Numerical simulation and experimental research of cavitation nozzle based on equation curve

To further investigate and improve the cleaning ability of the cavitation nozzle, this paper proposes a new model that is based on the Helmholtz nozzle and with the quadratic equation curve as the outer contour of the cavitation chamber. First, the numerical simulation of the ﬂ ow ﬁ eld in the nozzle chamber was conducted using FLUENT software to analyze and compare the impact of the curve parameters and Reynolds number on the cleaning effect. Next, the ﬂ ow ﬁ eld was captured by a high-speed camera in order to study the cavitation cycle and evolution process. Then, experiments were performed to compare the cleaning effect of the new nozzle with that of the Helmholtz nozzle. The study results demonstrate that effective cavitation does not occur when the diameter of the cavitation chamber is too large. For the new nozzle, with the increase of the Reynolds number, the degree of cavitation in the chamber ﬁ rst increases and then decreases; the cleaning effect is much better than that of a traditional Helmholtz nozzle under the same conditions; the nozzle has the best cleaning effect for the stand-off distance of 300 mm.

where t is time; u is the velocity; σ is the surface tension coefficient; ρ is the density of the mixture; k is the interface curvature; p is the pressure; τ is the viscous shear stress; n is the unit normal vector to the surface S; δ(x) is the Dirac function.
The volume fraction of each phase satisfies: where α i is the volume fraction; α l is the liquid volume fraction; α g is the vapor volume fraction; ρ l is the liquid density; ρ g is the vapor density; μ is the viscosity of the multiphase flow; μ l is the liquid viscosity; μ g is the vapor viscosity.

Turbulence model
The simulation adopts the RNG k-ε turbulence model since it can better capture small-scale vortices at the end of the shear layer (Zhang & Chen ; Aishvarya et al. ).
The RNG k-ε equation as: where where n is the number of bubbles per unit volume; R B is the bubble diameter; ρ v is the vapor density; R is the mass transfer efficiency.
Ignoring the second-order term, surface tension term, and viscosity term in the Rayleigh-Plesset equation, the following can be obtained: where p B is the surface bubble pressure; p is the partial pressure of non-condensed gas; Then, the mass transfer efficiency expressed by the volume fraction is obtained: The final mass transfer efficiency of bubble evaporation and condensation is expressed as where α nuc is the gas core volume fraction; C e is the evaporation constant term; C c is the condensation constant term.

Finite element solution
In The mesh independence was verified prior to the numerical simulation of the cavitation.The commercial software ICEM was used for meshing the computational area.
The mesh type is the unstructured mesh because of its strong adaptability.Figure 2 shows a mesh schematic.The

Reynolds number analysis
There are many factors that affect cavitation, such as flow rate, viscosity, absolute pressure, surface tension, and the number of gas nuclei (microbubbles, solid particles) in the water, and so on.However, the two most important factors are pressure and flow rate, which usually define the cavitation number σ as follows:   nolds number, which is expressed as:

EXPERIMENT Cavitation cycle
To explore the cleaning performance of the new cavitation nozzle, an experimental system, illustrated in      and the atomization at the outlet are stable; only the turbulent kinetic energy changes slightly.

Experiment
The cavitation jet is expressed by Equation ( 14):

