Shock Wave Characteristics Research Articles

On the approximations to fractional nonlinear damped Burger’s-type equations that arise in fluids and plasmas using Aboodh residual power series and Aboodh transform iteration methods

Damped Burger’s equation describes the characteristics of one-dimensional nonlinear shock waves in the presence of damping effects and is significant in fluid dynamics, plasma physics, and other fields. Due to the potential applications of this equation, thus the objective of this investigation is to solve and analyze the time fractional form of this equation using methods with precise efficiency, high accuracy, ease of application and calculation, and flexibility in dealing with more complicated equations, which are called the Aboodh residual power series method and the Aboodh transform iteration method (ATIM) within the Caputo operator framework. Also, this study intends to further our understanding of the dynamic characteristics of solutions to the Damped Burger’s equation and to assess the effectiveness of the proposed methods in addressing nonlinear fractional partial differential equations. The two proposed methods are highly effective mathematical techniques for studying more complicated nonlinear differential equations. They can produce precise approximate solutions for intricate evolution equations beyond the specific examined equation. In addition to the proposed methods, the fractional derivatives are processed using the Caputo operator. The Caputo operator enhances the representation of fractional derivatives by providing a more accurate portrayal of the underlying physical processes. Based on the proposed two approaches, a set of approximations to damped Burger’s equation are derived. These approximations are discussed graphically and numerically by presenting a set of two- and three-dimensional graphs. In addition, these approximations are analyzed numerically in several tables, including the absolute error for each approximate solution compared to the exact solution for the integer case. Furthermore, the effect of the fractional parameter on the behavior of the derived approximations is examined and discussed.

Rapid and sparse reconstruction of high-speed steady-state and transient compressible flow fields using physics-informed graph neural networks

Explosion flow fields are characterized by shock waves with varying intensity and position (i.e., explosive loads), which are the primary causes of structural damage. Accurate and rapid prediction of explosive loads is crucial for structural blast-resistant design and daily security management. While existing empirical models and numerical simulation methods can capture the propagation characteristics of explosive shock waves, high-precision simulation requires a massive computational workload, which is insufficient to meet the fast computational demands of various explosive scenarios. To address this contradiction, this study constructed a sparse reconstruction model for two-dimensional explosion fields based on machine learning algorithms. The model utilizes sparse observational data to establish a mapping relationship to the distribution of the entire flow field. The model is built by a physics-informed graph neural network (PIGN). The graph neural network is employed to associate node features, while the physical network is utilized to control model convergence, aiming to enhance model performance. Using the constructed dataset, the PIGN model was tested. Performance and generalization capabilities of the model were assessed by comparing its results with numerical simulation. This evaluation analyzed the relative error distribution and error statistical results of the reconstructed flow field. The results indicate that the PIGN model can effectively reconstruct explosion fields, with an average error in the reconstructed flow field below 4%. Furthermore, when the number of probe points reaches 10, the average error of the flow field reconstructed by the model is close to 6%. This model not only provides a highly reliable distribution of explosion overpressure and pressure-time variations but also, with a well-trained model, accomplishes flow field reconstruction within 1 ms. It offers a novel approach for achieving rapid and reasonable prediction of explosion fields or two-dimensional compressible flow fields.

Effect of trapping of electrons and positrons on the evolution of shock wave in magnetized plasma: A complex trapped K-dV burgers’ equation

Our research demonstrates that positive and negative ions, along with trapped electrons and positrons, propagate as small-amplitude electrostatic shock waves in a magnetized collisions-less plasma. The trapped Korteweg-de Vries Burger's (T-KdVB) equation was derived by the widely recognized standard reductive perturbation technique (RPT), employing two separate scales of standard coordinates. The study examines the impact of temperature ratio on the shock wave characteristics of positive and negative ions, as well as positrons with electrons. The analysis and discussion focus on the impacts of negative ion number densities, as well as the number densities of electrons and positrons. The T-KdVB equations are responsible for governing rarefactive shock waves. Additionally, it is important to acknowledge that nonlinearity and dissipation play a key role in controlling the amplitude of the shock.

