In order to study the breakup and atomization mechanisms of the fuel-jet in air crossflow and realize the accurate and controllable atomization effect of fuel, this paper proposes a method by coupling the large eddy simulation method and the VOF to DPM method. The results show that the breakup and atomization of the fuel-jet are mainly caused by the Rayleigh–Taylor (R–T) unstable surface wave and the Kelvin–Helmholtz (K–H) instability. The density of fuel-particles is higher near the central region of the fuel jet trajectory, decreasing as distance from the central region increases. As the momentum ratio increases, the penetration depth of the fuel-jet column also increases, resulting in a more concentrated spatial distribution of fuel particles. Conversely, with a lower momentum ratio, the spatial distribution of fuel particles becomes more uniform. For the spatial distribution of fuel particles, a new mathematical model is established to characterize the spatial distribution boundary of fuel particles. At the same momentum ratio, the higher the air velocity, the smaller the average particle diameter. At the same *We*_{g} number, the higher the fuel-jet velocity, the higher the average particle diameter. The relative error of the average particle diameter between the experiment and numerical simulation is very small, which is 6.4%.

## I. INTRODUCTION

The fuel considered here is an aviation liquid fuel that jets into the super-combustion ramjet engine and atomizes into fuel particles. Fuel jet atomization is a complex multiphase flow problem, and the effect of fuel atomization is directly related to the subsequent combustion process, engine power performance, and exhaust gas emission index.^{1–6} The fuel atomization process is a multi-scale problem, and the fuel-jet column atomization process is mainly divided into primary atomization and secondary atomization. Primary atomization is predominantly driven by the air–liquid shear force, causing the fuel-jet column to gradually thin, bend, fluctuate, and eventually break up into liquid blocks and liquid filaments. Secondary atomization mainly refers to the further breakup of liquid blocks and liquid filaments into liquid particles. In order to reveal the atomization mechanism of the fuel-jet, at home and abroad, many scholars have conducted related research on the fuel-jet.^{7–11}

Wu *et al.*^{12} studied the breakup mechanism of a liquid column in subsonic air crossflow and studied the jet trajectories of water, ethyl alcohol, alcohol–water mixed liquid, and glycerol–water mixed liquid through experimentation. The influence of different momentum ratios on jet trajectory was studied, and the expression of drag coefficient was obtained from different liquid jet trajectories. Salewshi *et al.*^{13} numerically simulated the atomization and breakup of liquid jet in air crossflow by using LES and studied the formation of different types of vortices and their influence on the atomization and breakup of liquid column, as well as studied the droplet distribution law. Arienti *et al.*^{14,15} studied the jet trajectory and atomization of a water column through experimental and simulation methods, obtained the distribution law of droplet particle diameters, and compared the error of droplet particle diameters. Shao *et al.*^{16} analyzed the influence of liquid–air flow ratio on velocity field, liquid–air interface location, and vortex structure by simulations of liquid atomization. Pai *et al.*^{17} predicted the liquid jet heights, liquid jet trajectories, and wavelengths of the liquid–air interfaces by numerical simulations, demonstrating good consistency with published experimental results. Xiao *et al.*^{18} studied the atomization mechanism of water jet column in air crossflow. O’Rourke and Amsden^{19} proposed a new TAB method to simulate fuel-jet column atomization and derived some theoretical properties from the TAB method, and the calculation results show consistency with other methods and experiments. Heinrich and Schwarze^{20} studied fuel jet atomization process by the VOF to DPM method. Primary breakup was investigated using the VOF method, while secondary breakup was studied using the DPM method. Herrmann^{21} studied the fuel-jet column breakup and atomization into Lagrangian drops based on a multi-scale coupling model and analyzed the distribution law of droplet particles along the jet trajectory of the liquid column. Yoo *et al.*^{22,23} studied the relationship between penetration depth and momentum ratio, and penetration depths of liquid jet showed good consistency with experimental data. Eslamian *et al.*^{24} conducted experimental research on trajectory and atomization of water jet into subsonic crossflow. A detailed numerical analysis was performed to investigate the effects of pressure, temperature, and air velocity on jet trajectory, penetration depth, and trajectory shape; by analyzing the effects of air velocity and pressure on jet trajectory, the optimal jet trajectory and penetration depth were obtained.^{25} Based on the above, existing studies primarily investigate the breakup mechanism of liquid jet via two methods: experimental observation and numerical simulation. Experimental methods capture images of jet breakup at different instants, while numerical simulations employ Volume-of-Fluid (VOF) method, level-set method, coupled method (CLSVOF), and VOF-to-DPM approach. VOF-to-DPM is a numerical computation method developed in recent years. This approach uses the VOF method to compute the gas–liquid interface in the continuous flow region. When the continuous liquid breaks up to a certain threshold, it switches to the DPM to calculate the trajectories of liquid droplets. Using this method can reduce grid count and enhance computational efficiency.

