1 Introduction

The development of high-pressure waterjet technology has started over 60 years ago. Despite that, the research is still ongoing in order to improve the process efficiency and waterjet properties [1, 2]. Abrasive waterjet was applied to extend the capabilities of processing many hard and brittle materials [3, 4].

In study [5], the influence of abrasive water jet (AWJ) cutting parameters on hardened AISI 440C steel was investigated. By optimizing the AWJ cutting process parameters, a surface roughness of approximately 2.404 µm and a kerf taper of approximately 1.352° were achieved. The experimental results [5, 6] of AWJ cutting indicated that significant parameters affecting the surface roughness and kerf taper include the mixing nozzle, the distance of the nozzle from the workpiece, the feed rate of the head, and the thickness of the material being processed [6]. In study [7], a numerical simulation was conducted, implementing a three-dimensional AWJ flow model (air–water-abrasive) based on the Euler–Lagrange approach [7]. The numerical studies [7] evaluated the abrasive velocity recorded at the end of the focusing nozzle and the effects of AWJ operating parameters. Abrasive velocity increases with the decrease in abrasive flow rate and the increase in the diameter of the focusing nozzle. Additionally, an increase in the length of the focusing nozzle initially caused an increase, followed by a decrease, in the AWJ velocity. The research enabled the proposal of a modified model accounting for the characteristics of abrasive particles. Chinese researchers conducted studies on the AWJ for single particles [8]. Their research found that water pressure and abrasive flow rate positively correlate with effective cutting depth, while nozzle traverse speed and nozzle distance from the workpiece negatively correlate with effective cutting depth. Among the adjustable cutting parameters, nozzle traverse speed and water pressure have the most significant impact on cutting depth. The AWJ technology has also demonstrated its capabilities in processing composite materials [9, 10] and laminates [11].The addition of abrasive grains was applied for cutting or surface processing [12, 13]. Nowadays, the waterjet technology is used for (i) cutting out difficult shapes from metal sheets (which is faster and more versatile than other processing methods like milling) [14], (ii) processing of fire or explosion hazard materials [15,16,17], (iii) underwater works [18], and many more. For a vast variety of applications, it was necessary to find solutions allowing for an increase in the cutting pressure [19].

Poloprudský et al. [20] investigate the effect of the pulsating waterjet on the polished surfaces of AISI 316L stainless steel and the Ti6Al4V titanium alloy. The frequency of the waterjet pulsations was 21 kHz. The independent variable in the experiments was the exposure time of the processed materials to the pulsating waterjet. In order to distinguish deformation stages, a low frequency value was chosen. The sample surfaces were measured with a confocal microscope to characterize surface topography changes. In detail, surface examination was also performed using SEM microscope. Transmission electron microscope was used to detect changes of surface microstructure. For the Ti6Al4V titanium alloy, irregular surface dales as well as slip and crack lines were observed, while in the 316L steel, evenly distributed straight textures occurred. The near-surface dislocation density was measured quantitatively, reaching high values of 1014 m−2 in austenitic steel and even higher (up to 3 × 1015 m−2) in the titanium alloy.

An extension of this research was to analyze the results of subsurface deformation of materials [21] proper to the fluid droplets periodic action, each of constant volume, also distributed with spatial frequency. Sample kerfs were analyzed at stand-off distances in conditions of the dominant mechanism reducing the acceleration climax using starshaped trajectories to avoid eventual Doppler effects. Two types of these cavities were identified, blind and transitional, with diameters of a few micrometers. The results suggest that the side stream effect after droplet collapse causes extensive hydrodynamic tunneling in the material, which is much greater during intense periodic water droplet action, even at subsonic speeds.

Poláček et al. [22] presented a method for generating waterjet instability by applying coaxial air flow. That approach allowed for improvement of the surface processing, sanding, or material cutting. The experimental study concerned the processing of calcium blocks, both in water and air conditions. The improvement of material erosion by 17% related to steady stream was observed at a water pressure of 20 MPa and an air overpressure 0.01 MPa with the distance to the material surface set to 120 mm.

Pulsating waterjet penning is another method of material processing. In that technique, accelerated water drops impact the surface causing elasto-plastic deformations. Applications of this strategy were presented in works [23, 24]. Processed surfaces, with pressure 10, 20, and 30 MPa, were assessed with surface roughness parameters and microhardness tests. It was measured that the microhardness increased from 43.3 to 47 HV. Controlled impacts of water drops could lead to improved surface roughness and simultaneously increase surface strengthening without causing thermal damage.

