Splashing generation by water jet impinging on a horizontal plate

Abstract

The report of COVID-19 virus in municipal wastewater raises the question of whether viruses can become airborne during wastewater transport in sewer systems. The present work experimentally investigates a water jet impinging vertically onto a horizontal plate and the behaviours of the generated tiny droplets. Depending on whether the jet breaks into primary drops before the impingement, three regimes can be non-splashing, jet-splashing and drop-splashing regimes. The splashing ratio, i.e., the portion of jet flow rate becoming splashing droplets, ranges from 1% to 70% in the drop-splashing regime, while it remains less than 2% in the jet-splashing regime. For the splashing droplets, their size and velocity distributions follow log-normal laws. Their diameters are mainly in the range from 0 to 0.3 of the impact jet or drop diameter with the median less than 0.1. Their velocities mostly range from 0 to 3.0 times of the impact velocity with the median around 1.0. The medians of both the dimensionless diameter and velocity of splashing droplets decrease with the impact Weber number. The ejection angles of splashing droplets obey a bell-shaped distribution with the maximum around 70° and the median ranging from 16° to 30°.

Keywords: Drop, Horizontal plate, Impingement, Size and velocity distributions, Splashing droplet, Splashing regime, Water jet

Nomenclature

Introduction

COVID-19 virus is around 100 nm in size and spreads primarily via respiratory droplets released when someone coughs, sneezes, or talks (Bar-On et al., 2020, Asadi et al., 2020). However, COVID-19 virus has recently been detected in municipal wastewater, hence, there is significant concern whether the viruses can become airborne and transport as virus-laden droplets in wastewater systems during the transport of wastewater (Cheung et al., 2020, Gormley et al., 2020). In wastewater systems, the falling and impact processes of sewage in drop structures are expected to be the main source of droplet generation (Granata et al., 2011, Adriana Camino et al., 2015). The falling sewage breaks into jets and drops depending on the flow rate and falling height (Ma et al., 2016a). While the breakup of ideal liquid jet has received considerable attention, there is relatively little understanding on their impact and splashing characteristics. Knowledge of liquid jet impact and splashing is also important in many industrial applications including quenching, cooling, coating, and cleaning processes.

A continuous liquid jet starts to break into primary drops at a certain distance from the nozzle, which is commonly referred to as the breakup length. The breakup of liquid jet can be divided into the dripping, Rayleigh, first wind-induced, second wind-induced and atomization regimes (Dumouchel 2008). Different breakup regimes correspond to various breakup structures, primary drop sizes, and correlations of breakup length with jet velocity at nozzle exist (Birouk and Lekic 2009).

Depending on whether a liquid jet breaks into primary drops before impingement, the impingement takes the form of impact by continuous jet or primary drop trains. Lienhard et al. (1992) observed the splashing produced by a continuous jet impact when liquid velocity is sufficiently large, and found that the amount of splashing droplets is closely related to the jet Reynolds number and the nozzle-to-plate distance. Trainer (2016) reported that when the jet Weber number is smaller than 1500, the impingement of a continuous jet hardly generates splashing whereas the impingement of primary drop trains always results in splashing. The detailed splashing onset conditions for the impingement of a continuous jet and primary drops still remain unclear. As the splashing droplets are small and fast-moving, the determination of their amount, size and velocity is challenging, and no comparison has been made for these splashing characteristics during continuous jet and primary drop impingement. For the impingement of primary drop trains, Zhan et al., 2018, Zhan et al., 2021 correlated the amount of splashing droplets with the impact Weber number and impact frequency of the primary drops. They also reported that the size distribution of the splashing droplets obeys the log-normal law with the maximum value reaching approximately 25% of the impact drop diameter.

The impact of a single drop has received a wealth of scientific interest (see reviews by Liang and Mudawar 2016, and Josserand and Thoroddsen 2016). Two kinds of splashing phenomena are the prompt splashing where small droplets are ejected directly from the advancing liquid-substrate contact line, and the corona splashing where the lamella detaches from the substrate and small droplets are ejected from the raised rim (Rioboo et al., 2001, Yarin, 2006, and Ersoy and Eslamian 2020). The outcome of single drop impact is mainly dependent on the impact parameters (impact drop diameter Dp and velocity Up) and the material properties of the liquid (density ρ, viscosity μ, and surface tension σ) (Mundo et al., 1995). For the splashing onset by single drop impact, a number of predicting equations have been developed using the impact Weber, Wep=ρUp2Dp/σ, and Reynolds number, Rep=ρUpDp/μ, (Yarin and Weiss 1995, Vander Wal et al., 2006). It is indicated that the splashing is encouraged for large, dense, low surface tension drops impacting at high velocities (Aboud et al., 2020). For the size and velocity of the splashing droplets, one of the first studies was presented by Stow and Stainer (1977). They reported that the splashing droplet size is influenced by the kinetic energy and surface tension of the impact drop and the surface properties of the target. For the splashing droplets produced in the corona splashing, Wu (2003) assumed that their size distribution satisfies the log-normal law and established analytical formulas for the parameters characterizing the distribution curve. Davidson (2002) proposed that increasing surface tension increases the diameter of the splashing droplets. Based on Motzkus et al. (2009), increasing liquid viscosity has the same effect on the diameter of the splashing droplets as surface tension. Faßmann et al. (2013) demonstrated that the diameter of the splashing droplets decreases while the velocity increases with the increase in impact velocity.

