Research on Multiphase Flow and Nozzle Wear in a High-Pressure Abrasive Water Jet Cutting Head

: Research on the mixing process of gas–liquid–solid multiphase ﬂow in a high-pressure abrasive water jet (HP-AWJ) is of great signiﬁcance in improving the performance of water jet cutting. In this paper, the Euler method-VOF model, a computational ﬂuid dynamics (CFD) simulation method, is used to solve the multiphase ﬂow of air–water in an abrasive water jet (AWJ). The Euler–Lagrange method is further used to study the multiphase ﬂow of abrasive particles. The method considers the shape factor of the particle, uses the Rosin–Rammler function, and deﬁnes an effective model for the particle-to-wall wear model. By solving the velocity of the continuous phase and particulate phase in the AWJ cutting head, the problem of nozzle wear caused by particles is studied. Finally, the exit velocity of the AWJ and particle wear are investigated by varying the model’s parameters. The results show that the double abrasive tube model effectively improved the problem of one-sided wear inside the nozzle, and the tangential velocity of the air affected the acceleration process of the abrasive inside the nozzle, with smaller abrasive velocity resulting in less wear on the model. Simultaneously, the effects of the abrasive inlet tube and mixing chamber size on abrasive exit velocity and nozzle wear are analyzed. The results obtained provide valuable guidance for addressing the multiphase ﬂow mixing issues in the AWJ, improving the abrasive acceleration process and extending the nozzle’s lifetime.


