On the numerical modeling of supercooled micro-droplet impact and freezing on superhydrophobic surfaces
Introduction
Only in the US, airframe icing is responsible for more than 50 incidents and the loss of more than 800 lives between 1982 and 2000 [1]. Ice on aircrafts, roadways, and wind turbines are probably the most serious meteorological hazard facing the associated industries. Most of the ice accretion on airframe is due to supercooled droplets in clouds which are located at altitudes below 2400 m which aircrafts frequently have to pass during takeoff and landing. The resulting in-flight icing of supercooled liquid water (SLW) droplet can take place on aircraft wing, tail, engine or instrument and lead to decrease the aerodynamic performance which could result in a lack of control or loss of thrust and constitute a major safety and security issue. In addition to the small SLW droplets (smaller than 50 m), the supercooled large droplets (SLD) with sizes larger than 50 m have been the focus of many researches. Better understanding of droplet impact dynamics including spreading, splashing, and recoiling on surfaces with various wettabilities from hydrophilic to superhydrophobic is necessary to predict the ice accretion on aircraft components.
The pioneering work on droplet spreading and solidification has been carried out by Madejski in 1976 [2]. His analytical approach provides an estimation of the spreading diameter (or the degree of flattening) during solidification by combining the Stefan problem and a simple radial flow assumption. His model based on a 2D axisymmetric flow of the velocity field has been improved in [3] using a more suitable approximation for both the velocity field and the dissipation. Those works were only concerned with metal droplet solidification and do not address the water freezing. There are very few publications addressing water droplet solidification apart from the early work by Anderson et al. [4], [5], [6] based on geometrical analysis. More recently a geometrical model has been developed to analyze the singularity at the tip of a frozen water droplet [7], [8], [9]. These models neglect the droplet impact as well as the spreading; in addition, they cannot predict the concave ice-front evolution.
It is worth noting that recent experimental, theoretical, and numerical works on supercooled droplet freezing are focused on the micro-physical processes involved such as the pattern and growth of dendrite following droplet impact. The relevant parameters controlling ice dendrites growth within supercooled pure water is investigated theoretically and experimentally in [10]. A very good agreement has been found on dendrite tip velocity between the numerical simulations based on both volume of fluid and level set methods, the experiments and the marginal stability theory of Langer and Müller-Krumbhaar in [11]. In addition, by repeating numerous supercooled droplets impact, a statistical model has been derived in [12] to estimate the rate of heterogeneous nucleation. A nice review in the physics, hydrodynamics and thermodynamics involved in the supercooled water droplet freezing can been found in [13]. Although essential in understanding the underlying micro-physical processes involved in supercooled droplet freezing, the modeling of dendritic ice is not enough to capture the full picture of supercooled droplet impact and freezing; and there is still a challenge to derive a numerical model, retaining the relevant physics, capable of simulating the solidification of supercooled water droplet upon impact on a solid substrate. The present work focuses on modeling the dynamics of droplet impact and freezing and can be considered as complementary to the work in [13], [14] where the emphasis is on analyzing dendritic ice growth.
From a numerical point of view, for instance, the work by Pasandideh-Fard et al. [15] which relies on the enthalpy formulation in [16], [17] is one of the few works reported treating droplet impact and solidification, though their approach is based on the weak formulation method and is more suited for metal droplet as pointed out by these authors. Although their approach dealt with numerical modeling of droplet impact, their model still makes use of experimental data to describe the complex dynamic contact angle for the spreading and neglect the surrounding air presence considered as void. It is important to emphasize that air may be entrapped at droplet impact and modify the heat transfer inside the droplet [18], [19]. Recently in [20] under static condition, the freezing of a water droplets on hydrophobic and hydrophilic surfaces under rapid cooling condition was investigated. A single phase numerical simulation is also performed to determine the temperature field within the droplet, while the ice fraction is approximated through an energy balance equation. In [21], considering droplet dynamics, it is shown experimentally, using both infrared (IR) thermometry and high-speed imaging, the critical role played by the contact area in controlling water droplet freezing. The mechanism of supercooled water droplet freezing under the effect of airflow has been experimentally investigated in [22], and the effect of the surrounding environment in controlling ice crystallization mechanism is evidenced.
