An analytical-numerical coupled model of liquid droplet impact on solid material surfaces

Abstract

Impacts of liquid droplets on wind turbine blade surfaces, for example sea sprays, can result in material damage through erosion. In this study, we derive an explicit, closed-form analytical approximation for droplet impact and subsequent spreading on a solid surface in inertia-dominated regimes of large Reynolds and Weber numbers. The formulation extends an existing theoretical framework based on inviscid potential flow for a rising expanding disk in an infinite liquid domain. The modified solution provides full spatio-temporal pressure distributions and impact force histories on the impact surface over the entire impact duration, capturing both the early-time self-similar flow and the inertia-driven lamella spreading following the peak impact force. The predicted pressure and force profiles show good agreement with analytical, numerical and experimental results reported in the literature, including accurate reproduction of the well-known ring-shaped pressure distribution. Key quantities, such as the radial location and magnitude of peak pressure, as well as the timing and magnitude of the peak impact force, are predicted analytically with reasonable accuracy. To enable solid material erosion analysis, the analytical liquid-phase solution is coupled with a finite-element (FE) simulation for the solid response. This analytical-numerical coupled method (ANCM) eliminates the need to explicitly simulate droplet fluid dynamics, which is conventionally performed using smoothed particle hydrodynamics (SPH). As a result, for the purpose of material response analysis, the proposed approach achieves grid independence at substantially lower mesh resolutions and reduces computational cost by more than 97% compared to SPH-based simulations, while maintaining or improving numerical accuracy.

Figures (17)

• Figure 1: The problem of liquid droplet impact (downwards at speed $u_0$) on a solid surface (a) has been found to be analogous to the problem of a thin rising expanding solid circular disk (upwards at speed $u_0$) in an infinite mass of liquid (b). The blue arrows in (b) denote the radial expansion of the disk with the wet radius. The figure is reproduced after Philippi1 and Hao1.
• Figure 2: Positions of the wet radius $a$, the separation point $r_{sep}$, and the pressure-maximum radius $r_{max}$ at times $t=5\times10^{-2}$ (a), $t=3\times10^{-1}$ (b), $t=6\times10^{-1}$ (c), and $t=10^{0}$ (d) are shown overlaid on the streamlines of flows around a flat, rising, expanding disk. The dashed line indicates the droplet interface, analogous to the droplet impact on a solid surface.
• Figure 3: (a) Temporal evolution of the radial positions of the separation points $r_{sep}$, the pressure maximum radius $r_{max}$, and the Wagner wet radius $a(t)=\sqrt{3t}$. Also superimposed are the contact-line evolutions from experiments Gordillo2 and numerical simulations Hao1, for a liquid droplet with kinematic viscosity $20$ cSt and impact velocity $1.93$ ($Re=212$). (b) Fractional contribution to the impact force from the pressure acting in the radial interval between $r_{max}$ and $r_{sep}$ (i.e. the green-shaded region in the inset), normalised by the total impact force within $r_{sep}$. The inset in (b) shows the radial pressure distribution at $t=0.5$.
• Figure 4: (a) Comparison between the predicted radius $r_{max}$ (red arrow and dashed line) and experimentally observed contact-line positions from Riboux2016 for an ethanol droplet of radius $R_0=1.03$. The dimensionless times (from top to bottom) are $t=0.016$, $0.049$, $0.082$, $0.164$, $0.246$, $0.328$, and $0.491$. The left column shows the corresponding numerical results in Riboux2016. (b) Comparison between the predicted radius $r_{max}$ (orange line) and experimentally observed droplet morphologies from Quintero for a water droplet of radius $R_0=1.53$. The dimensionless times corresponding to labels $b$1 to $b$9 are $t=0$, $0.4$, $0.7$, $0.9$, $1.2$, $1.4$, $1.9$, $2.4$, and $2.9$. See Quintero for other coloured lines.
• Figure 5: Fractional contribution of the Bernoulli constant $b(t)=1/2$ to the impact pressure, normalised by the peak impact pressure $p(r_{max},0,t)$, as a function of time $t$. The expression used to compute this fraction is shown in the figure, with $a(t)=\sqrt{3t}$.