As a first step towards our understanding of possible virus transport in real sewage systems, the present study experimentally investigates the splashing generated by water jets impacting vertically onto a horizontal plate. The effects of nozzle diameter, jet velocity at nozzle exit and nozzle-to-plate distance are taken into consideration. The flow behaviours caused by the impingement of continuous jet and primary drops are visualized. The thresholds of various splashing regimes are studied. In each splashing regime, the splashing ratio and the size and velocity of the splashing droplets are measured and analyzed.

Methodology

The experimental apparatus is shown schematically in Fig. 1 . Experiments were conducted at room temperature of 20 °C and the tap water was used. A water jet was produced by a nozzle mounted on the bottom of a supply tank, and the vertical falling jet impacted onto a horizontal plate. Three circular nozzles were used with a diameter D 0 of 2.0, 6.0 and 8.0 mm. These nozzles were made of copper and well curved for a smooth outflow. The water head in the supply tank was controlled by a movable overflow weir to produce a mean jet velocity at the nozzle exit U 0 varying from 1.59 to 3.17 m/s. These D 0 and U 0 testing ranges are selected because typically the initial velocity of falling sewage is less than 3 m/s and it breaks into jets and drops with diameters mostly around 2–4 mm. The corresponding jet flow rate Q 0, and Weber number We 0 and Reynolds number Re 0 at the nozzle exit were in the range of 5.0⩽Q0⩽155.4×10-6 m^3^/s, 70⩽We0=ρU02D0/σ⩽1,053, and 3,182⩽Re0=ρU0D0/μ⩽24,689. The target plate made by plexiglass was circular with a diameter of 30 cm, larger than the maximum diameter of the generated corona sheet during the jet impingement. The nozzle-to-plate distance l was varied from 100 to 600 mm. Table 1 summarizes the main experimental parameters.

Fig. 1 Experimental (a) nozzle and target plate; (b) droplet collection device.

During the experiments, the falling and impacting processes of the jet were recorded by a high-speed camera (Phantom v211, Vision Research) coupled with a Nikon AF Micro NIKKOR 60 mm f/2.8D lens and illuminated by a diffuser-LED (SP-E-360D SpectroLED, Genaray) in a shadowgraph configuration. Videos were taken at a resolution of 1280 × 800 pixels at a speed of 2000 frames per second and an exposure time of 2–10 μs. The spatial resolution was 300 μm/pixel for recording the falling jets/drops and 80 μm/pixel for the splashing droplets, given the much smaller size of the splashing droplets.

The obtained video images were analyzed using the software, ImageJ (Schneider et al., 2012). The size and displacement of the jet and primary drops were measured using the tools of “Line” and “Measurement” in ImageJ. For the splashing droplets, background subtraction and binarization were applied to the video images, extracting the moving splashing droplets on the focal plane with sharp edges. The tool of “Analyze Particles” was utilized to find the edge of the splashing droplets and measure their sizes. The plugin of “TrackMate” (Tinevez et al., 2017) was used to determine the velocity of the splashing droplets using particle tracking velocimetry (PTV). For each jet impact condition, more than 200 splashing droplets were randomly selected, and their size and velocity were measured as soon as they are detached from high-speed video images (within 2.5 ms).

For the splashing rate Qs, a hygroscopic paper (WypAll X80, Kimberly-Clark) was tailored to a shape of isosceles triangular with the vertex angle of 20° and the leg length of 150 mm and was used for capturing the splashing droplets (see Fig. 1(b)). The difference in the weight of the hygroscopic paper was measured over a given time, then, Qs was evaluated by assuming that the splashing rate was axisymmetric. The hygroscopic paper was horizontally placed 20 mm above the plate with its sharp corner close to but not in contact with the jet. From the videos taken by the high-speed camera, it was confirmed that the hygroscopic paper captured the splashing droplets and was out of touch with the film flow on the plate generated during the jet impingement. This measurement was repeated for more than three times for each impact condition, and it was verified that the splashing rate was fairly insensitive to the capturing duration. The measurement variation in Qs was within ±10%.

Results and discussion

Jet impact and splashing regimes

As the jet impact and splashing phenomena depend strongly upon the prior jet deformation and breakup, the jet breakup length Lb must be studied first. Herein, Lb is defined as the shortest length from the nozzle to the first breakup position measured from a period of 2 s of high-speed video images. According to the experiment of Zhan et al. (2020), L b becomes fairly constant when the measurement duration is larger than 0.5 s. The difference between the measurements from 0.5 s and 2 s video images was found to be less than 5%. In the present study, the jet initially has smooth surface and then develops nearly axisymmetric disturbances at some distances after the nozzle exit. The disturbances grow as the jet moves downstream, leading to successive detachment of the primary drops as soon as the disturbance amplitude reaches the jet radius. The jet breakup length Lb is observed to increase with the initial jet velocity U 0, which agrees with the observations of the Rayleigh regime in the literature (Dumouchel 2008). The linear jet stability analysis for the Rayleigh regime (e.g., Mansour and Chigier 1994) indicated that the dimensionless breakup length Lb/D 0 is related to We 0 ^0.5^(1 + 3Oh 0), where the Ohnesorge number Oh0=We00.5/Re0=μ/σρD00.5. For a water jet, 1+3Oh0≈1 as 3Oh 0 is less than 1% when D0>1.0 mm. Therefore, Lb/D 0 of the water jet is plotted in Fig. 2 as a function of We 0 ^0.5^. Our experimental data are compared with that from Trainer, 2016, Mansour and Chigier, 1994. It is found that the breakup length Lb increases from 50 to 140 times of the nozzle diameter D 0 when 100<We0<4400, and all the data can be well predicted by the following simple equation

Fig. 2 Breakup length of water jet.

Eq. (1) is compared with the correlation from Trainer (2016) who used the same definition of Lb

It is indicated that Eq. (1) has better agreement with these experimental data.

