension, these slight deviations are reasonable to some extent. Consequently, the self-similar solutions for $P_{max }$ and $\kappa _{max }$ exhibit a degree of credibility in the high superheat regime. Note that the above graph illustrates the correlation between one variable ( $P_{max }(t)$ and $\kappa _{max }(t)$ ) and another variable ( $h_{min }(t)$ ), with the initial input parameters ( $U_0$ and $\Delta T$ ) implicitly indicated in the results. We will then investigate the direct connection between the input and output quantities, referring to the context below as well as in figure 12.

Figure 9.

Variation of maximum vapour pressure $P_{max }$ and maximum interfacial curvature $\kappa _{max }$ as functions of the minimum vapour thickness $h_{min }$ , all normalized using new scaling factors provided in the right column of table 2. These impact scenarios correspond to high superheat conditions, with $U_0 = 0.34, 0.37, 0.39, 0.42\, \mathrm {m\, s^-{^1}}$ and $\Delta T = 300, 400, 500, 600\,\mathrm {K}$ . Both panels exhibit a remarkable agreement, with numerical data collapsing onto single curves.

Next, we provide a prediction for $h_{min }$ in the high superheat regime. As the droplet approaches the liquid pool, the initially disregarded capillary pressure ( $P_\sigma$ ) at the spreading front, denoted as the second term on the left-hand side of (4.3), cannot be ignored. This is because the dimensionless curvature at the neck scales as $\tilde {\kappa }_{max } \sim \tilde {h}_{min }^{-7/3}$ while the dimensionless maximum pressure at the neck increases as $\tilde {P}_{max }\sim \tilde {h}_{min }^{-2/3}$ . Consequently, the growth rate of the capillary pressure ( $\tilde {P}_{\sigma } \sim \tilde {\kappa }_{max } \sim \tilde {h}_{min }^{-7/3}$ ) surpasses that of the maximum vapour pressure ( $\tilde {P}_{max } \sim \tilde {h}_{min }^{-2/3}$ ). As a result, surface tension becomes dominant in regularizing the neck region as it approaches $h_{min }$ . Therefore, it is necessary to retain the complete set of equations (4.3), and the simplified (4.15) is no longer applicable.

To verify this quantitatively, we record the time evolution of $P_\sigma = \sigma \kappa _{max }$ and $P_{max }$ at $U_0 = 0.394\, \rm ms^-{^1}$ , $\Delta T = 120 \ \mathrm {and} \ 400\,\mathrm {K}$ ( $We = 8$ , $\mathcal {E} = 1.06 \times 10^{-6} \ \mathrm {and} \ 3.55 \times 10^{-6}$ ), as presented in figure 10(a). We observe that, during the initial falling stage, vapour pressure dominates over capillary pressure, making the surface tension term negligible. However, after $t = 0.36\mathrm {ms}$ , $P_\sigma$ significantly increases and becomes comparable to $P_{max }$ . Furthermore, figure 10(b) highlights the role of capillary forces in preventing contact by showing the time histories of the minimum film thickness when the surface tension is normal ( $\sigma = 1.7\times 10^{-2}\,\mathrm {N\, m^-{^1}}$ ) and two orders smaller ( $\sigma = 3.4\times 10^{-4}\,\mathrm {N\, m^-{^1}}$ ). Clearly, by reducing the surface tension coefficient, we observe that $h_{min }$ decreases much faster even at very high superheat of $\Delta T = 400\,\mathrm {K}$ , indicating that the growth of vapour pressure alone cannot prevent droplet–pool coalescence, but capillary pressure can. In this transitional stage, the self-similar solution, which does not include capillary effects, breaks down. Thus, it seems that capillary pressure should be responsible for preventing the Leidenfrost droplet from contacting the liquid pool in the early impact stage.

Figure 10.