Experimental investigation on the characteristics of the shock wave emitted by the cavitation bubble near the air bubble

This study explores the mitigation of cavitation damage in hydraulic engineering through air entrainment. The primary aim is to experimentally analyze the shock wave characteristics emitted by cavitation bubbles adjacent to air bubbles affixed to a tube nozzle. The schlieren optical system is utilized to visualize the shock wave, while a hydrophone measures its pressure. Experiments are conducted on cavitation bubbles induced by the spark-generated method in the vicinity of air bubbles, varying the dimensionless distances and sizes of the air bubbles. The results indicate that (1) The introduction of an air bubble noticeably changes the morphology, kinematic behavior, and shock wave features of the cavitation bubble. (2) Four distinct shock wave patterns are identified based on the quantity and shape of the shock wave, with variations in the cavitation bubble's collapsing behavior and shock wave characteristics across different patterns. (3) The dimensionless distance γ and size δ exert significant influence on the shock wave's quantity, pressure peak, shape, and energy. With γ decreases or δ increases, the shock wave quantity increases while the shock wave intensity decreases. This investigation of the interaction between cavitation bubbles and air bubbles is essential for elucidating the mechanism through which air entrainment mitigates cavitation damage.

Formation and propagation characteristics of a weak shock wave in maglev tube

The propagation of the weak shock wave (WSW) to the tunnel exits and their radiation as micro-pressure waves (MPWs) may cause sonic booms or structural resonance of buildings, posing potential hazards to humans, animals, and buildings in the exit's environment. The characteristics of the WSW and sonic booms of a maglev train/tube coupling model were studied based on the two-dimensional axisymmetric unsteady Reynolds average Navier–Stokes turbulence model. In the later stage of a MPW, the formation mechanism, geometry, and kinematic characteristics of compressible vortex rings (CVRs) were systematically analyzed. The inertial effect causes the initial wavefront to gradually transition from a Gaussian-shape waveform to a triangular waveform during its propagation, eventually coalescing into a WSW. The overpressure, density jump, and shock Mach number at the WSW location all increase with the increasing train speed, while the WSW thickness decreases accordingly. The formation distance of the WSW is inversely proportional to the amplitude of the initial wavefront gradient, and the WSW directly causes the occurrence of the exit sonic boom. The MPW amplitude has significant directionality with a largest value in the axial direction. Within the speed range of 450–700 km/h, the sound pressure level of the MPW exceeds the hearing threshold and even reaches the feeling threshold. The evolution of CVRs includes primary CVR, secondary CVR, and Kelvin–Helmholtz vortices. Primary CVR has the greatest impact on the axial MPW among them. The occurrence of CVRs will cause a second small noise level other than the sonic boom.

Deformation characteristics of a ring-stiffened cylindrical shell induced by shock waves and coalesced bubbles in double-charge underwater explosions

This study investigates the deformation characteristics of a ring-stiffened cylindrical shell induced by shock waves and coalesced bubbles in double-charge underwater explosions. A numerical model for coupling underwater multi-point explosion loads with the cylindrical shell is established by the Arbitrary Lagrange Euler method, and underwater explosion experiments with double charges are also conducted. The numerical model's effectiveness is validated by comparing shock wave superposition characteristics, bubble coalescence processes, and bubble pulsation periods with the experimental results. Based on the numerical model, the influences of detonation intervals and layout angles of charge on shock wave superposition effects, temporal and spatial distribution characteristics of shock waves, and the evolution process of coalesced bubbles in underwater explosions with double charges are explored. Additionally, the deformation characteristics of cylindrical shells induced by shock waves and coalesced bubbles for double charges with different detonation intervals and layout angles of charge are analyzed. The results indicate that double charges can induce more substantial deformation on the cylindrical shell at a specific detonation interval than a single charge with equal total mass. During the shock wave phase, the cylindrical shell's deformation decreases nonlinearly with an increased layout angle. Conversely, during the bubble load phase, the deformation of the cylindrical shell shows an approximately linear decrease with an increase in layout angle. A critical angle exists, below which the impact of double charges on the cylindrical shell is more substantial when detonated with an interval than simultaneous detonation.

Mesoscopic Kinetic Approach of Nonequilibrium Effects for Shock Waves.

