1 Introduction

When two solid colliding surfaces are wet or partially wet, a stress/pressure singularity is generated at the contact zone and cavitation can be produced [1, 2]. The nucleation, growth and particularly the collapse of cavitation bubbles between the two solid surfaces is an extremely violent event that accelerates the surface damage [3,4,5]. This has practical relevance in many industrial and natural processes such as filtration, agglomeration, bonding, wet granular flow, spray coating, to name a few [6]. The method based on the impact between solids to produce bubbles has been the target of numerous theoretical/numerical and experimental studies that have shed light on different mechanisms to explain this phenomenon. When two solid surfaces in contact and immersed in a fluid layer are separated from each other at high velocities, a drastic decrease in pressure (tensile pressure) occurs in the contact zone. The pressure can reach values lower than the vapor pressure and cavitation is generated. As suggested by some experimental works [7,8,9,10,11,12,13,14], the bubbles thus created presumably occur during the rebound stage, moments after the impact.This type of cavitation due to the fast separation of solids in a liquid (CSSL) has also been investigated from a theoretical perspective using lubrication models [15]. Under certain conditions, it is also possible to observe cavities around the contact zone at the moment of impact. This phenomenon was studied experimentally by Kleine et al. [16]. In this work the authors implemented a schlieren system to show how an elastic stress wave emanates from the contact zone (post-microseconds after the contact is initiated), thus explaining the formation of the observed bubbles. From a theoretical perspective, there are interesting approaches such as the one developed by Thorne [17], which in principle could be used to predict the pressure field (characterized by negative values developed a few moments after the impact) at a certain distance from the contact zone. This type of cavitation due to tensile waves is called here in abbreviated form as (TWC). Another plausible mechanism to produce cavitation is to subject a fluid to a shear stress as proposed in [18]. However, it does not appear to play a dominant role in explaining cavitation in Newtonian fluids employing spheres to impact solid surfaces immersed in thin layers of liquid [9]. On the other side, the elastohydrodynamic collision model has shown that a part of the kinetic energy is stored as elastic deformation energy allowing one of the involved solids to rebound, even without solid-solid contact [19]. Thus, the cavitation structure by a collision with apparent solid-solid contact shows remarkable differences with that from solid-solid contact, such as distinct onset conditions, evolution speeds and bubble size distributions [6, 8]. Some of these differences have been shown in dimensionless maps to facilitate their interpretation [6, 10, 20]. These maps are also useful tools to delimit and understand the onset of cavitation, its evolution and the acoustic and light emissions from bubble collapse in a simple way. This knowledge is key to understanding and determining the parameters involved in the control of the impact-induced cavitation phenomenon. Nevertheless, such maps are rare in the open literature. The aim of this research consists of constructing a map with experimental measurements to identify zones (conditions) of cavitation and non-cavitation when two bodies immersed in water collide. This can be useful in the development of potential applications based on cavitation induced by collision between solids such as: cavitation machining [21] or surface treatment (controlled erosion). It is also of interest to find out the collision events in which photon emission occurs. This may have applications in water treatment where light could be used to activate photocatalysts [22]. We performed an experimental parametric study to construct a map in which the onset of collision-induced cavitation is identified with a threshold line defined in terms of a cavitation number established with a lubrication model. In this dimension parameter map, different regions were identified where cavitation does not occur, in those where cavitation is only produced by the separation of solids in the liquid (CSSL) and in which cavitation caused by tension waves is generated in addition to CSSL (TWC - CSSL). Furthermore, the events in which luminous emissions were detected are shown. The tests were performed with a mechanical apparatus equipped with a spherical impactor, immersed in water together with the sample to be impacted, which allows easy variation of collision velocities and hydrostatic pressure.