When the nozzle-to-plate distance l is less than the breakup length Lb, continuous jet impingement is observed. Based on whether splashing is generated, the continuous jet impingement can be further classified into non-splashing and jet-splashing regimes, and their typical snapshots are shown in Fig. 3 (a) and (b). In the non-splashing regime, the jet still has smooth surface or weak disturbances before impingement. The impact jet spreads radially on the plate, creating a film flow usually with smooth free-surface. Sporadic surface waves may occur on the film flow, but they have small amplitude and always remain stable.

Fig. 3 Typical snapshots of the experiments with D0=6.0 (a) non-splashing regime (We0=432, l/D0=26.7), (b) jet-splashing regime (We0=617, l/D0=98.3), and (c) drop-splashing regime (We0=432, l/D0=98.3).

In the jet-splashing regime, the jet disturbances become significant when reaching the plate and they are strongly amplified after jet impingement, leading to the creation of an annular upraising wave that originates near the impingement point and travels radially outwards. The wave continues to upraise and sharpen, becoming a sheet bounded by a rim at the crest. During the travel process of the wave, holes firstly appear in the sheet and rapidly expand, finally leading to complete breakup of the sheet and detachment of the rim. The rim is unstable and subsequently breaks up into relatively larger splashing droplets compared to those produced by the sheet breakup. Lienhard et al. (1992) demonstrated that the impingement of the second wind-induced jet is susceptible to splashing and considered the reason as the jet surface disturbances driven by the liquid-side pressure fluctuations caused by the turbulence. Our experiments indicate that for the Rayleigh jet with surface disturbances driven by capillary instability, its impingement can also create splashing when the falling distance is long enough to have significant disturbances.

When l⩾Lb, the primary drops have already been detached from the jet before reaching the plate, and their impingement always produces splashing. Correspondingly, this phenomenon is named as the drop-splashing regime. Fig. 3(c) presents typical snapshots of primary drop impingement. The impact drop spreads out radially and a corona-like liquid lamella emerges in the direction normal to the plate. The rim-bending disturbances at the top of the lamella become unstable and consequently form several cusps, which then become the sources of multiple finger-like jets. The breakup of the finger jets produces plenty of tiny splashing droplets. Visualization shows that the liquid film on the plate generated by primary drops impingement has a thickness significantly smaller than the drop diameter. Therefore, the phenomenon of primary drop impingement could be treated as a superposition of single drop impingement on wet surface (Macklin and Metaxas 1976). The splashing morphology is similar to the corona splashing described by Rioboo et al. (2003), while the prompt splashing is not observed. According to Okawa et al. (2021), in the case of single water drop impact onto a quiescent liquid film, the prompt splashing occurs at lower Weber number and generates smaller splashing droplets than the corona splashing. One possible reason for not observing prompt splashing during the drop-splashing regime is that the majority of the primary drops in our experiments have high enough Weber number. Additionally, the relatively limited spatial resolution in the present study also makes small droplets from the prompt splashing difficult to be recorded.

In the present study, the jet length at the onset of splashing, Ls, is defined as the nozzle-to-plate distance where any splashing droplets are observed. As comparison, Bhunia and Lienhard, 1994, Trainer, 2016 studied the jet splashing characteristics and defined L s as the distance when 5% of the total impinging liquid is splattered. The reason is that the present study concerns the possible generation of virus-laden droplets, in particular these splashing droplets with small sizes as they potentially be transported by sewer ventilation despite with small contribution to the entire splashing rate.

Visualization in the present study and the literature demonstrate that once the primary drops are detached from the jet, their impingement always produces splashing, hence, there exists Ls⩽Lb. Fig. 4 (a) shows the dimensionless onset of splashing Ls/D 0 with a larger D 0 corresponding to a slightly smaller L s/D 0. Compared with the experimental data from Bhunia and Lienhard, 1994, Trainer, 2016, our data for D0=2.0 mm are in close agreement while that for D0=6.0 and 8.0 mm are smaller when We00.5>20. Note that there is difference in the definition of the onset of splashing used by our work and Bhunia and Lienhard, 1994, Trainer, 2016. For We00.5⩽20, visible splashing and 5% of the total impinging liquid splattered occur together at the nozzle-to-plate distance l=Lb, while for We00.5>20, visible splashing occurs at shorter l than 5% of the total impinging liquid splattered. In addition, the nozzle aspect ratio (the ratio of nozzle length to diameter) decreased from 2.7, 2.6 to 2.2 for the nozzles of D0=2.0, 6.0 and 8.0 mm, respectively, which can affect the jet turbulence and surface disturbances at the nozzle exit and therefore L s (Birouk and Lekic, 2009).

Fig. 4 (a) Dimensionless onset of splashing, and (b) regime map for non-splashing, jet-splashing, and drop-splashing.