Introduction
Water jet cutting technology (high-pressure water erosion) was first discovered in the mid-19th century for rock cutting and mining applications [1], and has the advantages of no thermal deformation and a wide machining range [2,3]. Many years later, around the 1980s [4], to improve the material removal rate and processing efficiency of the water jet, abrasives as a medium were introduced into the water jet, in which abrasives such as alumina, silicon carbide, or garnet were mixed in the pure water jet to improve process efficiency [5,6]. In addition, an abrasive water jet is considered a green processing technology and shows promising potential in various other industries [7][8][9][10], including milling, micro-machining, car manufacturing and maintenance, joint replacement surgery, etc. This manufacturing technology operates on the principle of mechanical erosion, whereby the abrasives are propelled by high-pressure fine jets to ultimately cut through the target material [11,12]. When the water pressure increases, the kinetic energy transferred to the abrasives increases, leading to high-speed particle impact and consequent nozzle wear. An increase in the nozzle outlet diameter will lead to a decrease in cutting speed and precision, further affecting the cost of AWJ machining [13]. Therefore, the influence of jet parameters on nozzle wear and the modeling of nozzle wear has always been the key focus Therefore, the influence of jet parameters on nozzle wear and the modeling of nozzle wear has always been the key focus of research. The AWJ cutting head operates by a three-phase mixed flow of water, air, and abrasives [14]. The working principle is as follows: when high-pressure water enters the mixing chamber through the orifice, a negative pressure relative to atmospheric pressure is created inside the nozzle [15], the air and abrasive are sucked into the mixing chamber by the Venturi effect [16], and the flow rate of the abrasive is provided by the abrasive application device, and the number of abrasives can be adjusted during the mixing process. Finally, the high-pressure water, air, and abrasives in the mixing tube enter the focusing tube for further acceleration, mixing, and outflow. Figure 1 shows a schematic diagram of the AWJ cutting system (a) and the nozzle of the cutting head (b). The nozzle, as an essential component of the AWJ system, plays a crucial role in improving processing quality and reducing energy consumption [17]. In the past few decades, scholars have conducted numerous studies on the basic structure and characteristics of the jet at the outlet of the AWJ. However, nozzle wear is inevitable, and even small changes in the nozzle's geometry can significantly affect cutting performance. Although the development of new wear-resistant materials has effectively increased the lifetime of AWJ nozzles, they are not up to the demands of high-hardness abrasive machining [18]. In addition to the material properties, the structure of the nozzle is also an important factor affecting the nozzle's lifetime. Various experimental studies have been conducted to optimize the nozzle's structure in order to improve the wear resistance of the nozzle [19,20]. The small characteristic size of the AWJ nozzle, high velocity, and the fact that the erosion of the abrasive is heterogeneous make the experimental study of water jet and particle wear problems inside the nozzle rather difficult [6,17]. With the development of computer software and high-performance hardware, the use of a numerical simulation strategy to establish the AWJ model and study the multiphase flow characteristics inside the nozzle plays a crucial role in solving the above problems. The nozzle, as an essential component of the AWJ system, plays a crucial role in improving processing quality and reducing energy consumption [17]. In the past few decades, scholars have conducted numerous studies on the basic structure and characteristics of the jet at the outlet of the AWJ. However, nozzle wear is inevitable, and even small changes in the nozzle's geometry can significantly affect cutting performance. Although the development of new wear-resistant materials has effectively increased the lifetime of AWJ nozzles, they are not up to the demands of high-hardness abrasive machining [18]. In addition to the material properties, the structure of the nozzle is also an important factor affecting the nozzle's lifetime. Various experimental studies have been conducted to optimize the nozzle's structure in order to improve the wear resistance of the nozzle [19,20]. The small characteristic size of the AWJ nozzle, high velocity, and the fact that the erosion of the abrasive is heterogeneous make the experimental study of water jet and particle wear problems inside the nozzle rather difficult [6,17]. With the development of computer software and high-performance hardware, the use of a numerical simulation strategy to establish the AWJ model and study the multiphase flow characteristics inside the nozzle plays a crucial role in solving the above problems.
Liu et al. [21,22] used a two-dimensional axisymmetric model to study the three-phase flow at the nozzle outlet by CFD methods. Since the mixing chamber and focusing tube were missing in the model, the method can quickly calculate the velocity of the continuous phase at the outlet and the particles inside the nozzle. Prisco et al. [23] simulated the generation and discharge process of air-water mixed flow inside the AWJ nozzle. The dynamic flow characteristics inside the water jet nozzle and fundamental knowledge about the dynamic characteristics of high-speed flow were obtained under steady-state, turbulent, two-phase flow conditions. However, the methods were based on the Euler-Euler approach and did not consider the solid particle phase. Yang et al. [24] conducted a numerical simulation to investigate the multiphase flow of the AWJ inside the cutting head under pressure conditions above 300 MPa and discussed the effects of abrasive physical properties, orifice size, and operating pressure on particle trajectories. Mostofa et al. [25] studied the flow characteristics of the AWJ on the inner surface of the nozzle and the effect of the water jet on the wear rate of the nozzle surface. By varying the mass flow rate of the abrasive and the shape factor of the particles, the velocities of the water jet flow, air and abrasive, and the erosion inside the focusing tube were studied. Hou et al. [2] used the CFD simulation method of solid-liquid two-phase flow in the Eulerian method to numerically simulate the velocity field inside and outside the AWJ nozzle. The results showed that the jet velocity increased abruptly at the corner inside the nozzle and then remained stable inside the focusing tube. Narayanan et al. [26] established a mathematical model of the three-phase flow in the AWJ cutting head and improved the model by considering the effects of particle size distribution and breakage on energy transfer. Through extensive experiments and simulation verification, the average particle velocity of the cross-section at the focusing tube outlet was verified. Basha et al. [27] investigated the flow characteristics of three-phase flow inside the AWJ nozzle using CFD technology and confirmed that CFD can be used to explain the relationship between nozzle parameters and AWJ performance. Long et al. [17] studied the water-air flow and particle motion inside the AWJ nozzle under a condition of 450 MPa based on three-dimensional Euler-Lagrange numerical simulations. It was found that a longer focusing tube can reduce the circumferential motion of particles, and the reduction in the particle shape factor can improve the particle acceleration process. In addition, the influence of particle density and diameter on the velocity was analyzed. Qiang et al. [15] proposed a comprehensive numerical model for three-phase flow in an abrasive cutting head and investigated the particle acceleration and trajectory in the jet field of an AWJ cutting head. By changing the particle inlet position, particle inlet angle, and convergence angle of the focusing tube in the model, the particle kinetic energy and the nozzle wear were observed. Pozzetti et al. [28] proposed a numerical method to study the erosion phenomenon inside the focusing tube of the AWJ to capture the erosion effect of the particle flow due to the cumulative impact phenomenon in the focusing tube. The CFD-DEM multi-scale algorithm was combined with the erosion model proposed in the literature, which provided insights into the nozzle wear phenomenon. However, the method used in this study had lower fluid velocities and only analyzed erosion in a specific model, which lacked generality. Riha et al. [29] studied the multiphase flow of water, air, and abrasives inside and outside the nozzle through a cutting head model with accurate geometric shapes. Detailed experimental and numerical results were provided as a reference for subsequent research. However, the study did not include the issue of particle wear on the nozzle.
To date, there have been numerous studies on multiphase flow and nozzle wear in the AWJ cutting head. They show that numerical simulations are an important prerequisite for obtaining good results, but all the above numerical simulations have been oversimplified. Solid-liquid or air-water flow models were used, or incomplete models with only a focusing tube or a 2D model were used. The HP-AWJ cutting head is a critical component of the AWJ system and is responsible for mixing and accelerating the abrasive in the mixing chamber, which is typical for multiphase flow problems. The cutting head nozzle is the first target for erosion wear, which significantly reduces the lifetime of the nozzle. Erosive wear has been difficult and expensive to capture through purely experimental studies. In short, a better understanding of the multiphase flow issues within the AWJ nozzle plays a critical role in ensuring the optimal cutting performance of the machine. Meeting all current and future challenges will require a major effort and an effective combination of scientific and engineering approaches. In this paper, ANSYS Fluent 2020 R2 is used to simulate the three-dimensional model of the air-water-abrasive three-phase flow inside the HP-AWJ cutting head under operating pressures of 200-450 MPa. The roundness of particles and particle size distribution are considered, and an effective erosion model is proposed. The multiphase flow model is solved by the two-way coupling method, which can accurately predict the velocity of solid particles and erosion/accretion on the nozzle wall. Due to the above two factors affecting the cutting performance and lifetime of the AWJ nozzles, the single abrasive tube model and the double abrasive tube model are adopted. Furthermore, various parameters such as the inlet/outlet size, the focusing tube length, the abrasive tube inlet size, the mixing chamber diameter, and the length are varied to study the effect of the nozzle's structure on particle erosion. This study attempts to reveal the effects of the AWJ cutting head model on particle velocity and nozzle wear.

Methods
Generally, AWJ cutting heads have been divided into single-or double-abrasive tube models. The optimum ratio of the inside diameter between the orifice and focus tube is approximately 1:3. Common orifice sizes are 0.28 mm and 0.33 mm, corresponding to focus tube inner diameters of 0.76 mm and 1.02 mm, respectively, which can affect cutting performance and kerf size. Standard focus tube lengths are typically 76.2 mm or 101.6 mm. The formation of the AWJ involves the interaction between air, high-pressure water, and solid abrasives, which is a typical problem of gas-liquid-solid multiphase flow. For this phenomenon, the air and high-pressure water jet can be treated as a continuous phase in the multiphase flow simulations, and the Volume of Fluid (VOF) model based on the Euler method can be used as a widely applicable method to solve this problem [17]. Since the volume fraction of solid abrasive particles is around 10%, the Euler-Lagrange method is further used to solve the particle phase in the AWJ [30]. In this paper, the VOF model based on the Euler method is first used to solve the continuous phases (air and water) in the AWJ cutting head. Then, the Discrete Phase Model (DPM) based on the Euler-Lagrange method is used for the two-way coupling between the granular phase and the continuous phase. This method provides a reasonable solution to the complex multiphase flow in the AWJ.