2 Theory

A cavitation number based on the lubrication model is established to characterize cavitation induced by underwater impacts. The phenomenon under consideration consists of a metallic sphere hitting a plane solid surface and then bouncing off it, both bodies being immersed in an incompressible Newtonian fluid with axial symmetry and no elastic or inertial effects [7, 9, 10, 12]. Figure 1 shows a close-up view of the impact region, where R, \(\delta\), h, d and \(H= h-d+R-(R^2-r^2)^{1/2}\) are the sphere radius, water column height, the minimum separation between the sphere surface and the solid plane, penetration depth and the height of the sphere at distance r, respectively. So, the radial velocity component is expressed as [9]

$$\begin{aligned} u_r(r,z)=\frac{r 3 V_r}{H(r)^3} \left( \left( z-\frac{H(r)}{2}\right) ^2-\left( \frac{H(r)}{2}\right) ^2\right) \end{aligned}$$
(1)

where r, z and \(V_r\) are the radial and axial directions, as well as the rebound speed of the sphere, respectively. In fact, the rebound speed is usually estimated by the product of the coefficient of restitution e and the impact speed \((V_r=e V_i)\) [23]. The coefficient of restitution e is a measure of the energy loss during impact. The value of e is between \(0 \le e \le 1\), when (\(e = 0\)) a perfectly plastic collision is assumed (total energy loss) while (\(e = 1\)) suggests a perfectly elastic collision (no energy loss). However, e in a liquid is also highly dependent on R and \(\delta\) [24]. As \(\delta\) increases, e decreases because some of the impact energy is also transferred to the liquid phase. In order to study the influence of the impact speed \(V_i\) and the water column heigh \(\delta\) on the cavitation onset, R is going to be constant.

Fig. 1
figure 1

Schematic figure of the impact region showing the sphere radius R, water column height \(\delta\), minimum separation of the sphere with the wall h, depth of indentation d, height of the sphere at distance r during contact \(H= h-d+R-(R^2-r^2)^{1/2}\), contact radius \(r_c\) and the respective Young’s modulus (\(E_1\), \(E_2\)) and Poisson’s ratios of both solids (\(\nu _1\), \(\nu _2\))

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The pressure distribution in the liquid is given by

$$\begin{aligned} \frac{\partial P(r)}{\partial r}=\mu \frac{\partial ^2 u_r}{\partial z^2} \end{aligned}$$
(2)

where \(\mu\) is the dynamic viscosity [9]. Solving Eq. 2 with the boundary conditions, \(\partial P(0)/\partial r =0\) and \(P(R)\approx P_{atm}\), results in

$$\begin{aligned} P(r)=\frac{3eV_i\mu }{h-d+R}\left( \frac{R^2-r^2}{H(r)^2}\right) +P_{atm} \end{aligned}$$
(3)

where \(P_{atm}\) is the ambient pressure. Cavitation is likely to occur where \(P(r)<P_v\), where \(P_v\) is the vapor pressure. Neglecting the hydrostatic pressure of the water column, we can establish

$$\begin{aligned} Ca=\frac{P_{atm}-P_v}{\frac{3eV_i\mu }{h-d+R}\left( \frac{R^2-r_c^2}{H(r_c)^2}\right) } \end{aligned}$$
(4)

as an indicator of cavitation onset due to the liquid stress/pressure singularity upon impact. The \(r_c\) parameter is the radius of the contact surface given by the Hertzian theory of non-adhesive elastic contact as

$$\begin{aligned} r_c=0.5\left( \frac{3RF}{2E}\right) ^{1/3} \end{aligned}$$
(5)

where F is the impact force and E is related to Young’s modulus and Poisson’s ratio of the two solids by [25]

$$\begin{aligned} \frac{1}{E}= \frac{1-\nu _1^2}{E_1}+\frac{1-\nu _2^2}{E_2} \end{aligned}$$
(6)

while the penetration depth is calculated by

$$\begin{aligned} d=\left( \frac{2F^2}{E^2R}\right) ^{1/3} \end{aligned}$$
(7)

Thus, we propose Ca as a dimensionless number to determine the cavitation induced by the collision between two solids immersed in water. Note that the elastic properties of the solids, the nature of the fluid, and the impact velocity are the main parameters considered in the calculation of Ca. The numerator in Eq. (4) represents the pressure drop required to initiate cavitation, as proposed in the traditional cavitation number [1]. However, the denominator in the current formulation is the tensile stress when the sphere starts its rebound. When the tensile stress exceeds the pressure difference of the liquid (\(Ca<1\)), cavitation is likely to occur. On the other hand, when \(Ca>1\) the pressure difference is large enough to balance the tensile stress generated during the rebound stage of the sphere.

