A combined experimental–numerical study towards the elucidation of spray–wall interaction on step geometries

ABSTRACT In this research, a coupled numerical–experimental approach is employed to investigate the spray–wall interaction on step geometries with slope angles from 0° to 90° as well as impinging on and off the step geometry. Spray contours are used to compare the numerical and experimental results and, in general, good agreement between them is found. The results from the model are used to provide deeper knowledge about the macroscopic and microscopic behavior of the impinging spray influenced by step geometries. The numerical results provide a three-dimensional perspective that reveals the deflection of the flow by the geometry along the slope and to the sides. Additionally, the quantitative results show the mass distribution over time on droplet behavior (such as moving droplet mass, wall film mass and evaporation mass) for the whole domain, but forward and backward directions separately. A strong dependency on the impinging position, but a smaller effect of the slope angle, on spray propagation is found. Consequently, these impinging positions have a strong influence on spray behavior in the forward and backward directions regarding moving droplet mass and the wall film mass. In general, this study reveals the severe consequences of step geometry on impinging spray behavior.


Introduction
Spray-wall interaction is a complex fluid dynamic phenomenon that appears in many practical applications such as in agriculture for distributing pesticides or for watering (Hilz & Vermeer, 2013;Miller & Butler Ellis, 2000), in industry processes (Gharsallaoui et al., 2007;Kurt & Schulte, 2006), painting (Poozesh et al., 2018) or cooling (Gao & Li, 2018), in direct injection combustion engine for injecting liquid fuel into cylinders (Fansler et al., 2020;Moreira et al., 2010) and in safety control for extinguishing fire (Wang et al., 2020). Interestingly, the falling raindrop and its interaction with a leaf surface can be found everywhere in nature (Dorr et al., 2015;Gart et al., 2015). For all these processes, sprays can be continuous (e.g. in agriculture or natural rainfall) or transient (e.g. in combustion engines). When interacting with a surface, it is of great interest to understand how sprays propagate along surfaces of varying geometry, and how droplets react when impacting especially under the influence of evaporation. In some applications, a wall film is desired (as in coating or painting) while in other applications it is not desired but happens (e.g. in combustion engines) (Fansler et al., 2020). The fuel spray community has conducted much research about impinging sprays and their parameters. For instance, the key parameters are the fluid properties (Yu et al., 2016), the distance between the spray nozzle and the surface, the injection pressure (Schulz & Beyrau, 2019), the surface roughness (Luo et al., 2017), the inclination of the impinged surface (Akop et al., 2013), and vortex formation at the propagation front of the impinged spray at sharp impinging angles (Suh et al., 2007). The high-speed Mie scattering imaging technique has been used extensively to visualize highly transient spray behavior as a free spray as well as in the impinged stage (Fansler & Parrish, 2015). Previous studies scrutinized spray characteristics under the control of injection timing, ambient pressure, the distance between the nozzle and an impinging surface, surface properties such as roughness and temperature, as well as liquid film thickness and its formation (Luo et al., 2018(Luo et al., , 2017Pan et al., 2019;Zhao et al., 2017;Zhu et al., 2017). However, most research investigating impinging sprays consider a flat surface or, in some special engineering cases, with slopes or corners that affect the interaction with the spray. Thus, Shim et al. (2009) studied an spray impinging into a cavity with different slope angles. It was revealed that the radial propagation decreases with an increasing slope angle, and the spray in the cavity becomes denser with liquid films growing on the walls. Moreover, Zhang et al. (2018) investigated liquid film separation at expanding corners caused by spray-wall interaction. Their results showed that expanding corners reduce the radial penetration but increase the vertical penetration and, with an increasing corner angle, film separation is promoted. Additionally, Steinberg et al. (2019) and Steinberg and Hung (2020) scrutinized step geometries with progressively increasing slope angles (from 0°to 90°) with different impinging scenarios of a tilted fuel spray. Their results showed a strong dependence on the impinging location (on-or offgeometry) as well as of the geometry slope angle on the spray propagation behavior. The geometry is able either to suppress or to boost the propagation after impingement with drastic consequences for the evaporation process. Furthermore, owing to the tilted spray, the backward and forward propagation behaviors show fundamental differences. However, a geometrical slope enlarges the backward propagation significantly, and