The Continuous Phase
For the special cases of the HP-AWJ involving high turbulence and supersonic flow, the VOF model within the Eulerian multiphase flow is used to calculate the air-water flow. The VOF model solves a single set of momentum equations for immiscible fluids throughout the computational domain while tracking the volume fraction of each fluid to simulate multiphase flow. The continuity equation for the volume fraction of the q th phase can be expressed as: where ρ q is the physical density of the q th phase, v q is the velocity of the q th phase, . m pq is the mass transfer from the q th phase to the p th phase, . m qp is the opposite case of mass transfer, and S α q is the source term with a default value of 0, which can also be specified as a constant or a custom mass source term.
In the Euler model, the volume fraction of the phase is calculated based on the constraint that the sum of the volume fractions of all phases equals 1, and the volume fraction equation is not solved. Multiphase flow is described as an interpenetrating continuum that incorporates the concept of phase volume fractions, denoted as α q . The volume V q of the q th phase is defined as: where the above equation is satisfied under the condition that the sum of all volume fractions is equal to 1.
When a given volume of water is pressurized to create a high-velocity water jet of water, Bernoulli's principle gives: where h 1 = h 2 , P at << P, v 0 >> v tube is the approximate velocity of the exiting water jet, which is obtained as: In practice: where P at and P are the atmospheric pressure and the inlet pressure of the high-pressure water, in units of MPa, ρ w is the density of water (Kg/m 3 ), v 0 and v tube are the effective velocity of the high-pressure water jet at the orifice outlet and the flow in the focus tube (m/s), v w is the effective velocity of the water jet (m/s), h 1 and h 2 are the heights of the fluid position in (mm), and ψ is the compressibility factor, which is the momentum loss due to wall friction, fluid flow turbulence, and compressibility of water, given as [31]: where L = 300 MPa and n = 0.1368 at 25 • C. The mass flow rate of air . m a and the air velocity v a are expressed as: where Q is the flow rate of the air, D a is the diameter of the air inlet tube, and ρ a is the density of the air.

The Disperse Phase
The Lagrange method is used to solve the problem of particles, which focuses on studying the variation of the motion parameters of a particle as it moves from one point to another in a flow field. On the other hand, the Euler method focuses on studying the variation of the mass point in a flow field over time. Obviously, the Lagrange method is more suitable for describing abrasive motion, while the Euler method is more suitable for describing fluid motion. The DPM model is based on these two methods to simulate the fluid and the particle, using the Euler method to describe the fluid motion and the Lagrange method to describe the particle motion. The force analysis of the abrasive gives equations in the Euler coordinate system (taking the y-direction in a Cartesian coordinate system as an example): where f y is the additional acceleration acting on the particles, which includes forces such as Magnus force, Saffman force, etc., f D (v − v p ) is the drag force experienced by the particles per unit mass, and f D is the drag force function, which is given by: where v is the velocity of the continuous phase, v p is the velocity of the particles, ρ p is the density of the particles, d p is the diameter of the particles, C D is the drag coefficient [32], Re p is the relative Reynolds number of the particles, and µ is the viscosity coefficient of the continuous phase; g y (ρ p − ρ)/ρ p is the combined force of gravity and buoyancy acting on the particles per unit mass, where g y is the gravitational force along the y-axis in the coordinate system and ρ is the density of the continuous phase.
Due to the relatively large density and size of the abrasive compared to water and air, the value of f y is ignored in the simulation. In addition, the particles in the cutting head are assumed to be inert, and the forces associated with heat or mass transfer are not considered. It is undeniable that the particles are sucked into the mixing chamber by the action of the high-pressure and high-velocity jet at the abrasive inlet tube. A random walk model is used to predict the turbulent diffusion of the particles since the volume fraction of the particles is around 10% [30,33], which is influenced by the continuous phase. Furthermore, a two-way coupling method is used to calculate the interaction between the particles and the continuous phase, taking into account the coupling effect between the water, air, and the particle phase, while ignoring the particle-particle interactions. These assumptions have also been adopted in some previous studies [2,16,34,35].
In AWJ machining, particles are released at the entrance of the abrasive tube and enter the mixing chamber, where they are accelerated by the continuous phase. For the obtained mass flow rate of water, the mass flow rate of abrasives, the mass flow rate of air m a , and the corresponding velocity v w , v a , v p of each phase, the conservation of kinetic energy (conservation of kinetic energy before and after three-phase flow mixing) [31,36] is used, considering that the mass flow rate of air and its velocity are negligible. The mass flow rate of the abrasive is negligible compared to the continuous phase; the abrasive enters the mixing chamber with an initial velocity of zero and is further accelerated by the focusing tube, and only a portion of the kinetic energy of the water is transferred to the particles during this mixing process [37]. The final abrasive and water velocities are nearly equal, giving the particle velocity as:

Distribution and Erosion of Particle Phase
The particles are not perfectly spherical in shape, which is very variable in reality. Two shape factors are considered in this study: the sphericity (S P ) and roundness (S R ) of the abrasive as proposed by Wadell [38,39], which are parameterized to control the shape factors of the particles, and their respective formulas are given below, ranging from 0 to 1: where b p is the width of the particle under the orthographic projection, l p is the length of the particle under the orthographic projection, d circle is the diameter of the circumscribed circle of the particle under the orthographic projection, r corner is the corner radius of the particle, d p is the geometric mean particle diameter, and N corner is the number of corners on a particle.
Furthermore, the size of the abrasive is non-uniformly distributed and typically follows a normal distribution with respect to particle size. The Rosin-Rammler distribution function can be used to describe the distribution of abrasive particle diameters [39].
where R d is the volume fraction of particles with a diameter larger than d, where d is the particle diameter, d is the mean particle diameter, and n is the distribution index. A larger value of n indicates a more uniform distribution of particle sizes. The lifetime of a nozzle depends on its geometric shape, and a reasonable geometric shape can effectively reduce abrasive erosion [40,41]. The DPM erosion model can effectively predict the nozzle wall wear: where R erosion is the erosion rate of the particles on the wall (Kg/(m 2 ·s)), C(d p ) is a function of the particle diameter, and f (α) is a function of the collision angle α, where α is the angle between the particle trajectory and the normal vector of the wall surface element, b(u) is a function of the relative velocity u, and A face is the surface area of the collided wall unit. In addition, the function for α can be expressed as: In fact, the definitions of C(d p ), f (α), and b(u) should reflect the influence of particle material and wall material properties. With C(d p ) = 1.559e −6 B −0.59 F φ and b(u) = 1.73, the erosion rate model that can reflect the relationship between the particle model and the collision angle during the erosion process is: (16) where B is the Brinell hardness and F φ is the shape factor of the particles.

Geometry and Mesh
In order to meet practical applications and commercial requirements, the geometry shape of the cutting head has been simplified in the design. Figure 2 shows a 2D diagram of the conventional physical model of the double abrasive tube nozzle. The parameters of the geometric model used in this paper are shown in Table 1, and the geometric parameters are designed based on actual conditions. The double abrasive tube nozzle model has two sand inlet tubes, which are symmetrically distributed on the left and right sides of the mixing chamber, while the single abrasive tube nozzle model has only one sand inlet tube. The origin of the coordinate system in the model is set at the center of the orifice outlet, and L is defined as the distance from the orifice flowing into the mixing chamber to the outlet of the focusing tube. A 3D model of the nozzle model is created for computational fluid dynamics (CFD) using Solidworks. Meshing is performed using ICEM CFD, as shown in Figure 3. Hexahedral meshing is used as it has been shown to give better results than tetrahedral meshing [42]. the mixing chamber, while the single abrasive tube nozzle model has only one sand inlet tube. The origin of the coordinate system in the model is set at the center of the orifice outlet, and L is defined as the distance from the orifice flowing into the mixing chamber to the outlet of the focusing tube. A 3D model of the nozzle model is created for computational fluid dynamics (CFD) using Solidworks. Meshing is performed using ICEM CFD, as shown in Figure 3. Hexahedral meshing is used as it has been shown to give better results than tetrahedral meshing [42].

Boundary Conditions and Other Parameters
The mesh model is imported into the ANSYS CFD solver Fluent for further analysis and solution of the multiphase flow problem involving gas, liquid, and solid phases in the nozzle. The boundary conditions for the model included three inlets and one outlet. The inlets consisted of two mixed inlets for air and abrasives and one high-pressure water inlet. The bottom of the model served as the outlet for the three-phase flow. All

Boundary Conditions and Other Parameters
The mesh model is imported into the ANSYS CFD solver Fluent for further analysis and solution of the multiphase flow problem involving gas, liquid, and solid phases in the nozzle. The boundary conditions for the model included three inlets and one outlet. The inlets consisted of two mixed inlets for air and abrasives and one high-pressure water inlet. The bottom of the model served as the outlet for the three-phase flow. All inlets are set as pressure inlet boundary conditions, while the outlet is set as a pressure outlet boundary condition. Double precision is enabled to improve the accuracy of the simulation data. The gravitational acceleration is set to 9.81 m/s 2 , and the atmospheric pressure is set to 101,325 Pa. High-performance computing resources are used to accelerate computational efficiency.
Assuming that the fluids are all incompressible viscous substances since the Reynolds number of the fluid flow is greater than the critical value, the realizable k-ε model in the high Reynolds number turbulence model from the K-epsilon is used as the control equation [43]. The general form of the transport equations for turbulence kinetic energy k and turbulence dissipation rate ε could be expressed as [44]:

•
The turbulent kinetic energy equation: • The turbulent dissipation rate equation: where σ k = 1.0, σ ε = 1.2, C 2 = 1.9, x i and x j denote the distances in the i and j directions, v is the velocity vector (m/s), P is the fluid pressure (Pa), µ t is the turbulent viscosity, and ρ is the fluid density (kg/m 3 ). For the VOF model, liquid water with a density greater than air is set as the primary phase. At the water inlet, which represents the nozzle orifice, the volume fraction of water is set to 1 and the volume fraction of air is set to 0 in the boundary conditions. At the air and abrasive inlet, the boundary pressure is set to the atmospheric pressure, and air and abrasive particles are naturally entrained into the mixing chamber by the Venturi effect. The volume fraction of the air is set to 1 in the boundary conditions. After mixing, the fluid flows out of the outlet at a pressure set to the atmospheric pressure. All wall boundaries are set to the no-slip velocity condition, while the other settings are kept at default. In the solution process for the continuous phase, the first-order scheme and second-order upwind schemes are used for the mass conservation equation and the momentum equation, and the turbulence closure equation is compared to the first-order scheme to save computational time and obtain more reliable results. The pressure-velocity coupling follows the SIMPLE scheme, with the PRESTO! algorithm used for pressure approximation. The volume fraction is calculated using the QUICK scheme. The convergence criteria for all equations are set to 10 −5 , and relaxation factors are adjusted accordingly to ensure rapid convergence of the model. For the DPM model, after obtaining the two-phase flow, further tracking and investigation are conducted using an erosion/acceleration model for 50,000 iterations. All particles are injected with no initial velocity and are perpendicular to the plane of the abrasive inlet, entering the mixing chamber. The outlet boundary for the discrete phase is set to "escape", while the remaining walls are set to "reflect", meaning that once the particles reach the outlet boundary, their trajectory tracking calculation stops, an elastic collision occurs when the particle reaches the wall boundary, and the particle continues to move, but the trajectory has changed. To achieve higher accuracy in the velocity distribution calculation, a two-way coupling method is used to calculate the interaction between particles and the continuous phase, as the motion of abrasive particles will affect the flow field shape of the water and air mixture [29]. Based on the above equations, initial conditions, and boundary conditions, a closed system was formed, which can compute the explicit solution of the steady-state multiphase mixture in the solved region. Figure 4 shows the streamlined plot and particle trajectories for the three-phase flow coupling at 400 MPa.  In addition, the following assumptions were made when modeling the AWJ multiphase flow in this study: 1. Water is in a continuous phase and is incompressible. 2. The sphericity and roundness of the abrasive are both 0.8 and the average size of the abrasive is 0.135 mm, with a diameter range of 0.1-0.16 mm. The diffusion coefficient is 5.9 and the abrasive mass flow rate is 0.007 g/s, with a density of 2600 kg/m 3 . 3. There is no heat exchange between the particles and the water in the flow, and the temperature remains constant.

Mesh Independence Verification
The quality and quantity of the mesh determines the accuracy and computational cost of numerical simulations. High-quality meshes are a prerequisite for accurate simulation calculations. In this study, a hexahedral structured mesh is used to ensure mesh quality. In theory, higher mesh density in numerical simulations results in more accurate calculations. As the number of grids approaches infinity, the calculated results will converge to the true solution. Therefore, the number of grids will affect the accuracy of the calculation. However, increasing the number of grids also increases the computational cost. Therefore, the calculation accuracy and time cost should be taken into consideration when determining the number of grids [45].
The research model used in this study is a double abrasive tube model, with the In addition, the following assumptions were made when modeling the AWJ multiphase flow in this study:

1.
Water is in a continuous phase and is incompressible.

2.
The sphericity and roundness of the abrasive are both 0.8 and the average size of the abrasive is 0.135 mm, with a diameter range of 0.1-0.16 mm. The diffusion coefficient is 5.9 and the abrasive mass flow rate is 0.007 g/s, with a density of 2600 kg/m 3 .

3.
There is no heat exchange between the particles and the water in the flow, and the temperature remains constant.

Mesh Independence Verification
The quality and quantity of the mesh determines the accuracy and computational cost of numerical simulations. High-quality meshes are a prerequisite for accurate simulation calculations. In this study, a hexahedral structured mesh is used to ensure mesh quality. In theory, higher mesh density in numerical simulations results in more accurate calculations. As the number of grids approaches infinity, the calculated results will converge to the true solution. Therefore, the number of grids will affect the accuracy of the calculation.
However, increasing the number of grids also increases the computational cost. Therefore, the calculation accuracy and time cost should be taken into consideration when determining the number of grids [45].
The research model used in this study is a double abrasive tube model, with the following parameter settings: θ = 45 • , α = 24 • , d 0 = 0.33 mm, d 1 = 4.5 mm, d 2 = 1.02 mm, D a = 3 mm, l 1 = 7 mm, and l 2 = 76.2 mm. Detailed grid partitioning strategies are described in Section 3.1. Numerical simulations of the air-water flow inside the nozzle are performed with different numbers of grids, including 0.7 million, 1 million, 1.5 million, and 1.8 million grids, using the simulation parameters listed in Table 2 as the CFD settings for the nozzle model. The velocity distribution of the two-phase flow at the nozzle outlet is compared within a simulation time of 10 s, and all the results meet the desired accuracy requirements. In order to achieve stable numerical calculations with high accuracy while saving computational resources, the final choice for the grid quantity of the computational model is 1 million.

Results and Discussion
In the trajectory diagram of the PH-AWJ multiphase flow (Figure 4), when the highpressure water passes the orifice, a high-speed water jet is formed, and the orifice converts the potential energy of the water pressure into kinetic energy. The calculation results of Equation (5) correspond to the velocity of the high-pressure water at the orifice. As the pressure increases, the velocity of the high-speed water jet increases accordingly under the influence of the pressure gradient, and the velocity at the nozzle outlet will be slightly lower than at the inlet. However, for all the nozzle models, the flow shape of the high-velocity water is always the same as the inlet pressure increases. When the high-pressure water jet exits the water nozzle under 400 MPa, its velocity can reach up to 900 m/s. At the same time, when the high-speed fluid enters the mixing chamber, a negative pressure is created inside the chamber, which creates a swirling effect (Venturi effect), entraining air and abrasives into the mixing chamber for preliminary mixing and contact with the water jet. The air forms a vortex motion around the high-speed water jet, generating a tangential velocity. During the suction process, the particles are accelerated along the length direction of the abrasive tube into the interior of the mixing chamber by the combined effect of gravity and the Venturi effect. Finally, the abrasive enters the focusing tube from the mixing chamber and undergoes secondary acceleration.