3 Experimental setup

Figure 2a shows a schematic of the experimental setup used in this study. The mechanism to generate cavitation consists of a steel sphere (\(\phi _{sph}=2R=25.15\) mm, 0.0567 Kg, \(E_1=112\) GPa, \(\nu _1=0.23\)) coupled to a cylindrical steel shaft (\(\phi _{sh}=12.8\) mm, 0.2253 Kg, \(E_1=112\) GPa, \(\nu _1=0.23\)) that was accelerated in a free-fall condition. This assembly was placed inside a rectangular glass container (60.4 x 60.4 x 110 mm) to allow visualization of the impact process and cavitation. One aluminum sample (60.4 x 60.4 x 12.6 mm, 0.280 Kg, \(E_2=69\) GPa, \(\nu _2=0.33\)) was firmly fixed at the base of the container (see Fig. 2c). A mechanical gripper system, to adjust the sphere height (maximum height 180 mm), and a trigger to activate its motion were also used. Furthermore, the mechanism was equipped with a steel frame and a bronze bushing to guide the vertical movement of the assembly and reduce oscillations (see Fig. 2b). The impact force was measured with a PVDF film sensor (DT1-028K, 10 M\(\Omega\), 0.001 - \(10^{9}\) Hz, dimensions 12.19 x 30 mm) placed between the lower face of the sample and the upper face of the base of the container. In addition, an ET Enterprises photomultiplier (model 9354KB) to measure light emissions was used. All analog signals were digitized using an oscilloscope Tektronix TD55104B. The cavitation dynamics, \(V_i\) and \(V_r\) were captured using a Phantom camera V7.3 with its focusing optics (SIGMA APO MACRO DG, 180mm 1:3.5) and using a 50 W halogen lamp (see Fig. 2a).

Fig. 2
figure 2

a Schematic of the experimental setup. b Device to generate cavitation by impacts. c Close-up view of the steel sphere, rectangular glass container, and aluminum sample

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3.1 Materials and methods

All experiments were performed at room conditions (Patm = 79.7 kPa and T = 23 \(^{\circ }\)C). The container was partially filled with distilled water to vary the hydro-static pressure (\(P_{v}=2.3\) kPa, \(\rho \approx 1000\) kg/m\(^3\), \(\mu \approx 10^{-3}\) Pa s, 0.2 \(\mu\)S/cm, 0.1 ppm). Furthermore, it was attached to a solid base to reduce any movement. The water column height \(\delta\) was changed from 10 mm to 90 mm in increments of 10 mm. The impact force and \(V_i\) were varied by increasing the initial distance L between solids (see Fig. 2c) from 1 mm to the \(\delta\) value, with increments of 1 mm. Due to the changes of \(V_i\) and \(\delta\), it was possible to change the coefficient of restitution e. The value of e was estimated by dividing \(V_r\) by \(V_i\) [23]. The PVDF film sensor was calibrated using the ball drop method, so the sensitivity constant was estimated in 0.0009 V/N [26]. The Phantom camera was operated at 100000 fps and 265 x 64 pixels resolution. In each experiment, the surface of the sphere was cleaned to remove trapped bubbles. Finally, cavitation onset, cavitation dynamics, \(V_i\) and \(V_r\) were analyzed using the Tracker software [27]. To calculate Ca it is important to take into consideration the following aspects: 1) The arithmetic mean roughness of both solids was measured with a laser surface roughness meter, resulting in \(h=12\) \(\mu\)m. 2) The impact force F was measured with the PVDF sensor.