Based on Fig. 2, Fig. 4(a), a regime map for non-splashing, jet-splashing, and drop-splashing is shown in Fig. 4(b) in the form of l/Lb versus We 0 ^0.5^. The data of l/Lb=Ls/Lb from the present experiment and from Bhunia and Lienhard, 1994, Trainer, 2016 as well as the line of l/Lb=1 that distinguishes between the impingement of continuous jet and primary drops are plotted for describing the thresholds between various splashing regimes. For the data of Ls/Lb from Bhunia and Lienhard and Trainer, their values of Lb are estimated by Eq. (1) as only Lb values were provided, and they can be well fitted by a logistic correlation in terms of Ls/Lb asymptotic with respect to 1 and 0 when We 0 ^0.5^ tends to 0 and 100

For our experimental data of Ls/Lb, segment fitting seems reasonable considering the relatively limited range of We 0 compared with that of Bhunia and Lienhard, 1994, Trainer, 2016

As shown in Fig. 4(b), the drop-splashing regime occurs beyond l/Lb=1. In this region, the nozzle-to-plate distance l is larger than the jet breakup length Lb, therefore, the primary drops are detached from the jet before impingent, always generating splashing. The yellow region between l/Lb=1 and l/Lb=Ls/Lb corresponds to the jet-splashing regime where splashing is produced by the unbroken jet impingement. The non-splashing regime falls in the blue region of l/Lb<Ls/Lb. Utilizing the different definitions of the splashing onset causes a transition (green region) between the non-splashing and jet-splashing regimes, and in this condition, although the splashing has been observed because of the continuous jet impingement, less than 5% of jet flow rate is turned into the splashing droplets. Note that when We00.5>15, the non-splashing, jet splashing, and drop-splashing regimes are observed successively with increasing l, while for We00.5⩽15, direct transition from non-splashing to drop-splashing regimes is obtained as l increases, without generating the jet-splashing regime.

Splashing characteristics in drop-splashing regime

In the drop-splashing regime, the splashing characteristics, including the splashing rate and the size and velocity of the splashing droplets, are mainly determined by the diameter Dp, velocity Up and frequency f of the impact primary drops. Ma et al. (2016b) showed that before breakup into primary drops, the jet is simply accelerated by gravity, and the jet velocity Uj is correlated to the falling height l as

where g is the gravitational acceleration. After the jet breakup when l>Lb, the air friction acting on the primary drops cannot be neglected due to the much larger surface area of the drops (Sallam et al., 2002). Zhan et al. (2018) developed an equation for predicting the velocity Up of the primary drops

Herein, C1=9.1×10-4CD/Dp for water jet, with CD being the dimensionless drag coefficient. Zhan et al. (2018) experimentally had CD=0.58 for Re 0 from 2000 to 10,000. Ub represents the jet velocity where jet breakup occurs and can be calculated by substituting l=Lb into Eq. (6). Fig. 5 (a) compares our experimental measurements of Up with Eq. (7) predication. It is found that Eq. (7) predicts the primary drop velocity within an error of ±15%.

Fig. 5 (a) Comparisons of the predicted and measured velocities of the primary drop; (b) correlation for the primary drop diameter.

For Rayleigh regime jet, Zhan et al. (2020) suggested that Dp is proportional to the jet diameter at the breakup point, Db=2Q0/πUb0.5 from the mass conservation. Our experimental values of Dp are plotted against the calculated values of Db in Fig. 5(b). It can be seen that Dp correlates well with Db within an error of ±10%

Variation of the drop impact frequency f with the nozzle-to-plate distance l is described as f remains zero when l⩽Lb and increases asymptotically to the maximum impact frequency fmax with the increase of l from Lb to Lmax, and finally it becomes fairly constant for l>Lmax. Compared with the (minimum) breakup length Lb, the maximum breakup length Lmax represents the point where the jet is consistently observed to break into the primary drops. In this consideration, Zhan et al. (2018) established a correlation for the dimensionless impact frequency f/fmax with the dimensionless falling height (l-Lb)/(Lmax-Lb) as

where k is a fitting parameter, Lmax is assumed to be in proportion to Lb. Our experimental data of f/fmax are plotted against (l − Lb)/(Lmax − Lb) in Fig. 6 . Herein, f is measured by image analysis, and fmax is calculated by fmax=6Q0/πDp3 based on the mass conservation for a jet, which agrees with the measured value by analyzing the high-speed video images at sufficiently downstream from the nozzle (Zhan et al., 2020). It is shown that Eq. (9) achieves fairly good agreement with the data (R2=0.90) when k=1.02 and Lmax=1.77Lb.

Fig. 6 Correlation for the dimensionless impact frequency f/fmax.

Splashing ratio

The splashing ratio Qs* is defined as the portion of jet flow rate Q 0 turned into splashing droplet rate Qs

In the drop-splashing regime, the splashing comes from the impingement of successive primary drops formed following the jet breakup, therefore, the splashing ratio per primary drop impact is written by

where Vs is the splash volume per impact, and Vp denotes the primary drop volume. Given that the splashing is promoted by the inertia of the drop while prevented by the liquid surface tension, it is expected that *Qs**(fmax/f) can be expressed as a function of the Weber number of the impact primary drop Wep.

Fig. 7 shows the splashing ratio per impact Qs**(fmax/f) plotted against Wep in the drop-splashing regime. It also depicts a direct comparison between the splashing generated by the vertical jet impingement onto a horizontal plate and the horizontal jet impingement onto a vertical plate to reflect effects of the impact angle of a jet. The data of vertical jet are from the present experiments and Zhan et al. (2018). For the horizontal jets studied by Wassenberg et al., 2019, Kim et al., 2020, only Qs values were provided, but we can simply assume fmax/f=1 and therefore Qs∗=Qs∗fmax/f. It is because their images of jet breakup and impact show that the nozzle-to-plate distances l are several times larger than the jet breakup lengths Lb, exhibiting that the impact frequency f of the primary drops have already reached fmax.

Fig. 7 Splashing ratio per impact *Qs**(fmax/f) for vertical and horizontal jet impact in drop-splashing regime.