yields to a more balanced liquid distribution between the forward and backward directions. Both papers provide an engineering basis for the scope of the current article, in which step geometries are investigated by numerical methods. Nowadays, numerical investigations are used in various fields of fluid dynamics such as swirl flow inside annular geometries with changing cross-sectional areas (Shakeel & Mokheimer, 2022), double droplets splashing on a wall film (Yuan, Peng, et al., 2021), cavitating jet behavior depending on valve seat structures (Yuan, Zhu, et al., 2021) and air pollution by particles of different size (Issakhov et al., 2019). Additionally, many investigations use a combined numerical approach with experimental verification (Asnaashari et al., 2016;Baumann et al., 2020;Benferhat et al., 2019;Iannetti et al., 2016;Liu et al., 2020). Hence, spray-wall interactions have also been investigated using modeling techniques. Mundo et al. (1995) investigated single droplets of different size, velocity, impingement angle, viscosity and surface tension impinging on a dry flat surface. This work was expanded to polydisperse near-wall spray flow by Mundo et al. (1998). The result was an empirical correlation with the non-dimensional parameter K, K = OhRe 1.25 (where Oh is the Ohnesorge number and Re is the Reynold's number using the droplet velocity component normal to the wall), which determines whether a droplet completely deposits (K ≤ 57.7) or partially splashes into secondary droplets (K > 57.7). This model includes the main physical parameters, such as the droplet diameter before impinging, the liquid properties and the impingement kinetics. In another work, Yarin and Weiss (1995) studied splashing droplets from a monodisperse train of droplets on a surface wetted by preceding droplets within the train. In their experiments, the frequency of the droplets was varied and measurements of the secondary droplet size and their mass were conducted. Their results showed that a splashing event happens if a dimensionless impact velocity u exceeds around 17. Mathematically, the dimensionless impact velocity u is described as where U 0 is the velocity component normal to the surface of the impinging droplet, σ is the surface tension, and ρ and μ are the density and dynamic viscosity of the liquid, respectively. The frequency of the impinging droplets f can be used to infer the constant distance between two in-flight droplets within a droplet train, denoted with D 0 , and is determined by f = U 0 / D 0 . In this model, it is apparent that U 0 and f are the driving factors, and droplet size plays no role. Findings from both research works were included by O'Rourke and Amsden (2000) in developing requirements for splashing droplets and the prediction of secondary droplet characteristics for the KIVA simulation code. O'Rourke and Amsden (1996) proposed an extension of their previous model that expands the single droplet model due to Mundo et al. (1995) and modifies the model by Yarin and Weiss (1995) to calculate the splashed mass after droplet impact. The established model differentiates between several droplet behavior in a spray-wall interaction such as droplet splashing and film spreading, as well as considering the ejection of liquid from the valve seat area when the valve shuts. As a result, a good qualitative match of the liquid film locations between the model and laser-induced fluorescence experiments was found (O'Rourke & Amsden, 2000). The code is included in the commercial software CON-VERGE (Richards et al., 2020), which has been widely used to investigate spray-wall interaction and also allows further sub-models to be added. For instance, Zhao et al. (2017) experimentally and numerically investigated the impingement of a diesel spray on a flat plate. Their results showed that the model due to O'Rourke and Amsden reflects the overall behavior of the impinged spray well, especially the axial penetration of both the bulk spray and at the wall. However, the spray height after impingement is predicted to be much lower by the model than that shown in measurement results. One explanation by the authors suggests that the momentum of the splashed droplets, which are moving in the opposite direction to the incoming spray, is under-predicted because the model might under-predict droplet velocities or the splashed mass. Recently, Torelli et al. (2020) used the O'Rourke and Amsden model (O'Rourke & Amsden, 2000) as a reference model for comparison with a modified Stanton and Rutland model (Stanton & Rutland, 1996, 1998a, 1998b, which considers the train of droplets approach due to Yarin and Weiss (1995). Measurement data from experiments with a single plume spray impinging on a flat surface were used to compare the simulation data. The results show an over-estimation of axial and radial penetration and a strong under-estimation of the propagation height by both models. Both the original model and the modified Stanton and Rutland model show little improvement when compared with the experimental results, especially in combination with an increasing surface roughness.
