Three-Dimensional Lattice Boltzmann Analysis of Droplet Impact and Freezing Dynamics on Cold and Ultra-Cold Surfaces

Liquid droplets interacting with cold solid surfaces appear in an unusually wide range of settings, such as in aircraft icing, but similar processes arise in heat exchangers, power transmission systems, and marine structures operating in cold or polar environments. In all of these cases, the basic event is: a droplet impacts a surface and what follows, however, depends on how inertia, surface tension, viscous losses, and heat transfer compete over very short timescales. If freezing intervenes early enough, even familiar impact scenarios can evolve in unexpected ways. Despite sustained interest over several decades, the coupled dynamics of droplet deformation and liquid–solid phase change are still not fully resolved. Much of the difficulty lies in the timing. Impact-driven spreading and retraction unfold over milliseconds, while solidification may initiate locally on comparable or even shorter timescales when surfaces are strongly supercooled. Once freezing begins, the flow field and thermal field cease to be separable, and intuitive extensions of room-temperature impact theory often fail. Experiments have clarified many aspects of the problem, but their limitations become apparent precisely in the regimes of greatest interest. At modest supercooling, high-speed imaging can track droplet outlines and contact line motion with reasonable confidence. As temperatures decrease further, however, the interior of the droplet rapidly becomes optically inaccessible, and the freezing front itself is difficult to observe directly. Small, uncontrolled variations in surface microstructure or chemistry can also exert an outsized influence on the outcome, particularly on hydrophobic or superhydrophobic substrates where contact times are short. As a result, it is often challenging to disentangle intrinsic physics from experimental variability. Numerical modeling is therefore an attractive complement, though it brings its own complications. Traditional CFD approaches based on the Navier–Stokes equations require explicit interface tracking and empirical representations of the solidifying region. In enthalpy–porosity methods, for instance, momentum suppression in the mushy zone is governed by parameters that are rarely known a priori and are frequently tuned to specific datasets. This undermines confidence when extrapolating beyond the calibration range. Stability is an additional concern: realistic water–air density ratios and low viscosities are notoriously difficult to handle in three-dimensional simulations without numerical damping. For these reasons, the lattice Boltzmann method has gained attention as an alternative framework. Its mesoscopic formulation handles evolving interfaces naturally and couples more cleanly to enthalpy-based phase-change descriptions. That said, much of the existing lattice Boltzmann literature on droplet freezing has relied on reduced density ratios or elevated viscosities to maintain stability. Extending these models to conditions that genuinely reflect droplet impact and freezing in air remains a nontrivial step—and one that motivates the present line of work.

To this end, two closely related studies published in Physics of Fluids and Heat Transfer Engineering by Dr. Yunjie Xu, Pro. Linlin Tian, Pro. Chunling Zhu, and Professor Ning Zhao at Nanjing University of Aeronautics and Astronautics developed a fully three-dimensional lattice Boltzmann framework for investigating droplet impact and freezing. The new framework was designed to operate under near-realistic conditions, resolving large water–air density ratios while maintaining numerical stability at low viscosities. Beyond the flow solver itself, the model accounts explicitly for volumetric expansion during liquid–solid phase change and incorporates a quasi-dynamic contact angle formulation, which allows advancing and receding behaviors to be represented without imposing a single, static wettability condition. In the first study, the authors focused on droplet impact and freezing on cold solid surfaces with moderate wettability. Through systematic validation against experimental observations, the model demonstrated an improved ability to capture contact line motion, interfacial deformation, and freezing-front evolution, addressing limitations that have constrained earlier numerical approaches. Building on this validated foundation, the follow-up work extended the same methodology to ultra-cold superhydrophobic substrates, a regime that remains difficult to probe experimentally. These simulations revealed that freezing can intervene during the retraction stage, fundamentally altering droplet dynamics by suppressing rebound, pinning the contact line, and, under certain conditions, inducing breakup at impact velocities well below those observed at room temperature.