The Effect of Model Infrastructure on AWJ Performance
Through simulation calculations, the internal movement mode and process of the abrasive can be better analyzed. Based on the results, structural optimization designs and parameter designs can be proposed to ensure the acceleration performance of the abrasive, minimize collisions and abrasive damage inside the nozzle, reduce unnecessary energy consumption, and finally increase the output energy of the AWJ to improve the processing performance. At the same time, the results of the numerical simulation can effectively show the flow behavior of high-pressure water, air, and abrasives, as well as the movement of abrasives in the mixing chamber and focusing tube.  Table 3 analyzes the cutting head outlet velocity and maximum wear rate at different operating pressures for Models 2 and 3. The results show that the nozzle outlet velocity and the maximum wear rate increase with increasing pressure. However, at the same operating pressure, the outlet velocity and maximum wear rate of Model 3 are higher than Model 2. This is because Model 2 has an additional abrasive compared to Model 3, which causes a larger volume of air to be entrained into the chamber due to the Venturi effect. As a result, the exit velocity of Model 2 is reduced by approximately 0.4% under the same conditions. In addition, the single abrasive tube in Model 3 causes one-sided wear on the conical section of the focusing tube, resulting in a higher maximum wear rate of Model 3 compared to Model 2 under different pressure conditions.  Figure 5 analyzes the effects of water jet velocity (a), particle velocity (b), and nozzle wear (c) for the four models at an operating pressure of 400 MPa. Due to the convergentdivergent structure of the abrasive nozzle, where potential energy has been converted into kinetic energy, the jet velocity reaches a peak at a certain point near the orifice outlet and then decreases slightly inside the nozzle. For different models at the same pressure, the peak jet velocity remains relatively constant. However, the jet is inevitably mixed with air, causing the velocity ratio of particles to water to be less than 1. Due to the higher amount of air mixed into the jet in Model 2, the outlet velocity of the water jet is lower than in Model 3. As the length of the focusing tube increases, the high-pressure water is fully accelerated while causing changes in the pressure drop within the fluid domain, resulting in a smoother change in jet velocity for Model 1. The kinetic energy of the high-pressure water jet is significant, and the core region of the jet can be maintained for a long distance within the nozzle. The reasonable length of the focusing tube will fully accelerate the abrasives in the tube, resulting in higher particle kinetic energy. However, in the double abrasive tube model, abrasives entering the mixing chamber will cause particle collisions, further increasing the wear rate in this area. Meanwhile, the double abrasive tube model reduces the occurrence of one-sided wear, which reduces energy loss due to particle wall collisions in the mixing chamber and allows better acceleration of particle velocity in Model 2. When the diameter ratio (d 0 /d 1 ) decreases, the water jet may form an atomization zone in the mixing chamber and focusing tube prematurely. In Model 4, the changes in the nozzle structure have reduced the outlet jet velocity, and the smaller nozzle size d 2 makes it difficult for the abrasives to enter the smaller focusing tube. When influenced by the tangential velocity of the air in the mixing chamber, most abrasives exhibit a circular motion in the mixing chamber, creating a circumferential velocity in the particle stream. The above phenomenon allows the abrasive to quickly reach a stable axial velocity in the nozzle while increasing wear in the mixing chamber. The processing performance of the AWJ depends on the outlet kinetic energy of the particles. At the same operating pressure, the differences in particle outlet velocity among the four models were small. However, different nozzle structures can increase the wear rate inside the nozzle, which further affects the lifetime of the AWJ cutting head. The design of the double abrasive tube model can effectively reduce the wear of the focusing tube, and an appropriate length of the focusing tube can ensure sufficient acceleration of the abrasive, resulting in higher machining and erosion capabilities. The following sections provide a detailed description of the performance effects on the nozzle by varying the particle inlet and mixing chamber size in a double abrasive tube model, in order to compare the effects of different model dimensions.

The Effect of Particle Inlet Size
In Figure 6a, the variation of water jet velocity at 400 MPa of pressure is analyzed for different particle inlet tube sizes Da, with other model parameters set to θ = 45°, α = 24°, d0 = 0.33 mm, d1 = 4.5 mm, d2 = 1.02 mm, l1 = 7 mm, and l2 = 76.2 mm. In the air-water flow, with all other parameters held constant, increasing the size of the particle inlet tube Da results in a larger volume of air being entrained into the model, resulting in an increase in the energy dissipation dissipated to the air. In the initial stage of mixing, the velocity in the core region of the jet reaches approximately 895 m/s for all models. Due to the large velocity and pressure gradient between the air and water phases, as the air enters, a tangential velocity is formed around the water jet as the air enters, which gradually reduces the initial kinetic energy of the jet. However, the variation in jet velocity at the outlet is small for all models.