Previous numerical work focused mostly on spraywall interaction on a flat plate; however, the interaction of a spray with a more complex surface structure is a highly three-dimensional process that can be seen in many practical industrial applications. Hence, in this research, the spray-wall interaction with a more complex surface structure is investigated numerically. Owing to the highly three-dimensional impinging process and the lack of microscopic experimental results, validation against experimental data remains a daunting task. Spray contours from a side view are used to validate the numerical model. Thus, this article should serve different purposes: to make a thorough analysis of a comparison between experimental and numerical results; to identify possible issues when modeling complex geometries for spray-wall interactions; and to gain more insight into spray propagation from a three-dimensional point of view-droplet behavior in the forward and backward directions as well as liquid film development.

Experimental setup
The experimental results used in this work come from the authors' previous research on spray impingement on step geometries (Steinberg et al., 2019;Steinberg & Hung, 2020). In these works, the main purpose was to investigate how a step geometry influences spray propagation from a side view.
High-speed Mie scattering measurements were conducted in a constant volume chamber (Figure 1). Liquid n-hexane was injected at 15 MPa for a duration of 1.5 ms through a custom-made single-hole injector with a spray angle of 30°, an inner orifice of 0.2 mm and a lengthto-diameter ratio of 1.5 into an environment with 20°C and an ambient pressure of 101 kPa. The spray nozzle was mounted 50 mm above the impinging plate made of polished sapphire with a diameter of 110 mm, thickness 3 mm and a surface roughness R a of 29.8 nm. The geometries were manufactured by fused silica with a height of 10 mm and a surface roughness R a of 9.1 nm. In order to capture the spray behavior, the spray droplets were illuminated via Mie scattering by a continuous 2 × 4 red light LED array panel (Visual Instrumentation TM Model 900750), and recorded by a high-speed camera (Phantom ® V1210) synchronized with the injection logic pulse signal. The LED array provided continuous backlit illumination. In this application, a camera lens with focal length 105 mm and an aperture of f/2.8 was used. An injector logic controller triggered the injector. The 5 V TTL trigger signal was input to an LaVision TM programmable timing unit (PTU) that activated the highspeed camera at 20,000 frames per second with a frame exposure time at 7 μs. Therefore, the time between two images was 0.05 ms. The recorded images were 12-bit   grayscale images with 4096 intensity levels and a resolution of 896 pixels in width and 656 pixels in height, with a resolution of 12.39 pixels/mm. The resulting images revealed the impinging spray in side view, and in-house MATLAB ® code was used to postprocess the images so that the forward and backward propagation could be scrutinized ( Figure 2). In this investigation, seven cases were considered depending on the slope angle of the geometry and the impinging position ( Figure 3). Figure 4 depicts the post-processed experimental images. The red contour shows a light intensity of 40, which corresponds to 1% of the maximum light intensity, and the spray boundary is easily distinguishable from the background noise with an average value of less than 20. The images reveal the development of the spray structure over time. In the on-geometry cases (Figures 4(a), 4(b) and 4(c)), the spray actively propagates in both directions, while strong vortex motion at the tips accompanies the propagation. From the images, it seems that the liquid mass splits almost evenly into both propagation directions. Instead, in the off-geometry cases (Figures 4(d), 4(e) and 4(f)), the liquid distribution is extremely unbalanced. The propagation shoots up the slope and a vortex motion develops at the propagation front. However, the other side is very degraded with respect to the liquid mass but also from the point of view of the behavior, where a small vortex appears only very briefly.
Owing to how the experiments were conducted in the authors' previous work, experimental data were collected from a side view to validate the numerical results. Hence, the purpose of this investigation is to reveal the transient spray impingement behavior further by using a validated numerical model to gain a deeper understanding of spray propagation in three dimensions and on the microscopic droplet level.

Modeling the spray-wall interaction
The numerical simulation was conducted with the CON-VERGE code (Version 3.0) where most of the models are already embedded (Richards et al., 2020) or can be added via user-defined functions. The compressible Reynoldsaveraged Navier-Stokes equations with the RNG κ − model are applied to solve the gas phase Eulerian form. The mass transport equation given by and the momentum transport equation is where the overline denotes the ensemble mean and the tilde sign represents the Favre (density-weighted) average. To enclose the momentum equation, the Reynolds stress term is modeled by where the turbulent kinetic energy k, turbulent kinetic viscosity μ t and mean strain rate sensor S ij are given by with C μ as a model constat, and Two transport equations are introduced to solve the turbulent kinetic energy k: and the dissipation of turbulent kinetic energy ε where S is a user-defined source term and set as zero in this study. S s represents the source term, which accounts for the interaction with the discrete spray phase. The C ε1 , C ε2 and C ε3 terms are model constants. R ε is calculated by and All constants in the turbulence model are set to default values to avoid obtaining biased results. For boundary conditions, the Launder and Spalding wall model is employed to describe the boundary layer. Additionally, the Dirichlet condition with specified pressure values is applied for the outer flow.
