Micro-PIV study on the influence of viscosity on the dynamics of droplet impact onto a thin film

Abstract

This work presents a systematic experimental study of droplet impact onto a wet substrate. Four different silicone oils are used, covering a range of Reynolds number between $116< \text{Re} <1106$ at two different initial wall film heights. The objective is to characterize the temporal and radial evolution of the velocity field within the crown crater by means of micro-PIV. Our findings show that the velocity field has the structure of an axisymmetric stagnation point flow with decaying strength a(t). The latter exhibits an exponential decay and can be explained in terms of the exponential decay of the pressure force exerted by the impacting droplet onto the wall film. In this context, the commonly accepted functional dependence $a(t) \propto t^{-1}$ represents only the first-order Taylor approximation of the exponential decay and has therefore only a limited temporal validity. The analysis also corroborates the existence of an inertial regime concerning the velocity field for ${\text{Re}} > 270$. This is not observed at lower Re numbers due to the increased pressure losses caused by the extensional (normal) strain during the radial spreading of the lamella. To validate these findings a holistic approach is chosen, which combines numerical results, analytical solutions and experimental data from literature. In particular, by using the continuity equation, it is shown that the experimental decay of the wall film height can be reconstructed from the velocity measurements. Consilience of results from different approaches provides a robust validation of the micro-PIV data obtained in this work.

Graphical abstract

1 Introduction

Droplet impact onto a thin liquid layer (film) is a well-known phenomenon occurring in our everyday life. Even though an impact of a single droplet appears mundane at first sight, it is a fundamental process that plays an important role in many natural and industrial processes such as soil erosion, application of pesticides, spray coating and printing technologies, spray cooling of micro-electronic devices or in propulsion systems in air and spacecrafts (Yarin 2006). Optimizing such applications requires a detailed knowledge of the droplet impact process.

The complexity of a droplet impact onto a thin liquid layer is shown in Fig. 1, which shows the rapid displacement and deformation of the liquid droplet and wall film. A variety of parameters affect the outcome of the droplet impact, such as the initial droplet diameter $D_0$, impact velocity $U_0$, initial film height $H_{f,0}$ and the liquid properties (surface tension $\sigma$, dynamic viscosity $\upmu$, density $\rho$). In this work, we investigate single-component, normal droplet impacts, where the Reynolds number $\textrm{Re}=\rho U_0 D_0/\upmu$, the Weber number $\textrm{We} = \rho U_0^2 D_0 / \sigma$ and the dimensionless film height $\delta =H_{f,0}/D_0$ define the impact conditions. The dimensionless time $\tau$ is defined as $\tau =t U_0/D_0$ with time t.

Fig. 1

Schematic of a single droplet ($D_0,U_0,\rho _d,\upmu _d,\sigma _d$) impact onto a thin liquid film ($H_{f,0},\rho _f,\upmu _f,\sigma _f$). The droplet is seeded with tracer particles ($D_p,\rho _p$). The crown crater consists of the liquid layer of the droplet, denoted as droplet lamella, as well as the liquid of the initial film, denoted as film lamella. The Region of Interest (ROI) represents the measurement volume, which is limited by the Depth of Field (DOF) and the Field of View (FOV)

This wide parameter space leads to a manifold of possible impact outcomes, such as deposition, transition and crown splashing. A comprehensive overview can be found in Liang and Mudawar (2016). In most cases, researchers investigated the macroscopic impact morphology and kinematics, i.e., the transformation of the primary droplet into a crown, its propagation and the formation of secondary droplets. Moreover, in order to predict certain macroscopic flow features, e.g., the crown wall propagation in radial direction, great effort has been put into simplified descriptions of the whole process via analytic modeling. In this context, Yarin and Weiss (1995) proposed an asymptotic solution for the axisymmetric inviscid flow within the crown crater, which covers the area from the impact point to the crown wall as shown in Fig. 1 (on the right). For sufficiently large times t, they modeled the radial velocity as $u_r(t) = B r /(1 + B t)$, where r denotes the radius and B is a positive constant. Note that this velocity profile automatically satisfies the inviscid momentum equation under the assumption of incompressible flow and negligible pressure gradient. These assumptions lead to a constant radial velocity over the film thickness. A rearranged version of this one-dimensional inviscid and incompressible model was given by Kittel et al. (2018), which reads

$$\begin{aligned} u_r(t) = \frac{1}{c \frac{D_0}{U_0}+t}r = a_{\text {YW}}(t) \, r \end{aligned}$$

(1)

with a dimensionless constant c. Due to this reformulation the constant B is now linked to at least some of the initial impact parameters, yielding $B=U_0/(c D_0)$. Lamanna et al. (2022) instead set the constant to $c = 1/ \lambda = f(\textrm{We},\textrm{Re},\delta )$, resulting in the estimation $B= \lambda U_0 / D_0$. The parameter $\lambda$ was derived empirically by Gao and Li (2015). Following the approach of Yarin and Weiss (1995), Lamanna et al. (2022) employed the parameter $\lambda$ to estimate the kinetic energy transmitted to the crown wall, thereby recovering the well-known $\sqrt{t}$-dependence to describe its propagation. Finally, the inclusion of viscous losses led to a very good agreement with experimental data over longer time periods (i.e., $5< \tau < 30$) and a wide range of impact conditions and test fluids (Lamanna et al. 2022). Philippi et al. (2016) performed a detailed numerical simulation of a single droplet impact onto a dry wall ($\textrm{Re}=5000$, $\textrm{We}=250$) by employing an incompressible Navier–Stokes solver. Due to adaptive mesh refinement procedure, they were able to fully capture the features of the boundary layer, leading to a very accurate description of the short-time dynamics of the impact process (i.e., $\tau \le 1$). During this time span, their numerical findings showed that the impinging droplet induces an unsteady stagnation point flow. The inviscid self-similar impact pressure and velocities were shown to depend solely on the self-similar variables $(r/\sqrt{t},z/\sqrt{t})$. Roisman et al. (2009) developed an analytical solution for an unsteady laminar viscous flow in a spreading liquid film. As asymptotic (outer) boundary condition, Eq. 1 was chosen with $c=0$. This approach represents one of the first examples of including boundary layers effects in droplet impact problems. Building upon this analysis, Roisman (2009) concluded that viscous effects influence the evolution of the lamella thickness only at late stages of the spreading.

Despite the noteworthy progress, it remains unclear whether the temporal velocity decay in Eq. 1 (i.e., the 1/t-dependence) remains valid beyond the short-time dynamics of the impact process. This is due to a number of concomitant factors. First, the available analytical solutions all rely on Eq. 1 as boundary condition for the outer flow (see e.g., Roisman et al. 2009; Roisman 2009). Second, the absence of reliable experimental data prevents a clearer picture of the long-time evolution (i.e., $\tau > 2$) of the flow velocity within the crown crater. Such measurements are extremely challenging due to the difficult optical access to the sub-millimeter measurement volume of the spreading lamella, the unsteadiness and the short duration ($\mathcal {O}(10^{-3}$s)) of the overall impact processes. Third, accurate velocity data from highly resolved numerical simulations can only be obtained with great computational effort (see e.g., Fest-Santini et al. 2021). This explains why the available studies, reporting a detailed asymptotic analysis of the velocity and pressure field, are extremely rare and limited to the short-time dynamics (i.e., $\tau < 1$) (Philippi et al. 2016).

More recently and thanks to the rapid development of high-speed imaging technologies over the last years (Cheng et al. 2022), experimental investigations have become increasingly more capable to resolve the relevant spatio-temporal scales, thereby offering a great time benefit if a large parameter space has to be investigated. The most common experimental technique for flow visualization is the particle image velocimetry (PIV), which employs tracer particles to obtain velocity information of a flow field (Raffel et al. 2018). In classical PIV, the tracer particles are illuminated by a planar laser light-sheet (typically 1–2 mm thick). The particles outside of the light-sheet are not illuminated and thus the measurement volume is well defined as a cross-sectional cut of the flow. In microfluidic applications, instead, the measurement volume itself is in the sub-millimeter range and, therefore the so-called volume illumination is typically applied, leading to the concept of micro-PIV (Santiago et al. 1998). In this case, the measurement depth is defined by the depth of focus of the microscope objective rather than by the thickness of the laser sheet (Lindken et al. 2009; Wereley and Meinhart 2010). In both PIV approaches, the measurement procedure is similar and consists in making consecutive recordings (image frames) of the tracers with a time separation $\Delta t$ in between. The image frames are then divided into interrogation windows and one velocity vector is obtained for each interrogation window by performing a cross-correlation of the image frames. More details on the post-processing procedure and the strategies to increase its accuracy can be found in Raffel et al. (2018).

With reference to droplet impact scenarios, both classical PIV and micro-PIV have been applied with minor differences, since the measurement volume is essentially determined by the thickness of the radially spreading lamella, which is in all cases in the sub-millimeter range. Hereafter, a short overview is presented on velocity measurements within a radially spreading lamella performed with both PIV approaches. The overview includes studies on droplet impact onto both dry and wet surfaces, covering the regimes of deposition and splashing. A very consistent picture can be derived from these measurements, thus corroborating the previous statement on the equivalence of velocity measurements with the classical and micro-PIV optical configuration.