A shock wave is a flow phenomenon that needs to be considered in the development of high-speed aircraft and engines. The traditional computational fluid dynamics (CFD) method describes it from the perspective of macroscopic variables, such as the Mach number, pressure, density, and temperature. The thickness of the shock wave is close to the level of the molecular free path, and molecular motion has a strong influence on the shock wave. According to the analysis of the Chapman-Enskog approach, the nonequilibrium effect is the source term that causes the fluid system to deviate from the equilibrium state. The nonequilibrium effect can be used to obtain a description of the physical characteristics of shock waves that are different from the macroscopic variables. The basic idea of the nonequilibrium effect approach is to obtain the nonequilibrium moment of the molecular velocity distribution function by solving the Boltzmann-Bhatnagar-Gross-Krook (Boltzmann BGK) equations or multiple relaxation times Boltzmann (MRT-Boltzmann) equations and to explore the nonequilibrium effect near the shock wave from the molecular motion level. This article introduces the theory and understanding of the nonequilibrium effect approach and reviews the research progress of nonequilibrium behavior in shock-related flow phenomena. The role of nonequilibrium moments played on the macroscopic governing equations of fluids is discussed, the physical meaning of nonequilibrium moments is given from the perspective of molecular motion, and the relationship between nonequilibrium moments and equilibrium moments is analyzed. Studies on the nonequilibrium effects of shock problems, such as the Riemann problem, shock reflection, shock wave/boundary layer interaction, and detonation wave, are introduced. It reveals the nonequilibrium behavior of the shock wave from the mesoscopic level, which is different from the traditional macro perspective and shows the application potential of the mesoscopic kinetic approach of the nonequilibrium effect in the shock problem.

Comparative analysis of blast load transfer and structural damage in multi-cabin structures during air and underwater explosions

This study presents comprehensive experimental and numerical findings regarding the impact of the fluid medium on blast load transfer and structural damage mechanisms in multi-cabin structures subjected to air and underwater explosions. The experiments demonstrated that the structure primarily underwent a comprehensive dynamic response during an air explosion. In contrast, underwater explosions induced severe localized damage, with fluid properties significantly influencing the shockwave intensity. The observed plastic deformation, shear tearing of the structure, and shock wave characteristics in the experiment were successfully predicted through numerical simulations. Additionally, the numerical simulation unveiled distinct mechanisms of pressure changes in the cabin, attributing air and underwater explosions to transmitted pressure waves and compression shocks of the outer plate, respectively. The high-intensity shock waves generated by underwater explosions and bubble loads sequentially impacted the structure, resulting in more pronounced structural damage compared to the weaker shock wave effect of an air explosion. This study provides valuable insights for comprehending and predicting the response of multi-cabin structures in diverse explosion scenarios.

Investigating the characteristics of internal discharge shock waves in gas‐insulated switchgear under varied gap configurations

AbstractGas‐insulated switchgears (GIS) are crucial components of high‐voltage power transmission and distribution systems. Internal discharges within GIS have garnered significant attention in the field of power engineering. This study investigates the characteristics of internal discharge shock waves in GIS under an air pressure of 0.3 MPa and three different discharge gap conditions: 1, 1.5, and 2 mm. High‐speed shadowing techniques are used to analyse the propagation speed, morphology, and post‐wave parameters. The study findings reveal that although the characteristics of internal discharge shock waves in GIS noticeably change with the gap size, they also exhibit a consistent trend: as the gap size increases, the initial shock wave accelerates, intensifying the discharge. Simultaneously, the rate of attenuation rises, with the shock wave becoming weaker after ≈30 µs. Furthermore, the post‐wave parameters follow a similar pattern, with an increase in gap size leading to higher parameter values but also a faster decay rate. The parameters decay more rapidly between 10 and 20 µs, slow down between 20 and 30 µs, and ultimately stabilize after around 30 µs. The results of this study hold significant theoretical and practical implications for the monitoring, diagnosis, and prevention of internal discharges in GIS.

Coupled models for propagation of explosive shock waves in cylindrical and spherical geometries

The propagation of shock waves in different geometries is crucial in engineering and scientific applications. A comprehensive model is developed to elucidate the hydrodynamic growth and decay of shock waves in cylindrical and spherical geometries by using the strong shock wave assumption. This model takes into consideration the conservation equations governing mass, momentum, and energy, thereby allowing for an accurate description of the coupled behavior between the piston and shock wave propagation. In contrast to the localized analysis employed in previous self-similar methods, this model incorporates the finite sound wave velocity to introduce the concept of retarded pressure on the piston surface. Consequently, the proposed model offers a multitude of advantages by providing a complete set of dynamic information concerning the trajectories, velocities, and accelerations of both the piston and shock wave. Furthermore, an asymptotic analytical solution is derived to describe the decay of shock waves in cylindrical and spherical geometries. To validate the theoretical analysis and illustrate the propagation characteristics of shock waves in these specific geometries, thorough comparisons are conducted. These findings contribute to the advancement of our understanding of shock wave dynamics in various physical systems, particularly in the field of plasma physics.