Another method for increasing the efficiency of water jet technology is generating the water pulses. The theoretical analyses of pulsating water jet, reported by Heymann [25] and Hung [26], showed the cumulation of mechanical energy on the hard material surfaces as a result of hydrodynamic high-frequency impacts of fast moving water drops [26, 27]. The effect of a discontinuous jet was considered to increase water pressure multiple times compared to continuous flow, as proven by Smith and Kinslow [28]. In addition to the high maximum pressure values, the erosive effect of the pulsating water jet has also been improved due to the increased impacts of cyclic shock waves occurring in the area of contact on the material.

Raj et al. [29] interpreted the process of generation of pulses for PWJ with mathematical equations and made comparison all techniques that have been reported so far, for creating pulses used in generating pulsating waterjets. They showed mathematical methods helpful in comprehending the manners of generation of pulsating water jet that is grounded on generation of pulses and transmission of created waves to the nozzle outlet.

One method for generating water pulses involved applying ultrasonic disruptions to the water flow within the nozzle, initially reported by Vijay et al. [30, 31] and further developed by Foldyna et al. [32, 33]. As a result, the water jet had significantly greater energy, leading to increased efficiency of the jet machine. Using this method to generate water pulses resulted in a water jet with regular pulsations, strongly affecting the surface of the material. The cyclical impacts of the pulses provided a much greater cumulative effect than that produced by a continuous flow of water. For this reason, this method is also used in the study of surface treatment processes and material cutting. The use of tubular nozzles for generating pulses in a jet was initially described by Nebeker et al. [34, 35] and further developed by Chahine et al. [36]. However, few of these solutions have found a production application.

Also, Hloch et al. [37] studied the erosion transition onto aluminum surface created by the periodic action of a tangentially acting pulsed water jet (PWJ) with a spatial frequency of f = 20 kHz. The erosion effect was observed as a function of standoff distance z (mm) along an inclined and constant trajectory (at different standoff distances under 100 MPa jet pressure and constant traverse velocity). Hydrodynamic erosion effects, such as elastic and plastic deformation, were observed using a scanning electron microscope and compared with acoustic emission (AE) time records recorded during the impact. Differences in the erosion phenomenon under various conditions were observed in terms of material failure, crater dimensions, and surface and subsurface features.

Presented work was created to explore the possibilities of designing the pulsating water jet by means of modified vortex chamber. The idea of a modified chamber without additional moving components, designed for required jet operating properties, was also presented in works [38, 39].

The paper presents new data on the influence of the geometry of self-excited heads for generating a pulsating water jet, which enhances the efficiency of machining processes. By optimizing the parameters of such heads, a pulsating water jet with increased interaction dynamics and beneficial erosive properties is achieved. The results have direct industrial applications, enabling the optimization of the design and operating parameters of pulsating heads for surface machining and material erosion. This can lead to increased efficiency and precision in various industrial sectors.

2 Simulation studies of a cutting head model designed for the surface treatment with a pulsating waterjet

In order to increase the efficiency of hydro-jet surface treatment, a new model of a self-excited pulse head was developed. In the search for an effective tool for hydro-jet machining and cleaning of technical surfaces, an attempt was made to develop self-excited pulsating heads. This required studies on the influence of the number of side holes and their positioning relative to the head’s swirl chamber on the generation of high-pressure water jet pulsations. Previous studies cited in the article for heads intended for drilling holes [39, 40] demonstrated that the most favorable configuration is having the side holes positioned perpendicular to the vortex chamber. Therefore, in the first stage of the research, the head for surface machining was analyzed in terms of the number of side holes positioned perpendicularly to the vortex chamber. Such simulation studies were conducted for a consistent range of diameters (0.6–1.2 mm) for both types of nozzles: the inlet nozzle d1 and the outlet nozzle d3, while keeping other dimensional parameters of the self-excited pulsating head constant.

As indicated by the data in the table shown in Fig. 1, most dimensions of this head are significantly reduced compared to the previously studied head for drilling holes [40]. This reduction is due to the necessity of miniaturizing the head intended for surface machining, which requires manual operation. Nevertheless, the findings from earlier studies were utilized, maintaining more suitable proportions between the interconnected geometric parameters of such heads. For example, the height of the vortex chamber was selected according to its diameter, and the diameter of the head’s side holes was chosen based on the height and diameter of the vortex chamber. Additionally, the relationship between the diameters of the inlet and outlet nozzles and their heights was kept considering the practical proportion d1:h1 = 1:3, at which the highest flow velocities of the water jet through the nozzles are achieved. Moreover, based on previous studies [40, 41], a fixed value of the vortex chamber attack angle λ = 120° was chosen, considering that both lower and higher values of this angle resulted in unfavorable formation of annular water vortices in the vortex chamber. The solution was examined in terms of the influence of the number of holes on the surface cleaning process. The principle concerning the location of the side holes perpendicular to the chamber was used in all such heads, regardless of their application [39,40,41,42]. The simulation tests were carried out for the same range of diameter variation (0.6–1.2 mm) of both types of nozzles: inlet d1 and outlet d3 at a constant pressure of 15 MPa and maintaining the unchanged values of the other dimensional parameters of the self-excited pulse head. In the article, water jet head designed for the surface processing was tested in terms of the parameters variability presented in Fig. 1.