Despite the research efforts by scholars at home and abroad, the atomization mechanism of the fuel-jet is not clear enough. Large eddy simulation (LES), VOF to DPM, grid adaptive, and time adaptive methods are used to simulate the breakup and atomization processes of the fuel injected into the high-speed air crossflow, and the mechanisms of fuel-jet column breakup and atomization are studied. The relationships between the average diameter of fuel droplets, jet trajectory, and different physical parameters are analyzed, and the influences of different parameters on fuel-jet column breakup and atomization are discussed.

## II. METHODOLOGY

### A. Theoretical models

The rationale of the current two-phase-flow LES formulation is as follows: a filter function is used to separate large-scale eddies and small-scale eddies in LES model, only large-scale vortices are simulated by the model, while small-scale vortices are approximated by a mathematical model that has an effect on large-scale vortices.

^{6,18,22}

*U*and

*P*are velocity and pressure, respectively,

*μ*is the kinematic viscosity,

*g*

_{i}denotes the gravitational acceleration, $H\varphi $ denotes the Heaviside function,

*ϕ*denotes the LS function, and Δ denotes the filtration scale.

^{18}

*σ*is the surface tension coefficient,

*k*is the interface curvature, and

*n*is the interface normal vector.

^{11,18}

### B. Computational modeling and boundary conditions

^{−9}s, with a refinement level of 4, an initial grid size of 0.2 mm, and a minimum grid size of 0.0125 mm. The simulation model is depicted in Fig. 1. The diameter of the fuel-inlet column is 0.4 mm, the length of the inlet pipe

*L*= 3 mm, and the size of the calculation domain is 40 × 5.0 × 20 mm

^{3}, as shown in Fig. 1. Table I shows the parameters of the four numerical simulation cases, and Fig. 2 shows the adaptive grid. Figure 3 shows the numerical simulation flow chart. The dimensionless parameters are

^{17}

. | u_{air} (m/s)
. | u_{oil} (m/s)
. | q
. | We_{g}
. | We_{l}
. |
---|---|---|---|---|---|

Case 1 | 90 | 12 | 11 | 153 | 1728 |

Case 2 | 60 | 12 | 25 | 68 | 1728 |

Case 3 | 75 | 8 | 7 | 106 | 768 |

Case 4 | 75 | 10 | 11 | 106 | 1200 |

Case 5 | 75 | 12 | 16 | 106 | 1728 |

. | u_{air} (m/s)
. | u_{oil} (m/s)
. | q
. | We_{g}
. | We_{l}
. |
---|---|---|---|---|---|

Case 1 | 90 | 12 | 11 | 153 | 1728 |

Case 2 | 60 | 12 | 25 | 68 | 1728 |

Case 3 | 75 | 8 | 7 | 106 | 768 |

Case 4 | 75 | 10 | 11 | 106 | 1200 |

Case 5 | 75 | 12 | 16 | 106 | 1728 |

## III. ANALYSIS OF RESULTS

### A. Breaking up and atomization process of the fuel-jet column

The LES and the VOF to DPM methods are used to numerically calculate breakup and atomization of the fuel jet into the air crossflow using the commercial software STAR CCM+. In this simulation, air is the first phase, while fuel is the second phase. The calculation time is 2 *µ*s, with a minimum time step of 10^{−9} s. Figure 4 shows the velocity field in the cross section z = 0, the isosurface of the fuel volume fraction at 0.1, and the spatial distribution of fuel-particles when Weber number *We*_{g} = 153 and momentum ratio q = 11. It is observed from Fig. 4 that the high-speed air interacts with the fuel-jet column from the round hole. Under the impact of high-speed crossflow air, the flow direction of fuel-jet column gradually deviates, forming a distinct fuel flow track. This results in the formation of a high-speed area above the fuel track and a low-speed area below it. As the height of the fuel-jet column increases, it gradually evolves from a circular to an elliptical and thin film shape. Subsequently, it further breaks up into branching fuel filaments and fuel particles, with the liquid filaments ultimately breaking into fuel particles. The fuel particles formed are distributed on both sides of the fuel-jet trajectory and gradually spread outward. Due to the interaction between the crossflow air and the fuel-jet column, the velocity of air in the y-direction results in more fuel particles being distributed above the computational domain and fewer below.