(a) Time evolution of the maximum vapour pressure ( $P_{g, max }$ , solid lines) and capillary pressure ( $P_{\sigma }$ , dashed lines). (b) Minimum vapour thickness as a function of time, shown for cases with the standard $\sigma$ (solid lines) and reduced $\sigma /50$ (dash-dotted lines). These results correspond to impact conditions with $U_0 = 0.394\, \rm m\, s^-{^1}$ and two distinct superheats, $\Delta T = 120 \,\mathrm {K}$ (green colour) and $400\,\mathrm {K}$ (yellow colour).

Therefore, we assume that the film thickness reaches its minimum value when the two pressures are of comparable magnitude, namely $P_{\sigma } \approx P_{max }$ ( $We^{-6/5} \mathcal {E}^{1/5} \tilde {\kappa } \sim \tilde {P}_g$ in dimensionless form). By substituting (4.24) and (4.26) into this equation, we obtain the minimum film thickness as follows:

(4.27) \begin{equation} \frac {h_{min }}{R_0}\sim We^{-28/25} \mathcal {E} ^{13/25}. \end{equation}

The derived scaling law offers valuable insights into the impact of various parameters on $h_{min }$ in the high superheat regime of the Leidenfrost problem. In dimensional terms, the minimum thickness adheres to a scaling relationship of $h_{min } \propto U_0^{-56/25} \Delta T^{13/25}$ , a formulation that closely aligns with the numerical results obtained in the high superheat regime, as demonstrated in figure 5. Notably, this expression exhibits a remarkable similarity to the relationship $h_{min } \propto U_0^{-2}\Delta T ^{1/2}$ derived by Chantelot & Lohse (Reference Chantelot and Lohse2021), despite the adoption of different characteristic scales (refer to table 2).

4.3. Reconstruction of the self-similar solutions at moderate superheat regime

Now, let us focus on the moderate superheat regime, where the influence of evaporation on the thin vapour film is substantial but not overwhelmingly dominant. It is important to emphasize that we obtain (4.27) in the high superheat regime based on two assumptions: first, the evaporation-induced velocity becomes dominant in the falling stage; second, in the lubrication equation, the evaporation term is balanced by the pressure-induced flow. However, in this intermediate regime, these two assumptions may not be fully valid. First of all, we find that the previously derived characteristic scalings (see (4.6b)) no longer hold true. Our empirical observations, illustrated in figure 7(b), reveal that in this moderate superheat regime, i.e. at $\Delta T \lt 200\,\mathrm {K}$ , the central dimple height $h_d$ does not conform to the anticipated $\Delta T^{2/5}$ variation law. Consequently, we have to employ $P_g \sim \mu _v R_0U_0/h^2$ as the basis for characterizing vapour pressure, and the scaling relations established by Mani et al. (Reference Mani, Mandre and Brenner2010) and Mirjalili & Mani (Reference Mirjalili and Mani2020) remain valid, as given in (4.6a)–(4.9a). Then, we can re-non-dimensionalize the governing equations for liquid and vapour flow, and they are given as follows:

(4.28) \begin{equation} \frac {\partial \tilde {\boldsymbol {u}}}{\partial \tilde {t}} + St^{-1/3} \tilde {\nabla } \left | \tilde {\boldsymbol {u}}\right |^2 = -\tilde {\nabla } \tilde {P}_l ,\end{equation}

(4.29) \begin{equation} \tilde {P}_g + We^{-1}St^{-1/3} \tilde {\kappa }= \tilde {P}_l ,\end{equation}

(4.30) \begin{equation} \frac {\partial \tilde {h}^{\pm }}{\partial \tilde {t}} + S t^{-1 / 3} \tilde {u}_{l}^{\pm } \frac {\partial \tilde {h}^{\pm }}{\partial \tilde {x}} = \tilde {v}_{l}^{\pm } ,\end{equation}

(4.31) \begin{equation} \frac {\partial \tilde {h}}{\partial \tilde {t}}=\frac {1}{\tilde {r}}\frac {\partial }{\partial \tilde {r}}\left (\frac {\tilde {r}\tilde {h}^{3}}{12 } \frac {\partial \tilde {P}_{g}}{\partial \tilde {r}}\right )-St^{-1/3}\frac {1}{\tilde {r}}\frac {\partial }{\partial \tilde {r}} \left ( \frac {\tilde {r}\tilde {h} \left (\tilde {u}^+ + \tilde {u}^- \right )}{2}\right ) + \frac {\mathcal {E} St^{5/3}}{We \tilde {h}}. \end{equation}