Smith and Bertola were the first to apply time-resolved micro-PIV to a droplet impact experiment by coupling a high-speed laser into an inverted epifluorescent microscope (Smith and Bertola 2010, 2011). The objective was to investigate the effect of polymer additives on the rebound of impacting droplets onto a hydrophobic surface in the deposition regime. They studied pure water (Newtonian) and aqueous solutions of poly-ethylene oxide (non-Newtonian) over time period of $0.5<\tau <4$. For the spreading phase they found that, apart from considerable scatter, the radial velocity distribution for both fluids fulfills the inviscid solution of Yarin and Weiss (1995) (i.e., Eq. 1) and the constant c should be zero. Lastakowski et al. (2014) conducted PIV measurements of droplet impact onto hot (above the Leidenfrost point) and cold (room temperature) surfaces with ethanol and isopropanol-glycerol mixtures for $2<\tau <6$. In the case of a hot surface, they found a good agreement of the radial velocity with the analytically solution by Yarin and Weiss (1995). Following their nomenclature, this is equivalent to setting the constant $c = 0.5$. In contrast, for the cold case they observed that the radial velocity is much lower and decreases faster with time, while preserving the linear radial dependence. The authors explained this difference in terms of viscous decoupling (hot case) or coupling (cold case) between the drop and the substrate, respectively. Erkan and Okamoto performed similar studies of impinging water droplets on unheated (Erkan and Okamoto 2014) and heated (Erkan 2019) surfaces (25–250$^{\circ }$C) with focus on the early spreading phase ($0.15<\tau <1$). They analyzed the effect of Weber number ($\textrm{We}=4-5,10-15,27.6$) on the radial spreading velocity and fitted their data to Eq. 1 with two different values $c=0.3$ for $\textrm{We} = 4.9$ and $c=0.58$ for $\textrm{We} = 27.6$. Moreover, the experimental radial velocity distributions demonstrated linear behavior for the inner radial positions, whereas a nonlinear behavior was observed for the outer radial positions (i.e., $r>D_0/2$) owing to the vertically upward flow (see also Gultekin et al. 2023). For the unheated surface, Erkan (2019) compared the radial velocity distributions from the PIV measurements with the viscous analytical solution from Roisman et al. (2009) and found a significant disagreement in the temporal decay. Specifically, the temporal decay predicted by the viscous solution is faster than in the experiment. The authors attributed this disagreement to the action of an additional force, such as a radial pressure gradient, that persistently overwhelms the viscous forces, thus retarding the deceleration process. This is, indeed, an interesting remark since all available analytical models neglect the influence of the pressure gradient, even though it is well-known that an impacting droplet exerts a transient force on the target surface (Mitchell et al. 2019). Gultekin et al. (2020) used PIV measurements to investigate the spreading velocity within the droplet lamella for $0.5< \tau <2$. They covered both single and double droplet impacts onto dry heated and unheated surfaces (20–250$^{\circ }$C) for moderate Weber numbers ($40<\textrm{We}<190$). They observed that the surface temperature has little effect on the radial velocity distribution in the inner radial position ($\tilde{r} =r/D_0 \le 0.7$) and proposed $c = 0.48$ and $c = 0.53$ as fitting constants for Eq. 1.

In a recent study, Gultekin et al. (2023) investigated droplet impacts on dry solid surfaces over a wide range of $\textrm{We}$ numbers ($5<\textrm{We}<183$) and associated $\textrm{Re}$ in the range of $882<\textrm{Re}<5570$. In all cases, the radial velocity exhibited a linear dependence for all inner radial positions, albeit the initial slope varied with $\textrm{We}$ for $\tau = 0.5$. This indirectly implies that the estimation of the constant c must depend upon the impact conditions and fluid properties (i.e., $c = f(\textrm{We}, \textrm{Re}$).

Even less studies have been conducted for droplet impact onto wet surfaces. The first attempt dates back to the work of Ninomiya and Iwamoto (2012). They measured the velocity of the crater surface of an impacting milk droplet with black urethane foam, shaped like flakes of size $100\text{-}400 \, \upmu \text{m}$ as tracer particles. The first example of time and height resolved velocity measurements on dry and wet surfaces was presented by Frommhold et al. (2015). As test fluids water and ethanol were used. The authors employed particle streak photography, where streak images of fluorescent tracers in the drop and in the liquid film (if present) were recorded by a high-speed camera. The latter was attached to an inverted microscope with a narrow depth of field ($< 3 ~\upmu \text {m}$), that could be shifted vertically by a precise piezo-driven z-stage. By changing the focal plane, it was possible to scan through the flow with an effective spatial resolution of $5 ~\upmu \text {m}$, because tracer particles out of focus become blurred and hardly visible. Several experiments were performed for each impact condition in order to measure the radial velocity stepwise within separate layers from the substrate up to $40 ~\upmu \text {m}$. This procedure was justified by the high reproducibility of deposition experiments and by the high precision of the dispenser mechanism. The measurements took place during the elongation phase of the droplet. High-speed imaging of the impact process were obtained separately. They found that the maximum wall shear stress by impact on liquid films is significantly lower (up to one order of magnitude lower) than for the dry case. This finding is of great importance and implies that for impact on wet substrates deviations from the inviscid solution of Yarin and Weiss (1995) (i.e., Eq. 1) may not necessarily be caused by viscous losses.

This brief excursus on literature studies allows us to draw the following conclusions: First, most of the velocity measurements have been performed during deposition experiments, while hardly any data can be found for droplet splashing on a wet substrate. Second, all available data confirm that the radial velocity increases linearly with distance from the impact point (stagnation point). No general consensus, instead, is found on the temporal decay of the radial velocity. In this context, the remark from Erkan (2019) on the role of the pressure gradient in controlling the velocity decay requires further evaluation. Third, mainly low viscosity fluids were used (e.g., water or alcohols) with no systematic variation of the dynamic viscosity. These open questions motivate the present study, which aims to acquire reliable radial velocity distributions for droplet impact onto a wet surface also during a splashing event. In this case, due to the higher fall height, the impact point varies within a circular area of a few square millimeters. Consequently, the macroscopic visualization of a splashing event must occur synchronously with the velocity measurements on the microscopic scale, in order to assure that the small scale features are correctly embedded in the overall dynamics of a splashing event. The feasibility of this approach was evaluated in two preliminary studies from this group. As a first step, Vaikuntanathan et al. (2020) investigated the possibility to perform time-resolved micro-PIV by employing a high-intensity light emitting diode (LED) to constantly illuminate the measurement volume from above. Even though LED illumination has been already successfully applied in microfluidic devices (Hagsäter 2008), its application to a splashing scenario is not straightforward. For this purpose, a high-speed camera in combination with an inverted microscope was employed to record shadowgrams of the impact zone with a backlight optical configuration. As a second step, Bernard (2020) added an additional high-speed camera to capture the macroscopic view of the impact process. This enabled a precise determination of the actual impact conditions and dynamics. The overarching goal of these preliminary studies was to perform a proof-of-concept study on the feasibility of simultaneous micro and macro flow visualization of a splashing event. The present work builds upon the previous studies and performs a trade-off analysis for the optimal choice of mechanical and optical parameters (e.g., magnification, light intensity, tracer size) to assure a high reproducibility of the experiments and maximize the measurement time. Moreover, it includes a systematic variation of the impact Reynolds number by one order of magnitude ($\mathcal {O}(10^{2})< \textrm{Re} < \mathcal {O}(10^{3})$) to cover both the regimes of deposition and splashing. The Weber number, instead, is kept approximately constant ($\textrm{We} \approx 800$). The objective is to investigate how the transfer of specific momentum and its radial distribution are affected by gradually increasing the relative importance of viscous forces. The term specific momentum refers to the momentum per unit mass which represents a velocity e.g., $u_r(t)$ (Lamanna et al. 2022). In this context, a holistic approach to the analysis of the PIV data is undertaken, which foresees three parallel evaluation paths to verify the plausibility of the measurements. First, the experimental data are employed to understand the influence of fluid properties and wall film thickness on the deceleration experienced by the spreading lamella. Second, a single experiment is compared to the predictions from a direct numerical simulation (DNS). The comparison is not only limited to the radial velocity distribution, but includes also the decay of the film height. For the conservation of mass, the radial spreading of the liquid lamella is associated to a decrease in film height. Hence, its functional dependence upon time can be directly derived from the measured radial velocity profiles. A similar analysis can be also performed with theoretical models. Consilience of results between different approaches provides an indirect validation of the experimental measurements and associated conclusions.

The present paper is organized as follows. Section 1.1 discusses the link between the radial velocity and temporal evolution of the wall film height. It also shows how the choice of the initial velocity distributions (and associated simplifying assumptions) constrains the evolution of all other variable in the flow. Section 2 describes the experimental test rig and post-processing procedure of the PIV data. Section 3 briefly introduces the in-house numerical solver. Finally, Sect. 4 discusses the experimental data and the comparison with numerical simulations and theoretical models.