The spray in the environment uses a Lagrangian-Eulerian framework, where the Lagrangian perspective represents the moving spray droplets flying through an environment symbolized by an Eulerian reference system. Reynolds-averaged Navier-Stokes equations with the RNG κ − model are applied. The calculation progresses with variable time steps following the Courant-Friedrichs-Lewy condition which considers the flow velocity u, time step t and mesh size x. A variable time step algorithm with a minimum step of 0.000,01 ms is applied to balance the simulation accuracy and efficiency. In order to calculate the aerodynamic breakup of spray droplets, the Taylor analogy breakup (TAB) (O' Rourke & Amsden, 1987) was activated, in which an oscillating and distorting droplet is compared to a spring-mass system. The notime counter method (Schmidt & Rutland, 2000), combined with post collision (Post & Abraham, 2002), calculates the droplet-droplet collision behavior caused by bouncing, grazing and reflexive separation, and coalescence. Once a droplet reaches the surface, the spray-wall interaction is modeled by the O'Rourke and Amsden approach (O'Rourke & Amsden, 1996& Amsden, , 2000. Therefore, a droplet is categorized by its Weber number We 0 (with the variables ρ l as the liquid density, U 0 as the perpendicular velocity of the droplet to the surface, d as the droplet diameter and σ as the surface tension): and the non-splashing parameter E 2 where h represents the local film thickness and δ bl is the boundary layer height. The definition of the boundary layer height is and μ l represents the dynamic viscosity of the liquid. According to We 0 and E 2 , the droplet behavior is determined. A droplet rebounds on the surface in cases when We 0 < 5, meaning that the droplet maintains its size, shape and kinetic energy after the impact. When an impinging droplet has a We 0 > 5 and E 2 < E 2 crit , the droplet loses all of its kinetic energy because it cannot overcome the surface tension and sticks to the surface while maintaining its size and shape. In the case where E 2 ≥ E 2 crit and We 0 > 5, the droplet splashes on the surface causing a break-up of the droplet into smaller droplets and a wall film left on the surface. The mass fraction of those smaller droplets to the incident droplet is defined as m Splashed m Incident = 0.75, as done by Zhao et al. (2017Zhao et al. ( , 2018 for instance. Eventually, the change of the droplet radius while evaporating is modeled by the Frössling correlation (Amsden et al., 1989;Richards et al., 2020). Other key parameters are listed in Table 1.

Mesh configuration and simulation parameter
The computational domain is of cylindrical shape, 300 mm in diameter and 67 mm in height ( Figure 5 and Figure 6). The arrangement mesh is non-uniform to achieve a more efficient computational time but still being very accurate in important zones such as the spray cone and in the vicinity of the wall. Hence, the mesh consists of five different mesh sizes. It is finer in the nozzle zone, spray cone zone and impinging region, whereas outer regions of the domain have a coarser mesh. As the mesh size affects the outcome of the numerical simulation, a grid dependency analysis was conducted using three different mesh configurations. From a basis mesh, the mesh configuration is created where the mesh sizes are increased and decreased by 25%. Table 2 displays the mesh sizes for the zones in different configurations. However, the mesh size has an impact on the number of cells leading to a change of the computational time.
The computing resource of two 64 core AMD EPYC TM 7742 were used. The finer mesh possesses a much higher number of cells resulting in a substantial increase in computational time. Figure 7 shows the spray contours to compare the spray behavior depending on different mesh configurations. The finer mesh provokes a longer propagation in both the free spray and the impinged spray, while the coarser mesh shortens the spray propagation. After a thorough comparison, the basis mesh was found    to match the experimental results best: hence, the mesh sizes are the following: Nozzle (0.25 mm), Spray Cone (0.5 mm), Close Ambient (1.0 mm), Boundary Zone 1 (2.0 mm) and Boundary Zone 2 (4.0 mm). This mesh arrangement method leads to a total cell number of around 1.45 million.