It is indicated that *Qs**(fmax/f) increases with Wep both for the vertical and horizontal jet impingement. For the vertical jet, the data of this study and Zhan et al. (2018) achieve a fairly good agreement, and they are well fitted by the functions of Zhan et al. (2018)

In view of this, Eqs. (12), (13) can be used for predicting the splashing ratio of a vertical jet in the drop-splashing regime. For a horizontal jet, agreement is fairly good for the data of Kim et al., 2020, Wassenberg et al., 2019, and the following correlation is found to compare well with their experimental data with R2=0.91

From Eqs. (12), (13), (14) shown in Fig. 7, we can find that in comparison with the horizontal impact jet, the vertical impact jet produces much less splashing ratio when Wep<1300 while slightly more splashing ratio when 1300<Wep<3500, in the drop-splashing regime.

Size and velocity of splashing droplets

Visualization shows that the splashing droplets generated in the drop-splashing regime have various sizes and velocities. To explore the size and velocity distributions of these splashing droplets, examples of the probability density distributions of Ds/Dp and Us/Up are presented in Fig. 8 , where Ds and Us refer the diameter and velocity of a splashing droplet, respectively. The dimensionless diameter of the splashing droplets mainly locates in the ranges of 0⩽Ds/Dp⩽0.3 with the peak between 0.03 and 0.12, and the dimensionless velocity mainly locates in the ranges of 0⩽Us/Up⩽3 with the peak between 0.6 and 1.2. Zhan et al. (2018) observed that the size of the maximum splashing droplet is approximately 25% of the impact drop diameter, and Li et al., 2019, Burzynski et al., 2020 showed that the maximum velocity of the splashing droplets can be 3–6 times of the impact drop velocity, similar to our results.

Fig. 8 Probability density distributions of Ds/Dp and Us/Up in drop-splashing (a) D0=2.0 mm and Wep=550; (b) D0=6.0 mm and Wep=1159; and (c) D0=8.0 mm and Wep=1676.

Fig. 8 indicates that the distributions of D∗=Ds/Dp and U∗=Us/Up for the drop-splashing regime follow the log-normal distribution

where AD and AU are the histogram bin widths, Dc* and Uc* represent the median dimensionless diameter and velocity of the splashing droplets, respectively, and wD and wU denote respectively the log standard deviations of the dimensionless diameter and velocity distributions (characterizing distribution width). In Fig. 8, Dc* rapidly decreases from 0.095 to 0.038 while Uc* decreases slightly from 1.13 to 0.96, when Wep increases from 550 to 1676. wD and wU appear to have no direct relation with Wep and fluctuate within 0.47–0.61 and 0.36–0.49, respectively.

Correlations of these log-normal distribution parameters of Dc**, wD, Uc and wU with the state of primary drops at the impingement are studied as follows. As the morphology of the drop-splashing regime is similar to the corona splashing, a physical analysis of splashing droplet diameter and velocity for corona splashing is conducted in the Appendix, based on establishing and analyzing conservation laws between the impact drop, the liquid sheet and the splashing droplets. The expressions for Ds/Dp and Us/Up are derived as a function of the Weber number Wep and Reynolds number Rep of the impact primary drops

where C is the parameter determined experimentally.

Dc* that characterizes the median dimensionless diameter of the splashing droplets are compared with Eq. (17) in Fig. 9 (a). The experimental data from the present study and Zhan et al. (2021) regarding the drop-splashing regime as well as that from Stow and Stainer, 1977, Mundo et al., 1995, Li et al., 2019 regarding the single drop impingement are plotted for comparison. It is found that Dc* for the drop-splashing regime and the single drop impingement have a similar tendency with WepRep ^0.5^, and they can be predicted by Eq. (17) within the error of ±30%. According to Eq. (17) and Eq. (7) that correlates Up with U 0 and l, the two parameters D 0 and U 0 dominate the production of the finest droplet fraction that potentially evaporate to generate dry residues, and the number of finest droplets increases with decreasing D 0 and increasing U 0. In addition, increasing l also promotes the production of the finest droplet in some degree.

Fig. 9 Log-normal distribution parameters of (a) Dc* and (b) wD for drop-splashing regime and single drop impingement.

Uc* that characterizes the median dimensionless velocity of the splashing droplets are compared with Eq. (18) in Fig. 10 (a). Besides our experimental measurements, also included are the experimental data for the single drop impingement from Li et al. (2019). Uc* for the drop-splashing regime agree with that for the single drop impingement. Uc* is in proportion of (Wep/Rep ^0.5^)^−0.25^, showing a good agreement with Eq. (18). When C=1.7, Eq. (18) can well fit the data within an error of ±30%.

Fig. 10 Log-normal distribution parameters of (a) Uc* and (b) wU for drop-splashing regime and single drop impingement.

For wD and wU that characterize the log-normal distribution width of Ds/Dp and Us/Up, our experimental data are compared with that for the drop-splashing regime and single drop impingement from the literature in Figs. 9(b) and 10(b). It is found that wD and wU remain almost constant for drop-splashing regime and single drop impingement. For wD, the data from the present study, Stow and Stainer, 1977, Li et al., 2019 agree well with each other, while they are slightly smaller than the data from Mundo et al., 1995, Zhan et al., 2021. The majority of wD data falls within ±30% of wD=0.5. For wU, the data from the present study and Li et al. (2019) can be well fitted by wU=0.4 within an error of ±30%.

The ejection angle α of the splashing droplets is defined as the angle between their initial velocities and the horizontal direction. Visualization shows that in the drop-splashing regime, the maximum of α is always around 70°. Considering the corona splashing generated by a single drop impingement, Burzynski et al. (2020) experimentally observed that the majority of the ejection angles α are less than 75°, showing a close agreement with our results. The distributions of α in the drop-splashing regime are presented in Fig. 11 for D0=2.0, 6.0, and 8.0 mm with Wep=942, 1341, and 2133, respectively. The results show that the ejection angles are not they have a bell-shape distribution with the interquartile ranges covering 14°<α<47°. As Wep increases from 942 to 2133, the median of α decreases from 30° to 22°, while its mean decreases from 32° to 23.0°.