Considering the range of impact parameters in the numerical simulations where the Stokes number is very high ( $St\sim O(10^5)$ ), the leading-order equations in the moderate superheat regime are identical to (4.14)–(4.16), with the exception of the lubrication equation (4.17), which now becomes

(4.32) \begin{equation} \frac {\partial \tilde {h}}{\partial \tilde {t}}=\frac {1}{\tilde {r}}\frac {\partial }{\partial \tilde {r}}\left (\frac {\tilde {r}\tilde {h}^{3}}{12 } \frac {\partial \tilde {P}_{g}}{\partial \tilde {r}}\right ) + \frac {\mathcal {E} St^{5/3}}{We \tilde {h}} .\end{equation}

Figure 11.

Variation of maximum vapour pressure $P_{max }$ and maximum interfacial curvature $\kappa _{max }$ as functions of the minimum vapour thickness $h_{min }$ , all normalized using scaling factors provided in (4.33) and (4.35). These impact scenarios correspond to moderate superheat conditions, with $U_0 = 0.34, 0.37, 0.39, 0.44\, \mathrm {m\,s}^{-1}$ and $\Delta T = 40, 80, 120\,\mathrm {K}$ . Notably, we adjust the correlation from $\tilde {\kappa }_{max } \sim \mathcal {E}^{2/3}$ to $\tilde {\kappa }_{max } \sim \mathcal {E}^{1/2}$ in panel (b) to achieve better data collapse.

In this regime, it is not straightforward to derive a theoretical scaling law from (4.32) if all three terms are included. Therefore, considering the moderate superheat condition ( $40\,\mathrm {K} \lt \Delta T \lt 200\,\mathrm {K}$ ), we maintain the assumption that the evaporation term is balanced by pressure, as was done in (4.23). Subsequently, by employing the self-similar ansatz as previously outlined in § 4.2, we can derive self-similar solutions applicable to the moderate superheat condition, which are presented as follows:

(4.33) \begin{equation} \tilde {P}_{max } \sim \tilde {h}_{min }^{-2/3} We^{-1/3} St^{5/9} \mathcal {E}^{1/3}, \end{equation}

(4.34) \begin{equation} \tilde {l} \sim \tilde {h}_{min }^{5/3} We^{1/3} St^{-5/9} \mathcal {E}^{-1/3} ,\end{equation}

(4.35) \begin{equation} \tilde {\kappa }_{max } \sim \tilde {h}_{min }/\tilde {l}^2 \sim \tilde {h}_{min }^{-7/3}We^{-2/3}St^{10/9}\mathcal {E}^{2/3} .\end{equation}

In a similar fashion, we proceed to analyse the values of $P_{max }$ and $\kappa _{max }$ obtained from our numerical results in this moderate superheat regime, corresponding to different values of $U_0$ (i.e. 0.34 , 0.37 , 0.39 and 0.44 m s^–1) and $\Delta T$ (i.e. 40, 80 and 120 K), as illustrated in figure 11. However, the numerical results show discrepancies. On one hand, when attempting to rescale dimensionless $P_{max }$ using the scaling relation provided in (4.33), it becomes evident that the data cannot be collapsed onto a single curve, as depicted in figure 11(a). On the other hand, (4.35) implies that the dimensionless curvature evolves with the evaporation rate, following the correlation $\tilde {\kappa }_{max } \sim \mathcal {E}^{2/3}$ . However, our numerical results, as shown in figure 11(b), suggest that this relation is closer to $\tilde {\kappa }_{max } \sim \mathcal {E}^{1/2}$ . These discrepancies between numerical results and theoretical models can be attributed to two pivotal simplifications we made before. Firstly, we ignored the evaporation induced velocity when selecting the characteristic scales (4.6a)–(4.9a), where the vapour pressure was scaled as $P_g \sim \mu _v R_0 U_0/h^2$ . Notably, the value of $U_e/U_0$ is not so small for the moderate superheat conditions ( $40\,\mathrm {K} \lt \Delta T \lt 200\,\mathrm {K}$ ), as depicted in figure 6. Thus such simplification leads to an underestimation of the impact of evaporation in the respective characteristic scales and subsequent dimensionless normalization. Secondly, we disregarded the left-hand side in (4.32) and maintain the assumption that the evaporation term is fully balanced by the pressure term as was done in considering the high superheat regime. This assumption may not be entirely tenable, as illustrated in figure 8(a), where the evaporation term only dominates the squeezing term within a much narrower range around the spreading front for the condition of $\Delta T = 40 \,\mathrm {K}$ compared with $\Delta T = 400 \,\mathrm {K}$ . Thus the significance of evaporation has been overemphasized at this stage. As a result, it appears that in this moderate superheat regime, those neglected terms may play some roles, and ignoring any of them leads to inaccuracies.