The research team developed numerical framework across the two studies and built upon an enhanced cascaded lattice Boltzmann formulation, specifically designed to maintain stability at large density ratios and low viscosities. They introduced multiple entropic stabilizers to independently relax different orders of kinetic moments, mitigating numerical instabilities that typically arise in high-contrast multiphase flows and that approach enabled the faithful representation of water–air systems without resorting to artificial parameter inflation. The authors captured droplet deformation using a pseudopotential multiphase model, while the freezing process is resolved through an enthalpy-based thermal lattice Boltzmann scheme. It is important to mention that the model incorporates volumetric expansion during solidification, which is an effect that alters local flow fields near the freezing front. Moreover, the team introduced quasi-dynamic contact angle formulation to address contact line dynamics, which allowed the effective wettability to switch between advancing and receding states in response to contact line motion. They found the model to reproduce the canonical stages of spreading, retraction, and eventual arrest due to freezing in simulations of droplet impact on cold hydrophilic and hydrophobic surfaces. Moreover, quantitative comparisons against experimental measurements show close agreement in temporal droplet profiles and contact length evolution across a range of Weber numbers and surface temperatures. Notably, simulations reveal that freezing initiates preferentially near the three-phase contact line, where enhanced heat flux and prolonged residence promote early solidification. This localized freezing progressively pins the contact line, suppressing retraction and altering the final droplet morphology.

The authors extended their research to ultra-cold superhydrophobic surfaces, and found that at modest supercooling, droplets retain the familiar spread–retract–rebound sequence characteristic of superhydrophobic substrates. However, as surface temperatures decrease further, a thin ice rim forms at the droplet periphery during retraction. This rim acts as a mechanical constraint, fundamentally modifying the energy redistribution within the droplet. Under certain conditions, the frozen base induces droplet breakup at impact velocities well below those required on room-temperature surfaces. They also performed parametric studies which demonstrated that maximum spreading is relatively insensitive to surface temperature, remaining primarily governed by inertial and capillary forces. In contrast, retraction dynamics and contact time exhibit strong thermal dependence once freezing becomes appreciable. Increasing Weber number enhances spreading but also accelerates heat transfer at the interface, and lead to earlier freezing onset in ultra-cold regimes. Moreover, they conducted spatially averaged heat flux analyses which further clarified how thermal gradients evolve during impact, and showed distinct signatures associated with rim formation and bottom-up solidification.

The combined contributions of these two studies demonstrate their successful unification of droplet impact hydrodynamics with realistic freezing physics under conditions that challenge both experiment and computation for the first time in LB framework. Indeed, the authors move lattice Boltzmann modeling closer to genuine predictive utility for icing phenomena by demonstrating stable, three-dimensional simulations at large density ratios. Additionally, the work clarifies that freezing actively restructures the pathways through which kinetic and surface energies are redistributed and the emergence of ice rims, contact line pinning, and bottom-up solidification introduces new regimes of behavior that cannot be inferred from room-temperature impact dynamics alone. These findings are especially relevant for the design of anti-icing strategies, where surface coatings or textures are often optimized based on assumptions drawn from non-freezing conditions.

Practically, the findings suggest that superhydrophobicity alone is insufficient to guarantee droplet rebound under extreme cold. Even highly repellent surfaces can lose their effectiveness once freezing intervenes on timescales comparable to retraction. This realization has direct implications for aerospace and energy applications, where surfaces are routinely exposed to ultra-cold environments. Together, the two studies establish an effective coherent numerical framework for examining impact-freezing phenomena across thermal and wettability extremes. Furthermore, this integrated body of work of Dr. Yunjie Xu et al sets the stage for future investigations into surface patterning, inclined geometries, and transient thermal fields and enables systematic exploration of design variables that would be prohibitively difficult to isolate experimentally by establishing a validated numerical foundation.