Fig. 1
figure 1

Diagram of a self-excited pulsating head for surface treatment with the range of geometrical parameters

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The main purpose of the research was to select a pulsating head model with appropriate geometric parameters and to verify its suitability for the generating a pulsating waterjet with the highest possible outlet velocity from the tool, ensuring an appropriate distribution of velocity and pressure influencing the shape of the created vortex rings. Moreover, due to the technological possibilities of making such a head, the characteristic dimensions were selected, which are shown in Fig. 1.

Head models with 2, 3, and 4 side holes were used to perform the simulations. The holes were placed at angles 180°, 120°, and 90°, depending on the construction variant. For each model tested, the water flow velocities in the sections marked in Fig. 2 were analyzed, as well as the shape of water vortices generated in the vortex chamber the tool.

Fig. 2
figure 2

The geometry of the self-excited pulse head model with a different number of side holes located perpendicular to the swirl chamber, with marked measurement sections: a with two side holes, b with three side holes, and c with four side holes

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3 Flow simulation method

The influence of the tool’s geometric parameters on the pulsating water jet was simulated in SolidWorks Flow Simulation software.

For numerical calculations, a structural mesh was used in the boundary area and a tetragonal mesh for the remaining parts. The mesh was densified at the walls of the chamber model and in the area of water inlet and outlet. This allowed for a more accurate assessment of flow parameters, including velocity and pressure. Figure 3 shows created grid. The total number of elements was 860,000. A further increase in the number of elements did not change the simulation results.

Fig. 3
figure 3

Simulation model of the tool with two side holes (Inlet 2) perpendicular to the water flow direction

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The k-ε model was used to simulate the water flow through the self-excited pulse head. This model was used to simulate turbulent flows [43,44,45] and can be used to confirm the formation of vortices [46,47,48,49]. Its basic assumption is isotropic turbulent viscosity. This means that the ratio between the Reynolds stress and the mean strain rate is the same in all directions [45]. The values of the Reynolds coefficients for the simulated flows through the heads are shown in Table 1.

Table 1 Reynolds number for the analyzed models of self-excited pulse heads
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Numerical calculations were performed for the steady state flow. The water temperature was 293.16 K. In the simulation, the medium was an incompressible fluid in a continuous phase with a dynamic viscosity coefficient of 0.0010014 Pa*s and a density of 998.16 kg/m3. A pressure of 15 MPa was set at the inlet of the head, while an atmospheric pressure of 0.101325 MPa was applied at its outlet and side holes. The gravitational force was considered. The turbulence flow rate was set at 5%. The turbulence kinetic energy dissipation was set at ε = 1 W/kg and the turbulent kinetic energy k = 1 J/kg. There was no heat exchange with the environment, which was also assumed in [39]. The process was stopped automatically after achieving convergence criterion, which required at least 500 iterations [40]. Roughness of the side walls was not considered.

4 Simulation studies of a self-excited pulse head with different numbers of side holes

In order to assess the influence of the number of holes on the self-excited head work, the tests were carried for all designed variants. The parameters of the simulation model are presented in Table 2.

Table 2 The parameters of the simulation model
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In order to calculate the water flow velocity through the pulsating head as a function of the nozzle diameters, the general equation was formulated:

$$V=a\left({d}_{1}^{2}\bullet {d}_{3}^{2}\right)-b\;\text{exp}(c\;{d}_{3}^{2}+d\;{d}_{1}^{0.4})+\text{exp}(2.9\;{d}_{1}^{2}),$$
(1)

The coefficients were calculated by means of the least squares method and validated through a R2 value, as presented in Table 3.

Table 3 Water velocity function coefficients
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4.1 Results of a water flow simulation through the tool with two side holes

The simulation results for the head model with two side holes are presented in Fig. 4. Negative velocity values mean that the external medium was not transported inwards through the side holes. This phenomenon occurred when the diameter of the inlet nozzle was equal to or greater than that of the outlet nozzle (d1 ≥ d3).

Fig. 4
figure 4

The relationship between the diameter of the inlet nozzle d1 and the outlet nozzle d3 on the average velocity of the jet in the side holes—case of the tool with two side holes

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Based on the obtained results, it was found that for the head model with two side holes, the inlet of medium occurred when the diameter of the inlet nozzle was smaller than the diameter of the outlet nozzle (d1 < d3) [40]. The greatest value of inlet mass velocity through the side holes was obtained for d3 = 1.2 mm and d1 = 0.6 mm. The water outflow through those holes occurred when the inlet nozzle diameter d1II was greater than or equal to outlet nozzle diameter d3.