Consequently, to better match our numerical solution, we adjust the exponent from $2/3$ with $1/2$ in the following analysis. With this modification, we can still derive the expression for $h_{min }$ by equating the maximum vapour pressure with the capillary pressure, i.e. $P_{max } \approx P_\sigma$ ( $\tilde {P}_g \sim We^{-1} St^{-1/3} \tilde {\kappa }$ in dimensionless form). Substituting the scaling relation for $P_{max }$ and the modified relation for $\kappa _{max }$ into this expression, we obtain the following revised expression:

(4.36) \begin{equation} \frac {h_{min }}{R_0} \sim We^{-4/5} St^{-8/15} \mathcal {E} ^{1/10} .\end{equation}

This scaling relation provides an accurate prediction for $h_{min }$ in the moderate superheat regime, taking into account the nuanced interplay between evaporation and other influencing factors. The dimensional correlation $h_{min } \propto U_0^{-32/15}\Delta T^{1/10}$ can indeed accurately predict our numerical results for moderate superheat situations, as evidenced by the power exponents presented in figure 5. This robust agreement further reinforces the significance of the balance between gas pressure and surface tension in preventing the interfaces from approaching in the early stages of impact. Regrettably, we have to acknowledge that the scaling laws described in (4.33)–(4.35) serve as the approximations for the moderate superheat regime primarily due to the aforementioned two key simplifications. Besides, since these simplifications underestimate and overestimate the effect of evaporation respectively, the conclusion drawn for this moderate superheat regime cannot seamlessly transition to either low or high superheat conditions. Nonetheless, the methodology we employed allows for a coherent description of various superheat regimes from a physical standpoint. When the superheat is very high ( $\Delta T \gt 200\,\mathrm {K}$ ), the dominance of evaporation is evident in both the initial descent and subsequent spreading stage. As the superheat decreases to the moderate superheat regime ( $40\,\mathrm {K} \lt \Delta T \lt 200\,\mathrm {K}$ ), the effect of evaporation diminishes gradually in the initial stage but remains significant in the second stage as the vapour film thickness at the spreading front $h_{min }$ is considerably smaller than the minimum height of the central dimple $h_d$ . Finally, the low superheat regime, or the so-called isothermal regime, arrives ( $\Delta T \lt 40\,\mathrm {K}$ ) as the superheat decreases further, where the effect of evaporation in the whole impact process is negligible.

In summary, our analysis, based on lubrication and potential flow theories, has unveiled two distinct regimes characterizing the short-time dynamics of a Leidenfrost droplet impacting a superheated liquid pool. In the moderate superheat condition, phase-change effects are initially negligible during the droplet’s descent but gradually come to dominate the lubrication flow of the vapour film as the superheat increases. In this regime, the minimum film thickness at the spreading front scales as $h_{min } \propto U_0^{-32/15}\Delta T^{1/10}$ . On the other hand, in the high superheat condition, phase-change effects exert dominance throughout the entire impact process, from the initial falling stage to the final levitation stage. In this regime, we obtain $h_{min } \propto U_0^{-56/25} \Delta T^{13/25}$ as a corresponding scaling law. Importantly, in both regimes, the variation of the minimum film thickness with the falling velocity approximates $h_{min } \propto U_0^{-2}$ , which aligns remarkably well with our numerical results. Furthermore, the scaling laws also predict the observed tendencies of $h_{min } \propto \Delta T^{1/10}$ and $h_{min } \propto \Delta T^{1/2}$ in different regimes of our numerical simulations.