CONCLUSION
The cleaning effect of a new cavitation nozzle was studied through the numerical simulations and experiments and compared with that of the Helmholtz nozzle.Based on the study, the following conclusions are obtained: (1) A new cavitation nozzle with the equation curve energy at the moment of collapse, which strengthens the liquid flow and increases energy efficiency.Therefore, the cavitation jet has been applied in many industrial fields such as rock breaking and cleaning.Yuan et al. () proposed a composite nozzle combining a Venturi tube structure and a Helmholtz resonant cavity and compared it with a traditional Helmholtz nozzle.Duret et al. () studied the impact of the cavitation in turbulent two-phase flow on the hydrodynamics using the Coupled Level Set and Volume of Fluid method (CLSVOF) for interface capturing.Based on the interface capturing method, Pivello et al. () proposed a new computational method for simulating 3-dimentional (3D) two-phase flows, which can greatly reduce the difficulty of handling 3D interfaces.Cai et al. (Cai et al. ) studied the Strouhal number by varying the nozzle structure parameters; the Strouhal numbers showed qualitatively similar behaviors.Their overall trend is rising rapidly with a relatively small angle, declining after the surge, and then increasing slightly.They also inferred that self-resonance would not occur if the straight line is too long.Fang et al.Fang et al. () conducted numerical simulation and experimental research to further examine the performance of the Helmholtz selfoscillating water jet.They compared the morphological characteristics of the erosion surfaces of the cone nozzle and the Helmholtz nozzle and obtained the evolution law of cavitation.Liu et al. (Liu et al. ) evaluated the erosion performance of the 120 impact edge Helmholtz nozzle based on previous studies and analyzed the vibration mechanism through numerical simulations and experiments.Wu et al. (Wu et al. ) captured the details of the cavitation flow under different nozzle structures and working pressures using high-speed photography.By quantitatively evaluating the cavitation jet generated by the Helmholtz nozzle, they discovered that the cavitation intensity produced by a Helmholtz nozzle is significantly higher than that by a traditional conical nozzle, and the geometry of the nozzle has a great influence on the length of the cavitation cloud and the shedding period.Lee et al. () studied the impact of the orifice inlet geometry on the flow rate and cavitation characteristics of high-temperature hydrocarbon liquid jets and correlated the flow characteristics with the Reynolds number and the cavitation number in a macroscopic view.Alehossein et al. (Alehossein & Qin ) simulated the formation and collapse of cavitation bubbles in the jet by solving the Rayleigh-Plesset equation and concluded that cavitation bubbles have a greater impact on the cavitation jet.Li () summarized the previous researches on cavitation jets, introduced the mechanism of cavitation, measurement methods, and factors affecting cavitation, discussed the rationality of commonly used cavitation numbers, and explained the reason that cavitation jets have greater rock breaking ability than ordinary jets.Sun et al. (Sun et al. ) studied the atomization of high-pressure fuel injector nozzles; they used high-pressure fuel injector nozzles to atomize liquid and generate spray in the combustion chamber, finding that the internal flow of the nozzle, especially the cavitation, plays an important role in promoting the breakup of the liquid jet.Akira et al. () used the Eulerian-Lagrangian approach for bubble tracking to further study cavitation in the jet of diesel engine liquid injection, excellently reproduced the morphological characteristics when the cavitation collapsed, and quantitatively predicted the length and thickness of the cavitation area.Kozlova et al. () studied the self-excited oscillation of cavitation caused by the liquid flow in the double resistance pipe.Based on their study, the natural frequency of the resonant cavity mainly depends on the nature of the cavity and the conditions of outflow into the atmosphere, and the generation of different self-excited oscillation modes is directly related to the wave number formed along the length of the cavity.Giussania et al. () introduced the development of a single-fluid solver, which can accurately capture the evolution of the three fluids and the approximate volume of fluid (VOF).Piscaglia et al. () developed a dynamic two-phase VOF solver to study the physical characteristics of the primary jet breaking and flow transient caused by the nozzle geometry during the opening of the high-pressure ejector; they discussed the applicability constraints of the two-phase solver cavitation model in the simulation of flow inside the ejector nozzle.Peng et al. (Peng et al. ) added quartz sand particles to the underwater cavitation jet to enhance its cavitation strength and erosion ability and verified the enhancement through high-speed photography, cavitation noise measurement and erosion tests.The research of Yuan et al. (Yuan & Schnerr ) indicated that since cavitation is highly sensitive to the imposed boundary conditions, the simulations that limit internal problems are qualitatively and quantitatively incorrect, and cannot reveal the principles behind phenomena such as hydraulic flipping and super-cavitation.Ma et al. (Ma et al. ) analyzed the oscillation characteristics of the self-oscillating water jets produced by a series of Helmholtz nozzles with different structures through spectrum analysis, and studied the cavitation effect by analyzing the noise power spectrum.According to their study, the self-oscillating water jets produced by Helmholtz nozzles have highfrequency pressure oscillations and strong cavitation effects compared to those produced by organ tube nozzles or cone nozzles.To evaluate the self-excited oscillation intensity, mass loss and surface morphology of the eroded sample of the water jet, Liu et al. (Liu & Ma ) conducted erosion experiments with inclination angles of α ¼ 0 , 5 , 15 and 30 .Ahmed et al. () proposed and verified a numerical framework based on interface capture to study cavitation and external jet formation; they performed numerical simulations on the development of cavitation and supercavitation and qualitatively compared the liquid and vapor structures obtained in the experiments and simulations.Belkacem et al. (Belkacem & Huang ) studied the impact of different aspect ratios on nozzle jet atomization characteristics, using high-speed cameras to record spray patterns under various conditions and discussed the influence of the aspect ratio on cavitation and spray structure.Li et al. (Li et al. ) used mathematical software to solve the original Rayleigh-Plesset equations with different dynamic viscosities and obtained the approximate expression of bubble radius over time through curve fitting.Helmholtz and organ pipe nozzles are widely applied to clean oil tanks, ships, etc.Their cleaning effects (including cleaning area and cleaning speed) directly impact the labor hours, work intensity, and operational expense (Knapprt et al. ).Research found that the cleaning effect is associated with the standoff distance and the degree of cavitation inside the nozzle, and the cavitation plays a key role in cleaning object surfaces.The occurrence of cavitation mainly depends on the nozzle shape and structural parameters (Shen & Zeng ; Li et al. ).Many studies have been conducted to obtain the influence of structural parameters on the cavitation jet using methods combining theory and experiment.However, few studies have been performed to enhance the cavitation effect by improving the shape of the traditional nozzle.This paper simulates the traditional Helmholtz nozzle using FLUENT software.By varying the geometry of the nozzle chamber, the impacts of nozzle structural parameters on the cavitation effect were analyzed.A holistic experimental system was established to verify the impacts and dynamically capture the cavitation cycle of the nozzle chamber using a high-speed camera.MATHEMATICAL MODEL Governing equation Due to the cavitation, the model to calculate the flow field in the nozzle often employs the multi-phase cavitation model.The continuity equation and the momentum equation of the mixed phase are established based on multiphase flow calculation model: G k is the turbulent kinetic energy produced by the velocity gradient; α k and α ε are turbulent Prandtl numbers; C 1ε and C 2ε are constants.Cavitation model Cavitation jet involves mass transfer with phase change.The key consideration in establishing the cavitation model is mass transfer.The establishment of cavitation mass transfer is not a separate process, in which the cavitation model needs to be used as part of the balance equation to describe the generation and fragmentation of steam.In this study, the Zwart-Gerber-Belamri cavitation model (Zwart et al. ) was selected (see reference 30 for the detailed derivation process) to simulate multiphase flow or material transport in multiphase flow.The model assumes that the bubbles in the liquid have the same initial size, and the mass transfer efficiency is calculated based on the number of bubbles per unit volume.
the simulations of the flow field inside the nozzle chamber, the inlet pressure and outlet pressure are set to 1,101,325 Pa and 101,325 Pa, respectively.The first phase is water at room temperature, and the second is water vapor at the saturated pressure of 3,540 Pa.The computational process adopts the transient model; the discrete grid uses the first-order upwind scheme; the numerical algorithm is based on PISO (Pressure Implicit with Splitting of Operators).NUMERICAL SIMULATION Physical modelReferring to the existing cavitation nozzle structure and size, a new cavitation nozzle with the contour of the cavitation chamber as the one-dimensional quadratic equation y ¼ ax 2 þ bx þ c was proposed.After comprehensive consideration and orthogonal parameterization selection, the optimal parameter values were determined.Figure1shows the two-dimensional schematic diagram of the nozzle.d 0 is the water inlet; d 1 is the water outlet; l 0 and l 1 are the upper and lower flow channels, respectively; D is the diameter of the cavitation chamber.The specific parameters are listed in Table1, where the value of D changes with the equation curve parameters shown in Table2.Numerical simulations were performed for different equation curve parameters; the best equation curve parameters were obtained to determine the final contour shape of the nozzle cavity.
average velocity at the nozzle outlet was selected as the monitoring object.The average velocity changes at the nozzle outlet are presented in Table 3.The simulation of free flow was conducted during the mesh independence test.The inlet condition is set as the velocity inlet, and the water velocity is 20 m/s.The outlet condition is free flow.When the number of meshes increases from 18,251 to 431,859, the average speed stays roughly constant, and the maximum error is less than 0.16%, leading to negligible impact on the simulation results.Therefore, the number of meshes was set as 18,251 in this paper considering the computing load and other factors.Due to the large number of equation parameters and the limited space, Figure 3 only presents a part of selected results.As shown in the figure, when the chamber diameter D is too large, no obvious cavitation occurs, and the degree of cavitation in the chamber is low.When a ¼ À1=18 and b ¼ 2:0, the cavitation generated by the chamber profile enclosed by the equation curve under this parameter is obvious, and the degree of cavitation is high.Therefore, the equation curve y ¼ À1=18x 2 þ 2x þ 25 was selected as the contour curve outside the chamber.