In the conditions presented in Fig. 5, there was no medium intake through the side holes, while the lack of such flow led to the disappearance of the vortex rings in the chamber. That could cause the incorrect distribution of velocity vectors.

Fig. 5
figure 5

An example of the medium flow without vortex rings: a d1 = 1.2 mm and d3 = 0.6 mm; b d1 = 0.6 mm and d3 = 0.6 mm

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The water jet flowing through the head characterized by the geometry in Fig. 5a, with an inlet nozzle diameter d1 = 1.2 mm and an outlet nozzle diameter d3 = 0.6 mm, leads to the absence of suction of the medium through the side holes equal to d2 = 0.6 mm and the absence of vortex rings. The jet, after exiting the inlet nozzle diameter, disperses, and there are no distinct swirls in the water layers. As a result, there is nearly uniform pressure throughout the swirl chamber, causing water to flow out through the side holes, hindering the formation of vortex rings. The same phenomenon can be observed when d1 = d3 as shown in Fig. 5b. In such circumstances, there were no conditions for generating hydrodynamic pulses in the water jet.

4.2 Results of a water flow simulation through the tool with three side holes

The values of the water flow velocity for the head with three side holes are shown in Fig. 6. The meaning of the negative values in the results was that the external medium was not transported through the side holes. This effect occurred in cases where the diameter of the inlet nozzle was larger than that of the outlet nozzle d1 > d3.

Fig. 6
figure 6

The relationship between the diameter of the inlet nozzle d1 and the outlet nozzle d3 on the average velocity of the jet in the side holes—case of the tool with three side holes

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Analyzing the results in terms of medium inflow through the side holes, it can be concluded that for the head model with three holes, the flow occurred when the diameter of the inlet nozzle was smaller than the diameter of the outlet nozzle d1 < d3. On the other hand, the transport of the medium (water) through the side holes occurred when the diameter of the inlet nozzle was greater than or equal to the diameter of the outlet nozzle. The highest intake of the water flow velocity through the side holes was calculated when the diameter of the inlet nozzle was d1 = 0.6 mm and equal to the diameters of the side holes (d2 = 0.6 mm). The diameter of the outlet nozzle was d3 = 1.2 mm.

However, Fig. 7 presents two examples of velocity vector fields in the head model with three side holes, in which the external medium is not transported into the chamber through the side holes, because no hydrodynamic vortex rings were formed.

Fig. 7
figure 7

Examples of an unfavorable distribution of velocity vectors inside the chamber with three side holes: a d1 = 1.2 mm and d3 = 0.6 mm; b d1 = 1.1 mm and d3 = 0.7 mm

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As in the case of the head model with two side holes, the formation of hydrodynamic impulses in the water jet was influenced by the intake of the external medium through the side holes and the proper shape of the water vortices. Disadvantageous formation of vortex rings always occurs when the diameter of the inlet nozzle d1 is greater than or equal to the diameter of the outlet nozzle d3, so the medium did not inflow through the side holes. In the presented data in Fig. 7, where d1 > d3, negative values of the flow velocity of the medium through the side holes diameters were observed. For the case shown in Fig. 7a, the velocity is d2 =  − 92 m/s, while for Fig. 7b, it is d2 =  − 79 m/s. Increasing the inlet nozzle diameter compared to the outlet nozzle diameter results in a greater influx of the medium into the chamber, leading to less emission through the outlet nozzle and increased efficiency of discharge through the side holes.

4.3 Results of a water flow simulation through the tool with four side holes

In the next step of computer simulations, the water flow velocities in selected sections of the head model with four side holes were analyzed (Fig. 8). The values were obtained for the same nozzle diameter d1 = d3 = 0.6–1.2 mm. The other dimensional parameters were constant.

Fig. 8
figure 8

The relationship between the diameter of the inlet nozzle d1 and the outlet nozzle d3 on the average velocity of the jet in the side holes—case of the tool with four side holes

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The obtained results indicate, similarly to the previous analyses, that the intake of the external medium through the side holes occurs for the head models characterized by a smaller diameter of the inlet nozzle than the diameter of the outlet nozzle (d1 < d3). The highest flow velocity of the medium through the side holes was υ = 30.35 m/s for the model in which diameter of the inlet nozzle and the diameter of the side holes were equal d1 = d2 = 0.6 mm and the diameter of the outlet nozzle was d3 = 1.2 mm.

The analysis of velocity vectors in such a head showed that increasing the diameter of the inlet nozzle, and keeping the outlet nozzle diameter constant, led to the disappearance of the external medium intake, and thus the water vortices inside the chamber. Examples of velocity vector distributions in the head with four side holes, in which the medium was not transported through these holes and no hydrodynamic vortex rings were formed, are presented in Fig. 9.