The Effect of Particle Inlet Size
In Figure 6a, the variation of water jet velocity at 400 MPa of pressure is analyzed for different particle inlet tube sizes D a , with other model parameters set to θ = 45 • , α = 24 • , d 0 = 0.33 mm, d 1 = 4.5 mm, d 2 = 1.02 mm, l 1 = 7 mm, and l 2 = 76.2 mm. In the air-water flow, with all other parameters held constant, increasing the size of the particle inlet tube D a results in a larger volume of air being entrained into the model, resulting in an increase in the energy dissipation dissipated to the air. In the initial stage of mixing, the velocity in the core region of the jet reaches approximately 895 m/s for all models. Due to the large velocity and pressure gradient between the air and water phases, as the air enters, a tangential velocity is formed around the water jet as the air enters, which gradually reduces the initial kinetic energy of the jet. However, the variation in jet velocity at the outlet is small for all models. Machines 2023, 11, x FOR PEER REVIEW 15 of 20 Figure 6. The effect of particle inlet size on water jet velocity (a), particle velocity (b), and nozzle wear (c). Figure 6b analyzes the effect of particle inlet tube size on the average velocity of the abrasive inside the nozzle at an operating pressure of 400 MPa. All other parameters being equal, the abrasive achieves a higher exit velocity at Da = 2.5 mm, with a 16% increase compared to Da = 3.5 mm, and a 5% increase compared to Da = 3 mm. The reason for this phenomenon may be that more air enters the mixing chamber, reducing the accelerating effect of the water jet on the particles. On the other hand, as the particle inlet tube wears, resulting in a larger diameter, the exit velocity of the particles decreases, reducing the processing power of the jet. Figure 6c analyzes the effect of different particle inlet tube sizes on nozzle wear. When Da = 3.5 mm, the minimum wear occurs inside the nozzle, and the total wear from the orifice to the mixing chamber is the lowest at the mixing chamber position (L = 7). Although the position and angle of the abrasive inlet tube are the same, the model is affected by the Venturi effect in the initial stage of mixing, which causes air to be entrained inside the nozzle. The small variation in continuous phase velocity in each model and the changes in the vacuum caused by the Venturi effect were not significant, and the air velocity at the abrasive inlet remained relatively stable. When the air velocity at the particle inlet tube and the particle mass flow rate remain unchanged, the particles in the mixing chamber are affected by gravity and the Venturi effect. A larger particle inlet tube resulted in a lower initial particle velocity, further reducing the internal wear in the mixing chamber. However, more air is sucked into the interior of the nozzle, increasing the tangential velocity of the air, which, in turn, increases the total wear of the particles in the first half of the focusing tube. For smaller particle inlet tubes, higher particle velocities and mass flow rates will accelerate the wear rate of the abrasive inlet tube in the nozzle. When Da = 3 mm, the wear problem in the front half of the mixing chamber can be controlled by adjusting the particle mass flow rate and reducing the inlet angle of the abrasive tube. Figure 6b analyzes the effect of particle inlet tube size on the average velocity of the abrasive inside the nozzle at an operating pressure of 400 MPa. All other parameters being equal, the abrasive achieves a higher exit velocity at D a = 2.5 mm, with a 16% increase compared to D a = 3.5 mm, and a 5% increase compared to D a = 3 mm. The reason for this phenomenon may be that more air enters the mixing chamber, reducing the accelerating effect of the water jet on the particles. On the other hand, as the particle inlet tube wears, resulting in a larger diameter, the exit velocity of the particles decreases, reducing the processing power of the jet. Figure 6c analyzes the effect of different particle inlet tube sizes on nozzle wear. When D a = 3.5 mm, the minimum wear occurs inside the nozzle, and the total wear from the orifice to the mixing chamber is the lowest at the mixing chamber position (L = 7). Although the position and angle of the abrasive inlet tube are the same, the model is affected by the Venturi effect in the initial stage of mixing, which causes air to be entrained inside the nozzle. The small variation in continuous phase velocity in each model and the changes in the vacuum caused by the Venturi effect were not significant, and the air velocity at the abrasive inlet remained relatively stable. When the air velocity at the particle inlet tube and the particle mass flow rate remain unchanged, the particles in the mixing chamber are affected by gravity and the Venturi effect. A larger particle inlet tube resulted in a lower initial particle velocity, further reducing the internal wear in the mixing chamber. However, more air is sucked into the interior of the nozzle, increasing the tangential velocity of the air, which, in turn, increases the total wear of the particles in the first half of the focusing tube. For smaller particle inlet tubes, higher particle velocities and mass flow rates will accelerate the wear rate of the abrasive inlet tube in the nozzle. When D a = 3 mm, the wear problem in the front half of the mixing chamber can be controlled by adjusting the particle mass flow rate and reducing the inlet angle of the abrasive tube.

The Effect of Mixing Chamber Diameter
In Figure 7a, the variation of water jet velocity at 400 MPa pressure is analyzed for different mixing chamber diameters d 1 , with other model parameters set to θ = 45 • , α = 24 • , d 0 = 0.33 mm, d 2 = 1.02 mm, D a = 3 mm, l 1 = 7 mm, and l 2 = 76.2 mm. In the two-phase flow, with all other parameters held constant, increases in the diameter of the mixing chamber d 1 result in a larger diffusion area in the mixing chamber. The water jet will form an atomization zone earlier in the mixing chamber and focusing tube, which reduces the velocity at the jet outlet. When the value of d 1 is more than 4.5 mm, the change in kinetic energy of the initial stage core is not significant, and the jet is stabilized in the diffusion region and further accelerated in the focusing tube. Compared with the other models, when d 1 = 4 mm, the velocity at the outlet is 0.3% higher than in the other conditions.

