The experimental system was composed of four parts: 1) a propellant feed system, 2) a measurement and control system, 3) an image acquisition system, and 4) a bench system, as shown in Figure 1A. The propellant feed system was composed of a high-pressure gas source, gas conduit, liquid conduit, propellant tank, and other components. To achieve a smooth flow supply to the liquid injector, the propellant (water) storage tank was pressurized with compressed air. Meanwhile, the air was delivered by a high-pressure gas source to the gas injector through the conduit via the pressure-reducing valve. The pressure of the liquid and gas manifolds was measured and recorded using a pressure sensor: OHR-M2G-2-L-C-0-1.6 MPa with an accuracy of 0.5% FS. The mass flow rates of the water and pressurized air were measured and recorded using an LWGY turbine flowmeter and a Coriolis mass flowmeter (MFC608), with an accuracy of 0.5% FS, respectively. The in-line image acquisition system was composed of a high-speed camera, shading plate, and rectangular LED background light source (HLS-30, power: 250 W). The distance of each component could be adjusted to ensure a fine spray transient image. During the experiments, the recording rate of the camera was set as 31,862 fps, the exposure time was 1/95,586 s, and the resolution of the transient image was 800 × 600 pixels.

A gas-centered shear coaxial injector was studied in the experiment, as shown in Figure 1B, in which the outer injector was a coaxial annular aperture, and the inner injector was a central circular hole. The liquid fuel entered the injector through the coaxial annular aperture, and the gaseous oxidizer was fed through the central hole. They interacted with each other after spraying out from the injector. For the recessed injector, the gaseous oxidizer was ejected from the inner injector and expanded to compress the liquid fuel; thus, the oxidizer and fuel were initially mixed inside the recess chamber. The liquid fuel was jetted from the outer injector to form a liquid sheet near the injector outlet and then was broken into liquid blocks and droplets after a certain distance (breakup length) downstream. Recess is an important parameter which determines propellants’ mixing efficiency and even flame stability (Xu, 2016). Three injectors with different recess lengths were applied in the experiment. The recess length was the distance from the outlet section of the inner injector to the outlet section of the outer injector (Chu et al., 2021), as shown in Figure 1. The key geometrical parameters of the injector are listed in Table 1, and the RR is defined as follows: R R = L r d g .

In this study, the spray characteristics and self-pulsation of the gas-centered shear coaxial injector under different working conditions and recess ratios were mainly explored. To ensure the comprehensiveness of the experimental study, the operating conditions in the present experiments were denser, as shown in Table 2, whereas the errors of the gas and liquid mass flow rates were less than 4.2 and 6.7%, respectively. The m ˙ l is set from 30 g/s to 120 g/s with a step of 10 g/s, and m ˙ g increased from 4 g/s to 25 g/s with a step of 1 g/s.

To obtain the spray angle, self-pulsation frequency, and other information, 2000 images were taken for each working condition and processed by the in-home code.

The extraction of the spray angle was performed using the clustering method. First, all the spray images were processed to obtain the time-averaged image; second, the weighted image was obtained by the background, and a time-averaged image was multiplied with the weighted image to eliminate the effect of uneven illumination; third, the time-weighted filtering image was obtained by Gaussian filtering, which was followed by the calculation of the average grayness of each class based on the K-means segmentation algorithm (Arthur and Vassilvitskii, 2007) to obtain the background image classification, and moreover, the background binary image was obtained by morphological filtering. The boundary coordinates were found to extract the spray boundaries, calculating the left and right boundaries of the spray to fit the straight lines l ₁ and l ₂ . Due to the spray being completely broken and unfolded only in the downstream direction and to avoid the interference of the background (as marked by the red circle in Figure 2A), the fitted straight line was taken in the 50–75% interval downstream of the spray. Finally, the included angle between the two fitting lines was calculated by the program formula to obtain the spray angle, as shown in the following equation: l 1 : y 1 = k 1 x + b 1 ,   α 1 = arctan ⁡ k 1 , (1) l 2 : y 2 = k 2 x + b 2 ,   α 2 = arctan ⁡ k 2 , (2) α = 180 ∘ − | α 1 − α 2 | . (3)

The self-pulsation frequency was obtained by first subtracting the background to obtain the grayscale image after background removal, followed by the binarization of the image after background removal, which was then processed to obtain the binarized image. At a certain distance from the injector outlet, a measurement line was selected (which is 2 mm below the injector for this study), and the spray width information reflected by the white area on the measurement line was extracted to obtain the spray width of each image. Finally, the frequency involved in the self-pulsation of the spray was obtained by the time series of the spray width subjected to a fast Fourier transform (FFT) (Xu, 2016), as shown in Figure 2B.

The governing equations include the mass, momentum, and energy conservation equations, which are described by the governing equations of fluid motion. Incompressible fluid treatment should be conducted if the flow velocity of the liquid is low. Moreover, the incompressible gas treatment should be conducted when the flow velocity of the gas is lower. However, the density change can no longer be neglected if the flow velocity of the gas is larger (M _a > 0.15) (Luo, 2017). In that case, it should be treated as a compressible fluid, that is, the energy equation should be considered.

Conservation of mass equation: ∂ ρ ∂ t + ∇ ⋅ ( ρ U ) = 0.

Conservation of momentum equation: ∂ ∂ t ( ρ U ) + ∇ ⋅ ( ρ U U ) = − ∇ p + ∇ ⋅ ( τ = ) + ρ g + F .

Conservation of energy equation: ∂ ∂ t ( ρ E ) + ∇ ⋅ [ U ( ρ E + p ) ] = ∇ ⋅ ( k e f f ∇ T − ∑ j h j J j + ( τ = e f f ⋅ U ) ) + S h . Whereas: τ = = μ [ ( ∇ U + ∇ U T ) − 2 3 ∇ ⋅ U I ] , E = h − p ρ + U 2 2 .

The realizable k-ε model was chosen as the turbulence model, for which the constraint on the Reynolds stress was satisfied. Therefore, it can be kept consistent with the real turbulent flow in terms of the Reynolds stress. One of the advantages of this feature in the calculation is that the diffusion velocity of planar and circular jets can be simulated more accurately. Meanwhile, the calculated results could be more consistent with the real situation in problems such as the calculation of rotating flow, calculation of the boundary layer with a directional pressure gradient, and separation flow calculation.

The volume-of-fluid (VOF) method can 1) easily calculate complex phase interface change processes, 2) capture the particularly sharp phase interfaces, and 3) have good volume conservation and other advantages. Its disadvantages are that it may have difficulty in accurately calculating the curvature and physical quantities related to curvature, which results in an increase in the use of computational cost to capture the phase interface location in detail if the encryption of the grid at the phase interface becomes necessary. The level set method has the following advantages: 1) it can accurately calculate the curvature and curvature-related physical quantities, which does not need to reconstruct the interface; 2) it has stronger topological description ability, and 3) the phase interface calculated based on the level set method is smoother than that using the VOF method. Its disadvantage is that it faces difficulty in accurately simulating sharp interfaces with relatively poor volume conservation. CLSVOF combined the VOF method with the level set method, which has the advantages of those two methods. CLSVOF has an excellent capture of the phase interface in two-phase flow simulations (Wang et al., 2016).

The realizable k-ε model and the controlling equations of the CLSVOF are described in detail in the literature (Chu et al., 2020).

Owing to the multiscale characteristics of the atomization phenomenon, to fully collect the motion and breakage details of the liquid film and reduce the numerical errors in the numerical solution, the AMR method, which was based on the gradient of the liquid volume fraction (α), was used in this study to encrypt the mesh to accurately track the gas–liquid interface (Wang et al., 2021), as shown in Figure 3B. Refinement is realized when the following conditions are satisfied: h | ∇ α | > ε ′ , where h is the grid scale, and ε ′ is a constant with a small value. Here, ε ′ = 0.05 . In other words, the grids at the gas–liquid interface are refined. The mesh refinement interval was 10 time steps with three refinement levels.

A schematic representation of the calculation model is illustrated in Figure 3A. It has a gas injector bore diameter of 3 mm and a liquid injector bore diameter of 4.5 mm, among which the vertically downward direction was the positive X-axis, the coordinate origin was located at the center A (0, 0, 0) of the gas injector inlet, and the pressure monitoring point was located at Point B (80.2, 2.25, 0).

The inlet boundary condition of the mass flow rate was applied to the inlet of the gas and liquid, the outlet boundary was set as the pressure outlet boundary, and the wall surface was a non-slip wall boundary. The gas was simulated using air, and the liquid was simulated using water. The pressure discretization method (PRESTO) and the pressure-velocity coupling method (PISO) were applied. Furthermore, the second-order upwind scheme discretization was applied in the momentum and other equations, and the phase interface reconstruction technique (Geo-Reconstruct) was used. Moreover, the simulation accuracy of the wall part was improved by non-equilibrium wall functions. At the initial moment, all the fluid in the calculation region was air. The compressibility of air was considered, and its viscosity was calculated using the Sutherland method, as shown in the following equation: μ μ 0 = ( T L T L 0 ) 3 / 2 T L 0 + B T L + B ,

where B ≈ 110.4 K. The equation is applicable to air over a fairly large range ( T L < 2000   K ) .

For the acquisition of the self-pulsation frequency in the simulation, first, the pressure pulsation curve of monitoring point B (Case 0-80-5) was monitored, and then, its frequency was obtained through FFT (Chu et al., 2021), as shown in Figure 4A.

The effects of different recess ratios and gas-liquid mass flow rates on the spray characteristics were studied. The simulation working conditions are listed in Table 3.

Case 0-80-5 was selected for the grid independence study. Three different grids, which initially (before AMR) contain 330,000 (coarse), 630,000 (intermediate), and 930,000 (fine) grids, respectively, were utilized to check the dependence of the grid size. The pressure pulsation frequency at the monitoring point B on basis of those three grids is plotted in Figure 4B. Compared with the self-pulsation frequency based on experimental data (5,286 Hz), the frequency errors obtained from the coarse grid calculation (10.6%) were greater than those obtained from the intermediate (4.7%) and fine grid calculations (0.8%). To reduce the calculation cost and ensure its accuracy, an intermediate grid was used for the calculation in this study.

For injector without recess, the spray angle first stayed almost constant with the variation of m ˙ g (4 ≤ m ˙ g ≤ 6). However, as m ˙ g continued to increase, the spray angle dropped linearly. On the other hand, for injectors with recess (RR = 1.25 and RR = 5), the spray angle got larger with m ˙ g . Holding m ˙ l = 60   g / s , spray images and the changes of spray angles versus m ˙ g are shown in Figures 5A,B, respectively. Here, the gas–liquid ratio (GLR) of mass flow rates is defined as follows: G L R = m ˙ g m ˙ l .

For an injector without recess, self-pulsation occurred at a low GLR (the details of self-pulsation are described in section 4.2). The spray pulsations in the radial direction led to an increase in the radial distribution of the droplets, resulting in an enlarged spray angle. The self-pulsation phenomenon of the spray was enhanced with the increasing GLR (0.07 ≤ GLR≤0.10). However, because the spray pulsations were very intense, the increase in the spray angle was not particularly significant. In other words, the spray angles stayed almost constant. When the GLR continued to increase, the axial velocity of the central airflow increased sharply, and the strong ejecting effects of the central airflow made the liquid film outlet completely broken. At the same time, the strong ejecting effects made the liquid converge toward the axis, which resulted in the spray angle dropping linearly. Even though there were still self-pulsations in the first working conditions (0.10 < GLR≤0.17), the promotion effect of spray pulsation on the spray angle was inferior to the weakening effect of ejecting effects on the spray angle. Meanwhile, the promotion effect of the expansion of the central airflow outlet on the spray angle was inferior to the weakening effect of ejecting effects on the spray angle. A larger GLR led to a stronger ejecting effect of the central airflow and hence minimizes the spray angle.

In the current study, the Mach numbers at the outlet of the gas injector were greater than 0.15. In other words, the gas could not be treated as an incompressible flow. The gas expanded at the injector outlet, and the liquid sheet was directly affected by the gas flow. For the injector with recess, the center gas experienced two expansions, as shown in Figure 6A. The first expansion occurred in the recess chamber after the center gas flowed out of the inner injector. Thereafter, the gas got expanded once more downstream of the outlet of the outer injector. However, for the injector without recess, the center gas flow experienced only one expansion at the outlet of the inner injector.

Under conditions with a low GLR (0.07 ≤ GLR<0.13), spray self-pulsation was found for injectors with recess. However, spray angles for those injectors were relatively smaller than those for injectors without recess. It was concluded that the radial diffusion of the liquid film is limited by the recess chamber wall. At the same time, recess can lead to a strong gas–liquid interaction. In other words, the central airflow has strong momentum transfer to the liquid film in the recess chamber. These reduced the radial velocity of the liquid film at the injector outlet, which resulted in a reduction in the spray angle. With the increase in RR, the recess chamber wall limited the radial diffusion of the liquid film for a longer time, and the acting time of the central airflow with the liquid sheet was longer. The radial velocity of the liquid sheet was reduced more significantly, thus further reducing the spray angle. However, with an increase in the GLR, the secondary expansion of the central airflow at the outlet of the outer injector became more and more significant. The expansion gas transferred a greater radial velocity to the liquid sheet, which resulted in an increase in the radial velocity of the liquid sheet and a subsequent increase in the spray angle. However, under a low GLR, the secondary expansion of the central airflow was insignificant and contributed little to the radial velocity of the liquid sheet. Moreover, the above-mentioned observations were confirmed through simulation. The Z-directional velocity contours for case 1.25-60–4, case 1.25-60–15, and case 5-60–15 are shown in Figure 6B. The two airflow expansions related to case 1.25-60-4 were minimal, whereas the two airflow expansions related to case 1.25-60-15 and case 5-60-15 were significant.

When conducting the experiment under specific working conditions, the injector produced a self-pulsation phenomenon, accompanied by a loud scream, and the self-pulsation frequency increased with m ˙ g , as shown in Figure 7A. However, for a constant m ˙ l (such as m ˙ l = 80   g / s = 80 g/s), when m ˙ g reached a critical value (here m ˙ g = 8   g / s ), the self-pulsation phenomenon disappeared.

Similar results were obtained from the simulation calculations. As shown in Figure 7B, under m ˙ l = 60   g / s and the gas mass flow rates at 4, 7, 8, and 15 g/s, the experimental images are compared with the simulated spray images. A comparison between the experimental and simulated frequencies is shown in Table 4. The error between the simulated and experimental frequencies was relatively small, verifying the reliability of the simulation model. The gas–liquid interface in the simulation was represented by the iso-surface with α= 0.25 (Li et al., 2019).

When m ˙ g was small, the self-pulsation phenomenon was caused by the gas–liquid interaction at the injector outlet. The self-pulsation spray was formed in the shape of a “Christmas tree”. Moreover, the structure of the spray comprised “shoulder” and “neck” sections, which appeared alternately to form the self-pulsation phenomenon of the spray. As shown in Figure 7B, the spray pulsations were violent near the injector outlet, whereas the downstream pulsation was weakened. In other words, the self-pulsation phenomenon had a greater impact on the spray characteristics near the injector outlet and less impact on the downstream. With an increase in m ˙ g , the central airflow of the ejector function was enhanced, the spray gradually gathered toward the axis, and the intensity of the pulsations and breakages of the liquid sheet increased. When m ˙ g reached a certain value and above, the self-pulsation phenomenon disappeared, and strong aerodynamic fragmentation of the annular aperture liquid sheet was observed.

Meanwhile, a consistent pattern was observed in the experimental and simulation results; the frequency of self-pulsation increased with m ˙ l at a constant m ˙ g . The variation in the self-pulsation frequency with m ˙ l for fixed gas mass flow rates and the comparison between the experimental and simulated frequencies are shown in Figure 8A. A comparison between the experimental and simulated spray images for m ˙ g = 4 g / s and liquid mass flow rates of 40, 60, 80, and 120 g/s is shown in Figure 8B. The minimum error between the simulated and experimental frequencies was 0.80%, and the maximum error was 9.79%.

Analysis suggests that the injector self-pulsation resulted from the combined action of the pressure difference between the inside and outside of the liquid sheet, the surface tension of the liquid sheet, and the impact extrusion of the central airflow. For example, in case 0-60-4, the liquid sheet flowed out from the outer annular aperture injector outlet at 1/4T. The airflow in the center of the annular liquid film was fast, while the airflow outside the annular liquid film was approximately in a static state, as shown in Figure 9A. In accordance with the Bernoulli equation, the outer pressure of the liquid sheet was demonstrably higher than the inner pressure. Owing to the action of the pressure difference between the inside and outside of the liquid sheet, the liquid film of the annular aperture injector tended to shrink toward the central axis shortly after it flowed out of the injector outlet. As the liquid sheet underwent contraction, the “neck” and “shoulder” sections of the spray formed gradually at 2/4T. As the “neck” section blocked the central gas channel, gas accumulation occurred in the inner side of the annular liquid film and caused the pressure inside the annular liquid film to rise. The pressure difference between the inside and outside of the liquid film began to reverse. When the “neck” section shrank to its minimum size, the pressure difference between the inside and outside of the liquid sheet attained the maximum value (as shown in Figure 10B at 3/4T). Concurrently, the effect of surface tension of the liquid film during contraction caused the downstream liquid film of the “neck” section to become weak. Subsequently, under the combined action of the pressure difference and the impact extrusion of the central gas flow, the “neck” broke, and the gas channel opened (the process was also caused by the combined action of the K-H instability and R-T instability). Then, the spray entered the next pulsation cycle. Substantial momentum was transferred at the “neck” fracture owing to the “breaking out of the liquid film” of the central airflow (as shown in Figure 9A at 1T) such that the droplet and liquid block gained a large radial velocity, thus increasing the spray angle. The self-pulsation spray was formed in the shape of a “Christmas tree”. The droplet clusters were periodically produced and moved downstream with the spray.

Meanwhile, the flow characteristics during the cycle were evident, as shown in Figure 10, which shows the velocity contour, pressure contour, and streamline chart during one cycle of case 0-60-4. As the liquid sheet began to contract, the gas channel at the contraction narrowed at 1/4T, which resulted in an increase in the velocity of the airflow. However, the velocity of the central airflow and the inner part of the liquid film of the annular aperture was reduced by shearing. This resulted in the formation of two counter-swirl reflux zones at the gas injector outlet, with two vortices in opposite directions. The “neck” formation began at 2/4T, and the increasing length of the shearing effect between the central airflow and the inner side of the annular aperture liquid sheet resulted in an increase in the counter-swirl reflux zones. Thus, the two opposite vortices formed between the outlet of the gas injector and the inside of the liquid film increased, and this growth was apparent at 3/4T. Finally, when the “neck” section broke at 1T, the central airflow “broke out of the liquid film” with the opened gas channel. The two vortices in opposite directions formed in the inner part of the “neck” section dissipated rapidly. At the same time, two larger opposing vortices at the “broken” section owing to gas-liquid entrainment and some small and complex vortex systems were generated simultaneously.

When the m ˙ g increased, the central airflow was accelerated. The iterative speed of the above-mentioned processes, such as the contraction of the liquid sheet, was accelerated. This resulted in an enhanced frequency of self-pulsations, and a stronger momentum was transferred when the central airflow “broke out of the liquid film”. The spray pulsation within one cycle for case 0-60-8 is shown in Figure 9B, and its spray pulsation is similar to that of case 0-60-4. However, compared to case 0-60-4, when the “neck” broke at 1T, the action of “breaking out of the liquid film” provided a greater radial velocity to the droplets and liquid blocks at the “neck” for the increase in m ˙ g . It resulted in the spray width of the “neck” being larger than that of the “shoulder”. In other words, compared to case 0-60-4, the comparison of the spray width values for the two above-mentioned values was reversed.

The experimental results revealed similar changes in behavior for the injector with and without recess. In other words, when m ˙ g was low for a constant m ˙ l , the phenomenon of self-pulsation occurred and was accompanied by a scream, and the frequency of self-pulsation increased with m ˙ g until it subsided. However, for injectors with recess, the frequency of self-pulsation significantly decreased compared to those without recess. The variations in the self-pulsation frequency at different gas mass flow rates for m ˙ l = 40   g / s are illustrated in Figure 11A.

The central gas flow in the recess chamber was assumed to have produced the entrainment and ejection functions on the annular aperture liquid sheet. On the one hand, the friction between the gas and liquid hindered the flow of the liquid sheet, which prolonged the period of liquid sheet pulsation, that is, the frequency of self-pulsation decreased. On the other hand, the wall limited the radial velocity of the liquid sheet, which reduced the amplitude of radial pulsation downstream of the outlet, that is, the intensity of self-pulsation decreased. When RR increased, the friction between the gas and liquid further hindered the flow of the liquid sheet. Therefore, the frequency of self-pulsation was further reduced.

Numerical simulations of the injector with recession showed that it also underwent a similar self-pulsation cycle as the injector without a recession, as shown in Figure 11B, for the comparison of the spray images of cases 0-60-4 and 1.25-60-4. However, as mentioned earlier, as the recess chamber wall also limited the radial velocity of the liquid sheet, the radial pulsations downstream of the exit amplitude were reduced. This demonstrated that the width difference between the “shoulder” and “neck” of the spray decreased, that is, the spray angle decreased. The spray pulsation process in one cycle of the working condition (frequency of 4,354 Hz obtained from the simulation) is depicted in Figure 9C. In this period, the spray pattern was more distinguishable as a “string” shape than a “Christmas tree” shape.