1.1 Fundamentals of modeling the crown crater dynamics

This section reviews important fundamentals on the dynamics of the thin liquid lamella within the crown crater during the impact process. Starting point is the assumption of an axisymmetric stagnation point flow. In the potential flow region, the radial velocity component is constant over the height (block profile) and reads as

$$\begin{aligned} u_r(r,t) = a(t) r . \end{aligned}$$

(2)

The radial velocity increases linearly with the radius but decreases with time, as confirmed in several experimental studies, see e.g., Smith and Bertola (2010), Gultekin et al. (2023). The time decay is controlled by the parameter a(t), also denoted as the strength of the stagnation point flow. As shown later, this relation is also observed in the present study and Eq. 2 holds for the investigated time interval $1<\tau <5$ and radii $0<\tilde{r}<0.5$, where $\tilde{r}=r/D_0$ denotes the dimensionless radius. Assuming an incompressible liquid (i.e., $\rho = const$) and a vanishing angular velocity component ($u_{\phi }=0$), the height of the film h(t) can be derived by substituting Eq. 2 into the continuity equation ($\nabla \cdot {\textbf {u}} = 0$), yielding

$$\begin{aligned} 0 = \frac{1}{r} \frac{\partial }{\partial r} \left( r \, u_r \right) + \frac{1}{r} \frac{\partial }{\partial \phi } \left( u_{\phi } \right) + \frac{\partial }{\partial z} (w). \end{aligned}$$

(3)

Solving for the vertical velocity component w(t) leads to the well-known expression for the outer solution of a stagnation point flow

$$\begin{aligned} w(z,t) = \frac{\partial z}{\partial t} = -2a(t)z. \end{aligned}$$

(4)

With the choice of the coordinate system as shown in Fig. 1 (on the right), we can set $h = z$ and the temporal evolution of the height is thus given by

$$\begin{aligned} \frac{\partial h}{\partial t} = -2 a(t) h(t). \end{aligned}$$

(5)

Note that within the time interval, we are looking at, the crater surface can be considered to be flat ($h(t,r)\approx h_c(t)$) with the subscript c denoting the position on the symmetry axis. Equation 5 can be integrated provided the temporal decay of the strength of the stagnation point flow a(t) is known. The latter can be derived from the momentum equation in radial direction

$$\begin{aligned} \frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + w \frac{\partial u_r}{\partial z} = \nu \left[ \frac{\partial }{\partial r}\left( \frac{1}{r} \frac{\partial r u_r}{\partial r} \right) + \frac{\partial ^2 u_r}{\partial z^2} \right] - \frac{1}{\rho } \frac{\partial p}{\partial r}. \end{aligned}$$

(6)

Taking into account that $u_r$ does not depend upon z (see Eq. 2) and neglecting viscous forces, surface tension, gravity and pressure gradient, Eq. 6 simplifies to

$$\begin{aligned} \frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r}=0. \end{aligned}$$

(7)

The potential flow of an axisymmetric stagnation point flow enables the use of the separation ansatz $u_r=a(t)r$ (see Eq. 2), which inserted in Eq. 7 yields

$$\begin{aligned} \frac{\partial a(t)}{\partial t} + a^2(t) = 0. \end{aligned}$$

(8)

The latter admits the unique algebraic solution

$$\begin{aligned} a(t) = \frac{1}{const. +t}, \end{aligned}$$

(9)

which coincides with the solution proposed by Yarin and Weiss (1995) for the radial velocity distribution (see Eq. 1, $const. = c D_0/U_0$). By inserting Eq. 9 into Eq. 5 and integrating, the solution of Yarin and Weiss (1995) for the height profile can also be recovered

$$\begin{aligned} h(t)=\frac{\beta }{(c\frac{D_0}{U_0} + t)^2}, \end{aligned}$$

(10)

where $\beta$ is the constant of integration. In literature, Eqs. 1 and 10 are often referred to as the inviscid solutions. Based on the previous derivation, it is clear that both inviscid solutions can have only a limited spatio-temporal validity. In this work, they are referred to as 1/t and $1/t^2$ dependence. This is because alternative inertia-driven solutions can be derived by including the effect of the pressure gradient, as shown in more detail in Sect. 4.4. This is necessary because both in the classical stagnation point flow and for an impinging droplet, the re-direction of velocity from the vertical to the radial direction results in a pressure force that pushes the liquid lamella radially outwards. For the wall film height, corrections due to boundary layer effects have been proposed. In particular, it has been shown that the liquid film approaches a residual film thickness $h_{res}$. For impacts on dry substrates, Roisman (2009) proposed the following estimation for $h_{res}$

$$\begin{aligned} h_{res}= 0.79 \, D_0 \textrm{Re}^{-2/5}. \end{aligned}$$

(11)

Van Hinsberg et al. (2010) extended the previous correlation to impacts on wet surfaces by introducing a dependence upon $\delta$, yielding

$$\begin{aligned} h_{res} = \left[ 0.98 \, \delta ^4 + 0.79\right] \, D_0 \textrm{Re}^{-2/5}. \end{aligned}$$

(12)

As a matter of fact, a variation in the initial film thickness for $\delta \ll 1$ has almost no effect on the residual film thickness $h_{res}$. This finding was recently corroborated by Stumpf et al. (2022), who measured $h_{res}$ for droplet impacts onto a very thin liquid film ($0.02\le \delta \le 0.06$). In the present work, initial film heights of $\delta = 0.1$ and 0.2 are employed so that both estimations can be used without any appreciable difference.

2 Experimental method

2.1 Experimental facility

The main components of the facility are the droplet generation system, the optical recording system and the impact area. The experimental rig is schematically shown in Fig. 2 and allows the simultaneous visualization of the macroscopic impact morphology and the microscopic flow field within the crown crater. An example of the macroscopic and microscopic recordings is shown in Fig. 3. The radial velocity is measured with the micro-PIV method by employing only the microscopic view. As tracers polystyrol particles are used, whose properties are shown in Table 1.

Fig. 2

Schematic setup of the experimental facility. It consists of the droplet generation system, the optical recording system and the impact area. The fall height $H=0.57 ~\text {m}$ is defined by the distance between the needle and the film surface. The mirror angle $\alpha =21.2^\circ$ is held constant

In order to enable a precise micro-PIV analysis during post-processing, a homogeneous distribution of particles within the droplet and a good reproducibility of the seeding is of crucial importance. As the generation of such a homogeneous mixture can be challenging, the droplet seeding procedure is described in more detail hereafter. A syringe pump (0.01 ml/min, Legato 2110, kdScientific) feeds pure silicon oil through a tubing system towards a Drifton 1/2" needle. The seeding particles are injected into this flow at a connector in the vicinity of the needle (see Fig. 2) by using an additional syringe containing a prepared particle-oil mixture (0.5 g particles added to 40 ml oil).

Table 1 Properties$^1$ of polystyrol seeding particles

This two-needle system not only enables a targeted dosing of seeding particles, but also prevents particle sedimentation in the tubing system. Note that the particle concentration might vary slightly for each droplet impact experiment. This slight variation, however, does not affect the results of the micro-PIV analysis and is, therefore, assumed to be negligible. When the weight of the oil-particle mixture, accumulated at the tip of the needle, exceeds the force caused by surface tension, a droplet detaches. During the free fall, the droplet passes a light barrier, which triggers the imaging system. The light barrier consists of a 635 nm continuous laser (1 mW, Laser Components), whose circular beam profile is transformed into a laser sheet by employing a cylindrical lens.

The main components of the imaging system are two fully synchronized Photron Fastcam SA-X2 high-speed cameras (Macro SA-X2 and Micro SA-X2) and a microscope Axio Observer Z1 with a 5x objective of the company Carl Zeiss. The cameras have a CMOS chip (1024 x 1024) with a pixel size of 20 $\upmu$m. Both cameras are set to record 12-bit grayscale images at a frame rate of 12,500 fps. The macroscopic perspective operates in backlit mode and records the droplet impact from an inclined top view. As shown in Fig. 2, the Macro SA-X2 is positioned horizontally next to the impact area and observes the impact by means of a tilted mirror, in order to avoid blocking the droplets falling path. A high power LED (Cree, color temperature of 6500 K, viewing angle 120$^{\circ }$, 1,827 lm) is used to illuminate the impact area (pool). The LED is placed underneath the pool and shines diagonally upwards onto the tilted mirror. The Macro SA-X2 is equipped with a magnification zoom lens system (NAVITAR). The resulting optical resolution is 19.2 $\upmu$m/px in the horizontal direction and 28.5 $\upmu$m/px in the vertical direction due to the tilted mirror. The shutter time of the camera is set to $1 \times 10^{-5}$ s. The microscopic perspective also operates in backlit mode and records the droplet impact from a bottom view through the liquid layer. A Constellation 160B mini $28^\circ$ from the manufacturer Veritas (color temperature of 5000 K, luminous flux of 16,000 lm) is used as the light source. The light passes from top to bottom through the measurement volume, the liquid film and into the inverted microscope. The optical resolution of the microscopic perspective is 2.44 $\upmu$m/px and the shutter time of the camera is set to $3.75 \times 10^{-6}$ s. The impact area consists of a pool construction, in which the liquid wall film can be placed. The pool has an inner diameter of 44 mm and its bottom consists of a sapphire glass disk with a thickness of 1 mm. The disk surface is considered as plane (S/D-20/10, $\lambda$/4) (Bernard 2020). Sapphire glass was chosen to enhance the confocal-chromatic film thickness measurements. This technique benefits from larger differences in refractive index $\theta$ between the solid substrate (sapphire glass: $\theta _1=1.755$) and the investigated silicon oils ($\theta _2\approx 1.4$).

2.2 Experimental parameter space

The experimental parameter space consists of eight points, which result from investigating four different silicone oils (B5, B10, B20, B50) at two different film thicknesses, respectively. Droplet diameter $D_0$ and impact velocity $U_0$ are kept constant. The parameter space and the liquid properties are shown in Table 2. For each test condition of the parameter space, each experiment was repeated five times. An overview of all test conditions is shown in Table 3.

