Combined dynamic-kinematic validation of droplet-wall impact modeling

TL;DR

This paper addresses the insufficiency of validating droplet-wall impact using only maximum spreading by introducing a combined dynamic contact angle (DCA) model that blends the generalized Hoffman-Voinov-Tanner law with the Hoffman function to capture both spreading and receding. Using axisymmetric CFD with Volume-of-Fluid and velocity preprocessing to align with PIV data, it demonstrates that the generalized HVT law can closely predict $D_{max}$ (within $\leq 7\%$) but struggles with receding, while the Hoffman function better reproduces internal kinematics; the combined DCA leverages both strengths and yields improved receding dynamics while maintaining geometric accuracy. A $\big(\beta_{max},Ca_{char}\big)$ diagram is introduced to relate final spreading to internal flow, suggesting contact-line geometry could inform internal kinematics with sufficient data. Overall, the work advocates a robust validation strategy that integrates geometric and dynamic indicators, with practical relevance to spraying, cooling, and surface engineering.

Abstract

Many numerical studies validate droplet wall impact using only maximum spreading diameter, yet this metric alone cannot ensure correct droplet dynamics. We present a combined dynamic contact angle (DCA) model that merges the geometric accuracy of the generalized Hoffman-Voinov-Tanner law with the kinematic consistency of a Hoffman function-based approach, improving predictions of droplet spreading and receding. We simulate water-glycerol droplet impact on sapphire glass at Weber numbers 20 -- 250 and assess both contact angle formulations. Simulated radial velocity fields are processed in Python using SciPy and compared with Particle Image Velocimetry measurements in the longitudinal section of the spreading droplet. The Hoffman function-based model captures the main droplet kinematic trends and provides more consistent receding dynamics. The generalized Hoffman-Voinov-Tanner law matches the maximum spreading diameter within 7%. However, during receding, it shows a median absolute error in radial velocity up to three times higher than that of the Hoffman function-based solution. Average radial velocity and spreading velocity can differ from experimental trends even when maximum spreading is reproduced. These findings support validation combining geometric and kinematic metrics and motivate the combined model for predicting spreading and receding. Using the maximum spreading factor $β_{max}$ as the ratio of the maximum spreading diameter over the initial droplet diameter and the characteristic capillary number $Ca_{char}$ defined from the mean internal horizontal velocity at 300 micrometer above the substrate, we introduce a $(β_{max},\,Ca_{char})$ diagram to relate spreading characteristics to internal flow dynamics. We hypothesize that, given sufficient data, the contact-line geometry may be used to estimate internal kinematics.

Figures (13)

• Figure 1: Schematic drawing of the computational domain, geometrical parameters and mesh.
• Figure 2: Average values of internal flow velocities for different adaptation levels.
• Figure 3: Radial velocity data preprocessing: (a) - original adaptive mesh refinement velocity values, (b) - velocity values interpolated on a uniform mesh ($u_{\textrm{uniform}, i}$), (c) - velocity values interpolated on a uniform mesh ($u_{\textrm{uniform}, i}$, blue) with cubic spline interpolated values ($u_{\textrm{spline}, i}$, red) further than 75% of maximum distance from symmetry axis, (d) - velocity values interpolated on a uniform mesh ($u_{\textrm{uniform}, i}$, blue) with weighted velocity values ($u_{\textrm{spline}, i} \frac{u_{\textrm{spline}, i}}{u_{\textrm{uniform}, i}}$, orange).
• Figure 4: Comparison of the values of the average velocity inside the droplet with experimental data. The numerical results are linearly interpolated without any further modifications. Hoffman function-based simulations are presented at the first row (a, b, c). Simulations with generalized Hoffman-Voinov-Tanner law are presented at the second row (d, e, f). Simulations with $We_{imp}=20$ are presented at the first column of plot (a, d), $We_{imp}=80$ - the second column (b, e), $We_{imp}=250$ - the third column (c, f).
• Figure 5: Comparison of the values of the average velocity inside the droplet with experimental data. CFD data are interpolated and filtered as explained in Section \ref{['subsec:velocity-preproc']}, for consistency with the resolution of PIV. Hoffman function-based simulations are presented at the first row (a, b, c). Simulations with generalized Hoffman-Voinov-Tanner law are presented at the second row (d, e, f). Simulations with $We_{imp}=20$ are presented at the first column of plot (a, d), $We_{imp}=80$ - the second column (b, e), $We_{imp}=250$ - the third column (c, f).