A numerical model for the electrical and shock wave characteristics of underwater pulsed spark discharge

Underwater pulsed spark discharge has been widely used in industrial fields as a source of shock waves or acoustic waves, and numerical modeling of the discharge and pressure wave characteristics is necessary to improve the application performance. In this paper, a numerical model is proposed that couples the circuit equation, the mass and energy conservation equations, and a momentum conservation equation based on the Rankine–Hugoniot conditions. A tabulated wide range equation of state and conductivity data of water are used, and various physical processes during the plasma channel expansion are considered, including the energy flow and mass exchange between the channel and the surrounding water due to thermal radiation, evaporation, and condensation. The model self-consistently solves the circuit current and voltage, the plasma channel parameters including composition, temperature, conductivity, pressure, etc., and the pressure profile at a certain distance from the discharge channel. The calculated results show good consistency with the experimental measurements, and three sets of experimental results from other literature are tested to further verify the applicability and effectiveness of the model.

Numerical Analysis of Aerodynamic and Shock Wave Characteristics of Biconvex and Double-Wedge Shape Airfoils for Supersonic Flow

This present study describes the aerodynamic characteristics of supersonic flow over biconvex and double wedge airfoils using a finite volume method-based commercial CFD code Ansys Fluent. A steady-state RANS approach is used with SST k-ω viscous modeling. A series of simulations are conducted to analyze the characteristics of shock and expansion waves formed around the airfoils for Mach numbers ranging from 1.4 to 3.4 with varying angles of attack (α) from 0° to 20°. It is observed that the lift and drag coefficients both increase with the angle of attack for a fixed Mach number and decrease with the Mach number for a fixed angle of attack. Double wedge airfoil generates about 5% more lift at a low Mach number and 1% more lift at a higher Mach number compared to the biconvex airfoil. However, the biconvex airfoil generates lesser drag than the double wedge airfoil. The maximum value of the pressure coefficient (Cp ) is found to be 1.7 for biconvex airfoil and 1.4 for double wedge airfoil. The maximum value of the lift-to-drag ratio for biconvex airfoil is 7.63, occurs at 1.4 Mach number and 3.46° angle of attack, whereas the value for double wedge airfoil is 5.19 at the same Mach number with 4.47° angle of attack, which suggest that biconvex airfoil has a higher lift-to-drag ratio and gives a better aerodynamics performance. The shock waves start to detach after an angle of attack of 5° and the shock wave is fully detached at a 15° angle of attack for biconvex airfoil for Mach number of 1.4. For the same Mach number, the double wedge airfoil, the shock wave starts to form the same as the biconvex airfoil but the waves are fully detached at a lower angle of attack of 10°. With the increasing Mach number, the shock waves remain attached to the airfoil.

Underwater electrical explosion of metallic wire and tube: A comparison study

With an initial energy storage of approximately 53.5 kJ, experiments on underwater electrical explosions of metallic wire and thin-wall tube were conducted. Two sets of controlled experiments were designed, and the wire and tube for each set were of the same length, cross-sectional area, and material. Load voltage, circuit current, and shock wave pressure were measured and analyzed, and electric power, energy deposition, and energy deposition rate were also calculated and investigated. Experimental results indicated that the underwater electrical tube explosion (UETE) always has higher and earlier local resistance peak, voltage peak, and electric power peak, as well as faster phase transition and ionization process than the underwater electrical wire explosion (UEWE). In addition, the energy deposition rate from the beginning of discharge or the decrease in current to the voltage peak is significantly increased when replacing a wire with a tube. For the shock-wave characteristics, the peak pressure, energy density, and impulse of the shock wave generated by UETE are greater than that generated by UEWE. The peak pressure at ∼33 cm increased from 21.1 to 24.5 MPa in the first set and from 18.1 to 21.7 MPa in the second when a tube was used instead of a wire. These experimental results help us understand the physical process of UETE and provide an alternative methodology for load design in industrial applications.