Table 2 Physical properties of test liquids and impact parameters. The latter have a sample standard deviation of 4.4 %

2.3 Analyses of the droplet impact phases

As shown in Fig. 3, the droplet impact process is divided here into four phases: $\tau <0$, $0<\tau <1$, $\tau >1$ and $\tau \gg 1$. The pre-impact phase ($\tau <0$) is used to determine the initial droplet diameter $D_0$ and impact velocity $U_0$. The last ten images of the macroscopic view, preceding the droplet impact onto the film, are used to calculate the impact velocity. The droplet shadow in the microscopic view is only sharp and clear in the last two images before impact. The reason is that the focus plane is set at the interface between the pool bottom and film. This shadow is used to determine the droplet diameter.

Fig. 3

Time sequence of the drop impact of exp. No. 18 (B10, $\textrm{We}=803$, $\textrm{Re}=566$, $\delta =0.2$) recorded with combined macroscopic and microscopic imaging techniques, as shown in Sect.2.1. (a) Macroscopic image sequence of the droplet impact onto a thin wall film resulting into a splashing outcome and the corresponding microscopic images of the fluid flow within the crown base. (b) and (c) Schematics of the surface contour at two different times, representing the early appearance of the shadowgram (b) and the growing radius of the crater area (c), delimited by the crown wall

The phase shortly after impact ($0<\tau <1$) describes the initial deformation of the droplet and the formation of the liquid lamella. During this period the hemispherical outer shape of the droplet contour reflects the incoming light from the top LED. Thus, areas with high slope cast a shadow in the microscopic view, as shown in Fig. 3. In contrast, areas parallel to the pool surface transmit light, which visualizes the seeding particles as black dots ($\approx 7$ px, blurred by the little out of focus position). The synchronized macroscopic and microscopic visualization allows to connect the microscale processes to the overall splashing dynamics and facilitates the interpretation of the shadowgrams. During the third phase ($\tau > 1$), the droplet shape flattens and a kinematic discontinuity (alias the crown wall) is formed, which surrounds a crater with a smooth and plane surface. Finally, in the late stage of the droplet impact process ($\tau \gg 1$), the crown spreads radially and then recedes.

The impacting droplet entraps air that forms a bubble at the impact center (Chandra and Avedisian 1991; Mehdi-Nejad et al. 2003), which is present in all experiments. As shown in Fig. 3, it is represented by a larger circular shadow of approximately 26 px in diameter. Moreover, clusters of particles may also form, as shown in Fig. 3c. The effect of these clusters on the velocity field can be neglected. The microscopic images are used to perform a micro-PIV analysis and extract a two-dimensional map of the radial velocity within the crown crater. The analysis covers the time interval $1<\tau <5$. Earlier times cannot be analyzed due to the presence of the droplet shadow.

2.4 Post processing of the experimental data

This section briefly discusses the extraction of the radial velocity field $u_r(r,t)$, its slope a(t) as well as the procedure to derive the temporal decay of the film height h(t) from the velocity distribution. In general the velocity of each tracer particle depends on its radial, angular, and vertical position within the liquid layer as well as on time, i.e., $u(r,\varphi ,z,t)$. An in-house Matlab^® routine determines the velocity field $u_r(r,t)$ from the microscopic images. Hence, the measured velocity is averaged over the line-of-sight of the microscopic view, i.e., in z-direction $u(r,\varphi ,t)$. The depth of field (DOF) is of the order of 280 $\upmu$m (Bernard 2020) and sufficient to detect all particles in the liquid layer. In addition, the velocity field is averaged over the azimuthal angle $\varphi$. The following paragraphs describe how the routine determines the velocity field. First, a 2D flow field is extracted with the open source software PIVlab (Thielicke and Stamhuis 2014; Thielicke 2014; Thielicke and Sonntag 2021). PIVlab determines the velocity field in a Cartesian grid with $63 \times 63$ velocity vectors u(x, y, t) with the following settings. For image pre-processing, the CLAHE filter (window size 20 px), Wiener filter (window size 3 px) and auto contrast stretch are enabled. The PIV algorithm uses Fast Fourier Transformation (FFT) and four passes are chosen, namely, 256, 128, 64, 32 with a step size of 50%. A subset of the resulting vector field is shown in Fig. 4a. Second, the in-house routine transforms the velocity field into a cylinder coordinate system $u(r,\varphi ,t)$. To perform the coordinate transformation the coordinates of the center of impact are used. The center of impact (see Fig. 4b) is derived by calculating the intersection point of the streamlines, in analogy to the particle streak approach by Smith and Bertola (2011).

Fig. 4

Post-processing of the velocity field. The white circle in both images marks the center $\textrm{C}$ of the velocity field. It is derived by calculating the intersection point of the streamlines. (a) Example of the extracted vector field. The length of each vector represents the magnitude. (b) Two-dimensional map of the velocity magnitude field. Here, the annulus $\Delta r$ is exemplary plotted for one radial step. Moreover, the radius with the maximum velocity $u_{r,\textrm{max}}$ as well as the inner border cast by the crown wall are labelid. Test case: exp. No. 6 (B5, $\delta =0.2$) at $\tau = 1.03$

As shown in Fig. 4a, the detected center of impact is located closely to the entrapped air bubble, which forms at the point of droplet impact. Since the droplet spreading is point symmetric with respect to the center of impact the velocity field is shown by $u_r(r,t)$. For data reduction the velocity data is filtered with the following three criteria:

• Vectors located within the crown wall or at larger distance from the center are excluded.

• Vectors with angular deviation from the radial direction ($>10^\circ$) are excluded.

• Vectors within a discrete annulus $\Delta r$ are averaged.

The resulting magnitude field for one time step is shown in Fig. 4b. The outer (red) circular segment marks the inner border of the shadow cast by the crown. All velocity vectors beyond this border are excluded and therefore the magnitude is color-coded in dark blue. The angular deviation for each velocity vector is calculated by performing the scalar product with its own position vector from the center. As an example, the magnitude field shown in Fig. 4b has dark blue regions at the right and left border, where vectors with high angular deviation were excluded. The discrete annulus ($\Delta r=10$ px) is represented by the two black circles in Fig. 4b. All vectors with the origin inside the discrete annulus are averaged over the azimuthal angle $\varphi$. Vectors with a magnitude deviation greater than three standard deviations (SD) are excluded. The location of the averaged maximum velocity $u_{r, \text {max}}$ is indicated by the black circular arc in Fig. 4b. For further analysis, all velocity vectors with radius larger than the radius of $u_{r, \text {max}}$ are also excluded, as they are located in the crown wall and experience a vertical upward flow. This data reduction procedure is applied to all time steps.

In Fig. 5, the spatially averaged velocities ($u_r(r,t)<u_{r,\text {max}}(t)$) are shown as a function of radial position and time, respectively. In agreement with previous studies, the velocity data exhibit a linear increase with radius and a nonlinear decrease with time. This representation allows also to identify the early time limit of our measurement technique. Note that the maximum attainable velocity in r-direction $u_{r,\text {max}}(t)$ depends upon the spreading of the droplet itself, which is a function of time and impact conditions (i.e., $\textrm{Re}, \, \textrm{We}, \, \delta$). For a specific time (e.g., $\tau =1$), this dependency of the maximum radial velocity upon impact conditions was also reported by Roisman et al. (2009) for different Weber numbers and constant $\textrm{Re}$.

Fig. 5

Example of a radial velocity field $u_r(r,t)$ extracted from the micro-PIV data. The different representations highlight (a) the linear increase of the velocity with radius and (b) the nonlinear decrease with time for fixed radii. Test case: exp. No. 6 (B5, $\delta =0.2$) for the range $1<\tau <5$

Based on the above considerations, a region of interest (ROI) is introduced to assure an adequate comparison among the different experiments. Specifically, the analysis of the velocity field is restricted to the area extending from the impact point to the radius $\tilde{r}\le 0.5$, as shown in Fig. 6a. This definition of the ROI guarantees that even the experiment with the lowest $u_{r, \text {max}}$ provides sufficient data points to achieve statistical significance for the determination of the slope $a(t) = \partial u_r(r,t)/ \partial r$. The latter is derived by performing a linear fit of the velocity data at a given t, as shown in Fig. 6b. The parameter a(t) [s$^{-1}$] represents the decay rate of the radial velocity. Its functional dependence upon time enables a meaningful comparison with numerical simulations and theoretical models.

Fig. 6

Example of data reduction to extract the decay rate a(t) from the radial velocity data. (a) Linear fit within the region of interest. (b) Derived function a(t) and comparison with an exponential fit. The coefficients $c_1$ and $c_2$ are shown in Table 3. Test case: exp. No. 6 (B5, $\delta =0.2$) for the range $1<\tau <5$

As shown in Fig. 6b, the experimental decay of the parameter a(t) can be well approximated by fitting an exponential function to the experimental data, yielding

$$\begin{aligned} a(t) \approx c_1 \exp (-c_2 t) . \end{aligned}$$

(13)

Following the analysis shown in Sect. 1.1, Eq. 13 can be employed to derive the decay of the wall film. Specifically, by inserting the above mentioned exponential function in the continuity equation (Eq. 5) and integrating yields the following relation for the film height

$$\begin{aligned} h_c(t)&= c_3 \exp {\left[ 2c_1 c_2^{-1} \exp {(-c_2 \,t)}\right] }. \end{aligned}$$

(14)

The constant of integration $c_3$ is determined from Eq. 12 by setting the residual film thickness $h_{res}$ as asymptotic boundary condition, yielding

$$\begin{aligned} \lim _{t \rightarrow \infty }{ h_c(t) } = c_3 = h_{res}. \end{aligned}$$

(15)

A list of the coefficients $c_1$, $c_2$ and $c_3$ can be shown in Table 3 for each experiment.