Figure 3 |
Figure 3 | Contours of liquid phase volume fraction in different structural parameters.

Figure 5 |
Figure 5 | Volume fraction of liquid phase in cavity diameter.

Figure 10 |
Figure 10 | The different moments in the turbulent kinetic energy period.

Figure 11
Figure 11 presents the cleaning effect.The figure demonstrates that as the cleaning target distance increases from d ¼ 100mm to d ¼ 600mm, the cleaning area gradually increases, but the depth of the etch pit gradually decreases.The water jet plays the key role in the cleaning process when the standoff distance is too small, because cavitation bubbles are not fully formed and collapse before reaching the target surface.As the nozzle-to-target distance increases, cavitation bubbles are fully developed and able to collapse on the target surface; therefore, the cleaning is mainly performed by the micro-jet and water jet generated by the collapse of cavitation bubbles.However, when the standoff distance becomes too large, the cavitation bubbles also collapse before reaching the target although they have sufficient time to be completely formed; the cleaning effect cannot be remarkably improved in this condition.As shown in Figure 11, when the target distance is 300 mm, the nozzle has the best cleaning effect.Figures 12 and 13 compare the cleaning effect of the new nozzle with that of the Helmholtz nozzle under the same conditions.It can be seen from the figures that both the cleaning area and etch pits of the new cavitation nozzle are significantly increased.Meanwhile, Figure 11 | Cleaning effect of target.

Figure 12 |
Figure 12 | Cleaning effect of two kinds of nozzle target surface.

Table 1 |
Structural parameters of nozzle

Table 3 |
Results of the mesh independence