Impact of Supercooled Drops onto Cold Surfaces

Abstract

Ice accretion resulting from the impact of supercooled water drops is a hazard for structures exposed to low temperatures, for instance aircraft wings and wind turbine blades. Despite a multitude of studies devoted to the involved phenomena, the underlying physical processes are not yet entirely understood. Hence, modelling of the conditions for ice accretion and prediction of the ice accretion rate are presently not reliable. The research conducted in this study addresses these deficiencies in order to lend insight into the physical processes involved. While presenting an overview of results obtained during the first funding periods of this project, new results are also presented, relating to the impact of supercooled drops onto a cold surface in a cold air flow. The experiments are conducted in a dedicated icing wind tunnel and involve measuring the residual mass after impact of a liquid supercooled drop exhibiting corona splash as well as the impact of dendritic frozen drops onto a solid surface.

1 Introduction

Icing of surfaces due to the impact of water drops at subfreezing temperatures, so-called supercooled large drops (SLD), poses a severe problem for transportation systems, as it may result in ice accretion on aircraft and roadways. This is also a frequent problem for power lines and wind turbines. Iced surfaces not only affect the proper and reliable function of the respective system; they can also represent a serious danger, since aircraft may crash, ships can capsize, iced roads result in traffic accidents and power lines and wind turbines may collapse as a consequence of additional loading by ice accretion. It is therefore of eminent importance that the physics of drop solidification upon impact onto surfaces are understood and are predictable in order to properly model the phenomenon in numerical simulations and to design surfaces and/or means to reduce or eliminate icing.

However, the problem is complex. For one, solidification is initiated by nucleation, which in itself is inherently complex and normally treated in a stochastic manner. Then there is a complex interaction between thermodynamics and hydrodynamics, involving conjugate heat transfer with the substrate and wetting phenomena with material properties which are temperature dependent. Furthermore, these processes can involve mixed phases in which liquid, liquid with dendrites and ice are all present. Finally, the state of the substrate must also be considered. If the impact target is already covered with ice, then the entire solidification process can occur instantaneously upon impact, without exhibiting any freezing delay.

All of these complexities have been the focus of research in this project over the past 12 years at the Institute of Fluid Mechanics and Aerodynamics at the Technical University of Darmstadt within the framework of the CRC-TRR 75. The present article attempts to summarize the results of this work, divided into two main chapters: Chapter “Interactive Visualization of Droplet Dynamic Processes”, dealing with work performed during the first two funding periods from 2010 until 2017 and Chapter “Development of Numerical Methods for the Simulation of Compressible Droplet Dynamics Under Extreme Ambient Conditions”, dealing with on-going research until the completion of the CRCTRR 75 in 2021. Further review articles of the work performed in the framework of this project can be found in [1, 2].

2 Nucleation and Freezing of Supercooled Large Drops (SLD)

The impact of a supercooled large drop (SLD) onto a surface is largely unaffected by the temperature of the substrate before nucleation occurs, exhibiting only minor dependencies related to the change of viscosity at low temperature [34]. Thus, a large body of literature addressing the hydrodynamics of drop impact onto dry surfaces with no phase change is directly applicable [27, 42]. However, once nucleation occurs, the rapid propagation of dendrites within the drop inhibits further hydrodynamic spreading and/or receding. Therefore, the nucleation instant and the associated freezing delay is of utmost importance in defining the final iced area and is an essential ingredient into any physics-based model describing ice accretion.

Once solidification starts as a result of the nucleation of ice embryos, a fast propagation of a cloud of dendrites throughout the bulk occurs, during which the remaining liquid portion of the drop warms to the freezing temperature. This rapid propagation of dendrites is followed by a slower, planar solidification corresponding to the Stefan problem. This process of solidification is depicted in Fig. 1. In the framework of this project, these phenomena have been examined in detail, experimentally, numerically and theoretically. Numerous articles summarize the main developments achieved, and two topics illustrating the main physical phenomena will be discussed in this Chapter: nucleation and freezing delay; and the three stages of solidification within the drop. The reader is referred to the other published material from this project for further details [1, 4, 5, 18, 19, 24, 29, 33,34,35,36,37].

Fig. 1

Reprinted with permission from [35]. Copyright 2017 by the American Chemical Society

Three stages of solidification of a supercooled drop in the vicinity of a solid wall: (1) heterogeneous nucleation and spreading of a thin ice layer on the substrate, (2) dendritic freezing of the bulk liquid, (3) freezing of the remaining water at $T_\text {m}$. Note that the orientation of the dendrites in 2 and 3 is only schematically represented.