The Effect of Mixing Chamber Diameter
In Figure 7a, the variation of water jet velocity at 400 MPa pressure is analyzed for different mixing chamber diameters d1, with other model parameters set to θ = 45°, α = 24°, d0 = 0.33 mm, d2 = 1.02 mm, Da = 3 mm, l1 = 7 mm, and l2 = 76.2 mm. In the two-phase flow, with all other parameters held constant, increases in the diameter of the mixing chamber d1 result in a larger diffusion area in the mixing chamber. The water jet will form an atomization zone earlier in the mixing chamber and focusing tube, which reduces the velocity at the jet outlet. When the value of d1 is more than 4.5 mm, the change in kinetic energy of the initial stage core is not significant, and the jet is stabilized in the diffusion region and further accelerated in the focusing tube. Compared with the other models, when d1 = 4 mm, the velocity at the outlet is 0.3% higher than in the other conditions.  Figure 7b analyzes the effect of the mixing chamber diameter on the average velocity of the abrasive at an operating pressure of 400 MPa. In the initial mixing stage, the velocity of the abrasive at d1 = 4 mm is lower than the other two structures, because the acceleration of the abrasive depends on the initial velocity of the water jet in the continuous phase. Due to the small variation of the velocity of the air-water two-phase flow in the continuous phase for the other two groups of models, the velocities at the outlet of the abrasive are basically the same for both groups of models. Compared to other parameters, the abrasive outlet velocity in a smaller mixing chamber diameter is 6% lower than in other models under the same acceleration environment. Figure 7c analyzes the effect of different mixing chamber diameters on nozzle wear. Due to the lower particle velocity associated with a smaller mixing chamber diameter, this model has less wear on various parts of the nozzle than other models at the same operating pressure. In the computational model, the convergence angle at the focusing tube remains constant. When the abrasive enters the focusing tube, a smaller mixing  Figure 7b analyzes the effect of the mixing chamber diameter on the average velocity of the abrasive at an operating pressure of 400 MPa. In the initial mixing stage, the velocity of the abrasive at d 1 = 4 mm is lower than the other two structures, because the acceleration of the abrasive depends on the initial velocity of the water jet in the continuous phase. Due to the small variation of the velocity of the air-water two-phase flow in the continuous phase for the other two groups of models, the velocities at the outlet of the abrasive are basically the same for both groups of models. Compared to other parameters, the abrasive outlet velocity in a smaller mixing chamber diameter is 6% lower than in other models under the same acceleration environment. Figure 7c analyzes the effect of different mixing chamber diameters on nozzle wear. Due to the lower particle velocity associated with a smaller mixing chamber diameter, this model has less wear on various parts of the nozzle than other models at the same operating pressure. In the computational model, the convergence angle at the focusing tube remains constant. When the abrasive enters the focusing tube, a smaller mixing chamber diameter leads to a shorter length at the convergence angle position, resulting in a lower tangential velocity of the particles and a stable axial acceleration. In contrast, increasing the diameter of the mixing chamber leads to a longer length at the convergence angle position, and the particles receive a higher axial acceleration, which is also affected by the tangential velocity of the air, and the particles generate a circumferential velocity in the mixing chamber, resulting in an increased wear rate at this position. As the fluid moves forward in the focusing tube of the nozzle, the tangential velocity of the air decreases, further reducing the wear on the wall. Therefore, the first half of the focusing tube experiences more wear, while the wear near the outlet position gradually decreases, which is consistent with the actual damage of nozzles in practical applications. In general, the wear of the mixing chamber and the focusing tube is inevitable. As the mixing chamber wears, resulting in a larger diameter, it will affect the exit kinetic energy of the abrasive and, at the same time, increase the wear on the focusing tube, but as the diameter of the mixing chamber increases, it will have less effect on the exit velocity of the abrasive, and the wear inside the focusing tube will continue to increase.

The Effect of Mixing Chamber Length
In Figure 8a, the variation of water jet velocity at 400 MPa pressure was analyzed for different lengths l 1 of the mixing chamber, with other model parameters set to θ = 45 • , α = 24 • , d 0 = 0.33 mm, d 1 = 4.5 mm, d 2 = 1.02 mm, D a = 3 mm, and l 2 = 76.2 mm. In the two-phase flow, the velocity decreases with increasing distance from the nozzle outlet due to energy dissipation and energy transfer to the air. The jet velocity at the outlet is reduced by varying the length of the mixing chamber in the model's structure. Under the same pressure conditions, the model with the extended mixing chamber shows a 0.4% decrease in velocity at the outlet compared to the other models. Theoretically, as the length of the mixing chamber in the model increases, the water jet will continue to transition from the initial core region to the diffusive flow region, resulting in a gradual reduction in the kinetic energy of the water jet. Figure 8b analyzes the effect of the mixing chamber length on the average velocity of the abrasive within the nozzle at an operating pressure of 400 MPa. With the abrasive inlet tube height keep constant, increasing the mixing chamber length changes the position of the focusing tube, resulting in a shift in the secondary acceleration position of the particles in different nozzles. Under the influence of the continuous phase of the jet, the particles in all models accelerate suddenly after the focusing tube, which is also the reason why the particle velocity in the first half of the model at l 1 = 11 mm is generally lower than in other models. However, the longer mixing chamber is 7% more efficient at accelerating particles than the other models at the same operating pressure. Figure 8c shows the effect of different mixing chamber lengths on nozzle wear. This is determined by calculating the total wear at the position from the orifice outlet to the location of the focusing tube in order to assess the wear problem in the first half of the model. As the length of the mixing chamber increases, the internal volume increases and the particles are accelerated uniformly throughout the mixing chamber. At this point, the circumferential velocity of the particles exceeds the tangential velocity of the air, causing the abrasive to move in a circular motion within the longer mixing chamber, accelerating the wear on the end wall of the mixing chamber. At the same time, with the convergence angle unchanged, the full acceleration of the particles in the mixing chamber causes further blockage at the convergence angle of the nozzle and exacerbates the wear in the first half of the focusing tube. When l 1 = 9 mm, the length of the mixing chamber is just enough to fully accelerate the particles, with the tangential velocity of the air being greater than the circumferential velocity of the particles. This allows the particles to enter the focusing tube of the nozzle in an orderly fashion for secondary acceleration, further reducing the occurrence of wear problems. This also demonstrates the importance of designing the mixing chamber to the appropriate length. length results in a 7% higher acceleration efficiency of the particles compared to other models. 5.
The lower the velocity of the abrasive inside the nozzle, the less wear on the model. The tangential velocity of the air and the circumferential velocity of the abrasive affect the wear inside the mixing chamber and the acceleration of the particles.