Figure 5 represents the velocity distribution and pressure distribution in the vicinity of the liquid phase. As shown in Fig. 5, the fuel surface is extremely unstable, forming unstable fluctuations on the surface of the fuel-jet column. The high-speed crossflow air is resisted by the fuel-jet column. Some airflow bypasses both sides of the fuel-jet column, flowing downward, while the rest flows upward along the fuel-jet column, forming a high-pressure area in front of the fuel-jet column and a low-pressure area behind it. The ejected fuel-jet column bends under the dynamic force formed by the joint action of the high-pressure and the low-pressure. At the same time, a large normal velocity gradient is formed in front of and behind the fuel-jet column. Under the action of this velocity gradient, unstable waves are generated on the surface of the fuel-jet column. The unstable waves generated on the surface of the fuel-jet column are called Rayleigh–Taylor (R–T) waves. With the gradual bending of the fuel-jet column, the amplitude and wavelength of the unstable wave gradually increase. At the same time, the fuel-jet column gradually transforms into filiform fuel particles and spherical fuel particles, which then fall off from the fuel-jet column. Under the action of air crossflow, the breaking zone of the fuel-jet column always occurs in the wave-valley of the R–T unstable wave, where the fuel column gradually breaks up and atomizes, eventually forming a large number of fuel particles. Under the action of the tangential force at the air–liquid interface, the liquid filament gradually overcomes the surface tension and breaks into fuel droplets.

Figure 6 represents the velocity and pressure contour maps at the air–liquid interface of the fuel-jet column. It is seen from Fig. 6 that there is a large pressure gradient between the windward side and leeward side of the fuel-jet column. Under the effect of the pressure gradient, the fuel-jet column gradually changes from a circle to an ellipse and further becomes a liquid film. The surface tension and viscous force of the liquid film cannot overcome the shear force from the velocity gradient of the gas–liquid interface, resulting in the air flow passing through the R–T unstable wave valley. As a result, the liquid film breaks up and atomizes. The breakup of the fuel-jet column caused by the R–T unstable wave is the main form of breakup and atomization of the fuel-jet column. Namely, the R–T instability results in the breakup of fuel-jet column. The R–T instability fluctuation is a process of gradual thinning and breakup of the fuel-jet column. At the same time, at the gas–liquid interface between air and fuel, a very large tangential velocity gradient will be generated, leading to the generation of Kelvin–Helmholtz (K–H) unstable waves and the formation of wrinkles on the surface of the fuel-jet column. These wrinkles gradually develop to both sides and above the fuel-jet column, forming liquid filaments that eventually break into fuel droplets. Due to the existence of K–H unstable waves, liquid filaments on the fuel surface were gradually peeled off. The K–H instability fluctuation and fragmentation are accompanied by the whole process of fuel-jet column injection.

Figure 7 represents the velocity contour map of the section with y = 0 mm, which can further reveal the breaking process of the fuel-jet column. Figure 7(a) represents the velocity contour map of the section with y = 0 mm. As shown in Fig. 7(a), there is a large velocity gradient on the windward side of the gas–liquid interface. Under the shear force generated by the velocity gradient, the fuel-jet column gradually produces bending, folding, and fluctuation. Figures 7(b)–7(d) are the velocity vector diagrams of A, B, and C zones in Fig. 7(a), respectively. Figure 7(b) is the velocity vector diagram of the upstream zone of the fuel-jet column. As shown in Fig. 7(b), the velocity of high-speed air on the windward side of the fuel-jet column gradually decreases and changes direction and rises along the air–liquid interface. At the same time, vortex phenomenon appears on the windward side, and the vortex on the windward side will lead to instability of the fuel-jet column. Figure 7(c) shows the breakup of the fuel-jet column. This area is where the primary breakup of the fuel-jet column occurs. It can be seen from Fig. 7(c) that with the development of the surface wave, the amplitude of the surface wave gradually increases and vortices are generated in multiple wave valleys. The vortices lead to opposite speeds appearing on the windward and leeward sides of the fuel, accelerating the deformation and fragmentation of the fuel. Figure 7(d) shows the velocity vector diagram of the secondary breakup of fuel-jet column. It can be seen from Fig. 7(d) that under the action of high-speed air crossflow, the vortex further strengthens, with large vortices decomposing into smaller ones and the liquid film further thinning, eventually breaking into smaller liquid blocks and filaments. The high-speed air penetrates the liquid film from multiple zones and drives the liquid droplets to flow downstream. From the above analysis, it is shown that vortex is the main reason for the breakup and atomization of the fuel-jet column.