Fig. 9
figure 9

Distribution of the velocity vectors for tool with four side holes, in which the proper vortex rings did not occur. a d1 = 0.6 mm and d3 = 0.6 mm. b d1 = 1.2 mm and d3 = 0.6 mm

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Proper selection of all geometric parameters of the pulse head could guarantee that the water flowing out of it will be a pulsating jet.

5 Analysis of results

The analysis of various models of the self-excited pulsating head showed that in the heads with three and four side holes, in most of the analyzed cases, there is no favorable formation of vortex rings, which determine the generation of hydrodynamic pulses in the water jet. In addition, the heads with four and three holes were characterized by a lower velocity of the external medium (water) inflow, compared to the head with two holes. Due to the above reasons, the models with four and three side holes were omitted in the further analysis of this issue. For the further study, the model with two side holes was chosen.

From the examined group of the remaining models of the self-excited pulse head with two side holes, the most advantageous solutions in terms of the shaping of water swirls and the velocity of the liquid external medium entering through them were selected. Exemplary results of the chosen model analyses are presented in Fig. 10.

Fig. 10
figure 10

Distribution of flow velocity vectors and pressure values inside the vortex chamber with two side holes. a d1 = 0.6 mm and d3 = 0.8 mm. b d1 = 0.9 mm and d3 = 1.1 mm

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Due to the advantageous shape of the hydrodynamic vortex rings and the inlet of the medium through the side holes, the diameter of the inlet nozzle d1 should be smaller than the diameter of the outlet nozzle d3. The results presented in Fig. 11 show the velocity vector distributions for the head model meeting this condition (d3–d1): d3 > 30%.

Fig. 11
figure 11

Distribution of flow velocity vectors for tool with two side holes. a d1 = 0.7 mm and d3 = 1.1 mm. b d1 = 0.6 mm and d3 = 1.1 mm

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Only for structures with appropriate diameters of the nozzles does the inlet of the medium through the side holes takes place. Therefore, further analyses were carried out only on the self-excited pulse head model with two side holes perpendicular to the vortex chamber.

6 Analysis of the inlet nozzle slenderness influence on the water jet velocity

Initial simulation tests confirmed that the velocity is also influenced by the slenderness of the inlet nozzle. It is determined by the ratio of its height (h1) to diameter (d1). For those reasons, computer tests were carried out to determine the best slenderness of them. The analyses were carried out at a constant pressure of 15 MPa.

It was found that at low values of the h1/d1 ratio, the highest speeds were obtained both at the inlet and the outlet of the tool. The influence of the inlet nozzle slenderness (for the ratio h1/d1 = 0.8; 2.5; 3; 5) on the different water flow velocities in the measurement cross-sections is shown in Fig. 12.

Fig. 12
figure 12

Change of the inlet and outlet water flow velocity relative to the slenderness of the inlet nozzle; d1 = 0.6 mm; d3 = 0.8 mm; d2 = 0.6 mm; D = 3.2 mm; H = 6.2 mm

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The highest velocities of the medium at the head outlet were obtained with the minimum ratio h1/d1 = 0.8. However, for these values, the highest velocities of the fluid inlet through the side holes of the head did not occur.

The influence of the inlet nozzle slenderness on the velocity of the medium flowing through the side holes is shown in Fig. 13.

Fig. 13
figure 13

Change of the water inlet velocity in the side holes relative to the slenderness of the inlet nozzle; d1 = 0.6 mm; d3 = 0.8 mm; d2 = 0.6 mm; D = 3.2 mm; H = 6.2 mm

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Analyzing the obtained results, it was found that the highest speed equal 42 m/s of the external medium inflow through the side holes, ensuring the formation of vortices generating hydrodynamic impulses in the water jet, was obtained with the ratio h1/d1 in the range of 2.5 to 3.

7 An influence of the side holes diameter on the water jet velocity

The values of the water flow velocity in the inlet nozzle as a function of the diameter of the side holes d2, with the remaining parameter constant, are shown in Fig. 14.

Fig. 14
figure 14

The influence of the diameter of the side holes on the velocity of the medium in the inlet nozzle: a d1 = 0.6 mm, d3 = 0.8 mm. b d1 = 0.6 mm, d3 = 1.2 mm

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The diameter of the side holes (d2) had a large effect on the intake of the external medium. This determined the shaping of the vortex rings which cyclically blocked the flow of the waterjet.