Concept of spray contours
Owing to the differences between experimental and numerical approaches to obtaining the spray images, the contour needs to be constructed differently. The experimental contour (Figure 8(a)) is created from high-speed Mie scattering, which is a three-dimensional projection onto a two-dimensional plane. A fixed threshold is applied on the spray image, which is more advantageous for the present purpose as it allows a direct comparison between the images. After a thorough examination of the light intensities present in the images, a fixed threshold value of 40 was chosen, which represented approximately 1% of the maximum light intensity for a 12-bit grayscale image. The low threshold value satisfies two main purposes. Firstly, the background noise in the image with an average value of less than 20 is filtered out. Secondly, fine details of the spray boundary are captured because the light intensities, particularly at the propagation tip, are low due the reduction of droplet size during the impinging process and the interaction of droplets with ambient air. However, it is possible that some droplets become so small in the area around the propagation tip that their light intensity does not meet the threshold of 40 and thus are not recognized by the contour. Hence, owing to this limitation, the propagation might appear shorter than it is in reality. The images of the numerical results depict the location of the parcels (Figure 8(b)) in the three-dimensional volume, which is projected onto a two-dimensional plane. Hence, all parcels are considered to construct the numerical contour. The contour is established by using a MATLAB code that connects the boundary parcels. The advantage compared to the experimental contour is that it does not matter if the parcel size is included in the contour. However, in this technique, single parcels that are located away from the bulk flow expand the contour, for example, with spikes on the forward propagation (Figure 8(b)).
Owing to the different contour techniques, it is expected that the spray boundaries might differ in some areas when compared against each other. However, the contour plots remain a valuable tool for assessing the overall spray behavior. Once the contour for each time step is established, it is overlaid to compare the results of the numerical model with the experimental images. In this work, the term time step refers to the time after the start of injection.

Comparison of experimental and numerical contours
For the comparison between experimental and numerical results, the contours for the free spray, the flat case, as well  as the two 60°slope angle cases (on-and off-geometry) are analyzed. The propagation behavior of the 30°and 90°s lope angles follows a similar trend as in the 60°cases. Figure 9 shows the evolutions of spray propagation over time.
To establish the image series for the free spray (images 1-6), two contours in their impinging moment were taken as a starting point with a time step of 0.1 ms backwards. As the numerical propagation takes slightly longer to develop, there is a gap of 0.15 ms between the methods due to the mechanical delay (opening response) of the solenoid valve in the injector. However, it is much more important that the same behavior can be temporally compared in both contours while accounting for the time difference. It is clear from image 2 that the experimental and numerical contours match very well, and only in the last two images does the experimental contour becomes slightly broader. The mixing process of droplets and stronger ambient air, resulting in an expansion of the experimental spray boundary, is the major cause of the larger experimental contour.
For cases where the spray impinges on the surface in Figure 9, the image series starts with the moment of impinging of both contours. As the time period after impingement is much longer, a time step of 0.2 ms was chosen to assess the behavior of the contours. In the flat case (images 7-12), the experimental results start with a slightly broader contour until image 10, where the forward propagation of the numerical contour matches the experimental contour. The reason is a deceleration of the experimental contour with time, while the numerical contour does not slow down visibly. Another reason could be that droplets at the propagation tip disperse and mix with the ambient air leading to a very low light intensity, which it might not be possible to detect clearly. Although the backward propagation of the numerical contour progresses faster, the general behavior still matches the experimental contour well.
In the 60°on-geometry case (images 13-18), the impingement takes place close to the upper edge of the geometry, which greatly separates the flow. Hence, the slightest fluctuation in the spray before impinging has strong consequence for the propagation. The spray cone of the experimental contours is slightly shifted towards the backward direction, meaning that more droplets go in the backward direction. Instead, the spray cone position of the numerical contour provokes more droplets to be deflected in the forward direction. Consequently, the experimental and numerical contours differ at the propagation tip as this impinging scenario is challenging for modeling. Although the behavior of the numerical contour does not match the experimental contour well, the behavior is reasonable and can be explained by the above reasoning. In the 60°off-geometry case (images 19-24), the experimental contour is broader at the beginning. As seen in the previous cases, the numerical contour tends to progress faster on both sides. Possibly, the difference is partly caused by the limitations of the imaging technique, which is unable to recognize very small droplets. Once both contours meet (image 23), there is very good matching between the experimental and numerical contours. The slope decelerates the forward propagation and this effect is more prominent in the experimental contour.