### B. Effect of different parameters on the atomizing process

Figure 8 depicts the jet trajectory of fuel under different cases in the positive direction of Z and X. From the analysis of the jet trajectory of fuel under different cases in the positive direction of Z, the trajectory and shape of the fuel-jet column vary with different momentum ratios. The jet height gradually increases with increasing momentum ratio. The breakup of the fuel-jet column is mainly caused by K–H and R–H instability waves. With an increase in the momentum ratio, the K–H unstable wave breaking form gradually weakens, while the R–H unstable wave breaking form gradually strengthens. When the momentum ratio is 25, the R–H instability is dominant, and the fuel-jet height is higher than others. With the development of R–H unstable surface waves, the fuel-jet column gradually breaks up and atomizes into droplets. Momentum ratio is an important factor on the breakup and atomization of the fuel-jet column. The Weber number has little influence on the fuel-jet trajectory in air crossflow.

It is observed from the jet trajectory of fuel under different cases in the positive direction of X that the surface of each fuel-jet column shows generation and development of R–T instability waves. R–T unstable waves also show some differences under different cases. The smaller the momentum ratio, the shorter the wavelength of R–T instability waves and the more intense the vibration of the fuel-jet column. Consequently, the fuel-jet column breaks rapidly, resulting in a shorter trajectory of the fuel-jet. With an increase in momentum ratio, the wavelength of R–T instability waves becomes larger, leading to a tendency for the fuel-jet trajectory to become longer.

### C. Fuel jet trajectory analysis

^{12,25–28}as shown in the following:

The simulation results of the fuel-jet trajectory under different cases are compared with Eqs. (11)–(15) and the revised equation of Wu *et al.* Eq. (16). Figure 9 shows the isosurface of liquid phase volume fraction at 0.1, the spatial distribution map of fuel particles, and the five curves calculated using Eqs. (11)–(16). It can be observed from Fig. 9 that the greater the momentum ratio, the greater the penetration depth. The momentum ratio has a great influence on the penetration depth and jet trajectory. According to the five empirical formulas, the fuel-jet trajectory shows good agreement between the Stenzler’s empirical formula and simulation. Equation (16) shows good agreement with the simulation results on the spatial outline of fuel particles. The theoretical results from the Gopala, Wang, and Wu formulas do not align well with the simulation results on the fuel-jet trajectory and the spatial outline of fuel particles. The distribution of fuel particles is similar, but the particle distribution density is quite different. In the liquid phase track area of fuel-jet column, the fuel particles are more densely distributed, and the farther away from the liquid phase track of fuel-jet column, the thinner the fuel particles. As the momentum ratio increases, the penetration depth of the fuel-jet column also increases, leading to a more concentrated spatial distribution of fuel particles. Conversely, with a smaller momentum ratio, the spatial distribution of fuel particles tends to be more uniform.

### D. Study of particle size distribution

Figure 10 compares the particle diameters with the same momentum ratio of 11. The momentum ratio for case 1 and case 4 is 11. It can be seen from Fig. 10 that there are far more atomized fuel particles in case 1 than in case 4. The number of atomized particles with a diameter of less than 90 *μ*m is larger in case 1 than in case 4. The number of atomized particles with a diameter exceeding 90 *μ*m is lower in case 1 than in case 4. In case 4, the proportion of larger diameter particles is larger than in case 1; therefore, a higher velocity of the fuel-jet column results in the production of smaller particles when the momentum ratio is the same.

Figure 11 compares the particle diameters at different fuel-jet velocities**.** In terms of the number of particles produced, case 5 has the most, case 4 is in the middle, and case 3 has the least. Case 5 produces more large diameter particles than the other cases. From the PDF of fuel particle diameter perspective, in case 4, the proportion of larger diameter particles is greater than in the other cases. The average particle size first increases and then decreases with the increasing velocity of the fuel-jet column.