4 Results and discussion

Figure 3a shows cavitation by separation of the solids in the liquid (CSSL). It is generated by the rupture of the water film, between the solids, when the sphere rebounds from the aluminum sample. Notice how an annular cloud of bubbles is generated around the contact zone after impact, between \(0 - 100\) \(\mu\)s. The cloud of bubbles rapidly grows in the confined space between both solids, from 100 \(\mu\)s to 400 \(\mu\)s. As the bubble cloud approaches its maximum size (8.8 mm diameter), protrusions around the cloud due to surface instabilities were formed [20]. After the growth stage, the bubble cloud starts its collapse stage (\(450 - 500\) \(\mu\)s). Some small bubble clouds are generated after the collapse at 550 \(\mu\)s (1). The CSSL then begins its rebound stage (3), and the small bubble clouds grown to their maximum volume (2). The sphere rebound is perceptible at 1.2 ms. The corresponding Reynolds numbers and the impact force magnitudes are \(Re=\phi _{sph}V_i\rho /\mu =O(10^3 - 10^4)\) and \(F=O(10^2 - 10^3)\) [N]. Note how the CSSL regimen is generated at the transition from laminar to turbulent flow, whit impacts moderately strong. On the other hand, Fig. 3b shows both tension wave cavitation and cavitation by separation of solids in liquid (TWC - CSSL). The TWC is generated before the CSSL as bubbles outside the contact zone (\(0 - 12\) \(\mu\)s). Sometime later (36 \(\mu\)s), the CSSL is generated in the impact zone. Noteworthy is the fact that the growth and collapse processes of the TWC are faster than those of the CSSL. The TWC growth time is about 276 \(\mu\)s, while the CSSL growth time is 420 \(\mu\)s. In fact, when the CSSL reaches its maximum size (10.2 mm in diameter), the cavities due to the tension wave are already collapsing at 420 \(\mu\)s. Subsequently, the CSSL collapses up to 600 \(\mu\)s and the rebound process extends to (612 \(\mu\)s). This regime where the two types of cavitation occur is characterized by light emissions from collapsing bubbles, Reynolds numbers in the range \(Re=O(10^4-10^5)\) and impact force magnitudes \(F>O(10^3)\) [N], note that the latter impacts are much stronger than those in which only the CSSL is observed.

Fig. 3
figure 3

High-speed images of the cavitation generated by the impact. a Cavitation by separation of solids in the liquid (L = 30 mm, \(\delta\) = 70 mm). This condition is characterized by an annular cloud of bubbles generated in the contact zone after impact. b Generation of both tension wave cavitation and cavitation by separation of solids in the liquid (L = 55 mm, \(\delta\) = 70 mm). This condition is characterized by bubbles generated outside of the contact zone and an annular cloud of bubbles generated around the contact zone

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Figure 4 shows the maximum radius of the CSSL as a function of (L) and (\(\delta\)). Note that a greater L distance generates a greater impact force and, consequently, an increase in the maximum radius of the CSSL (estimated by measuring the maximum diameter of the bubble cloud as shown in Fig. 3).In the region of \(\delta\) less than or equal to 50 mm and L less than or equal to 30 mm, it is observed that for a constant L the gradual increase of \(\delta\) tends to decrease the maximum radius of the CSSL. This is due to the fact that increasing the liquid column, and therefore the hydrostatic pressure, significantly counteracts the effect of pressure decrease due to the separation of solids after collision, inhibiting a large growth of the bubbles generated. In contrast, when L is greater than 30 mm, the impacts are intense, generating large pressure drops during rebounds together with larger bubbles, despite the fact that, at the same time, the liquid columns used tend to be larger.

Fig. 4
figure 4

Maximum radius of the CSSL as a function of the water column height (\(\delta\)) and the initial distance (L) between solids. As L increases, so does the impact force, generating bubble clouds of larger radii

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Figure 5 summarizes the experimental results as a dimensionless parameter map showing a threshold line \((Ca=1)\) that delimits the no cavitation \((Ca>1)\) and cavitation regions \((Ca<1)\). Note that Ca depends on e, so the mechanical properties of the solids, liquid properties and kinetic parameters are critical in determining the cavitation onset. Certainly, most events where no cavitation occurred are located in the no-cavitation zone or near the threshold (red symbols). Similarly, most CSSL (blue symbols) and TWC - CSSL (black symbols) events are in the cavitation zone. These results show that the proposed cavitation number effectively delimits the cavitation onset. The blue and black lines at the top and on the right side of the map represent the probability density curves of the CSSL and TWC - CSSL regimes, respectively. If the impact is of low energy, the CSSL regime will occur. On the other hand, if the impact is sufficiently energetic, it will lead to a regime where both TWC and CSSL are present. In this regime, light emissions are generated if the values of e and Ca are as low as 0.28 and 0.1, respectively. While it may appear that the values of both e and Ca are close between one regime and the other, the magnitude of the impact force required to generate one or the other regime can be significant.