Fig. 11 Distributions of ejection angle α in drop-splashing regime.

Splashing characteristics in jet-splashing regime

Splashing ratio

In the jet-splashing regime, the splashing characteristics are influenced by the velocity Uj and diameter Dj of the impact continuous jet. Fig. 12 compares our experimental measurements of Uj with the prediction of Eq. (6) with a difference less than ± 10%. Therefore, Eq. (6) will be used in calculating the jet velocity Uj, and the jet diameter Dj can be obtained by Dj=2Q0/πUj0.5.

Fig. 12 Comparisons of the predicted and measured jet velocities.

Fig. 13 plots the splashing ratio Qs∗=Qs/Q0 in the jet-splashing regime versus the Weber number of the impact jet Wej=ρUj2Dj/σ. Compared with the drop-splashing regime where Q s* varies from 1 to 70%, Qs* in the jet-splashing regime is significantly smaller with its value less than 2%. For a given jet, Qs* increases with Wej. As the jet Weber number We 0 at nozzle exit increases, the correlation of Qs* with Wej has a tendency of moving towards the positive direction of horizontal axis. In the jet-splashing regime, the splashing is caused by the jet disturbances impacting onto the plate. With decreasing We 0, the breakup length of Rayleigh jet decreases, which accelerates the growth of jet disturbances. The analytical correlation of jet disturbance amplitude established by Lienhard et al. (1992) shows that the jet disturbances increase as We 0 decreases within the range of 330<We0<1060 in Fig. 13.

Fig. 13 Splashing ratio Qs* in the jet-splashing regime.

Size and velocity of splashing droplets

The probability density distribution of Ds/Dj and Us/Uj in the jet-splashing regime are shown in Fig. 14 . The dimensionless diameter of the splashing droplets locates in the ranges of 0⩽Ds/Dj⩽0.36 with the peak between 0.03 and 0.09, and the dimensionless velocity locates in the range of 0⩽Us/Uj⩽2.1 with the peak between 0.6 and 1.2. Similar to that in the drop-splashing regime, the distributions of D∗=Ds/Dj and U∗=Us/Uj in the jet-splashing regime can be expressed by the log-normal function (see Eqs. (15), (16)). In Fig. 14, Dc* and *Uc**, representing the median of Ds/Dj and Us/Uj, decrease from 0.100 to 0.086 and from 1.02 to 0.87, respectively, when Wej increases from 1005 to 1224. As the parameters characterizing the log-normal distribution width for Ds/Dj and Us/Uj, wD is around 0.70 and shows no direct relation with Wej as in the drop-splashing regime, and as comparison, wU decreases from 0.42 to 0.25 with increasing Wej.

Fig. 14 Probability density distributions of Ds/Dj and Us/Uj in jet-splashing (a) D0=6.0 mm and Wej=1005; and (b) D0=8.0 mm and Wej=1224.

For the jet-splashing regime, since the splashing droplets are produced by the breakup of the wave sheet and rim shown in Fig. 3(b), the splashing droplet diameter should be comparable with the sheet thickness and rim diameter. An analysis about the impingement of a continuous jet with significant surface disturbances is shown in the Appendix, and the equations about the dimensionless sheet thickness hsh/Dj and rim diameter Dr/Dj are obtained as a function of the Weber number Wej and Reynolds number Rej=ρUjDj/μ of the impact jet

Fig. 15(a) compares the median dimensionless diameter Dc* of the splashing droplets with Eqs. (19), (20). It is found that Dc* proportionally increases with (WejRej^0.5^)^−0.5^. Our experimental data, although significant over Eq. (19), are fitted by Eq. (20) within an error of ±30%. Hence, Eq. (20) can be used for predicting Dc* for the jet-splashing regime, reflecting that majority of the splashing droplets are from the rim breakup.

Fig. 15 Log-normal distribution parameters of (a) Dc* and (b) wD for jet-splashing regime.

wD in the jet-splashing regime is plotted against WejRej ^0.5^ in Fig. 15(b). wD remains almost constant with all the data included within ±30% of wD=0.7, which is larger than wD=0.5 in the drop-splashing regime. The reason may be that the splashing droplets are produced by breakup of the rim and sheet in the jet-splashing regime, while the finger jets are the main sources of the splashing droplets in the drop-splashing regime. Multi-sources of splashing droplet creation in the jet-splashing regime bring about wider size distribution of the splashing droplets.

Same as that for the drop-splashing regime, the median dimensionless velocity Uc* of the splashing droplets for the jet-splashing regime is plotted against WejRej ^−0.5^ in Fig. 16 (a). The following equation is found to correlate our experimental data within an error of ± 30%

wU in the jet-splashing regime are plotted against WejRej ^−0.5^ in Fig. 16(b). It is found that wU significantly decreases with WejRej ^−0.5^ and all the data can be fitted by wU=2.5Wej/Rej0.5-1 within ± 40%. Compared with that in the drop-splashing regime, most of wU in the jet-splashing regime are smaller. The potential reason is that in the drop-splashing regime, the splashing droplets have various velocity characteristics at the different moments of drop impact and transformation process, leading to wider velocity distribution of the splashing droplets (Li et al., 2019, and Burzynski et al., 2020).

Fig. 16 Log-normal distribution parameters of (a) Uc* and (b) wU for jet-splashing regime.