Lastly, in figure 12, we present the dimensionless values of $h_{min }$ for these two distinct superheat regions, normalized using the scaling laws (4.27) and (4.36). These findings enhance our understanding of the intricate interplay between phase-change effects and the impact dynamics in Leidenfrost droplet scenarios.

Figure 12.

Dimensionless values of $h_{min }$ for the (a) moderate and (b) the high superheat regimes, normalized using the scaling laws (4.27) and (4.36).

4.4. Comment on the critical impact velocity and superheat

In reality, both intermolecular van der Waals forces and gas kinetic effects (for air at atmospheric pressure $p_{atm} \approx 10^5\, \rm Pa$ , the mean free path is approximately 70 nm) become significant and cannot be disregarded as the vapour film thickness decreases to the scale of 100 nm (Couder et al. Reference Couder, Fort, Gautier and Boudaoud2005; Thoroddsen et al. Reference Thoroddsen, Thoraval, Takehara and Etoh2012; Tran et al. Reference Tran, de Maleprade, Sun and Lohse2013). However, it is still a challenge to incorporate all these diverse effects in the DNS; here, we directly employ a fixed critical thickness $h_c$ to explore the potential coalescence criterion.

By substituting the critical value into the minimum film thickness $h_{min }$ in the derived scaling law (4.27) and (4.36), the corresponding phase diagram for the critical impact velocity and superheat can be obtained, as presented in figure 13, where the solid and dash dotted lines correspond to different critical thicknesses, with $h_c$ set at 200 nm and 100 nm, respectively. It can be observed that each solid and dash dotted line comprises three segments: an orange interval derived from the high superheat regime (4.27), a green interval from the moderate superheat regime (4.36) and a purple interval calculated based on the isothermal scaling derived by Mirjalili & Mani (Reference Mirjalili and Mani2020). These piecewise segments divide the parameter space into two regions: on the left side of the lines, where $h_{min }$ exceeds $h_c$ , coalescence of interfaces does not occur; while on the right side of the lines, $h_{min }$ is smaller than $h_c$ and this would result in the occurrence of coalescence. Additionally, we also depict the critical lines derived from (4.27) across all impact conditions as grey dashed lines in figure 13. Notably, they expands the coalescence domain compared with the prediction relying on the combination of (4.27) and (4.36).

It should be noted that the critical lines presented in figure 13 are not only influenced by the specific critical thickness $h_c$ , which depends on the intricate coalescence mechanics, but also by the working fluid’s physical properties. Consequently, this diagram is relatively qualitative in nature and cannot be seen as a universal criterion for predicting whether the coalescence occurs. Besides, the vapour film collapse throughout the entire impact process can be driven by multiple mechanisms (Chantelot & Lohse Reference Chantelot and Lohse2021), beyond the short-time contact specifically examined in this study. Therefore, a comprehensive understanding of all these mechanisms is essential for accurately predicting the dynamic Leidenfrost temperature.

Figure 13.

Phase diagram of the non-coalescence and coalescence regions for ethanol drops impacting onto a superheated silicon oil pool in the parameter space defined by impact velocity $U_0$ and superheat $\Delta T$ . The solid and dash-dotted lines demarcate the boundaries between these regions, estimated from the combination of scaling laws (4.27) and (4.36), with critical thicknesses of $h_c = 200\,\mathrm {nm}$ (solid) and $h_c = 100\,\mathrm {nm}$ (dash-dotted), respectively. Additionally, the grey dashed lines indicate the boundaries derived using only (4.27) across the parameter space.