2.1 Nucleation and Freezing Delay

The random nature of nucleation suggests that a statistical model should be possible to formulate by observing nucleation over a large number of identical experiments. Such a model for heterogeneous nucleation has been developed, based on a large number of impact experiments using double-distilled water drops and substrates at temperatures between $+17$ and $-17^{\,\circ }\text {C}$. Denoting $J_s(t)$ as the instantaneous rate of nucleation per unit area, the total average number of nucleation sites per drop can be estimated as

$$\begin{aligned} \lambda (t)=\int _0^{A_c(t)}\left( \int ^t_{t_w(A_c)} J_s \text {d}t\right) \text {d}A , \end{aligned}$$

(1)

where $A_c(t)$ is the wetted surface area at time t and $t_\text {w}$ is the instant when the given area element is wetted by the spreading drop for the first time. For long times after impact, $t\gg t_\text {w}$, the time of drop spreading can be neglected in comparison to the total time of nucleation and the expression for the average number of nucleation sites reduces to

$$\begin{aligned} \lambda (t) \approx \int _0^{t} J_s \text {d}t. \end{aligned}$$

(2)

Since the process of nucleation is completely random, the statistics of the number of active nucleation sites follows a Poisson distribution. Considering statistics of $N_0$ initially liquid drops impacting onto a cold substrate, the time at which the first nucleus appears in each drop is denoted $t_n$ and can be determined from the experiments. The number of all the liquid drops $N_{\text {liq}}(t)$, whose nucleation time is larger than t continuously reduces in time. The above analysis yields the following remote asymptotic relation between the relative number of liquid drops and the nucleation rate

$$\begin{aligned} \lambda _s(t) \equiv \int _0^t J_s \text {d}t= \frac{1}{A_c(t)}\ln {\frac{N_0}{N_{\text {liq}}(t)}} \end{aligned}$$

(3)

for times much longer than the time of drop spreading, where $\lambda _s(t)$ is the average number of nucleation sites per unit area.

The values of $\lambda _s(t)$ estimated from the experiments using the approximate expression (3) are shown in Fig. 2 for the impact of warm drops (left graph) and for impacts of supercooled drops (right graph) with various initial temperatures. In all cases, a time-dependent nucleation rate is observed. For the case of supercooled drops, the nucleation rate is relatively high during the first 35 ms after drop impact and is smaller for longer times. Moreover, a clear time lag for nucleation of approximately 20 ms is observed for oblique impact of warm drops (see the insert in Fig. 2 (left graph).

Fig. 2

Reprinted with permission from [37]. Copyright 2017 by the American Physical Society

Average number of nucleation sites per unit area as a function of time. Left graph: water drops at $14.3^{\,\circ }\textrm{C}$ impacting onto a cold, polished, aluminum substrate at $-17.0^{\,\circ }\textrm{C}$. Drop diameter is $3.09\,\textrm{mm}$ and the impact velocity is $4.09\,\mathrm {m/s}$. The impact angle is $30^\circ$. The contact temperature $T_\text {c} = -14.7^{\,\circ }$C is estimated using theory [28]. Right graph: supercooled water drops impacting onto a cold, sandblasted, glass substrate. The temperature of the substrate and the drop are the same. Drop diameter is $3.2\,\textrm{mm}$ and the impact velocity is $2.2\,\mathrm {m/s}$. The impact angle is $90^\circ$.

Omitting further details which can be found in the cited literature, the surface temperature is found to be crucial for the nucleation rate. For low substrate temperatures, the contact area and contact time between the liquid and the substrate are increased as a consequence of varying liquid properties; especially the liquid viscosity exhibits a significant increase for temperature below $0^{\,\circ } \text {C}$. A more viscous flow leads to increased contact time, resulting in a higher nucleation probability. Concluding, the nucleation rate depends on time and is highest in the short phase immediately after impact. Further experiments performed with double-distilled degassed water drops exhibit a clear dependence of the nucleation rate on the liquid gas content, revealing another influence independent of the surface temperature [37].

2.2 Three Stages of Solidification within the Drop

To investigate the freezing process within a SLD in contact with a cold surface, a unique Hele-Shaw facility was conceived, constructed and proven to represent well the freezing of a sessile drop [38]. This facility is pictured in Fig. 3, illustrating the possibility of observing the freezing process through the acrylic glass plates using a high-speed video camera. The Hele-Shaw cell is placed on a cooling plate which maintains a specific temperature down to $-30^{\,\circ }$C using an external chiller. A typical freezing process of a water drop in this facility is shown in Fig. 4, indicating the three stages of freezing, with distinctly different time scales.

Fig. 3

Reprinted with permission [35]. Copyright 2017 American Chemical Society

Schematic of the Hele-Shaw cell with an inserted drop.

Fig. 4

Reprinted with permission [35]. Copyright 2017 American Chemical Society

Observations of freezing process using Hele-Shaw cell. Top row—first two stages of solidification, supercooled to approx. $-15.8^{\,\circ }$C. Bottom row—Third stage of solidification of drop in top row. Approx. 19% of the liquid is already frozen at the beginning of this stage.

In the first stage, heterogeneous nucleation at the wall is followed by the tangential growth of a thin ice layer spreading over the metal substrate/water interface with a constant speed, which depends on supercooling. This initial ice layer has been well documented in [35] with comparison to existing studies [39]. In the second stage, the growth of dendrites arising from the layer instability is observed. While single dendrites are observed for smaller supercooling, the dendrite density increases with increasing supercooling, resulting in a dense front of dendrites for large supercooling. The spreading of the initial ice layer on the bottom locally causes solidification of the meta-stable liquid, leading to an asymmetric propagation of the dendrite front, as shown in the top row of images in Fig. 4 between $t=-16$ ms and $t=-10$ ms.

At the end of the second stage ($t=0$) only a portion of the initially supercooled drop is frozen and a lattice of dendritic ice fills the entire drop. The latent heat released during solidification has warmed up the water/ice mixture to thermodynamic equilibrium at the melting temperature. In the third stage ($t>0$) a stable freezing of the remaining water occurs. The freezing front in this stage moves in the opposite direction of the applied heat flux. This stage can be observed in the lower row of images in Fig. 4.

Although the final two stages of freezing have been previously identified experimentally [12], the modelling of these stages was unclear until recently. Building on the morphological instability of the initial tangential ice layer introduced by [22, 23], the analytic solution for the steady-state tip velocity and radius of growing dendrites from [10], as well as the principles governing the tip shape using Marginal Stability Theory (MST) introduced by [15,16,17], a unique tip growth velocity as a single valued function of the supercooling can be calculated. However, for larger supercooling the theory overpredicts the growth rate of dendrites, as shown in Fig. 5. Therefore an empirical model for the kinetic coefficient was developed, as outlined in the following. It represents a revised version of the model derived in [5].

Fig. 5

Reproduced from [6]. Copyright 2015 Elsevier Masson SAS. All rights reserved

Steady state tip velocity, $v_{\text{ t }}$, dependence on initial supercooling $\Delta T$ for $\lambda _{\text{ s }}/\lambda _{\text{ f }} =1$: computational versus MST-theoretical and experimental results; the results obtained by present empirical model accounting for kinetic undercooling at the solid-liquid interface exhibit good agreement with reference experiments over the entire range of initial supercooling.

In order to quantify the model term mimicking the kinetics-limited growth, the theoretical and experimental results displayed in Fig. 5 are comparatively analyzed. First of all, it is assumed that the steady-state dendrite tip velocity determined experimentally in [39], $v_{\text {n,exp}}$, is directly proportional to $\Delta T_{\text {T}}$

$$\begin{aligned} v_{\text {n,exp}} \propto \Delta T_{\text {T}}, \end{aligned}$$

(4)

with $\Delta T_{\text {T}}$ representing the total undercooling (consisting of viscous and kinetic fractions, $\Delta T_{\nu }$ and $\Delta T_{\text {kin}}$ respectively) at the solid-liquid interface, which is defined as

$$\begin{aligned} \Delta T_{\text {T}} = \Delta T_{\nu } + \Delta T_{\text {kin}}. \end{aligned}$$

(5)

In the analytical MST solution, the viscous undercooling is accounted for. In order to account for the kinetic undercooling, the ratio of the theoretical steady-state velocity $v_{\text {MST}}$ (related to the viscous undercooling only) to the experimentally determined one $v_{\text {n,exp}}$ (influenced in addition also by the kinetic undercooling), $|\tilde{v}_{\text{ n }}| = v_{\text {MST}}/v_{\text {n,exp}}$, is computed assuming its proportionality to the ratio of the viscous (capillary) undercooling to the total undercooling

$$\begin{aligned} \Delta T_{\text {T}} = \frac{\Delta T_{\nu }}{|\tilde{v}_{\text{ n }}|}. \end{aligned}$$

(6)

Inserting it into Eq. (5) and adopting a linear function for the kinetic undercooling, the following expression for the kinetic coefficient is obtained

$$\begin{aligned} k_{\text {kin}} = \frac{|\tilde{v}_{\text{ n }}|v_{\text {MST}} L \rho }{(1-|\Delta v_{\text{ n }}|)\kappa \sigma T_{\text{ m }}}. \end{aligned}$$

(7)

Here, $\sigma$ and $\kappa$ describe the interfacial energy of water and the curvature of the surface area, respectively. $T_\text {m}$ denotes the melting temperature of water and L represents its latent heat of fusion. The density of ice and water is assumed to be equal and is given by $\rho$. The preliminary results obtained at high supercooling degrees by accounting for the kinetic undercooling in the present computational model exhibit good agreement with the experimental data. The only limitation when using Eq. (7) is that the kinetic coefficient is directly dependent on the theoretical steady-state velocity; hence, on the initial supercooling degree. In order to find a coefficient value valid for all supercooling degrees considered, the relation between the kinetic coefficient and the theoretical steady-state velocity is introduced by applying the following relation:

$$\begin{aligned} k_{\text {kin}} = \xi \left( v_{\text {MST}}\right) ^{\frac{2}{3}}. \end{aligned}$$

(8)

The value of the coefficient $\xi$ representing a pre-exponential factor, is calibrated by reference to the experiment by [39] (see Fig. 5), amounts to ($\pi /11)^{2/3}$. Accordingly, the coefficient $\xi$ represents a dimensional quantity whose units corresponds to $\text {m}^{1/3}/ \left( \text {s}^{1/3} \text {K}\right)$. The kinetic undercooling can be formulated as follows:

$$\begin{aligned} \Delta T_{\text {kin}} = \frac{v_{\text{ n }}}{k_{\text {kin}}}. \end{aligned}$$

(9)

Assuming that the solid-liquid interface velocity, $v_{\text{ n }}$, corresponds to the theoretical value $v_{\text {MST}}$ obtained by neglecting the kinetic effects, the kinetic undercooling can be redefined, yielding in following equation:

$$\begin{aligned} \Delta T_{\text {kin}} = \frac{\left( v_{\text {MST}}\right) ^{\frac{1}{3}}}{\xi }. \end{aligned}$$

(10)

Thus, the appropriate quantification of the kinetic undercooling influence on the dendrite growth in the high supercooling range is provided. After accounting for the kinetic effects by considering the present approach, the computational results at high supercooling degrees follow closely the experimental results, Fig. 5. Further details of the numerical procedure used can be found in [5, 6].

3 Impact of Supercooled Large Drops Superimposed with a Cold Air Flow

Further investigations were devoted to the investigation of supercooled drops impacting under the influence of a cold air flow. These investigations involve fluid drops impacting with a corona splash as well as drops which have completed the second solidification stage (growth of dendrites in the bulk) before impact. The associated experiments have been conducted in a newly developed icing wind tunnel which is described in detail in the following section, followed by sections describing the experimental results and analyses.

3.1 Test Facility

The wind tunnel is a vertical open return blower wind tunnel placed inside a cooling chamber, as depicted in Fig. 6. After passing the inlet section and nozzle, the flow enters a square test section with a side length of $140\ \text {mm}$, before entering a diffusor leading to a radial fan. The flow exits the fan into the cooling chamber. The velocity profile across the test section is highly uniform, reaching velocities up to $U_\text {air} =40\ \text {m/s}$ with a turbulence intensity of $\text {Tu}=0.5{\%}$.

Supercooled drops are generated from above the tunnel inlet using a syringe needle inside of the wind tunnel and protected from the airflow by a shroud pipe. In order to minimize the probability of solidification inside of the water supply, the drops are produced from de-ionized water (Milipore ®). After detaching from the syringe, the drops accelerate due to gravity until entering the air flow, where they are further accelerated by the air flow. In the test section, the drops impact onto a surface placed horizontally in the wind-tunnel. The impact is captured using a high-speed camera (Photron SA-X2) with background illumination.

Fig. 6

Reprinted from [7]

Icing wind tunnel placed inside of a cold chamber for investigating the impact of supercooled water drops. Some components are not shown to scale.

The cooling chamber maintains ambient temperatures down to $T_\text {amb}=-20^{\,\circ }\text {C}$; thus determining the temperature of the airflow, the drops and the impact target. All temperatures range within $\pm 1.5^{\,\circ }\text {C}$ of the chamber temperature. Furthermore, the ambient air is fully saturated, avoiding any temperature variation due to drop evaporation. In order to quantify the supercooling of the drops, the temperature is continuously measured by a thermocouple inside of the syringe needle. Additionally, the temperatures of the surface and the air flowing through the test section are monitored during the experiments.

The drop impacts onto a flat aluminum target. To examine the deposited liquid or ice on the surface after drop impact, some additional features have been incorporated into the target, as depicted in Fig. 7. A heating wire wraps around the cylindrical aluminium target in order to melt any residual ice remaining on the target. This is especially helpful in retroactively determining the volume/mass of the deposited fluid, since after heating the possibly frozen drop it becomes a sessile, liquid drop. To examine the residual ice and/or sessile drop in more detail, the target can be rotated, during which images from numerous viewing angles are captured. This technique, employing multiple views, yields a high precision measurement of the remaining liquid volume, elaborated in more detail below.

Fig. 7

Reprinted from [7]

Rotatable impact surface for the investigation of residual mass, enabling multiple viewing angles.

3.2 Impact of Mushy Frozen Drops

Supercooled drops are in a meta-stable state and their freezing depends on the occurrence of nucleation sites. An analysis of the nucleation probability after impact conducted previously [37] indicated that the highest nucleation probability occurred shortly after impact. However, the number of drops with a distinct freezing delay remains non-negligible. Thus, different icing scenarios have to be considered.

An onset of freezing long after the impact results in a solid spherical cap (truncated sphere) with an elevated top due to freezing expansion, arising from the solidification of the receded, sessile drop. The influence of fluid temperature and surface properties were investigated for this case in previous studies [6, 38]. However, if a shorter delay arises between impact and solidification, other freezing scenarios are possible. One frequent scenario is solidification shortly after impact. A freezing during the receding of the drop fixes the fluid in its current position and leads to a larger surface area covered with ice. The solidified area is therefore strongly linked to the impact hydrodynamics and the surface properties as described in [42]. In fact, dendritic solidification of the drop might occur even before impact if nucleation arises due to contaminants or disturbances beforehand. The short timescale of the first dendritic solidification stage and the significantly slower second stage then lead to an impact of a mushy, frozen drop. Moreover, the dendritic freezing may occur for supercooled liquid remaining on the surface after impact, which also yields a mushy phase. The impact and spreading of a ‘mushy’ phase, or dendritic frozen drop, remains unexplored to date and is one focus of the present study.

Although to the authors’ knowledge no study has addressed the impact of dendritic frozen drops, the impact of two-phase drops has been the topic of several investigations in the recent past. A review of compound drop impact is found in [2], although the investigated drops mainly involved two liquid phases. A study of the impact of a sand/water mixture drop was conducted in [20]. A more recent study from [14] is devoted to the impact of a partially frozen binary drop composed of hexadecane and diethylether. However, the solidification of these drops occurs uniformly from the outside to the centre during their descent.

In the present study, the impact of dendritic frozen drops is examined, whereby their ice fraction is varied systematically by the degree of supercooling of the initially liquid drop. In order to characterize the plastic flow behaviour of the impact, an existing model for the impact of a semi-brittle sphere is adapted [31]. Using this model and information gained from the high-speed recordings—the residual height after the impact of the mushy drop, the impact velocity and the ice fraction—the yield strength of the mushy phase is estimated. The results of this study can be found in [8].

3.2.1 Method

The drops used in this study all had a diameter of $D_0=3$ mm and their impact velocity was varied in the range 3.8–5.6 $\text {ms}^{-1}$ by varying the flow velocity in the wind tunnel. The initial supercooling of the drops was varied in the range −11.5 to −7 K. To promote and ensure dendritic freezing before impact, several small ice crystals were pinned to the end of the shroud pipe. Since the typical time for dendritic freezing of the drop, $\mathcal {O}(10\,\text {ms})$, is much shorter than the time it takes the drop to reach the impact target, $\mathcal {O}(100\,\text {ms})$ after nucleation, all drops in the present study have finished the dendritic stage of supercooled solidification upon impact. Hence, all impacting drops exhibit a temperature of $0^{\,\circ }\text {C}$, since thermodynamic equilibrium has been reached.

In order to determine the volume ratio of ice in the drop, the ice fraction $\xi _{\text {ice}}$ is obtained from an energy balance in the drop [9]. Considering the temperature dependent properties of the fluid, the ice fraction is obtained from the initial supercooling after [33] as

$$\begin{aligned} \xi _{\text {ice}}=\frac{\rho _w c_{p,w} \Delta T}{\rho _{\text {ice}} L + \Delta T (\rho _w c_{p,w}- \rho _{\text {ice}} c_{p,\text {ice}})}. \end{aligned}$$

(11)

Here, $\Delta T$ represents the initial supercooling of the liquid in Kelvin and L the latent heat of fusion. Furthermore, $\rho _w$ and $c_{p,w}$ are the density and heat capacity of the supercooled liquid respectively, and correspondingly, $\rho _{\text {ice}}$ and $c_{p,\text {ice}}$ the properties of ice at $0^{\,\circ }\text {C}$. The ice fraction of the mushy drops used in the experiments ranges between $\xi _{\text {ice}}(11.5\,\text {K})=15.0\%$ and $\xi _{\text {ice}}(7\ \text {K})=9.4\%$. The specific heat capacity at constant pressure ($c_p$) is used instead of the specific heat capacity at constant volume ($c_v$), since during freezing the pressure remains constant, but not the volume.

One quantity characterizing the mushy phase of drops with various dendritic ice fractions is the yield strength of the ice/water mixture. To estimate the yield strength, a model describing the impact and fragmentation of ice particles is adapted from [31]. The particle is considered as a semi-brittle sphere being crushed upon impact. The yield strength is obtained from a force balance normal to the surface and the dimensions of the crushed particle when it comes to rest. Throughout the impact event of a mushy frozen drop, the yield strength Y is assumed to be constant. This way, it is related to the impact velocity and the remaining height of the particle by

$$\begin{aligned} \frac{1}{Y} = -\frac{2}{\rho U_0^2} \ln {\left[ \frac{(2-\delta _{\text {max}})^2 (1+\delta _{\text {max}})}{4}\right] }, \end{aligned}$$

(12)

with $\rho$ being the density of the particle and $U_0$ its impact velocity. The remaining dimension in the vertical direction is denoted by the maximum of a dimensionless displacement $\delta _{\text {max}}$. It is obtained from the particle height $h_{\text {max}}$ and the initial radius $R_0$ as $\delta _{\text {max}}=2-({h_{\text {max}}}/{R_0})$. A more detailed derivation of the model is given in [31].

3.2.2 Results and Discussion

The ice dendrites developing inside the drop during the first freezing stage cause a significant change of its material properties and its impact behaviour. A pure liquid drop upon impact forms a thin lamella spreading over the surface [42]. In contrast, the flow in a mushy frozen particle corresponds more to a plastic flow behaviour, with an increased deceleration of the flow during impact. For the investigated mushy frozen particles, the deceleration is sufficient to cause a nearly cone-shaped ice agglomeration at the impact position when the mixture comes to rest. This behaviour is visible in the image sequence shown in Fig. 8.

Fig. 8

Image sequence of a mushy particle impacting with $U_0=4.2\ \text { m/s}$ originating from a drop with an initial supercooling of $\Delta T_0= 9.2\ \text {K}$

In the first image before the impact occurs, the dendritic state of the drop is already noticeable. The first-order refracted light from the background illumination passing through a pure liquid drop would form a bright spot in the middle of the drop. However, the dendrites in the mushy particle cause a diffuse light scattering within the drop, resulting in an evenly dark appearance of the drop. In the first instance after the impact, small fragments leave the spreading mushy frozen particle, similar to the prompt splash of a liquid drop. However, their appearance suggests a mixture of water and broken dendrites, i.e., ice. Subsequently, the above-mentioned deceleration of the ensuing ice/water mixture initiates an accumulation of the mushy phase in the center. Eventually, the mushy phase comes to rest and forms a shape similar to a cone, as shown in the last two images of Fig. 8. The shape remains undisturbed on the surface while the remaining fluid in the mixed phase finishes its solidification.

The dimensions of the remaining cone are obtained from images captured while rotating the target. The analysis of the resulting geometry indicate two main dependencies. First, a higher impact velocity leads to a decreasing height of the remaining shape due to higher inertial forces during impact. Second, the height of the remaining shape alters with the initial supercooling of the drop, which corresponds to different ice fractions according to Eq. (11). This influence suggests a change of the mushy particle properties. With regard to the model of [31], this change manifests itself in a different yield strength of the mushy phase. With the aid of Eq. (12), a yield strength $Y(U_0,\delta _{\text {max}})$ for each impacting particle as a function of its impact velocity and residual height is obtained. In Fig. 9, this yield strength is plotted over the volume ice fraction $\xi _{\text {ice}}$.

Fig. 9

Yield strength determined from the experimental results for residual height plotted over the ice fraction of the corresponding mushy particle. The drawn line indicates the general trend

The experiments reveal a clear tendency towards higher values of Y for increasing ice fraction; however, the scatter of the data indicates significant uncertainty in the exact value of Y and a larger amount of data would contribute to improvement of the predictive capability indicated by the data. Nevertheless, this data presents a novel step in understanding the physics and dynamics related to the impact of mushy frozen drops and their properties.

3.3 Residual Mass of Drops Impacting with a Corona Splash

The impact of a supercooled drop on a surface increases the probability of solidification of the fluid [37]; however, as long as the drop is not contaminated before impact it usually remains liquid during the impact. Considering the timescales of the kinematic and spreading phase of the impact [26] and the propagation of dendrites in the initial solidification stage [39], in general the impact stages are significantly faster. Hence, the first stages of a supercooled drop impacting as a liquid are not affected directly by the solidification of the fluid at the beginning. However, the flow following the first contact determines the distribution of fluid on the surface and influences where it eventually solidifies. Hence, the following section deals with the spreading dynamics of a liquid drop impacting with a co-flow and the mass remaining on the surface for situations in which a splash occurs.

3.3.1 Materials and Methods

Using the icing wind tunnel described above, experiments have been carried out with drop impact velocities in the range $U_\text {0}= 0$ to $10.5\ \text {m/s}$ at supercoolings in the range $\Delta T_{\text {0}}=-5$ to $-13\ \text {K}$, keeping the drop diameter constant at $D_{\text {0}}=3\,\text {mm}$.

To conduct the measurement of deposited mass, the target is first heated to melt any ice which had formed after the splash. When melted, the liquid recedes into a spherical cap. Several images of this sessile drop are then captured from different viewing angles by rotating the target; hence, any deviations from axisymmetry or off-centering can be detected and accounted for, increasing the precision of the determined volume. The images are used for a three-dimensional reconstruction of the drop shape. This reconstruction is visualized in Fig. 10. The red lines in this figure correspond to elliptic contours obtained from different viewing angles; the black dots lie on the interpolated surface. From this point cloud, a triangulation of the enclosed volume is performed. The images are calibrated using metal spheres exhibiting a precision of ${<} 0.01\ \text {mm}$ in diameter. During the calibration, a reconstruction of the sphere volume with a maximum deviation of $0.2\ \upmu \text {l}$ is achieved. Thus, the method enables a reconstruction of drop volumes with an uncertainty of $\pm 0.2\ \upmu \text {l}$.

Fig. 10

Point cloud of drop (black) reconstructed from the framework of points (red) obtained from multiple viewing angles

3.3.2 Results and Discussion

Considering the impact of a drop with a superimposed co-flow, the flow of the impacting fluid is also affected by a deformation of the drop as it approaches the surface. An example of drop deforming in an airflow with $U_\text {air}=25\,\text {m/s}$ is shown in the right image of Fig. 11. In comparison to a spherical drop (left image), the drop exhibits a flattening of its front close to the surface. This deformation is most likely attributed to co-flow stagnation pressure arising on the target surface.

Fig. 11

Drop deformation before impact due to acceleration in the surrounding air flow. The left drop impacts without a co-flow. The right drop is accelerated by an air flow with $U_\textrm{air}=25$ m/s. Both drops are of the same volume ($V_0 \approx 14.1\,\upmu \text {l}$) [7]

In order to investigate the effect of the deformation on the flow developing on the surface, the curvature of the drop needs to be quantified. For this purpose, the curvature of the drop front, $\kappa$, is determined from the inverse of the average radius in the immediate vicinity of the first point of contact before impact. Hence, a flattened drop front leads to an increase of average radius and to smaller values of $\kappa$. Furthermore, the spreading lamella flow on the surface is considered in terms of the evolution of the wetted area as introduced by [26]. In this work, the time dependent evolution of the radius of wetted area r is found to grow proportional to $\sqrt{t}$, resulting in an expression for the dimensionless radius of the wetted area $r^+=r/R_0$ as

$$\begin{aligned} r^+=b\sqrt{t^+}. \end{aligned}$$

(13)

Here, $t^+=t U_0/D_0$ denotes the dimensionless time and b a dimensionless proportionality constant. In the investigated drop impacts of the present study, the radius of the wetted area is tracked by an image recognition software. Assuming a correlation according to Eq. 13, the proportionality constant b is determined from a least squares fit to the obtained data for each impact. The resulting values are plotted in Fig. 12 with regard to the corresponding values of dimensionless curvature $\kappa R_0$. It is apparent that the values of b increase with decreasing curvature. Thus, a flattened drop front (smaller $\kappa$) causes a faster spreading of the fluid on the surface, i.e. an increased spreading velocity of the fluid on the surface.

This perception leads to the hypothesis that the flow during the kinematic impact phase is not dependent on the average radius $R_0$, but on the curvature $\kappa$. Replacing $R_0$ in Eq. 13 leads to a correction of the proportionality constant given by

$$\begin{aligned} b=\frac{b_0}{\sqrt{\kappa R_0}}. \end{aligned}$$

(14)

Here $b_0$ denotes the constant for a spherical drop ($\kappa R_0=1$) which is marked in Fig. 12 with a dashed line. The solid line in the same figure represents a least squares fit of Eq. 14, which exhibits good agreement with the experimental data. The value of $b_0$ for a spherical drop is determined as $b_0=2.12$, which agrees well with the value obtained by [26], being $b_0=2.05$. Note that the original value given in their work is obtained with a different scaling, which requires a conversion of the value with the factor $1/\sqrt{2}$ in order to compare it to the findings of the present study. The curvature correction enables the incorporation of drop deformation effects in models describing the flow developing during impact, for instance with respect to the splashing of a drop as in [25]. A detailed description of the adaption of this model with regard to drop deformation can be found in [7].

Fig. 12

Proportionality constant b as a function of the dimensionless curvature ($\kappa R_0$) of the impacting drops lower surface [7]

Besides the drop spreading on the surface, some liquid will splash. This re-emitted liquid will end up somewhere else, possibly not even on the target [42]. The amount of deposited mass is of particular interest, since this directly influences the amount of ice accretion. Splashing of a drop impacting onto a dry surface has been a topic of exhaustive reviews in the past [11, 41, 42]; however, little work has succeeded in quantifying the ejected and/or deposited mass during the splash event. Some studies have attempted to estimate the mass of secondary drops emitted during impact of a spray. For instance, in [13] experiments were conducted using a phase Doppler system to measure this quantity. A summary of empirical models devoted to the ejected secondary droplet mass is given in [21]. In the present study, the deposited mass after impact of a supercooled drop undergoing corona splash is experimentally determined using the technique of melting the residual ice and viewing the resulting sessile drop involving several camera images.

A water drop at $20^{\,\circ }\text {C}$ impacting with a velocity and diameter considered in this study would deposit on the surface without a splash. However, supercooled drops exhibit a significant change in fluid viscosity; more precisely, the viscosity increases by a factor of 3.3 for a temperature change between 20 and $-15^{\,\circ }\text {C}$. This increase leads to a corona splash of the drop upon impact. According to [30], the threshold of this regime is defined by a critical Ohnesorge number $\text {Oh}_\text {crit}=0.0044$ ($\text {Oh}= \mu / \sqrt{D_\text {0} \rho \sigma }$), whereby a corona splash is predicted for values larger than $\text {Oh}_\text {crit}$. Considering the change in fluid properties due to supercooling, this condition is fulfilled for all investigated drops. Since the Ohnesorge number is independent of the impact velocity, this transition to the corona splash regime is solely accountable for by the fluid properties and the drop diameter. Considering splashing of supercooled water drops ($T<0^{\,\circ } \text {C}$), drops smaller than $D_{\text {0}}=2.2\,\text {mm}$ will always exhibit a corona splash. Moreover, larger droplets will transition to this regime for a relatively small increase in supercooling.

Although all observed drops impact with a corona splash, the extent of the splash differs with temperature and air flow velocity, i.e. impact velocity. This change is already observable qualitatively from the high-speed recordings. Figure 13 shows the instant before the thin film of the corona breaks up for the highest and lowest temperatures and impact velocities respectively. Considering the two upper images, a slight influence of the temperature on the splash is noticeable. The crown of the corona extends farther away from the impact position just before breakup. With respect to the time of the breakup after impact, the increased extent can presumably be explained by a more stable film during crown formation. The stabilization of the film is most likely caused by the increased viscosity due to the lower drop temperature.

A higher impact velocity will increase the extent of the corona distinctly. When comparing the two images on the left of Fig. 13, a change due to an increased impact velocity is observable. For the higher velocity the film of the corona spreads faster. Note, the asymmetry of the corona is most likely caused by the air flow onto the surface and the drop deformation before impact. An additional increase of supercooling increases the spreading of the corona even more. Again, the uprising film is stable for a longer period of time, providing more time for its expansion.

Fig. 13

Comparison of the corona extent in the incident before break up from four drops ($D_{\text {0}}=3\ \text {mm}$) with different drop temperatures and impact velocities. a $U_0 \approx 4.2 \,\text {m/s}$, $\Delta T_\text {0} \approx -5^{\,\circ } \text {C}$; b $U_0 \approx 10.5 \,\text {m/s}$, $\Delta T_\text {0} \approx -5^{\,\circ } \text {C}$; c $U_0 \approx 4.2 \,\text {m/s}$, $\Delta T_\text {0} \approx -13^{\,\circ } \text {C}$; d $U_0 \approx 10.5 \,\text {m/s}$, $\Delta T_\text {0} \approx -13^{\,\circ } \text {C}$

The changes apparent from the qualitative analysis suggest that for higher supercooling more fluid enters the film during splash; hence, a subsequent breakup would result in more mass being ejected from the drop. In order to experimentally verify this hypothesis, the fluid volume remaining on the surface after impact is quantified with the drop size estimation method described above. Since the qualitative analysis suggests a residual volume change related to the film thickness of the corona splash, the origin of the film (lamella) lifting of the substrate surface is assumed to be a crucial influencing factor for the residual volume $V_\text {res}$ on the surface.

To describe the lamella lift-off, the model of [25] considers a force balance of the aerodynamic lift force acting on the lamella and the surface tension of the fluid. This ratio is introduced as the dimensionless parameter $\beta >\beta ^*$, where $\beta ^*$ describes the critical value for the occurrence of a splash. While the value of $\beta$ in the definition by [25] depends on the Reynolds number, Ohnesorge number, the mean free path of the surrounding gas and the average radius $R_0$ of the drop before impact, the values presented in the following are obtained with an adaption to the model due to drop deformation before impact as described above. For a detailed description of the deformation, their consequences for the liquid flow on the surface and the adaption in the model, reference is made to [7].

Fig. 14

Residual volume in relation to original drop volume plotted over the splashing parameter $\beta$ defined after [25]. The impact parameters account for splashing well in the corona splashing regime ($\text {Oh}>0.0055$)

Considering the residual volume $V_\text {res}$ of a single impact with regard to the corresponding values of $\beta$, a clear trend is visible when considering drop impacts with a well-developed corona splash ($\text {Oh}>0.0055$). As shown in Fig. 14, the residual volume decreases with increasing values of $\beta$. The trend agrees well with values obtained in [3] marked with squared symbols in the figure. Hence, considering that $\beta$ monotonically increases with the impact velocity $U_0$, the drop diameter $D_0$ and the fluid viscosity $\eta _l$, an increase in each of these quantities will lead to a decrease in residual volume $V_\text {res}$ on the surface after a corona splash. Here, the influence of $U_0$ is the strongest factor with $\beta \propto U_0^2$ in the range of the present values. Despite the visible trend in the data, the physical scatter in the data suggests some missing or unaccounted for influence in the parameter $\beta$.

A consideration of the residual volume obtained for a constant value of $\beta$ confirms this interpretation of the results. In Fig. 15, the data determined for impacts with $\beta \approx 0.24$ are shown as a function of the Ohnesorge number $\text {Oh}$. The decreasing trend of $V_\text {res}$ with higher $\text {Oh}$ suggests an underrepresented influence of the viscous forces ($\sigma \approx \text {const.}$ in the present study) in the parameter $\beta$ in order to predict the remaining fluid after the splash. Since the determination of $\beta$ is mainly based on the hydrodynamics at the very beginning of the splash, the missing viscosity influence possibly takes effect at later stages of the splash, where the lifted film is already developing into a corona.

Fig. 15

Residual volume in relation to original drop volume plotted over $\text {Oh}$ for a constant value of the splashing parameter $\beta \approx 0.24$

Especially a combination of viscous effects with the overriding air flow, which is in no way captured in the quantity $\beta$, seems likely. The highest $\beta$ values correspond to the highest impact velocities $U_0$, which are themselves linked to the air flow velocity. Hence, the asymmetric splashing for high air flow velocities (see Fig. 13) suggests an influence of the air flow, a supposition which requires further investigation. Nevertheless, the quantification of the residual volume and the dependence on the splashing parameter $\beta$ represents a significant step towards measuring the mass remaining on the surface after a supercooled drop impacts on a dry surface.

4 Conclusion

The results presented above address the multitude of physical phenomena related to the impact and solidification of supercooled large drops onto cold surfaces. These results are particularly relevant to ice accretion in the aviation industry, but they elucidate fundamental phenomena which can be found in many other icing situations. The ultimate goal of this research is to improve predictive capability of ice accretion, and for this the experimental observations made in this study have been used to develop physics based models describing the respective hydrodynamics and thermodynamics involved.

The freezing of supercooled large drops occurs in three stages—nucleation, dendritic freezing, bulk freezing—whereby a significant freezing (nucleation) delay can occur even after impact onto a substrate. Depending on the freezing stage upon impact, the hydrodynamics of drop spreading can be strongly affected, influencing the area over which the ice is formed on the surface. Of particular importance for predicting ice accretion is to know the amount of volume of liquid and/or ice left on the surface, after some portion is splashed. For this, a novel method of experimentally determining this residual volume has been introduced. The experimental results obtained using this method were then used to adapt an existing model for residual mass, leading to a new theoretical relation between residual mass and impact parameters. Future work will expand the range of impact parameters experimentally investigated and consider necessary modifications to the derived models.