In general, by directly comparing the experimental and numerical contours, the overall propagation behavior is captured well. The differences between experiment and the model have been explained well, and they will be considered further when analyzing other impinging results such as a three-dimensional perspective as well as the droplet level.

Velocity distribution at the end of injection from different viewpoints
The velocity of the droplets and its distribution are two important factors influencing the propagation of the impinging spray. Figure 10 displays the velocity magnitude of the droplets from different viewpoints at the end of injection (1.50 ms). It was chosen to analyze the end of the injection time step for two reasons. The first is that the propagation is very advanced, which allows a thorough investigation of the flow structure during the impingement process. Secondly, the spray cone is still intact so the flow has not significantly decelerated, which gives information about the flow structure. Additionally, as seen in Figure 11, a change in behavior happens at the end of injection. For instance, the liquid spray mass is beginning to decline strongly, and the wall film mass and evaporation mass change from a linear behavior to a reducing growth with time.
The side view images display droplets with very high velocity in the spray cones, while droplets that have impinged are likely to have much lower velocity. Moreover, the spray cone fluctuates, which has strong consequences for the liquid distribution in the on-geometry cases. For the flat case and the on-geometry cases, the forward direction exhibits large vortex behavior at the propagation tip, which is responsible for the large spray height, while in the off-geometry cases, the vortex behavior is much smaller. In all cases, during the vortex motion at the propagation tip, droplets rise up and their velocity decreases to a very low magnitude.
The bottom view reveals a very different behavior depending on the impinging position. In the flat case as well as the on-geometry cases, the forward direction is pronounced with high velocity droplets that drive the propagation. Differences occur in the backward direction where, in the on-geometry cases, the impact on the slope provokes higher velocity droplets that push the propagation in the backward direction. However, in the off-geometry cases, the droplets possess a much lower velocity in general. In all cases, when the spray cone impacts on the geometry or on the surface, there is an area with very low velocity droplets, which indicates the development of a wall film. In the on-geometry cases, an additional wall film is established at the transition from the slope to the bottom surface. On the contrary, when impinging on the bottom surface (the off-geometry cases), an additional wall film evolves on the slope.
From the back view, the impinging position on the upper geometry surface of the on-geometry cases is slightly higher and wider compared to the flat case. At the slope, the propagation narrows down with an increasing slope angle. An interesting behavior occurs at the 90°s lope angle, where the flow on the slope is narrower than the propagation in the corner of the transition from the slope to the bottom surface. In the off-geometry cases, the propagation shoots up along the slope and is also deflected to the sides. Finally, the three-dimensional view illustrates most of the aspects mentioned and gives an overall view of the spray-geometry interaction.

Mass distribution for moving droplets, wall film, and evaporation over time
The mass of moving droplets, the wall film, and evaporation in the domain over time are the key factors in most spray-wall applications. Thus, the left graph (Figure 11(a)) depicts the mass of droplets over time that are currently moving through the domain and are not bounded by a wall film. The middle graph (Figure 11(b)) displays the droplet mass over time that is contained in a wall film. The right graph shows the evaporated mass over time (Figure 11(c)). In all three graphs, while it is very apparent that impinging position is the driving factor for differences in the behaviors, the slope angle does not significantly alter the behavior. At marker 1 for the on-geometry cases, the spray already impinges on the geometry as the vertical distance to the nozzle is 10 mm shorter than in the other cases. Afterwards, the mass still increases in a wavy behavior that could come from fluctuations in the spray cone. For the flat case and the off-geometry cases, impingement occurs on the surface at marker 2, where a change in the increase of moving droplet mass becomes visible. But while the moving droplet mass in the flat case still increases until 1.25 ms (marker 3), in the off-geometry cases the moving droplet mass abruptly declines after reaching the geometry slope at 0.9 ms. Shortly before the end of injection, the mass of the moving droplets in all cases decreases owing to the closing of the injection event, and the decline becomes faster at the end of injection. The declination rate in the on-geometry cases is very similar to that of the flat case (marker 4), while the off-geometry cases have a steeper declination rate (marker 5). From the 2.75 ms time step, the flat case and the on-geometry cases follow the same behavior (marker 6), meaning that an influence by the geometry is not present any more. At the end of the time frame, the mass of the moving droplets is at 0.425 mg, which is 29% of the maximum of the mass in the flat case, and 34% of the maximum in the ongeometry cases. The off-geometry cases drop more, and at marker 7 their behavior deviates slightly because of the slope angle. From around the 3.0 ms time step, the mass  of moving droplets stagnates at a low level, approximately 1 mg, which corresponds to only 7% of its maximum value.