For the ejection angle α in the jet-splashing regime, it is found that the maximum α can reach about 70°, similar to that in the drop-splashing regime. Distributions of α are presented in Fig. 17 . Measurements from four cases are shown, D0=6 mm with Wej=1068 and 1182, and D0=8 mm case with Wej=1434 and 1640. α in the jet-splashing regime follows a bell-shaped distribution with the interquartile range covers 8°⩽α⩽34°, which is significantly smaller than that in the drop-splashing regime. Comparison between these cases shows that the median of α decreases from 24° to 16° while its mean decreases from 24° to 18° when Wej increases from 1068 to 1640.

Fig. 17 Distributions of the ejection angle α in jet-splashing regime.

Summary and conclusions

In the present study, the generation of splashing droplets by water jet vertically impingement onto a horizontal plate is experimentally explored. The key findings including the splashing regimes and the splashing characteristics in various splashing regimes are summarized as follows.

Depending on whether the jet breaks into primary drops before the impingement, three regimes can be non-splashing, jet-splashing and drop-splashing regimes. The non-splashing and jet-splashing regimes occur during the continuous jet impingement and they are distinguished according to whether splashing is generated. As soon as the primary drops are detached from the jet at some distance below the nozzle exit, they already achieve sufficient velocity for producing splashing after impingement (Rioboo et al., 2003). The regime map of jet impingement is established dependent on the Weber number We 0 at the nozzle exit and the nozzle-to-plate distance. Increasing We 0 and nozzle-to-plate distance both contribute to the splashing generation. Note that the jet-splashing regime only occurs for jets with We00.5>15, while for jets with We00.5⩽15, the direct transition from non-splashing to drop-splashing regimes is obtained as the nozzle-to-plate distance increases.

In the drop-splashing regime, the splashing ratio ranges from 1% to 70% and exponentially increases with the Weber number Wep of impact primary drops. Compared with the horizontal impact jet, the vertical impact jet produces much less splashing ratio when Wep<1300 while slightly more splashing ratio when 1300<Wep<3500. The produced splashing droplets are small and fast-moving, and their size and velocity distributions follow the log-normal law. Their diameters are mainly in the range from 0 to 0.3 of the impact drop diameter with the peak between 0.03 and 0.12. Their velocities mostly locate from 0 to 3.0 times of the impact velocity with the peak between 0.6 and 1.2. As the log-normal distribution parameters, Dc* and Uc* characterizing the median diameter and velocity of the splashing droplets both decrease with Wep; wD and wU characterizing the distribution width of the splashing droplet diameter and velocity remain almost constants of 0.5 and 0.4 respectively. The ejection angle of the splashing droplets follows a bell-shaped distribution with the maximum around 70°, and the median ranges between 22° and 30° and decreases with increasing Wep.

In the jet-splashing regime, the splashing ratio remains less than 2% and is significantly smaller than that in the drop-splashing regime. For the splashing droplets, their size and velocity distributions follow the log-normal law. Their diameters are mainly in the range from 0 to 0.36 of the impact jet diameter with the peak between 0.03 and 0.09. Their velocities are in the range from 0 to 2.1 of the impact velocity with the peak between 0.6 and 1.2. Dc* and Uc* both decrease with increase in the Weber number Wej of impact jet. With increasing Wej, wD remains almost a constant of 0.7 while wU significantly decreases. The ejection angle of the splashing droplets follows a bell-shaped distribution with the interquartile range smaller than that in the drop-splashing regime. The median of the ejection angle ranges between 16° and 24° and decreases with increasing Wep.

Our established correlations for the splashing onset and the splashing characteristics in the jet-splashing and drop-splashing regimes make the prediction possible by using the nozzle diameter, jet velocity at nozzle exit, and nozzle-to-plate distance. While this study was initially motivated by the possible generation of virus laden aerosols in sewer systems, the current work focused on much larger sizes of droplets, which have a great chance of settling before they evaporate to produce airborne aerosols. The current work, however, is a first step towards a better understanding of possible virus transport in complicated sewer systems. In addition, the generated splashing droplets in drop-splashing regime have much larger splashing ratio, smaller size and larger velocity of splashing droplet, and larger ejection angle, therefore could potentially carry viruses and be transported by the sewer ventilation, causing potential risk to operation and maintenance personnel. In future studies, characteristics of the aerosols (smaller than 50 μm) generated by jet impingement should be analyzed, on the other hand, virus surrogates should be added into test liquid as the tracers of virus, and their influences on splashing morphology, their concentrations in splashing droplets/aerosols, and their airborne lifetime and settlement should be carefully studied to tackle Covid-19 airborne resuspension threat by liquid jet fragmentation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The authors are thankful for the strong interest and guidance from Dr. Rasha Maal-Bared (EPCOR Water Services Inc.), Mr. Bert van Duin (the City of Calgary), Mr. Kingsford Amoah (the City of Red Deer), Dr. Lyndon Gyurek (Alberta Environment and Parks), and Dave Kryiak and Nicole McLellan (Stantec Consulting Ltd.). Financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada and the International Postdoctoral Exchange Fellowship Program by the Office of China Postdoctoral Council (Grant No. 20190025) is gratefully appreciated. The authors also would like to thank Perry Fedun for his technical assistance.

Analytical study of splashing droplet diameter and velocity for drop-splashing and jet-splashing regimes

Drop-splashing regime

For the drop-splashing regime, its morphology is similar to that of the corona splashing. Therefore, physical analysis about the splashing droplet diameter in the drop-splashing regime is conducted based on Wu (2003) for the corona splashing by single drop impingement, and further analysis is added about the splashing droplet velocity.

Fig. A1 schematically depicts the liquid transformation progress of drop impingement and splashing. The impact drop is first transformed into a corona sheet, and the flow instability at the corona top produces finger jets along the circumferential direction which finally disintegrate into splashing droplets.