3 Numerical simulation

This section briefly describes the in-house multiphase flow solver Free Surface 3D (FS3D), employed for the direct numerical simulation (DNS) included in this work. A more detailed description and its validation for droplet film interactions can be found in Fest-Santini et al. (2021), Rieber and Frohn (1999), Steigerwald et al. (2021). FS3D solves the equations for mass and momentum conservation

$$\begin{aligned} \partial _t \rho + \nabla \cdot \left( \rho {\textbf {u}} \right) = 0, \end{aligned}$$

(16)

$$\begin{aligned} \partial _t \left( \rho {\textbf {u}}\right) + \nabla \cdot \left( \rho {\textbf {u}} \otimes {\textbf {u}} \right) = \nabla \cdot \left( {\textbf {S}} - {\textbf {I}} p \right) + \rho {\textbf {g}} + {\textbf {f}}_{\gamma } \end{aligned}$$

(17)

on finite volumes, where $\rho$ denotes the density, ${\textbf {u}}$ denotes the velocity vector, p the static pressure, $\textbf{g}$ the acceleration due to gravity, ${\textbf {S}}$ the shear stress tensor and ${\textbf {I}}$ the identity matrix. The term $\textbf{f}_{\gamma }$ models surface tension forces at the interface between the gaseous and the liquid phase. The interface is captured by using the volume-of-fluid (VOF) method, which introduces an additional scalar variable f with $0 \le f({\textbf {x}},t) \le 1$, representing the liquid volume fraction in each control volume (Hirt and Nichols1981). The VOF-variable f is advected within the computational domain by solving an additional transport equation.

In the present investigation, we are interested in a comparison of the height-averaged radial velocity within the lamella between the simulation and experimental data. At this point one has, however, to keep in mind that only the droplet is seeded with particles and that the experimentally evaluated velocities thus only belong to the layer of droplet liquid. As the traditional VOF-variables cannot be used to distinguish between different liquids, we use the multi-component framework of FS3D, in which additional VOF variables $\psi _i = V_i/V$ are introduced, representing the volume fraction of species i inside the liquid phase. The scalars $\psi _i$ are advected simultaneously and in a closely coupled way to f. In this way we can identify both droplet and film liquid. For more details about FS3D the reader is referred to Eisenschmidt et al. (2016) and Steigerwald et al. (2021).

The computational setup is identical to the one shown in Steigerwald et al. (2021) and is therefore only shortly described. We simulate a quarter of the impact scenario within a cubic computational domain with an edge length of $7D_0$. A perfect spherical droplet exhibiting an initial velocity $U_0$ towards the film is initialized in a distance of $2D_0$ above a quiescent film. The domain is discretized rectilinearly with $1024^3$ grid cells. In the impact region, the grid resolution corresponds to 256 grid cells per droplet diameter $D_0$. The simulated impact scenario corresponds to exp. No. 18, which uses silicon oil B10. The impact conditions are shown in Table 3. The properties of the surrounding medium are set to those of ambient air. Figure 7 shows a comparison of the impact morphology between the exp. No. 18 (top) and the numerical result (bottom) for several points in time.

Fig. 7

Comparison of the impact morphology between exp. No. 18 (B10, $\delta = 0.2$) and the DNS for several points in time. The minimal temporal mismatch is due to the finite sampling rate in the experiments and numerical simulations

The growth and the shape of the numerical reproduced crown is very similar to the experimentally observed one. The experimental figures show that the impact scenario lies within the deposition regime as no liquid fingers occur at the crown rim. Furthermore, the seeded particles within the droplet liquid are visible as tiny dots within the crown crater and the crown wall. The impact regime is well reproduced by the numerical simulation. The only visible difference between the simulation and the experiment is the shape of the crown rim as it does not show the bulges observed in the experiment. The reason for their absence in the simulation is a premature disintegration of the crown rim, which takes place slightly before $\tau =1.0$. At this point, the crown wall is too thin for a proper interface reconstruction. This leads to an artificial ejection of mass, which is then missing in the rim. However, this difference is irrelevant, since the crown top, which is the most difficult feature of drop film interactions to reproduce numerically, is not of interest in this study. As we will show in the following, the grid resolution is indeed sufficiently fine to extract precisely velocity data from the lamella with respect to both the wall film and the droplet liquid layer.

4 Results and discussion

This section discusses the PIV experimental results in the framework of a holistic approach that includes a comparison with predictions from DNS, theoretical models and experimental measurements of the wall film thickness. Consistency of results from different approaches provides a robust and indirect validation of the PIV measurements. The analysis focuses on the temporal evolution of the central film height h(t), the radial velocity $u_r(t)$ and its decay rate a(t). The implications of our experimental findings on the required improvements of theoretical models are also discussed.

4.1 Comparison of the total film height

This section presents a comparison between the experimentally derived film height, numerical predictions and literature data that include both experiments and theoretical results. Experimental data mainly exist for droplet impact onto dry walls, where the decay of the droplet central peak was measured by means of the Fourier transform profilometry technique. A detailed description of the experiments can be found in Lagubeau et al. (2012). Similarly to this work, the authors systematically varied the dynamic viscosity of the test fluids, resulting in a variation of the impact Reynolds number by two orders of magnitude (i.e., $\mathcal {O}(10)< \textrm{Re} < \mathcal {O}(10^3)$). Lagubeau et al. (2012) identified a self-similar inertial regime, which is independent from $\textrm{Re}$, followed by a viscous regime, where the minimal thickness is limited by the growth of the boundary layer. Consequently, the residual film thickness increases with decreasing $\textrm{Re}$, in agreement with the findings from van Hinsberg et al. (2010). Our hypothesis is that these findings should be valid also for droplet impact onto a wet surface, since the latter experiences significantly smaller wall shear stresses (Frommhold et al. 2015). To verify this assumption, Fig. 8a compares the studies by Lagubeau et al. (2012) ($\textrm{Re} = 2690$), Roisman et al. (2009) ($\textrm{Re} = 1068$) and Eggers et al. (2010) ($\textrm{Re} = 400$) on droplet impacts on dry surfaces with one test case from this work (B10, $\delta = 0.2$, $\textrm{Re} \approx 550$). For the B10 test case, both DNS and the experimentally derived film height (i.e., Eq. 14) are included. To compare the data adequately, each height shown in Fig. 8a is non-dimensionalized with $D_0+H_{f,0}$. Additionally, Fig. 8a shows also the theoretical line $h_{freefall}$ of the droplet apex of a free fall.

Fig. 8

Temporal decay of the top central point of the drop surface. Reference test case: B10, $\delta = 0.2$, $\textrm{Re} \approx 550$. For the reference test case, the predictions from DNS and the experimental correlation (Eq. 14) are compared to (a) literature data on dry wall impact and to (b) semi-empirical models, based on Eq. 10 or Eq. 18 and fitted to the DNS data. The respective fitting parameters ($\hat{c}_1,\hat{c}_2,\hat{c}_3, \hat{\beta }_1, \hat{\beta }_2, \hat{c}_4$) are calculated with the least square method. For Eq. 14, the constants $c_1$, $c_2$ and $c_3$ are averaged from exp. No. 16-20 (Table 3). Experimental dry wall data are from Lagubeau et al. (2012) ($\textrm{Re} = 2690$) as well as simulation data from Roisman et al. (2009) ($\textrm{Re} = 4010$) and Eggers et al. (2010) ($\textrm{Re} = 400$). The dashed line $h_{freefall}$ indicates the free fall propagation

As expected, despite the large variation in $\textrm{Re}$ numbers, all data collapse into a single curve in the inertial regime. Only towards the end of the measurements do small differences become visible, as the viscous regime is slowly entered. As shown in the inset of Fig. 8a, the simulation data from Eggers et al. (2010) ($\textrm{Re} = 400$) are the first to depart from the inertial solution. This is in agreement with the experimental findings of Lagubeau et al. (2012), who found that the highest the $\textrm{Re}$ number the later the viscous regime is entered. In particular, for $\textrm{Re} \approx 2100$, Lagubeau et al. (2012) found that the viscous regime is entered in the range of $5< \tau < 6$ for impact onto dry walls. For impact onto a wet surface, due to the reduced wall shear stress (Frommhold et al. 2015), the start of the viscous regime is delayed. This explains why our B10 test case ($\textrm{Re} \approx 550$) matches so well the high Reynolds data ($\textrm{Re}=2690$) from Lagubeau et al. (2012) for dry wall impact both with respect to the DNS and the empirically derived composite exponential function (i.e., Eq. 14). This agreement among experiments and numerical simulations from different authors indirectly corroborates not only the accuracy of Eq. 14, but also the procedure to recover the temporal evolution of the central film height from the empirically determined decay rate a(t). The small deviations observed in the time interval $1< \tau <2$ are due to the different evaluation of h(t) between the DNS and the experiment. The height h(t) of the DNS is defined directly by the reconstructed surface, whereas the height from the experiments is derived with Eq. 14. In addition, as discussed in section 4.2, the simulation can evaluate both liquid layers, unlike the experiment where only the droplet liquid is measured.

Having established that the film height decay occurs predominantly in the inertia-controlled regime, it is interesting to compare the widely accepted $1/ \tau ^2$-dependence with our DNS and experimentally derived results. This comparison is shown in Fig. 8b, which includes two variants of the Yarin and Weiss solution and additionally a non-dimensionlized composite exponential fit of the DNS data, defined as

$$\begin{aligned} h(t)/D_0=~\hat{c}_3~\exp {(\hat{c}_1~\exp {(-\hat{c}_2 \tau )})} \end{aligned}$$

(18)

All three curves are fitted for the same period of time $1<\tau <5$ to the DNS data. As shown by the bold dash-dotted line, the numerical data can be fitted very accurately by using Eq. 18, thus corroborating the proposed functional dependence upon time. If no constrains are attached to the fitting parameters ($\hat{c}_1,\hat{c}_2, \hat{c}_3$), the corresponding L2 norm evaluated by means of the Matlab^® function ’lsqcurvefit’ is small $\Vert \text {residual} \Vert _2 = 4.938 \cdot 10^{-3}$. The other two fits were obtained by using the Yarin and Weiss solution of the film height with one free parameter $\hat{\beta }_1$ $(c=0)$ or two free parameters $\hat{\beta }_2$ and $\hat{c}_4$, respectively. Even though the latter (i.e., $\hat{c}_4 \ne 0$) provides a better fit to the DNS data, the validity of the ($\tau ^{-2}$) dependence is temporally limited. Indeed, for $\tau >3$ a significant deviation from the DNS data is observed. As pointed out already in Sect. 1.1, the ($\tau ^{-2}$) dependence for the height decay was derived by assuming an unbounded, asymptotic velocity ($u_r = a(t) r$) constrained only by the requirements of mass and momentum conservation. Consequently, as the liquid lamella expands radially without any limitation, the film height must rapidly decrease to zero, in order to fulfill the above mentioned requirements. In reality, the liquid lamella is accelerated by the pressure force created by the impinging droplet. This pressure force is not constant, but decreases in time after having reached its peak value. The viscous forces, on the other hand, become increasingly dominant with decreasing wall film height and eventually bring the spreading liquid lamella to rest. The composite exponential function, found empirically in this work, reflects this complex interplay between decaying pressure force and increasing relative importance of viscous losses. To conclude this section, we explicitly point out that both time dependencies, namely $\tau ^{-2}$ or the composite exponential function, are not valid for the early times (i.e., $\tau <1$), where the droplet central peak follows the freefall regime as shown in Fig. 8a.

4.2 Analysis of the radial velocity field

In the following we compare the experimentally measured radial velocity field with results from direct numerical simulations. In the experiments, only the liquid droplet is seeded with tracer particles. Hence, for a meaningful comparison, the numerical data are also averaged over the portion of the lamella that corresponds to the thickness of the liquid drop layer. Figure 9 shows the radial velocity $u_r(r,t)$ as function of r for selected points in time and a very good agreement is found among the corresponding numerical predictions.

Fig. 9

Comparison between numerical and experimental radial velocity distributions $u_r(r,t)$ for selected time steps. Test case: exp. No. 18 (B10, $\delta =0.2$). The DNS data are averaged over a portion of the lamella corresponding to the thickness of the liquid drop layer and written out at a time interval of $\Delta \tau =0.1$. The experimental data, instead, are obtained at a fixed frame rate, which corresponds to a inter frame time interval of $\Delta \tau _{exp}\approx 0.13$. This explains the small time difference between the numerical and experimental data

This demonstrates that two-dimensional velocity fields can be measured with great accuracy with the described micro-PIV setup. The PIV results show a low level of scattering and the DNS accurately reproduce the observed temporal decay of the radial velocity. The latter increases linearly with radius and its slope remains almost constant as long as the velocity field is not influenced by the flow at the crown base. In such a case, the slope starts to deviate owing to the vertical upward flow into the crown wall. This phenomenon is also observed in the experiments, where the deviations from the linear profile start at slightly smaller radial distances in comparison to the DNS data, especially during early times. These differences are caused by the evaluation routine, since the extraction of the radial velocity becomes more difficult near the crown wall as shown in Fig. 4b.

Fig. 10

Evaluation of the slope a(t) from DNS data by employing different portions of the liquid lamella: droplet liquid, film liquid and both liquids (i.e., total height). Test case: exp. No. 18 (B10, $\delta =0.2$). For comparison, the ($\tau ^{-1}$) dependence (i.e., Eq. 1) with two different c values ($c=0$, $c=0.25$) is plotted together with experimental slopes, evaluated from exp. No. 16–20

To evaluate the effect of the wall film thickness on the averaging of the radial velocities, the DNS data have been additionally averaged over the total thickness of the lamella (both liquids) and over the thickness of the wall film only (denoted as film liquid). The results of this exercise are shown in Fig. 10 for the range $0\le \tilde{r}\le 0.5$. As reference test case, exp. No. 18 is chosen (B10, $\delta = 0.2$). As can be seen, the slopes obtained by averaging on the droplet-liquid lamella and on both liquids lamella decrease in a very similar way, because the lamella consists mainly of droplet liquid. The decrease of the slope of the film liquid lamella is stronger than that of the droplet liquid lamella due to wall effects. For comparison, the evaluated slopes from exp. No. 16–20 are also shown. The experimental data nestle closely to the slope of the droplet liquid up to $\tau \approx 3$. For later times, in the experiments the temporal evolution of the slope approaches and follows closely the slope of the full lamella. This occurs due to the height decay of the liquid lamella so that even the droplet layer weakly experiences the retarding effect caused by viscous losses. Apart from these small deviations, the overall good agreement between the numerical and experimental data confirms once again the accuracy of the velocity measurements. For comparison, the decay rate predicted by the ($\tau ^{-1}$) dependence (i.e., Eq. 9) are also shown in Fig. 10 for two different values of the constant c. The direct comparison of the decay rate a(t) highlights even more clearly the limited validity of the ($\tau ^{-1}$) dependence, alias an inviscid solution without pressure gradient. The latter can provide reasonable estimations only in the early times (i.e., $1< \tau < 2$), as shown in Sect. 4.1.

4.3 Comparison of a(t) with analytical models

The preceding discussion highlighted the complex interplay between the pressure force, generated by the impinging droplet, and the shear stresses in the near-wall region. This system of counteracting forces de facto controls the temporal decay of both the radial velocity $u_r(t)$ and the wall film height h(t). However, it is still unclear which of the two components plays the dominant role and essentially dictates the exponential decay of the slope a(t) in the first place and subsequently of all other variables. The answer to this question is not straightforward, since both the pressure force for droplet impact on wet substrates and the viscous forces are not known a priori. Moreover, the viscous forces are not only dependent upon the fluid viscosity, but also upon the velocity gradient normal to the wall, which varies with the impact conditions. In order to obtain an estimation of the viscous losses and evaluate their effect on the decay of the slope a(t), the following procedure is applied. As a first step, the theoretical (inviscid) model from Yarin and Weiss (1995) is taken as a reference (i.e., Eq. 1). As a second step, a viscous correction is applied to the inviscid solution, in order to evaluate how viscous losses affect the temporal decay of the function a(t). The details of this evaluation procedure are explained hereafter.

As a starting point, we point out that Eq. 1, the $\tau ^{-1}$ dependence, has been widely used in literature. Many authors simply adjusted the constant c to obtain an improved match to their own experimental or numerical data. Hereafter, three different options are considered, namely the values $c=0$, $c=0.25$ and $c=2.49$, respectively. The corresponding decay rate is denoted with $a_{YW}(t)$ as a reminder that they refer to the inviscid solution from Yarin and Weiss (1995). The corresponding profiles are shown in Fig. 11 together with the experimentally derived decay rate a(t) for the test case exp. No. 6 (B5, $\delta = 0.2$). As can be seen, higher values of the c parameter lead to a less steep decay of the function a(t) due to the fact that the start value is significantly quenched. For completeness, the reasons for selecting the values of the constant c are summarized hereafter. The value $c=0$, for instance, was chosen by Smith and Bertola (2010, 2011), who studied Newtonian and non-Newtonian droplets impacting onto a dry hydrophobic surface. The value $c=0.25$ was proposed by Roisman et al. (2009), who investigated inertia dominated axisymmetric drop collisions onto a dry substrate and onto another sessile droplet. The correlation with $c=0.25$ was validated in the range $0.7<\tau <\tau _{\text {viscous}}$. The value for $\tau _{\text {viscous}}$ can be estimated with correlations from Roisman (2009) and Stumpf et al. (2022). The value $c = 1 / \lambda = 2.49$ is obtained from the empirical correlation, proposed by Gao and Li (2015) and applied to the test conditions of exp. No. 6 (B5, $\delta = 0.2$)

$$\begin{aligned} \lambda = 0.26 (\textrm{Re}^{0.05} \textrm{We}^{-0.07} \delta ^{-0.34}). \end{aligned}$$

(19)

As explained in Sect. 1, Lamanna et al. (2022) proposed to use this empirical correlation (i.e., Eq. 19) to estimate the kinetic energy effectively transmitted to the radially spreading lamella, deprived of impact losses. The latter are associated to dissipative effects induced by the strain rate during the deformation of the impinging droplet and by the conversion of kinetic energy into surface (potential) energy.