In the wall film mass (Figure 11(b)), marker 8 shows an offset due to the different impinging times for impact on and off the geometry. After the propagation reaches the geometry slope in the off-geometry cases, the wall film behavior deviates from the flat case (marker 9), and the geometry slope provokes a stronger increase in wall film mass. Instead, the on-geometry cases follow a very similar rate of increase to the flat case (marker 10). With the end of injection for all cases, the linear increase of the wall film mass levels off except for the flat case and on-geometry cases where the level stagnates much earlier (marker 11) at approximately 1.725 mg. Eventually, the wall film mass in the off-geometry cases increases to approximately 2.2 mg, which corresponds to 28% compared to the flat case (marker 12).
The evaporated mass (Figure 11(c)) is not affected by the geometry shortly before the end of injection (marker 13). Afterwards, the behavior of the flat case and the ongeometry cases on one side and the off-geometry cases on the other side starts deviating (marker 14). The end of injection causes a slower increase of the evaporated mass in general. However, while the evaporated mass in the flat case and the on-geometry cases still increases strongly, in the off-geometry, the evaporated mass is hindered. At the end of the time frame in Figure 11(c), the evaporated mass in the flat case and the on-geometry cases are at around 0.85 mg while the off-geometry cases are at 0.69 mg, which corresponds to a reduction of 19%. Compared to the wall film mass over time, there is a very strong linkage between the wall film mass and the evaporated mass, as it is very likely that an extensive wall film hinders a quick evaporation.

Backward and forward domains
In order to evaluate the differences between the forward and backward directions as well as the effect of the geometry, the whole analysis zone along a spray angle of 30°is divided into two domains representing a backward domain and a forward domain as shown in Figure 12.
The moving droplets in the on-geometry cases (Figures 13(a) and 13(b)) show strong fluctuations (marker 1) until the end of injection at 1.5 ms. A peak in the forward direction corresponds to a valley in the backward direction, and vice versa. This effect is due to fluctuations in the spray cone that are amplified by the step geometry edge, which splits the flow into a forward and backward direction. Afterwards, the backward direction shows a larger deviation (marker 2) and a strong decline (marker 3) from the flat case than is present in the forward direction. However, the mass of moving droplets is only under 0.05 mg from the 2.75 ms time step (the maximum in the flat case is approximately 0.69 mg and the maximum in the on-geometry cases is approximately 0.57 mg). In the forward direction, the moving droplet mass is still at approximately 0.4 mg at the end of the time frame (marker 4) while the maximum reaches 0.85 mg in the flat case and 0.79 mg in the on-geometry cases. In general, the mass of the backward moving droplets is lower than in the flat case, which is surprising as the geometry slope provokes an increase in backward propagation ( Figure 10). The forward direction shows only a slight decrease in moving droplet mass apart from the fluctuations and the faster decline after end of injection.
The moving droplets in the off-geometry cases (Figures 13(c) and 13(d)) depict a very different behavior from that in the cases before. As the backward direction propagation is not influenced by the geometry, it is almost identical with that in the flat case (marker 5). Marker 6 shows two increases that are only present in the geometry cases, but the reason for this is unclear as yet. In the forward direction, the behavior of the geometry cases deviates right after impinging on the surface, and already at one time step before the propagation reaches the geometry slope at around 0.9 ms. Once the geometry slope is reached, the difference from the flat case enlarges significantly (marker 7). Moreover, the strong decline starts one  time step earlier (1.20 ms) than in the flat case (1.30 ms) and it is more severe than it is in the flat case (marker 8). At the end of the time frame, the moving droplets of the off-geometry cases are at a very low level (approximately 0.073 mg) compared to the flat case, which is at approximately 0.4 mg. Surprisingly, the 30°case shows the lowest mass of moving droplets, although one would expect that, with a higher slope angle, the obstacle to propagation would increase resulting in a lower moving droplet mass. Figure 14 depicts the wall film behavior for the different geometries and impinging positions over time. The general behavior of the wall film is very similar for all cases. First, the wall film grows almost linearly until the end of injection, and afterwards the growth rate decreases until the wall film mass stagnates from approximately 2.5 ms.