The highest velocities (VI1 = 150 m/s and VI2 = 160 m/s for d1 = 0.6 mm, d3 = 1.2 mm; VI1 = 146 m/s and VI2 = 151 m/s for d1 = 0.6 mm, d3 = 0.8 mm) were obtained for side hole diameters d2 = 0.4 mm. Increasing the value to d2 = 1.4 mm caused a slight reduction (Fig. 15) in the velocity of the inlet nozzle. Thus, it was found that the change in the diameter of the side holes did not significantly affect the velocity value in the inlet and outlet nozzles of the tool.

Fig. 15
figure 15

The influence of the diameter of the side holes on the velocity of the medium in the outlet nozzle: a d1 = 0.6 mm, d3 = 1.2 mm. b d1 = 0.6 mm, d3 = 0.8 mm

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The highest water jet outflow velocities were obtained for the smallest diameter of the side holes d2 = 0.4 mm, as shown in Fig. 16. It was especially visible for the head parameters d1 = 0.6 mm, d3 = 1.2 mm, and d2 = 0.4 mm where the velocity in the outlet nozzle was VO1 = 96 m/s and VO2 = 84 m/s. On the other hand, for the diameters of the side holes equal to d2 = 1.4 mm, velocity decreased by 15 m/s at the intake to the outlet nozzle and by 22 m/s at its outlet.

Fig. 16
figure 16

Change of the additional water intake velocity in the side holes related to their diameter d2

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For the diameter d2 = 0.6 mm, both the speed and direction of the medium flow through the holes were advantageous due to the possibility of the vortices formation in the chamber. That improved the formation of hydrodynamic impulses in the waterjet.

8 Analysis of the vortex chamber shape affecting the water jet velocity

The results of computer simulations of the tool parameters, influencing the speed of the water flow in the individual sections of the head, are presented in Fig. 17. Based on the analysis of these values, it was found that the geometric parameters of the head did not affect the velocity of the flow in the inlet nozzle, because it depends on the nominal pressure and geometrical parameters within the inlet nozzle, which were kept constant in the analyzed case. However, the speed of the water jet in the outlet nozzle depends on the height of the vortex chamber H. At the lowest chamber height (H = 1.4 mm), the highest water jet velocities (120–130 m/s) were obtained, while at the height (H = 6.2 mm), the lowest jet velocities (73–81 m/s) occurred.

Fig. 17
figure 17

Water flow velocity in selected areas with different geometric parameters of the self-excited pulsating tool; the constant analysis parameters: d1 = 0.6 mm, d3 = 0.8 mm, d2 = 0.6 mm, h1 = 1.5 mm

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By analyzing the obtain values (Fig. 16), it was found that the highest speeds (20–21 m/s) of the external medium inlet through the side holes were obtained at the smallest heights of the vortex chamber H = 1.4 mm. On the other hand, the lowest speeds (0.5–3 m/s) were determined for the height H = 4.5–5.2 mm and the chamber diameter D = 6.2–7.4 mm.

The conducted analyses of the geometric head parameters, influencing the speed of the water jet and the formation of hydraulic vortex rings, confirmed that the formation of the jet pulses requires a high velocity of the medium inlet through the side holes of the tool.

9 The parameters and properties of the tool

The model of the self-excited head, selected based on computer simulations, was characterized by the following geometric parameters:

  1. 1.

    Inlet nozzle diameter d1 = 0.6 mm; the value was chosen due to the increase of the external medium flow velocity through the side holes of the vortex chamber.

  2. 2.

    The diameter of the side holes was d2 = 0.6 mm, because the application of a smaller diameter (0.4 mm) resulted in a reduction of the velocity of the external medium intake, while the larger side holes caused an unfavorable shape of the vortex rings.

  3. 3.

    The diameter of the outlet nozzle was d3 = 0.8 mm because its further increase in relation to the diameter of the inlet nozzle d1 = 0.6 mm contributed to the unfavorable distortion of the vortex rings in the chamber.

  4. 4.

    The height of the inlet nozzle was assumed as h1 = 1.5 mm, because it did not have a significant effect on the speed of the water flow in the tested range (the difference in velocity in the side holes for the tested range h1 = 0.5–3.5 mm varied in the range 2 m/s), while the lower height of the inlet nozzle adversely affects the formation of the vortex rings.

After research and analysis of the value of the water jet flow velocity and the shape of the vortex rings, the model of the self-excited pulsating head intended for surface treatment was selected. The characteristics of the geometric parameters of this model are presented in Table 4.

Table 4 Selected geometric parameters of the self-excited head model
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The velocity vector field and pressure values inside the selected model of the self-excited head are shown in Fig. 18.