Fig. 11

Comparison among the experimentally derived function a(t) and the predictions from different formulation of the inviscid model of Yarin and Weiss, obtained by specifying different value for the constant c, specifically $c=0$, $c=0.25$ and $c=2.49$. In addition, the viscous correction from Lamanna et al. (2022) (Eq. 20) is applied to two different formulation of the inviscid model, namely $c=0.25$ and $c=2.49$. Test case: exp. No. 6 (B5, $\delta =0.2$)

As a second step, the viscous correction developed by Lamanna et al. (2022) is applied to two inviscid formulations, corresponding to the potential flow solutions obtained by setting $c=0.25$ and $c=2.49$ in Eq. 1, respectively. The viscous correction is obtained by solving numerically the unsteady momentum balance equation in the radial direction for an axisymmetric flow. The boundary layer flow admits a self-similar solution and enables the calculation of a profile-averaged velocity $\bar{u}_{vis}$. The latter encompasses the momentum losses in the boundary layer. In non-dimensional terms, this can be expressed by the ratio $\lambda _1(t) = \bar{u}_{vis}/u_{r\infty }$, which measures the specific momentum reduction compared to the asymptotic velocity. Thanks to the self-similarity of the solution, this approach can be applied to any position within the impact zone, yielding

$$\begin{aligned} u_r=a_{\textrm{YW}}(t) \, \lambda _1(t) \, r = a_{\textrm{L}}(t) \, r{.} \end{aligned}$$

(20)

Here, the parameter $a_L(t)$ is introduced as a reminder that the potential flow solution has been adjusted to include the effect of viscous losses. The results of this exercise are also shown in Fig. 11 and lead to the following conclusions: First, as stated above, the relative importance of the boundary layer correction is strongly dependent upon the velocity gradient in the normal direction to the wall. For the case $c=2.49$, the strongly quenched values of the potential flow $u_r$ lead to very reduced viscous losses. A stronger effect is observed for the test case $c=0.25$ due to the higher values of the parameter a(t). Second, the inclusion of viscous losses provides only a correction to the potential flow (inviscid) solution. In other words, they do not dictate the functional dependence of the parameter a(t) upon time. The validity of this statement is verified in Sect. 4.4. Based on these findings, it follows that the exponential decay, observed experimentally, for the parameter a(t) must be associated to an exponential decay of the pressure force. First confirmations of this statement are provided by the numerical simulations of Roisman (2009) and by the experiments of Mitchell et al. (2019). In both works, the authors investigated the transient force profile of a low-speed droplet impinging on a dry wall at the center of symmetry. In particular, Mitchell et al. (2019) modeled the transient force as follows

$$\begin{aligned} F(t) = A_1 \sqrt{A_2 \tau } \, e^{-A_2 \tau }. \end{aligned}$$

(21)

Here, $A_1$ and $A_2$ are constants that depends upon the impact conditions. Mitchell et al. (2019) found that, for early times (i.e., $\tau \le 0.1$), Eq. 21 behaves like $\sqrt{\tau }$. This phase corresponds to the creation of a peak pressure force. For $\tau > 0.1$, the transient force exhibits an exponential decay with time. This finding is consistent with our measurement of the decay rate a(t) for $\tau > 1$. The interdependence between pressure and decay rate a(t) is further explored in Sect. 4.4.

4.4 Parametric study on the effects of $\textrm{Re}$ number

We vary systematically the dynamic viscosity of the liquid $\upmu$ and analyze the effect on the evolution of the slope a(t). For this purpose, the radial velocity distribution has been measured for eight different impact scenarios by varying the Reynolds number $\textrm{Re}\in \{116,280,554,1106\}$ and the film thickness $\delta \in \{0.1,0.2\}$, while keeping the Weber number constant (i.e., $\textrm{We} \approx 800$). For each impact scenario, the experiment was repeated five times, which resulted in 40 experiments. Details about each experiment can be found in the appendix (see Table 3). In Fig. 12, the radial velocity of the four most different test cases is presented, corresponding to the silicon oils B5 and B50 for $\delta =0.1$ and $\delta = 0.2$. In all cases, the radial velocity $u_r$ increases linearly with the radius regardless of the initial film thickness and Reynolds number. In this representation, the variations in the radial velocity distribution with the initial film thickness $\delta$ are hardly discernible. This is because the investigated film thicknesses $\delta$ induce small wall shear stress (Frommhold et al. 2015). In some cases, the effect of decreasing the Reynolds number can be visible even at earlier times (i.e., $0.6< t < 2.5$ m/s), resulting in a faster temporal decay of the radial velocity. This is clearly visible when comparing the B50 ($\textrm{Re} = 115$) and B5 ($\textrm{Re}= 1142$) test cases (e.g., for $\delta =0.2$). At $\tilde{r}=0.5$ and $t=1.56$ ms (pink marker), the radial velocity has already fallen below $u_r = 0.25$ m/s for B50, whereas for B5 the radial velocity is still $u_r \approx 0.4$ m/s. This behavior is shown even more clearly in Fig. 16 and is rather unexpected. Indeed, in the inertial regime (typically valid for $1< \tau < 2$), the velocity decay should be independent of $\textrm{Re}$ (see e.g., Eq. 1), because the shear strain rate does not play a significant role yet.

Fig. 12

Radial velocity fields for two different viscosities (B5, B50) and two different initial film heights $\delta =0.2$ and $\delta =0.1$

Fig. 13

Influence of viscosity and non-dimensional film thickness on the slope a(t). All eight test cases are plotted (exp. No. 1-40). Results for different Reynolds numbers are indicated by color and for different film heights by the line style. In addition, the $\tau ^{-1}$ dependence (i.e., Eq. 1 with $c=0$) is plotted as a dashed black curve. The $\tau$ values are calculated by averaging $D_0$ and $U_0$ over the exp. No. 1-40

Fig. 14

Semi-logarithmic plot of the slope a(t) from Fig. 13. The $\tau ^{-1}$ dependence (i.e., Eq. 1 with $c=0$) is plotted as a dashed black curve. The $\tau$ values are calculated by averaging $D_0$ and $U_0$ over the exp. No. 1-40

In order to better visualize the influence of $\textrm{Re}$ and initial film thickness $\delta$, the experimentally derived slopes a(t) of all 40 experiments are shown in Figs. 13 and 14 in a normal and in a semi-logarithmic scale, respectively. From both plots, the following main conclusions can be drawn. First, at $\tau \approx 1$, the starting value for the slope a(t) seems to be independent of $\textrm{Re}$ and $\delta$. Second, the decrease of a(t) with time is stronger for lower $\textrm{Re}$, albeit all curves tend towards zero. Third, the experimental results can be reasonably fitted using Eq. 1 (i.e., $\tau ^{-1}$ dependence) only for a short time period, e.g., with $c=0$ for B5 and $1<\tau <2$. Fourth, with increasing fluid viscosity, the period of validity of Eq. 1 decreases rapidly, as reported also by Bakshi et al. (2007). Fifth, starting from $\tau =1$ the slope a(t) shows a linear trend in the semi-log plot of Fig. 14, which indicates an exponential decrease. Moreover, the influence of the initial thin film height ($\delta =0.1,0.2$) on the slope a(t) is negligible. These findings are consistent with the inception of an inertial regime, where the velocity decay rate is independent upon $\textrm{Re}$ and $\delta$. However, around $\tau = 2$ all curves exhibit a slight kink. The latter is associated to the inception of the viscous regime, which leads to stronger deviations for thinner films due to increased relative importance of viscous losses. The only noteworthy exception to this behavior is observed for the B50 test cases, where a clear dependence upon $\textrm{Re}$ in the otherwise inertial regime is detected.

In the following, a detailed analysis is presented, aiming to provide a sound physical explanation to the above mentioned experimental findings. As a first step, Fig. 15 illustrates the variation of the fitting coefficients $c_1$, $c_2$ (see Eq. 13) and their ratio as function of the initial Reynolds number. The larger scatter of $c_1$ for small $\textrm{Re}$ is due to the significant scatter of a(t) at early times, as shown in Fig. 13. The fitting coefficients for each experiment are shown in Table 3. As can be seen, both empirical parameters decrease with increasing $\textrm{Re}$. The decay is nonlinear and seems to evolve towards an asymptotic value with increasing $\textrm{Re}$. Both coefficients are almost insensitive to variations in the initial film thickness $\delta$ and their ratio remains basically constant ($c_2/c_1 \approx 0.4$) over the parameter space investigated in this work. These functional dependencies are already an indicator that the empirically derived exponential function is indeed suited to describe the inertial regime, particularly for $\textrm{Re} > 300$. The constancy of the ratio $c_2/c_1$ seems to suggest a direct dependence upon the Weber number, which is held constant in the present work.