Characterization of internal discharge shock waves of gas‐insulated switchgears using shadowgraphy

AbstractThe gas‐insulated switchgear (GIS) is a vital component of a power system, and understanding the shock wave characteristics of its internal discharge is crucial to ensure the safe and stable operation of the entire system. In this study, a high‐speed shadowing technique is employed to obtain the shadowgraphs of a typical flow‐field evolution of sulfur hexafluoride (SF6) gas following the breakdown of a pin–plate gap inside a GIS. Experimental data were used to derive parameters, including the Mach number, propagation speed, distance of the discharge shock wave, and the temperature, pressure, density, and velocity of the SF6 gas after wave generation. With a 0.2‐mm discharge gap and 0.4‐MPa pressure, the discharge generates a spherical shock wave, centred at the contact between the discharge channel and electrodes, that propagates in all directions. The shock wave gradually weakens after 20 μs, propagating at near‐sonic speed, and its Mach number decreases from 2.92 to 1.15; similarly, its post‐wave parameters sharply decrease during the first 20 μs. Additionally, a significant reflection occurs when the shock wave propagates to the metal wall. The authors’ findings provide a valuable reference for detecting and diagnosing the internal discharge in GISs.

Molecular dynamics simulation of the heterostructure of the CoCrFeMnNi high entropy alloy under an impact load

There have been numerous experimental studies conducted on the CoCrFeMnNi high entropy alloys (HEAs) at the macroscopic level. However, it is challenging to quantitatively analyze the shock behavior of the HEAs from a microscopic level through experiments. In this study, we construct single-crystal, twin-crystal, multilayer, hole, and two-phase structures of the CoCrFeMnNi HEAs using the molecular dynamics method. The effects of impact loading on the microscopic level are investigated for CoCrFeMnNi HEAs with different structures. By analyzing the evolution of their microstructure and the changes in physical parameters, the response laws and propagation characteristics of shock waves in various heterogeneous of CoCrFeMnNi HEAs are revealed at the atomic scale.

Numerical analysis of the effect of ventilation door on the propagation characteristics of gas explosion shock waves

Ventilation door are commonly found in tunnels and other underground engineering ventilation structures, disaster periods using its explosion isolation, explosion relief, wind regulation characteristics for disaster prevention and mitigation is of great significance. This paper numerically simulates the propagation characteristics of the gas explosion shock wave in the nearby tunnel when the ventilation door are opened at different degrees, and analyzes the influence mechanism of the opening degree on the change law of the shock wave overpressure distribution in the nearby tunnel. The results show that the shock wave forms a strong turbulence area (high pressure area) on both sides in front of the ventilation door, and the area range and the overpressure value decrease with the increase of the opening degree; the ventilation door reduce the intensity of the shock wave, so that the overpressure behind the ventilation door decreases, and the smaller the opening degree, the lower the overpressure behind the ventilation door. The secondary explosion formed shock wave and the ventilation door reflected shock wave meet to form a stronger shock wave, which leads to different opening degrees of ventilation door, its before, after the roadway and after the bifurcation of the main roadway in the measured points of the overpressure change curve is different, the main difference is that the peak overpressure for the first wave or the second wave peak. The peak overpressure in the tunnel before and after the ventilation door decreases and increases respectively with the increase of the opening length, and the overall decay of the peak overpressure at 5 m and 10 m before the ventilation door is 49.56% and 4.04% respectively and only has an effect on the peak overpressure in main tunnel within 20 m from the bifurcation.

Study of the coupling characteristics of damage elements with cased charge

In this paper, the coupling characteristics of fragments, detonation products and explosive shock wave have been studied. Although a large number of studies have been conducted to discuss the damage effects of structures under the combined loading of blast and fragments impact, there is a lack of research on the coupling characteristics of damage elements and their mechanism of action. This study uses the law of conservation of energy to propose a more reasonable method to evaluate the velocity of fragment. In addition, the isentropic expansion theory of bare charge detonation products is used to explore the expansion characteristics of cased charge detonation products, and the expansion process of cased charge detonation products can be divided into three phases. The results show that the limiting expansion volume of detonation products with cased charge is smaller than that of bare charge due to the inhibiting effect of the casing. Taking into account the energy absorbed in the crushing of the casing, the shock wave produced by the explosion of the cased charge is equal to the shock wave produced by the explosion of a given amount of bare charge, and the corresponding formula is given. Then, based on the known performance parameters of each damage element, the arrival time of the damage element at the target structure is systematically analyzed, and the coupling of the damage element is divided into five phases according to the arrival time, and the range of damage effect of each coupling phase is analyzed in two cases. Finally, a numerical simulation based on Euler's method is performed, and the theoretical results agree well with the numerical simulation results.