Fig. 18
figure 18

Distribution of the velocity vectors (a) and pressure values (b) inside chosen configuration of the head model, where d1 = 0.6 mm, d2 = 0.6 mm, d3 = 0.8 mm, h1 = 1.5 mm, D = 6.2 mm, H = 3.2 mm, and SP = 1.4 mm

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The selected design of the self-excited pulse head should have a vortex chamber with a diameter of D = 6.2 mm and a height of H = 3.2 mm with an upper and lower annular surface width SP = 1.4 mm and an angle of attack λ = 120°. It should also have two side holes with a diameter d2 = 0.6 mm opposite each other and perpendicular to the swirl chamber, and an inlet nozzle with a diameter of d1 = 0.6 mm and a height of h1 = 1.5 mm and an outlet nozzle with a diameter d3 = 0.8 mm.

Based on the simulations results, it was concluded that the distribution of the flow velocity vectors and pressures inside the self-excited pulsating head depends on the nominal water pressure and the geometric parameters of the head.

10 Experimental tests

To verify the operational properties of the pulse head working in a water environment, experimental tests were performed. A test rig was built with appropriate technological equipment and specialized measuring apparatus.

The source of water with specified pressure and flow rate was a hydromonitor. A selected construction of self-excited pulsating heads was used to produce a high-pressure impulse waterjet.

The test rig was built to assess the shape of the waterjet, as presented in Fig. 19.

Fig. 19
figure 19

1 Stand for assessing the pulsed jet shape produced in a self-excited head in the water environment. 2 Example of eroded material caused by the pulsating water jet moved with velocity: (a) 18 mm/s, (b) 15 mm/s; in both cases, the pressure was 15 MPa, and the distance from the tool outlet to the processed surface was 15 mm

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Examples of erosion visual effects are presented in Fig. 19, showing differences in the depth of cut for continuous and pulsed jets.

The average eroding Assessment of the effectiveness of high-pressure water jet machining generated using was 4.23 mm (Fig. 19a) and 5.14 mm (Fig. 19b), while for the non-pulsation head, it was 2.18 mm (Fig. 19a) and 1.89 mm (Fig. 19b). The results of this research are in line with previous studies on concrete processing [50, 51] conducted using PWJ, generated by the hydrodynamic method [41, 52, 53].

The frequency of water pulses was measured using a force sensor. The principle of measuring the frequency of the waterjet concerns the placing of dynamometer in each range from the tool outlet, where a high-pressure impulse or continuous jet affects the sensor. As a result, the force effect on the sensor was determined. The measurement stand was presented in the Fig. 20.

Fig. 20
figure 20

Stand for measuring the water pulse frequency generated in the head working in the water

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An analysis of the images of the water jet shapes recorded with an ultrafast TV camera was carried out for nominal pressure of 15, 20, 25, 30, and 35 MPa. Examples of the shapes of the waterjet generated by the head working in the water environment are presented in Fig. 21.

Fig. 21
figure 21figure 21

The images of pulse formation in a water environment at different operating pressures

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The analysis of such images shows that the process of shaping each impulse begins immediately after the water emerges from the nozzle. As the distance from the nozzle increases, the fragmentation of this stream becomes more pronounced. Based on the review of numerous water stream images, typical distances from the nozzle were determined at which hydrodynamic impulses are formed. It was observed that such impulses form at a distance of about 15 mm from the nozzle. Hydrodynamic impulses, longer and particularly generated at higher water pressures, ensure increased erosiveness of the stream in the area 60 mm away from the outlet of the pulsating head. Images captured by an ultra-fast TV camera reveal the elongation of water impulses with an increase in nominal water pressure.

A piezoelectric force sensor was used to record the instantaneous pressure changes. The distributions of hydrodynamic impulses occurring in the pulsating jet were recorded, as shown in Fig. 22.

Fig. 22
figure 22

The values of pulsating water jet, as a result of forces acting on the area of 3.14 mm.2 (the distance from the tool outlet to the processed surface was 15 mm)

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The recorded values of the force of water jet acting on the processed surface, with an area of 3.14 mm2 in 40 ms, are shown in Fig. 22.

The analysis of the force exerted by the pulsating water jet on the machined surface of a 3.14 mm2 object in the shape of a circle with a radius of 1 mm revealed that an increase in the nominal water pressure by 5 MPa increases the average force exerted (Fig. 22). The deviation of the force exerted by the pulsating water jet changes non-monotonically. This is due to the formation of longer impulses at higher operating pressures (35 MPa).

Figure 23 shows the signal of the impulse strength of the pulsating water jet. The presented measurement results (Fig. 23) recorded in successive cycles during 20 ms.

Fig. 23
figure 23

The variable component of impulse strength amplitude of the water jet thrust in relation to the duration of impulses

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The signal recording in cycles enabled the analysis of the frequency and duration of pulses. The results of these analyses are presented in Fig. 24. Additionally, based on the image analysis, a visualization was created illustrating the change in velocity displacement from the forefront of successive (1, 2, 3, 4, 5, 6) water pulses shown in Fig. 24.