Figure 12 compares the particle diameters under different air crossflow velocities. The fuel-jet velocities in the three cases are 12 m/s, while the air velocities are 90, 75, and 60 m/s, respectively. From the number of particles produced, Fig. 12 shows that the higher the air crossflow velocity, the greater the number of small particles produced, and the lower the number of large particles produced. The higher the air crossflow velocity, the more the number of particles produced. From the PDF of fuel particle diameter, average particle size decreases with increasing air transverse velocity.

Table II shows the average particle diameters, standard deviations of particle diameters, and average velocities of particles under different conditions. From the average particle diameter, the order is case 1, case 5, case 3, case 4, and case 2. From the standard deviation of particle size, the order is case 1, case 4, case 5, case 2, and case 3. When the momentum ratio is 11 in both cases, the average particle size and standard deviation of case 4 are greater than those of case 1. In case 1, case 2, and case 5 with different air transverse velocities, the higher the air transverse velocity, the smaller the average particle diameter. The average particle velocities range between 10 and 20 m/s, with the maximum being 20 m/s for case 1 and the minimum being 14.5 m/s for case 3.

Working conditions . | Case 1 . | Case 2 . | Case 3 . | Case 4 . | Case 5 . |
---|---|---|---|---|---|

Average particle diameters (μm) | 60.8 | 97.2 | 73.8 | 83.4 | 72.8 |

Standard deviations | 20.2 | 30.6 | 34.4 | 41.9 | 37.7 |

Average particle velocities (m/s) | 20 | 15.3 | 14.5 | 15.5 | 17.1 |

Working conditions . | Case 1 . | Case 2 . | Case 3 . | Case 4 . | Case 5 . |
---|---|---|---|---|---|

Average particle diameters (μm) | 60.8 | 97.2 | 73.8 | 83.4 | 72.8 |

Standard deviations | 20.2 | 30.6 | 34.4 | 41.9 | 37.7 |

Average particle velocities (m/s) | 20 | 15.3 | 14.5 | 15.5 | 17.1 |

### E. Comparison between numerical simulation results and experimental results

Figure 13 shows the probability density function of the particle diameters in case 1, which follows a normal distribution. The average diameter of fuel particles obtained through experiments is 57 *µ*m, while the average diameter of fuel particles obtained through numerical simulation is 60.9 *µ*m. The relative error of the average diameter between the experiment and numerical simulation is very small, which is 6.4%. This shows that the large eddy simulation, VOF to DPM, grid adaptive, and time adaptive methods are effective in simulating fuel-jet in crossflow, and the fuel has been fully atomized.

## IV.CONCLUSION

The breakup and atomization of fuel-jet in the air crossflow were simulated using the LES, VOF to DPM, adaptive grid, and adaptive time methods. The breakup mechanism, jet track, and particle distribution law of fuel-jet column are expounded, and the following laws are obtained:

The main reason for the instability of the fuel-jet column is the generation of vortices. The breakup forms of fuel-jet column mainly include R–T instability and K–H instability. R–T unstable surface waves lead to unstable fluctuation and breakup of the fuel-jet column, while K–H unstable surface waves lead to wrinkles on the surface of the fuel-jet column and stripping and breakup of liquid filament.

Momentum ratio is an important factor affecting the jet trajectory of the fuel-jet column, the larger the momentum ratio, the higher the penetration height of the fuel-jet column.

In this paper, a new model is proposed that can more accurately predict the boundary of the spatial distribution of fuel particles. In addition, Stenzler’s empirical formula aligns well with the track of the fuel-jet column as obtained from simulation results.

All else being equal, the average particle velocity increases with increasing air transverse velocity, while the average particle size decreases with an increase in the air crossflow velocity. Furthermore, the average particle velocity rises with increasing velocity of the fuel-jet column. The average particle size initially increases and then decreases with increasing velocity of the fuel-jet column. In addition, the average particle velocity increases with increasing air transverse velocity. For the same momentum ratio, a higher air velocity results in a smaller average particle size and standard deviation, as well as a higher average particle velocity.

## ACKNOWLEDGMENTS

This work was supported by Shanghai Research Center of Engineering and Technology (Grant No. 20DZ2253200).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Rujun Wu (吴入军)**: Data curation (equal); Investigation (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). C**en Sun (孙涔)**: Conceptualization (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Writing – original draft (equal). **Yifei Gui (桂夷斐)**: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition*