Measurements and modelling of the residual mass upon impact of supercooled liquid drops

The wind tunnel is a vertical open return blower wind tunnel placed inside a cooling chamber, as depicted in Fig. 3. 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 back 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 inside the tunnels settling chamber using a syringe needle inside of the wind tunnel. They are protected from the airflow by a shroud pipe during their formation. After detaching from the syringe, the drops accelerate due to gravity until entering the air flow, where they are further accelerated by the flow. In the test section, the drops impact onto a flat aluminium target, placed horizontally in the wind tunnel. The impact is captured using a high-speed camera (Photron SA-X2) with background illumination.

In order to quantify the supercooling of the drops, the temperature is continuously measured by a thermocouple inside of the syringe needle, as introduced by Schremb et al. (2017a). Additionally, the temperatures of the surface and the air flowing through the test section are monitored during the experiments.

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. 4. 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 eventually frozen drop 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, elaborated in more detail below.

In order to estimate the deposited mass, the target is first heated to melt any ice which formed during or after the impact. When melted, the liquid recedes into a sessile drop, whose exact geometry is determined by the volume of the liquid and the substrate wettability. Several images of this sessile drop are then captured from different viewing angles by rotating the target. In this manner, any deviations from axial symmetry or off-centring are accounted for, increasing the precision of the determined volume. These images are used for three-dimensional reconstruction of the drop shape.

The drop apex is used as a fix point to align the various contours obtained from each image. After arranging and scaling the contours into a single bulk object, they are used to create a point cloud of the drop. For this purpose, an elliptical curve is fitted to horizontal slices of the constructed object. Thus, the curvature of the drop surface is included in the reconstruction. Any deviation from an elliptic surface would occur most likely immediately adjacent to the surface, caused by pinning of the contact line. However, the undisturbed receding of the drop further reduces the probability of strong deviations from a non-elliptic shape. From the dot cloud, a triangulation of the enclosed volume is performed. The method is calibrated using metal spheres exhibiting a precision better than \(10\,\upmu {\text {m}}\) in diameter. In order to estimate their volume, images between \(0^\circ\) and \(180^\circ\) are captured with an angular increment of \(30^\circ\). During this 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 \(\pm 0.2\,\upmu {\text {l}}\).

To illustrate the drop deformation, two drops immediately prior to impact are visualized in Fig. 5. The drop in the left photograph of this figure impacts approximately with its terminal velocity, i.e. without an airflow. The drop in the right photograph is additionally accelerated in a co-flow with \(U_{\mathrm {air}}=25\) m/s. Three features are immediately apparent. The drop is strongly deformed, exhibiting a flattening of the lower surface, a stronger curvature on the upper surface, and a slight asymmetry. These features become even more apparent in the picture sequences presented in Fig. 6, showing a drop with an impact velocity of \(U_0\approx 9.5\,{\text {m/s}}\) in an air flow with \(U_{\mathrm {air}}=25\,{\text {m/s}}\). Here, we assume that the aerodynamic drag changes from a vertically downward force acting on the drop far from the wall to a vertically upward directed force as the drop is decelerated in proximity to the wall. Correspondingly, the stagnation end of the drop is flattened and the wake end exhibits suction and a bulging, the stagnation and wake ends changing as the drop approaches the substrate.

The light vertical asymmetry observed in these photographs and those in Fig. 9 can be attributed to the fact that the drop does not always impact exactly at the stagnation point of the gas co-flow on the target surface. Falling a distance of approx. 70 cm from entering the air flow to impact, it is not possible to control the impact position better than \(\pm 0.5\) cm.

Associated with the deformed drop, the lower surface of the drop is flattened before impact. This shape change before impact presumably also affects the development of the expanding liquid lamella ejected from the drop impact. In order to quantify the effect of drop deformation before impact on this expanding liquid lamella, the radius of the drop as it spreads on the surface (wetted area) is evaluated.

Starting with the case of an impacting spherical drop as a reference, the radius of the wetted area during the initial kinematic stage of impact is given as \(r \propto \sqrt{t}\) (Yarin 2006). It is obtained simply by truncating a sphere at a radius decreasing linearly with time and examining the radius of the cut cross section (assuming constant drop velocity upon impact). This leads to the dimensionless relationwhereby \(r^+=r/R_0\) is the dimensionless radius, \(t^+=t U_0/R_0\) the dimensionless time and b a dimensionless proportionality constant. \(R_0\) is the radius of the spherical drop. Considering now the impact of a deformed drop, its wetted radius upon impact can be estimated using Eq. (1) with b being of the order \(\mathcal {O}(2)\). If now the radius of the lower drop surface differs significantly from the spherical value \(R_0\), then the value of b will also differ. Denoting the curvature (inverse radius) of the lower drop surface as \(\kappa\) and by applying a least square fit to the wetting radius growth with time obtained from the evaluation of the high-speed recordings, b can be determined for every impacting drop event. This experimental result is shown in Fig. 7, indicating that b decreases for increasing values of the dimensionless curvature \(\kappa R_0\), i.e. when the drop lower surface flattens before impact, the spreading of the drop on the surface after impact increases in velocity.

A first-order model for the quantity \(b\) is proposed aswhere \(b_0\) corresponds to the proportionality constant of a spherical drop marked with a vertical dashed line in Fig. 7. A least square fit of Eq. (2) with \(b_0=2.12\) is plotted in Fig. 7 as a straight, solid line. The value for \(b_0\) agrees well with Rioboo et al. (2002), who suggest a value of \(b_{0}=2.05\) for spherical drops (at \(\kappa R_0 = 1\)). 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.

Finally, inserting Eq. (2) into Eq. (1) leads to a relation for the wetted area r(t) no longer dependent on \(R_0\) but on \(\kappa\); thus, supporting the hypothesis that the initial flow on the surface is governed by the curvature of the lower drop surface instead of the radius representative of its volume, \(R_0\). The scatter of the data in Fig. 7 is likely influenced by the fact that the experimental values were obtained from a single projected side view of the impacting drop from the high-speed recordings. An observation of the spreading from the top or bottom perspective would presumably provide more consistent data of the average spreading velocity. However, Fig. 7 clearly reveals a dependency between \( b \) and the local drop curvature.

An accurate evaluation of the kinematics of spreading, accounting for the local drop curvature, is necessary for an improved description of the conditions for the corona development and splash, and thus the fluid volume remaining on the surface. The parameters taken into account are now introduced following closely the theory presented by Riboux and Gordillo (2014).

The supercooled water drops examined in this study exhibit a corona splash upon impact, which is not immediately self-evident. At temperatures around \(20\,^\circ {\text {C}}\) a drop impacting with comparable size and impact velocity would exhibit a prompt splash or deposition (Palacios et al. 2012). The change into the corona splash regime is attributed to the low temperature of the water drops. When water is supercooled, its viscosity increases significantly, for instance, the viscosity of water increases by a factor of 3.3 for a temperature change from \(20\,^\circ {\text {C}}\) to \(-15\,^\circ {\text {C}}\) (VDI and GVC 2006). According to Roisman et al. (2015), the threshold parameters for the prompt/corona splash regimes are well-defined using a critical Ohnesorge number \({\text {Oh}}_{\text {crit}}= \eta / \sqrt{\rho D_0 \sigma } = 0.0044\), whereby \(\eta\) is the liquid dynamic viscosity, \(\rho\) the liquid density, \(D_0\) the diameter of the impacting drop and \(\sigma\) the surface tension. 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 drops investigated in this study, as illustrated in Fig. 8. This figure shows the Weber number \({\text {We}} = \rho U_0^2 D_0/\sigma\) of all examined drop impacts plotted over the corresponding impact Reynolds number \({\text {Re}}=\rho D_0 U_0 / \eta\), where \(U_0\) is the impact velocity of the drop normal to the target. The critical Ohnesorge number is shown in this figure as a dashed line. The temperature of the examined drops ranges between \(T_0=-3.3\,^\circ {\text {C}}\) and \(T_0=-13.8\,^\circ {\text {C}}\).

The drop temperature, represented in Fig. 8 by the symbol colour, indicates that a change in temperature results in a significant change of the Reynolds number \({\text {Re}}\); however, the Weber number \({\text {We}}\) is hardly affected, since the fluid properties contained in \({\text {We}}\ (\rho\), \(\sigma )\) only change slightly. Since the Ohnesorge number is independent of the impact velocity, the transition to the corona splash regime is merely caused by the fluid properties and the drop diameter. As indicated by the arrow in Fig. 8, an increasing impact velocity entails a transition to higher \({\text {We}}\) and \({\text {Re}}\), yet no transition to the other splashing regimes occurs. Considering a change of \({\text {Oh}}\) due to drop size, a prompt splash/deposition of a supercooled drop (\(T_{\text {drop}}<0\,^{\circ }{\text {C}}\)) is only possible for large droplets. In fact, according to this threshold, supercooled drops smaller than \(D_{{\text {drop}}}=2200\,\upmu {\text {m}}\) will always exhibit a corona splash on a smooth surface. Moreover, larger droplets will also exhibit a corona splash for a relatively small increase in supercooling.