Fig. A1 Liquid transformation progress of drop impingement and splashing.

The conservation laws between the corona sheet and the finger jets are considered. Assuming that the average spacing between the adjacent finger jets is lf, the mass conversation can be written as

where hsh is the thickness of the corona sheet, Df is the diameter of the finger jet, and Ush and Uf are the liquid velocities inside the sheet and finger jet respectively.

The total energy including the kinetic energy and the surface energy is conserved during the transformation of the corona sheet to finger jets, yielding the energy conservation equation as

The momentum conservation equation is given as

Herein, psh and pf are the pressures in the sheet and finger jets respectively, and the Laplace formula provides their relationships with the atmospheric pressure pa as psh=pa and pf=pa+2ρσ/Df, respectively. Substituting the above formulas into Eq. (A3) leads to

The energy and momentum conservation equations of Eqs. (A2), (A4) can be simplified by using the mass conservation equation of Eq. (A1) to eliminate lf

Assuming Df and Uf are in proportion to hsh and Ush respectively, Eqs. (A5), (A6) can be solved to yield

According to Yarin and Weiss (1995), when the finger jets formed on the top edge of the corona sheet, the relationship between the local velocity Ush and thickness hsh of the corona sheet can be estimated by the Taylor formula

Therefore, both of Df and Uf can be directly related to hsh based on Eqs. (A7), (A8). The relationship between hsh and the state of the impact primary drop is analyzed by establishing the conservation equation linking the total energy of the primary drop and corona sheet.

A=π/6Dp31/hsh represents the side surface area of the corona sheet. Ed accounts for the energy dissipation in the creation of sheet against the viscosity, and it can be estimated by

where ts marks the characteristic time scale for the drop impingement, ϕ is the local energy dissipation, and ϕ¯ represents the average value of ϕ. By estimating ts=Dp/Up and ϕ¯=μUp/hsh2, Eq. (A11) can be reduced to

Substituting Eqs. (A9), (A12) into Eq. (A10) yields

where, Wep=ρUp2Dp/σ and Rep=ρUpDp/μ are the Weber and Reynolds numbers of impact primary drop. Thus,

It is known that the finger jets disintegrate into the splashing droplets because of the Rayleigh-Plateau instability, that is, the finger jets of diameter Df produce the splashing droplets with a diameter Ds=1.889Df. Combining Eqs. (A7), (A14) gives the correlation of the dimensionless splashing droplet diameter Ds/Dp with Wep and Rep

As Wep⩾230 in the present study, the above formula can be simplified as

with

Compared with (WepRep ^0.5^)^−0.5^, f 1(Wep ^2^/Rep) can be treated as almost a constant of 4.69 for 6⩽Wep2/Rep⩽140 in the present study. Hence, Eq. (A16) can be further simplified as

Combining Eqs. (A8), (A9), it can be shown that

Substituting Eq. (A14) into Eq. (A19), we obtain

As the velocity of the splashing droplets Us are closely related to that of the finger jets Uf, there should be the following correlation for the dimensionless splashing droplet velocity Us/Up

where C is the parameter determined experimentally.

Jet-splashing regime

For the jet-splashing regime, splashing is observed when large jet surface disturbances reach the plate as shown in Fig. A2 . It is considered that the splashing droplets are produced by the impingement of an ellipsoidal drop enclosed by the symmetric disturbances. Therefore, the analysis about the spherical drop impingement can be adopted for the ellipsoidal drop impingement after the following modifications.

Fig. A2 Sketch of impact jet in jet-splashing regime.

The ellipsoidal drop is elongated in the vertical direction and its projection in the horizontal plane is a circle. Assuming a spherical drop with the diameter of Dp has the same volume as the ellipsoidal drop, there is πDp3/6=πD12D2/6, where D 1 and D 2 are the minor and major axes of the ellipsoidal drop. Hence, D 1 and D 2 can be expressed as D1=aDp and D2=Dp/a2 respectively, where a is a constant in the range from 0 to 1. We assume that the impact jet diameter Dj equals the average width of the ellipsoidal drop

For the ellipsoidal drop impingement, the characteristic time scale shown in Eq. (A11) is expressed by ts=D2/Uj=4Dj/a3πUj, and Eq. (A14) can be rewritten to yield the correlation of the dimensionless sheet thickness hsh/Dj with the Weber number Wej=ρUj2Dj/σ and Reynolds number Rej=ρUjDj/μ of the impact jet

with

Bhunia and Lienhard, 1994, van Hoeve et al., 2010 indicated that when the jet disturbances are large enough for splashing, D 2 is expected to approximate to the optimum wavelength of Rayleigh jet, λopt=1.41πDj. Therefore, D2=Dp/a2=1.41πDj, yielding a=0.66. Eq. (A24) can be treated as a constant of 9.69 over the range of 25⩽Wej2/Rej⩽120 in the present study. Hence, Eq. (A23) can be further simplified as

For the rim at the crest of the liquid sheet generated by the impingement, our experiments show its diameter Dr is 2.8–4.4 times of the sheet thickness hsh. Experimentally, Negeed et al. (2011) gave Dr=2.86hsh, and Raux et al. (2020) indicated that Dr is around 3 times of hsh, showing close agreement with our result. Using the average value of the ratio between Dr and hsh in the present study, Dr/hsh=3.6, we obtain

As the splashing droplets are produced by the breakup of the sheet and rim in the jet-splashing regime, the dimensionless splashing droplet diameter Ds/Dj should be comparable with the dimensionless sheet thickness hsh/Dj and rim diameter Dr/Dj.

References