In cases where the spray impinges directly on the geometry (Figures 14(a) and 14(b)), there is an offset due to a shorter spray distance (marker 1). In the backward domain (Figure 14(c)), before the end of injection, it seems that the wall film is slightly more affected by the spray cone fluctuations, which decrease with increasing slope angle (marker 2). The wall film mass behavior of the 30°slope angle in the backward direction becomes very close to the flat case behavior, and the wall film increases with increasing slope angle (marker 3). At the end of the time frame, the wall film mass in the backward direction is slightly above 0.3 mg in the flat case, 0.31 mg in the 30°a ngle slope case, 0.345 mg in the 60°angle slope case, and 0.36 mg in the 90°slope angle case, which is an increase of 20% compared to the flat case. In the forward domain, the geometry does not affect the wall film development in general (marker 4) except the offset compared to the flat case (marker 1). However, the wall film mass, around 1.4 mg, is up to 4.5 times higher than in the backward direction.
In cases where the impinging happens off the geometry, the backward domain is not influenced at all by the geometry before the end of injection (marker 5), while afterwards the wall film mass behavior deviates slightly from the flat case (Figure 13(c)) (marker 6). However, in the forward domain, a separation in the wall film mass happens at the 0.9 ms time step once the propagation hits the geometry (marker 7) provoking a strong increase in wall film formation (marker 8). Consequently, the wall film grows to 1.89 mg, which corresponds a rise of around 34% compared to the flat case. Interestingly, a strong difference between the slope angles is not observed although the flow is much more deflected with increasing slope angle, as shown in the spray images ( Figure 10).

Conclusions
This research presents a coupled numerical and experimental approach to investigating the spray-wall interaction on step geometries with different slope angle of 0°to 90°as well as impinging on and off the geometry. Validation between the experimental data and the numerical model shows good agreement. Differences occur in cases where the spray impinges on the geometry because the upper geometry edge splits the flow sharply, which amplifies any fluctuation in the spray cone. Still, in these cases, the spray behavior is very reasonable. The results are analyzed for the whole domain but also separately for the forward and backward directions with respect to droplet behavior such as moving droplet mass, wall film mass and evaporation mass.
The results show that the impinging process with a step geometry is a highly three-dimensional process. For instance, in the off-geometry case with a 60°slope angle, the forward propagation shoots up the slope, while in the on-geometry case with a 60°slope angle, the backward propagation is guided by the flow, enabling a much longer propagation. In both the 90°slope angle cases, the flow is deflected to the sides, provoking severe horizontal propagation.
Analyzing the droplet behavior in the whole domain reveals that, in the off-geometry cases, the moving droplet mass is significantly reduced. At the same time, a substantial wall film develops on the slope and the overall wall film grows by 28%, leading to a reduction of the evaporated mass by 19% compared to the flat case. Instead, in the on-geometry cases, the impinging process on the geometry does not increase the wall film mass and does not hinder the evaporation process. In general, the impinging position has a major impact on the spray impinging behavior while the slope angle has an only minimal influence.
A new approach to analyzing the impinging spray behavior has been undertaken by splitting the flow domain into the forward and backward directions. It has been shown that, in the on-geometry cases, the wall film increases by 0.06 mg (corresponding to 20%) which only occurs in the backward direction. Instead, in the offgeometry cases, the wall film grows more severely by 0.49 mg (corresponding to 34%) and appears only in the forward direction.
In this study, the foundation is laid for understanding the macroscopic and microscopic behavior of spray impingement depending on step geometries. Spray propagation is examined without the effect of heat transfer. Hence, for future work, a heat transfer model could be included to consider the effect of wall temperature on spray impinging behavior. Spray propagation and its mechanisms have an impact on the heat transfer performance. These results should be valuable for practical applications such as spray cooling where the spray impinges on surfaces with slopes and edges. For instance, the on-geometry cases support propagation and, therefore, the wall film thickness is likely to be small enough to enhance the evaporation process and heat transfer. On the other hand, in the off-geometry cases, the wall film condenses at the transition from a slope to a flat surface. Thus, a higher wall film thickness may develop, which could hinder the evaporation process of the droplets. Additionally, further experiments will be designed to validate the transient spray movement on the impinging surface from a three-dimensional viewpoint due to significant spray propagation in the transverse direction.