Blake et al. [23] propose an approach to perform droplet freezing, however the model relies on the classical formulation in [16], [17]. In addition, the rapid phase change is not considered, the simulation are performed only after the recalescence phase. The same model has been used in [24] with the impact on inclined plane. However, this rapid growth phase is critical for droplet dynamics freezing as the nucleation and subsequent pinning occur during that phase. Zhao et al. [25] uses an interesting approach, based on lattice Boltzmann method (LBM), to model the impact and freezing of a saturated liquid droplet on a cryogenic spot. However, the validation with experiment is not provided. In addition, their solidification model relying on the solidus and liquidus temperatures would be more convenient for metal solidification [15]. Furthermore, the surface hysteresis effect is not considered. Unlike most of the models in the literature, the present work addresses the following challenges pertaining to supercooled droplet modeling which make it unique: (i) phase change for water at a fixed(defined) temperature, instead of assuming the freezing to occur over a range of temperature, (ii) the dynamic contact angle and hysteresis effect are accounting for in order to accurately capture droplet impact dynamics, (iii) heterogeneous nucleation which controls the pinning upon droplet impact and freezing will also be incorporated into our model.
We recently showed in [26] both theoretically and experimentally the critical role played by the surrounding air on supercooled water droplet dynamics impacting on superhydrophobic surfaces. Although the Volume of Fluid (VOF) model can capture the physics governing metal solidification phenomenon relatively well (albeit neglecting the air phase), the water freezing problem which presents much more severe discontinuity at the phase front, seems out of reach by the conventional technique based on the enthalpy formulation as pointed out in [15]. Finding a formulation for water droplet capable of addressing these limitations of the enthalpy method will be one of the aims of the present paper.
Since all the icing certification conditions are tested in flight condition and/or tunnel experiments mostly due to the cost, development of reliable numerical tools is necessary. Most of the codes for aircraft icing are based either on thin film approach, panel method, or over-simplified scenarios for droplet impact and solidification using the average mass and heat transfer balance at the surface to predict ice accretion and neglect the dynamics of droplet impact and freezing. In this study, a model of the impact and freezing of a supercooled droplet is developed by coupling the volume of fluid (VOF) method and a dynamic contact angle model accounting for the substrate hysteresis to track water-air interface. The phase change is handled through both the modified momentum and the enthalpy formulation of the energy equations. Finally, the nucleation theory via Gibbs free energy is used to control the SLW upon impacting on the substrate.
The paper is organized as follows: in Section 2, the governing equations to handle supercooled water droplet impact and freezing are presented and in Section 3 the numerical schemes and the model validation are provided. Numerical results and discussions are detailed in Section 4. Finally, conclusions are drawn in Section 5.
Section snippets
Governing equations
Supercooled droplet modeling is challenging due to the combining effect of a moving contact line, simultaneous heat transfer and phase change. We assume a stable freezing of the supercooled droplet and neglect the anisotropic effects. An isotropic surface tension seems sufficient to describe the mechanism of supercooled droplet freezing [22]. While the models focused on the dendritic growth [27] are local, the present paper provides a global framework where both the interaction with the
Numerical techniques
The governing equations Eqs. (9), (10), (11), (12), (13), (14), (15), (16), (17), (18) are implemented in OpenFOAM/C++ and discretized using a finite volume based volume of fluid method following the schemes detailed in Table 1.
Results and discussion
In order to determine if the simulation agrees with experimental observations, a comparative study was performed for a certain range of SLW droplet temperatures with various impact velocities as reported in the work of Maitra et al. [41]. The substrate is superhydrophobic with a static contact angle of 154° and an advancing and receding contact of 162°/148°, respectively. Fig. 5 shows the time evolution of the impact of 2 mm supercooled water droplet at −5°C. It is noticeable that icing layer
Conclusions
In this paper, a volume of fluid (VOF) based model is developed to undertake water droplet impact, spreading and freezing under supercooled conditions. The model is based on the enthalpy formulation of the energy equation, and an approximation of the ice fraction, as well as the classical nucleation theory. These equations by coupling the momentum and contact angle model enabled us to simulate the water freezing upon droplet impact on superhydrophobic substrate exhibiting hysteresis effect. The
Conflict of interest
The authors declare no conflict of interest.
Acknowledgements
Authors gratefully acknowledge the financial support from Natural Sciences and Engineering Research Council of Canada (NSERC). The authors are grateful to the anonymous reviewers, whose comments have helped improve the manuscript.
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