Fig. 24
figure 24

A Rate of occurrence of pulses of a given duration. B Visualization of changes in the speed of displacement of the front of subsequent (1, 2, 3, 4, 5, 6) water pulses

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The dominant frequency of the pulses with a period of 0.55 ms was observed at a pressure of 15 MPa (Fig. 24). However, for the remaining operating pressures, this frequency was around 1 ms.

The displacements from the forefront of sequentially recorded impulses via the ultra-fast camera indicate sudden momentary accelerations, confirming the formation of distinct water impulses.

Figure 25 presents the amplitude spectrum of the force impulse for the pulsating water jet generated in a water environment at different pressures: 15 MPa corresponding to a frequency of 1123 Hz, 20 MPa–863 Hz, 25 MPa–928 Hz, 30 MPa–960 Hz, and 35 MPa–879 Hz.

Fig. 25
figure 25

The value of the strength signal in the frequency domain for the pulsating water jet from the newly designed head

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Figure 25 presents the amplitude spectrum of the pulse force for a pulsating waterjet generated in the water environment at various pressures: 15 MPa with a corresponding frequency of 1123 Hz, 20 MPa with a frequency of 863 Hz, 25 MPa with a frequency of 928 Hz, 30 MPa with a frequency of 960 Hz, and 35 MPa with a frequency of 879 Hz.

The analysis of the force signal in graph 25 indicates that at higher operating pressures, there is an increase in intensity and a decrease in the frequency of impulses in the water stream. This effect is a result of the geometry of the swirling chamber and the increased force of water outflow from the head, leading to the generation of a smaller number of impulses.

The study of the hydrodynamic pulse occurrence in the waterjet, carried out with the use of a piezoelectric force sensor for the head operating in the water environment, showed the temporary, periodic increase of the pulsating water pressure.

11 Conclusions

The geometric features of the additionally created side holes and the water inflow initiated in them, resulting in the formation of vortices inside the tool chamber. Then, changes in the water flow pressure were induced, which caused cyclical blocking of the flow. As a result, pulsations of the waterjet were generated.

Based on the conducted research, the conclusions presented below were formulated.

For the geometric model of the self-excited pulse head with two, three, and four side holes perpendicular to the vortex chamber, the inlet of the medium through these holes occurred when the diameter of the inlet nozzle was smaller than the diameter of the outlet nozzle (d1 < d3).

The head models with three and four side holes were characterized by a lower amount of external medium flow (by 8% for 3 side holes and 37% for 4 side holes compared to 2 side holes) and a lower water flow velocity through them. In the case of these models, favorable formation of vortex rings, which determined the generation of impulses in the water stream, was not observed.

Based on the research of the water flow velocity and the shape of the vortex rings, the model of the head with two side holes was selected as the most effective in terms of using the phenomenon of jet pulsation.

The review of numerous images of impulses captured by an ultra-fast camera allowed the determination of typical distances for the formation of hydrodynamic impulses from the outlet nozzle. At lower pressures, this distance was 15 mm. However, at higher pressures, it increased to 60 mm, indicating an increase in the length of impulse formation and erosiveness within this range.

The variability of the deviation in the force exerted by the pulsating water stream is non-monotonic due to the increase in impulse length at higher operating pressures. This was observed at a pressure of 35 MPa, where a slight decrease of 0.068 was noted in the standard deviation.

The analysis revealed that at a pressure of 15 MPa, the dominant frequency of the impulses was 0.55 ms, whereas for other pressures, the oscillations had a longer period, around 1 ms. This indicates significant structural variability in the water stream dependent on the pressure level.

The recorded displacements of successive impulses confirm the existence of sudden momentary accelerations, which is a crucial aspect in the process of shaping these impulses. These observations are key to understanding the dynamics of the water stream concerning variable operational parameters.

At higher operating pressures, there is a noticeable increase in signal intensity accompanied by a simultaneous decrease in the frequency of impulses within the water stream. This effect primarily arises from the geometry of the swirling chamber and the greater force of water outflow from the head, leading to the generation of a smaller number of impulses.

The research is focused on the significant exploitation aspects of self-excited pulsating heads, particularly concerning their design, operational parameters, and their impact on impulse generation.

As a result of head optimization, a pulsating water jet with increased dynamics and erosive properties is obtained. The pulsating water jet can find application in various industries, including the processing of technical materials and the cleaning of their surfaces.

Further work will focus on expanding the scope of applications to include abrasive jet machining. The goal will be to develop self-excited pulsating heads with the introduction of abrasive material streams. Introducing such a modification will allow for the optimization of the erosion process, contributing to the improvement of the efficiency of material removal from processed surfaces. New application areas may include the processing of highly porous composite materials, inorganic materials, as well as applications in biomedical engineering.