Although all drops observed in the present study impact with a corona splash, the extent of the splash differs with temperature and air flow velocity, i.e. impact velocity. This change is observable qualitatively in Fig. 9, which shows the instant before the thin film of the corona breaks up for the highest and lowest temperatures and impact velocities, respectively. Comparing Fig. 9a, b, a slight influence of the temperature on the splash is noticeable. The crown of the corona of the colder drop extends farther away from the impact position just before breakup. We assume that the increased time until breakup and the ensuing increased extent of the splash originate from a more stable film during crown formation; the stabilization presumably being a consequence of the increased viscosity at lower temperatures.

A higher impact velocity will increase the extent of the corona distinctly. When comparing Fig. 9c, d, a change due to an increased impact velocity is observable. For the higher velocity, the film of the corona spreads faster. An additional increase in supercooling enhances 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. The asymmetries observed in the photographs of Fig. 9c, d can be attributed to drop deformation before impact and the wall tangential flow superimposed on the splashed drop.

According to the theory of Riboux and Gordillo (2014), the corona propagation during drop impact onto a dry smooth substrate is governed mainly by the inertia of the flow in the lamella, surface tension and aerodynamic effects acting on the propagation of the contact line. This propagation is described using the dimensionless parameter \(\beta\), defined in their work as the ratio of the aerodynamic lift force acting on the spreading lamella to the surface tension force. A critical value \(\beta ^\star \simeq 0.14\) is given as the splashing threshold. Hence, a splash occurs, if the impact parameters satisfy the condition \(\beta > \beta ^\star\). It should be noted that the impact parameters in this study also satisfy this condition of the corona generation on a solid substrate, caused by the aerodynamic stresses in the fast-propagating contact line.

In order to calculate \(\beta\), the instant \(t_e\) at which the lamella starts to lift from the surface is required. The fluid and gas properties and the dimensions of the lamella considered in the model at \(t=t_e\) are outlined in Fig. 10. To determine \(t_e\), Riboux and Gordillo (2014) provide an implicit correlation with regard to the impact conditions asHere, \({\text {Re}}_R\) and \({\text {Oh}}_R\) refer to the dimensionless Reynolds and Ohnesorge number, respectively. As indicated by the index \(R\), they are defined using the initial drop radius \(R_0\) as \({\text {Re}}_R=\rho U_0 R_0/\eta\) and \({\text {Oh}}_R=\eta /\sqrt{R_0 \rho \sigma }\). With knowledge of \(t_e\), the lamella thickness \(H_t\) and its velocity \(U_t\) in the moment of lamella lift-off can be calculated according toandThe parameter \(\beta\) is obtained from a balance of the lamella lift force and the fluid surface tension aswith \(\eta _g\) and \(\rho _g\) being the surrounding gas dynamic viscosity and density, respectively. The parameters \(K_l\) and \(K_u\) result from contributions to the lift force from lubrication and suction. While \(K_u\) is approximately constant (\(K_u \simeq 0.3\)), \(K_l\) depends on the mean free path of the gas molecules \(\lambda\) and \(H_t\). It is estimated using the relationNote, that Eqs. (4) and (7) are taken from the erratum published to Riboux and Gordillo (2014). Assume now that any deformation of the drop prior to impact also causes a change in wetting radius propagation and therefore in the spreading velocity of the lamella. Hence, the expression for the tangential velocity of the lifting sheet \(U_t\) must be adjusted with the correction term from Eq. (2), i.e. \(U_{t,c}=U_{t}/\sqrt{\kappa R_0}\). Additionally, the radius \(R_0\) is replaced by the inverse of the curvature \(\kappa\) in \({\text {Re}}_R\), \({\text {Oh}}_R\) and in the term for the sheet thickness \(H_t\) i.e. \(H_{t,c}=H_{t}/(\kappa R_0)\).