Fig. 15

Mean values of the empirical coefficients $c_1$, $c_2$ and their ratio as function of $\textrm{Re}$ for the eight impact scenarios (exp. No. 1-40). The error bars of $c_1$ and $c_2$ represent the min and max value of each test case

As a second step, the reciprocal of the empirically derived exponential function (i.e., Eq. 13) is approximated using a Taylor’s expansion, yielding

$$\begin{aligned} a(t)^{-1}&= \frac{1}{c_1} \exp {(c_2 t)} = \frac{1}{c_1} + \frac{c_2}{c_1} t +{\frac{(c_2 t)^{2}}{c_1 \, 2!}}+\cdots . \end{aligned}$$

(22)

Likewise the reciprocal of Eq. 1 reads

$$\begin{aligned} a_{YW}(t)^{-1}= const. + t. \end{aligned}$$

(23)

By comparing Eq. 23 with the Taylor series (Eq. 22), it is clear that the ($t^{-1}$)-dependence for the inertial regime represents the first-order approximation of the exponential function. Moreover, the slope of the first-order approximation is $c_2/c_1 \approx 0.4$ for the investigated cases and not one, as commonly assumed in Eq. 23 (or alternatively in Eq. 1). Note that Smith and Bertola (2011) found $c_2/c_1=0.81$ for a dry impact with $We=75$. Based on the previous remarks, this would imply that the slope of the first-order model for a(t) depends upon the Weber number only, alias upon the relative importance of inertial forces and surface tension. It is also clear that the use of a single constant in Eq. 23 cannot simultaneously satisfy the $\textrm{Re}$ and $\textrm{We}$ dependency observed experimentally in Eq. 22. This also explains why, in literature, the constant (c) in Eq. 1 was continuously adapted to fit different datasets. The higher order derivatives can only be neglected when $(c_2 t) \ll 1$. In the considered time interval $t = (0.6 - 3)$ ms, the terms $(c_2 t)^n$ with ($n>2$) are not negligible for our parameter space, thus explaining the faster decay of a(t) compared to the ($t^{-1}$)-dependence. As a third step, the momentum balance equation in direction normal to the wall (i.e., z-axis) is considered. Pressure is integrally transmitted through the boundary layer, so that the analysis can be simply restricted to the potential flow region, yielding

$$\begin{aligned} \rho \left( \frac{\partial w}{\partial t} + u_r \frac{\partial w}{\partial r} + w \frac{\partial w}{\partial z} \right) = - \frac{\partial p}{\partial z}. \end{aligned}$$

(24)

For an unsteady and axisymmetric stagnation point flow, the velocity components for the potential flow region are $u_r(r,t) = a(t) r$ and $w(z,t) = -2 a(t) z$ (see Eqs. 2 and 4). Inserting these expressions into the momentum balance yields

$$\begin{aligned} \rho \left( \frac{\partial w}{\partial t} + w \frac{\partial w}{\partial z} \right)&= - \frac{\partial p}{\partial z}, \end{aligned}$$

(25)

$$\begin{aligned} \rho \left[ - 2 a'(t) z +4 a^2(t)z \right]&= - \frac{\partial p}{\partial z}. \end{aligned}$$

(26)

Integration of Eq. 26, from the wall ($z = 0$) to an arbitrary height z, by separation of variable gives

$$\begin{aligned} \int _0^z [-2a'(t) + 4a^2(t)]\tilde{z} \, d\tilde{z}&= -\frac{1}{\rho }\int _{p_0(t,r)}^{p(t,r,z)} dp, \end{aligned}$$

(27)

$$\begin{aligned} \left[-a'(t) + 2a^2(t)\right] z^2&= \frac{1}{\rho } \left[p_0(t,r)- p(t,r,z)\right] \end{aligned}$$

(28)

with the boundary condition $p = p_0(t,r)$ at $z = 0$ and $p_0(t,r)$ denoting the stagnation pressure at the wall. Hence, for an arbitrary height position z, the temporal decay of a(t) must follow the exponential decay in time of the pressure. This conclusion is not only consistent with our experimental findings, but also physically plausible. The finite mass and momentum of the impinging droplet is inevitably associated to the rapid decay of the driving force for the motion of the lamella. The exponential decay of the pressure force has been corroborated both experimentally and numerically by Mitchell et al. (2019), Yu and Hopkins (2018), Yu et al. (2022) and Roisman et al. (2009). In physical terms, the limited temporal validity of the ($t^{-1}$)-dependence can be therefore ascribed to the omission of the pressure term, as pointed out already in Sect. 1.1.

Equally interesting insights can be derived by solving the unsteady momentum balance equation in radial direction and restricting it to the potential flow region. The exact derivation is presented in appendix. Hereafter, only the main findings are summarized. By inserting the potential flow solution $u_r = a(t) r$ with a(t) given by Eq. 13, it is found that the pressure profile in radial direction has the shape of a parabola with downward concavity. Its curvature decreases in time till it almost resembles a plateau. These theoretical evaluations, based on the empirically found exponential decay of a(t), agree qualitatively very well with the numerical simulations of Roisman et al. (2009). This indirectly provides an additional confirmation on the plausibility of the experimental findings and associated analysis.

One last aspect to clarify is the $\textrm{Re}$ dependency of the slope a(t). As shown in Fig. 14, for $\textrm{Re} \approx 100$ this occurs already at $\tau \approx 1$, when the lamella radial spreading is mainly controlled by pressure and inertial forces even for droplet impact on a dry wall (Mitchell et al. 2019). The close interdependence between a(t) and pressure established in this work enables us to provide a physical explanation for this behavior. Following Roisman et al. (2009), the pressure in an expanding liquid lamella can be expressed as

$$\begin{aligned} p = p_{\sigma } - 2 \upmu \left[ \frac{\partial u_r}{\partial r} + \frac{u_r}{r} \right], \end{aligned}$$

(29)

where $p_{\sigma }$ represents the additional pressure contribution due to the surface curvature, as predicted by the Young-Laplace equation (Roisman et al. 2009). Equation 29 implies that in presence of extensional (normal) strain due to the radial spreading of the lamella the effective pressure is reduced. At high $\textrm{Re}$ numbers, the pressure loss due to the extensional strain is negligible. Consequently, the slopes a(t) collapse to a single curve in the inertial regime, as shown in Fig. 14. Deviations appear as soon as viscous losses are no longer negligible in presence of significant shear strain rate. At low $\textrm{Re}$ numbers, instead, the pressure losses caused by the extensional strain are so high that the decrease in a(t) (alias in the radial velocity $u_r$) is immediately evident even outside of the boundary layer flow.

5 Conclusion

In this study, the simultaneous recording of the macroscopic droplet impact dynamics together with micro-PIV measurements within the crown crater is performed for the first time. Thanks to this optical configuration, it is possible to investigate the characteristic of the radial velocity distributions across all impact regimes, namely deposition, transition and splashing. The study additionally includes a systematic variation of the impact Reynolds number by one order of magnitude ($\mathcal {O}(10^2)< \textrm{Re} < \mathcal {O}(10^3)$), while $\textrm{We}$ is kept constant ($\textrm{We} \approx 800$). The overarching goal is to gain better physical insights into the complex interplay between pressure force, shear and normal stresses on the evolution of the radial velocity distribution within the crown crater.

In agreement with literature data, our measurements confirm the linear radial increase of the spreading velocity within the crown crater. A discrepancy, instead, is found on its temporal evolution, which exhibits an exponential decay. To explain this behavior, a holistic approach is chosen which foresees an integrated analysis among numerical simulations and analytical models. The following major conclusions are found: In the lamella’s velocity and height decay, it is possible to identify a self-similar inertial regime and a viscous regime. In the self-similar inertial regime, the decay is independent of $\textrm{Re}$ and $\delta$ , and is controlled by the exponential decay of the pressure force. The only noteworthy exception is found at low Reynolds number, where the extensional (normal) strain leads to enhanced pressure losses even outside of the boundary layer flow. The inception of the viscous regime corresponds to a kink in the exponential decay. The strength of this deviation depends upon $\textrm{Re}$ and $\delta$. The commonly used $t^{-1}$ dependence for the inertial regime represents only a first-order approximation of the exponential function and has therefore limited validity.

Appendix

1.1 Radial momentum balance

For an unsteady and axisymmetric laminar stagnation point flow, the velocity components for the potential flow region read as: $u_r(r,t) = a(t) r$ and $w(z,t) = -2 a(t) z$. Inserting these expressions into the momentum balance in radial direction yields

$$\begin{aligned} \rho \left( \frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + w \frac{\partial u_r}{\partial z} \right)&= - \frac{\partial p}{\partial r}, \end{aligned}$$

(30)

$$\begin{aligned} \rho \left[ a'(t) + a^2(t) \right] r&= - \frac{\partial p}{\partial r} . \end{aligned}$$

(31)

Integration of Eq. 31, from the stagnation point ($r = 0$) to an arbitrary position r, by separation of variables gives the following functional pressure dependency

$$\begin{aligned} p(t,r) = \frac{\rho }{2} a'(t) r^2 + \frac{\rho }{2} a^2(t) r^2 + c_4 \end{aligned}$$

(32)

with $c_4 = p(t,r=0) = p_0(t)$ denoting the stagnation pressure at the wall. Inserting the exponential decay for the slope $a(t) = c_1\exp {(-c_2 t)}$ yields

$$\begin{aligned} p(t,r)&= \frac{\rho }{2} c_1 c_2 \exp {(-c_2 t)} r^2 + \frac{\rho }{2} c_1^2\exp {(-c_2 t)}\exp {(-c_2 t)} r^2 + p_0(t), \end{aligned}$$

(33)

$$\begin{aligned} p(t,r)&= \frac{\rho }{2} c_1 r^2 \exp {(-c_2 t)} \underbrace{\left[ c_1\exp {(-c_2 t)} -c_2 \right] }_{<0} + p_0(t), \end{aligned}$$

(34)

$$\begin{aligned} p(t,r)&= -f(t)^2 r^2 + p_0(t). \end{aligned}$$

(35)

As can be immediately seen, the pressure profile in radial direction has the shape of a downward concave parabola. Its curvature decreases in time, due to the exponential decay of a(t). This finding is consistent with the numerical simulations of Roisman et al. (2009) and provides a physical explanation for the numerical simulations.

1.2 Experimental conditions

In this section, the experimental conditions and empirically derived fitting coefficients are listed for all experiments. In addition, a detailed comparison of the radial velocity distribution between the two most different test case is shown in Fig. 16.

Table 3 Experiment impact parameters

Fig. 16

Selected time steps of the radial velocity $u_r\le u_{r,\textrm{max}}$ for $\delta = 0.2$. The vertical black line marks the image boarder, which dependence of the impact center and varies for each experiment