On the characteristics of DA shock waves in a polarized electron-depleted plasma with a mixed nonextensive high energy-tail ion distribution

A first theoretical attempt is made to understand the characteristics of dust acoustic shock waves (DASWs) in a polarized electron-depleted plasma with a mixed nonextensive high energy–tail ion distribution and to examine the impact of the generalized nonthermal polarization force on DASWs in an electron-depleted dusty plasma. First, a short numerical investigation illustrating the behavior of the mixed nonextensive high energy–tail (or Cairns–Tsallis) ion distribution and the domain of validity of its two parameters (q and [Formula: see text]) is carried out. Next, the generalized nonthermal polarization acting on dust grain in electron-depleted dusty plasma is determined within the context of Tsallis’ statistical mechanics. The behavior of this generalized polarization force is significantly affected by the nonextensive character of the nonthermal ions. Specifically, we have shown that, for both space and experimental dusty plasmas, the magnitude of the polarization force (in the case where [Formula: see text]) decreases as the nonextensivity of the nonthermal ions becomes significant. As an application, we have analyzed the changes caused by the nonthermal nonextensive polarization force on the main characteristics of DASW (viz. phase velocity, polarity, amplitude, width, etc.). Numerical results showed that our plasma model supports rarefactive and compressive DASWs that are significantly affected by the effects of nonthermal ion nonextensivity and polarization force. In particular, we have shown that for large values of the nonthermal parameter [Formula: see text], the increase in q allows the transition from rarefactive to compressive DASW structures. The present theoretical investigation is helpful to fully understand electrostatic disturbances in space and experimental dusty plasmas.

Attenuation characteristics of shock waves in drilling and blasting based on viscoelastic wave theory

This study investigates the propagation and attenuation characteristics of shock waves produced by drilling and blasting. The focus is on a novel method for calculating the attenuation of the shock wave pressure. Based on the propagation of shock waves in a rod with a variable section and a Maxwell model for rock, theoretical solutions for the attenuation of the blast-induced shock wave pressure were derived and the main factors influencing the attenuation speed were analyzed. The attenuation law of shock waves for siltstone, limestone, and granite, with common borehole diameters of 76–150 mm and initial shock pressures of 1–6 GPa, was analyzed through numerical simulations. The results show that the shock wave falls off as the square root of the relative distance. The attenuation speed of the shock wave is mainly related to the borehole size, rock properties, and shock wave intensity. The attenuation speed increases with increasing initial shock pressure on the borehole wall, but decreases with increasing borehole radius. Compared with soft rock, the shock wave pressure attenuates faster in hard rock. Combining theoretical analysis and numerical simulations, an exponential attenuation coefficient was introduced to the attenuation formula. Statistical analysis shows that the exponential attenuation coefficient is linearly proportional to the initial shock pressure. Furthermore, a new method for calculating the attenuation of shock wave pressure was proposed, reflecting the comprehensive influence of rock characteristics and shock wave intensity. The new method is an improvement over existing methods, which only consider the effect of rock properties on shock wave attenuation. The reliability of the new method was verified against the results of field tests. The findings provide a theoretical basis for the optimization of blasting parameters and assessments of blast-resistance for structures.

Numerical simulation of angled-injected liquid jet breakup in supersonic crossflow by a hybrid VOF-LPT method

The breakup of angled-injected liquid jets in supersonic airflow is investigated numerically by a hybrid Volume of Fluid and Lagrangian Particle Tracking (VOF-LPT) method. A Multi-criterion adaptive mesh refinement (AMR) procedure and dynamic load balancing (DLB) algorithm are applied to improve the accuracy of interface and shock wave characteristics and reduce the use of computational resources and liquid mass loss. The flow characteristics of the spray field and penetration depth of the angled-injected liquid jet from the simulations agreed well with the experimental results. Under the supersonic crossflow conditions, the jet has momentum in the counter-flow direction that improves gas-liquid interactions. The penetration depth of the liquid jet increase with the increase of the injection angle. In particular, the penetration depth of the angled-injected liquid jet is given in the: y/d=0.12·sin(2θ/3)·(esin(2θ/3))3.185·q0.389(x/d)0.309. Moreover, the liquid jet at a larger injection angle has a larger spray spread angle and wider wake region due to the larger windward area. Furthermore, the total pressure loss of airflow increases with the injection angle increasing. Considering the total pressure loss for all injection conditions is lower than 14%, the total pressure loss caused by the injection angle increase can be negligible.