The mass ejected from the drop during a corona splash is strongly connected to the development of the liquid sheet lifting from the surface. As indicated in Fig. 9, the extent and shape of this sheet until it collapses varies with impact velocity as well as with liquid viscosity. Whereas an increase in viscosity is expected to result in a more stable film, an increasing impact velocity presumably lifts the sheet earlier. Both effects provide more time for fluid to enter the liquid sheet before emerging instabilities lead to its breakup. In order to analyse the residual volume on the surface as a function on impact velocity and liquid viscosity, the capillary number \({\text {Ca}}=\mu U_0 /\sigma\) is introduced. Figure 11 shows the residual volume fraction \(V_ {\text {res}}/V_0\) plotted in relation to \({\text {Ca}}\). It is apparent that for higher \({\text {Ca}}\) lower residual volumes occur. However, the large scatter for higher \({\text {Ca}}\) leaves great uncertainty in a clear decreasing trend. Considering the exclusion of drop size and shape in this empirical view, a more physical approach is thought to be more elucidating, which leads to the above-mentioned model from Riboux and Gordillo (2014).

The characteristics affecting the residual volume are combined in the parameter \(\beta > \beta ^\star\), as successfully used in Burzynski et al. (2020). Moreover, in contrast to the approach using \({\text {Ca}}\), the model provides the possibility to include changes conditional on the drop shape, as introduced earlier. In the present study, this hypothesis is followed by examining the measured residual volume as a function of \(\beta\) (Fig. 12) for Ohnesorge numbers \({\text {Oh}} > 0.055\), corresponding to a well-developed corona, far from the threshold for prompt splash. The data for \(V_{\mathrm {res}}/V_0\) correlate well with the values of the \(\beta\) parameter, supporting the model of Riboux and Gordillo (2014). The data are supplemented with values obtained by Burzynski et al. (2020), which agree well with the decreasing trend of the residual volume for increasing \(\beta\).

However, the scatter in the data from this study and data of Burzynski et al. (2020) represented in Fig. 12 by the error bars, indicate that some influencing parameters are still unaccounted for. The parameter \(\beta\) only characterises the impact during the very first stages of splashing. One can expect that at later stages of the lamella spreading further factors enter the problem, for instance the presence of Rayleigh–Taylor instabilities (Burzynski et al. 2020). Moreover, the air flow in the present experiments will also have an influence, which can only be quantified by performing additional experiments in which the drop impact point is systematically varied with respect to the air flow stagnation point on the target surface. Nevertheless, the results presented in Fig. 12, revealing a strong correlation between the factor \(\beta\) and the residual volume on the surface, is a valuable result, for which there are very few alternatives presently available in the literature.

From Eqs. (3) to (6), it is apparent that the functional dependence of parameter \(\beta\) on material properties and impact parameters is complex. Nevertheless, it is interesting to investigate whether the residual volume exhibits an additional systematic dependence on \({\text {Re}}, {\text {We}}\) or \({\text {Oh}}\). Many combinations have been investigated, and one representative example of this dependence of residual volume on \({ {Oh}}\) is shown in Fig. 13 for values of \(\beta \approx 0.24\). As expected, the residual volume decreases with increasing Ohnesorge number; yet, this trend is not universal since the influence of \(\beta\) visible in Fig. 12 is not included. In several of their studies, Riboux and Gordillo point out that under certain conditions, the parameter \(\beta\) can be represented by a combination of \({\text {Re}}, {\text {We}}\) or \({\text {Oh}}\) (Riboux and Gordillo 2014; García-Geijo et al. 2021). However, regarding the physics determining the residual volume, some influence of the forces represented by the Ohnesorge number is still unaccounted for in the parameter \(\beta\). Thus, no parameter combination is able to account for all influences responsible for the residing volume.

Nevertheless, in an attempt to illuminate to a greater degree the dependence of residual volume on impact parameters, the correlation between residual volume and the parameter \(\beta\) evident in Fig. 12 has been used to compute some dimensioned relations. To do this, first a polynomial fit to the data in Fig. 12 was determined which is shown in the figure as a dashed line. This fit was then used to compute the residual mass, given changes in various parameters entering the quantity \(\beta\). The results of this computation are shown in Fig. 14 for variations of the impact velocity, impact drop diameter and dynamic viscosity of the drop liquid. In all cases, an increase in these quantities leads to less residual volume. Note that a larger value of drop diameter as well as an increase in impact velocity has the same effect as a lower curvature of the impacting surface of the drop. Thus, a more flattened drop upon impact will result in a lower value of residual volume.