Fig. 5
figure 5

Dimensionless parameter map of cavitation generated by the impact between two solid surfaces in water. It is divided into two regions: the no cavitation \((Ca>1)\) and cavitation \((Ca<1)\) zones. The red, blue and black symbols correspond to no cavitation, CSSL and TWC-CSSL events, respectively. According to the density curves there is a transition from CSSL (blue line) to TWC - CSSL (black line) as e diminishes. The colored zone indicates the events in which photon emission was detected

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Figure 6 shows two transitions between different processes: 1) from the non-cavitation to the CSSL and 2) from the CSSL to the TWC - CSSL regimes as a function of e and \(V_i\). The threshold from no cavitation regime to CSSL has been estimated at \(e\approx 0.4\) and \(V_i\approx 0.25\) [m/s]. The change from CSSL to TWC - CSSL occurs in a transition zone between \(0.5<V_i<0.6\), with \(e\approx 0.25\). In this transition region, CSSL is always generated but TWC may not be generated. When \(V_i>0.6\) and \(e<0.25\), TWC - CSSL events are generated. Light emissions are only detected with the TWC - CSSL events when \(e<0.25\) and \(V_i>0.7\). In fact, as \(V_i\) increases and e decreases, the amplitude and number of light emissions increase.

Fig. 6
figure 6

Coefficient of restitution (e) and amplitude of light emissions as a function of the impact speed (\(V_i\)). The threshold from the no cavitation regime to CSSL is estimated at \(e\approx 0.4\) and \(V_i\approx 0.25\) [m/s]. A transition zone is observed in which there is a change from CSSL to TWC - CSSL regime (\(0.5<V_i<0.6\) when \(e\approx 0.25\)). The light emissions are detected when \(e<0.25\) and \(V_i>0.7\), symbols in blue

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At this point it is pertinent to comment on some of the possible applications that can be developed based on the cavitation phenomenon due to the collision of solids immersed in liquids. Among several possibilities, we visualize two interesting applications: cavitation machining and surface treatment. To achieve this type of processes it is necessary to cover important requirements: 1) To produce cavitation in a localized area. 2) And also, that the bubbles produced can collapse violently, thus generating shock waves and/or jets with the ability to erode or plastically deform the impacted material. We believe this is plausible with the device studied here and, of course, would be useful for developing micromachining techniques. Additionally, when the impactor (in our case a solid sphere) makes contact with another solid it can produce localized (point) permanent deformations. If such plastic point deformations (due to either cavity implosion, solid-solid collision, or a combination of both effects) are distributed over an area of the impacted material, their surface mechanical properties can be improved as demonstrated in peening processes [28]. It is worth noting that the level of plastic deformation due to solid-solid interaction and violent implosion of cavities could be controlled by a combination of the impact velocity and the hardness of the material from which the impactor is constructed, on the other hand, different etching patterns and/or surface roughness in the impactor could be used to produce different distributions and types of plastic deformations in the impacted material. These ideas, however, will have to be the focus of future research.

5 Conclusions

A dimensionless parameter map was constructed in which cavitation (caused by underwater collisions between two solids) and non-cavitation zones were identified and delimited with a threshold line for a cavitation number formulated with a lubrication model. The experimental observations were consistent with the theoretical prediction. Likewise, the usual coefficient of restitution can be used to establish the onset of cavitation, and it is also useful to estimate the transition from CSSL to TWC - CSSL. In the latter regime where the two types of cavitation occur, some events were identified in which light emissions from bubble collapses appear. We believe that an additional cavitation number as a function of the acoustic properties of the materials and the medium would provide greater certainty in determining the transition from CSSL to TWC - CSSL regimes. The information provided by the dimensionless parameter map of cavitation and the coefficient of restitution can be useful in the development of new and varied applications involving cavitation generated by the collision between solids. Interesting possibilities are envisioned: cavitation machining, water treatment, surface treatment of solids, etc.