Introduction

In high-speed propulsion systems, efficient fuel mixing plays a critical role in optimizing combustion. The configuration of fuel injectors, particularly the spacing between them, has a significant impact on the mixing dynamics of fuel jets with the surrounding airflow. Understanding how different injector configurations influence the fuel penetration, circulation formation, and interaction with free stream flow is essential for enhancing fuel-air mixing efficiency, which directly affects combustion stability and efficiency1,2,3.

Computational Fluid Dynamics (CFD) plays a crucial role in the modeling and analysis of fuel injection systems, particularly in high-speed propulsion applications. The use of CFD techniques enables the detailed simulation of complex fluid flow behaviors that are difficult or impossible to capture through experimental methods alone. In the context of the proposed injection system, CFD offers several advantages that are vital for optimizing the injector design and understanding the underlying physical phenomena4,5,6.

CFD provides detailed insights into the fuel-air mixing process, including the visualization of shock waves, circulation zones, and velocity gradients7,8,9. This level of detail helps engineers understand how different injector configurations—such as the gap between injectors—affect the flow patterns and overall fuel mixing efficiency. By simulating the entire flow field, CFD allows the identification of critical areas where fuel mixing can be improved, such as the regions behind the strut where circulation forms10,11,12.

This study investigates the effects of varying injector gaps on fuel mixing and circulation patterns behind a strut in supersonic flow. By comparing two injector configurations—one with a low gap and the other with a high gap—the study aims to highlight how these variations influence key phenomena such as the formation of circulation zones, jet penetration, and interaction with the free stream13,14,15. The behavior of the fuel jets is evaluated using Mach contours, fuel mixing zones, and vortex formation, with a particular focus on how changes in injector spacing impact the strength and location of circulation behind the strut. Additionally, the study explores how these factors affect fuel mixing efficiency and jet interaction over different downstream distances16,17,18.

Through a detailed analysis of the mixing zones and shock wave interactions, this research provides insights into the influence of injector spacing on fuel-air mixing, offering valuable guidance for the optimization and design of fuel injection systems in high-speed propulsion applications19,20,21.

Fig. 1
Fig. 1
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Strut with multi orthogonal injectors.

This investigation determines the fuel jet mixing and penetration of hydrogen behind the ramp injector at supersonic combustion chamber. Computational fluid dynamic is used for the modeling of the hydrogen sonic flow and supersonic free stream inside the combustor via solving RANS equations. Effects of jet space on the distribution of hydrogen jets are also investigated to reveal the role of the jet interactions inside the combustor with ramp configuration. Stream and mixing zone behind the ramp are also evaluated and compared to discriminate the efficiency of these two models. Three-dimensional model of these two configuration are used to present real flow structure inside the combustor.

Governing equations and applied methodology

Analyze of the real three-dimensional highly compressible flow is complex task since the contact of the free stream and fuel jet results in the complicated flow feature22,23,24. The modeling of the flow without reactions has been done by using RANS equations and assuming flow as ideal gas is almost acceptable for the fuel mixing of hydrogen in the combustor of scramjet25,26. Due to complexity of flow and formation of vortex, SST model of turbulence is highly recommended in the previous articles27,28. The main flow is air stream and secondary gas is hydrogen, consequently mass equation is also coupled with main RANS equations29. The energy equation is also solved due to formation of the expansion and shock wave in our model. The main equations of RANS and coupled transport and energy equations are fully presented in the available published articles30,31.

The main configuration of the jet configurations and the applied conditions on boundaries for the designated domain are exhibited in Fig. 1. The injectors are located behind the ramp and the angle of ramp is 30 degrees. The height of strut is 10 mm and the jets have located with equal distance in case 1. The gape (h) of the nozzles in the case 1 and case 2 are 3.3 mm and 5 mm, respectively. Free stream air flow with Mach = 2 and Pt = 2 bar is applied at inlet and the outflow is obtained via extrapolation of the flow at outlet. The hydrogen gas is injected with sonic velocity and total pressure of 3 bar. The symmetry is applied on the side of the domain to reduce the computational cost. The spanwise length of strut is half of overall width of domain. The diameter of the circular jets is 0.5 mm and the size of entire computational domain is 110 mm.

Fig. 2
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The grid generation.

The generated grids for the modeling of these two cases are exhibited in Fig. 2. The organized grid with variable resolution is considered for the model. The grid study is done with various cells to ensure the size of the generated grid is acceptable and attained results are not dependent to the number of the generated grid. As offered in Table 1, four grids are examined and the value of the average hydrogen gas concentrations are compared on the plane located 20 mm behind the injector for these grids32. The evaluation of the grid indicates that the grid 3 is acceptable since the increasing of the grid cells does not results in the meaningful change in the average of the hydrogen gas downstream of the injector. The Y + is also below 4.8 for the fine grid which is also reasonable for the SST turbulence model.

Table 1 Grid study.
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Results and discussion

The authentication of the numerical model is done by equating the normalized pressure behind the jet with experimental data Donohue et al.33. There is good agreement between experimental data and our achieved data (Table 2).

Table 2 Validation of the normalized pressure (P/Pinf).
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The evaluation of the mixing zone of fuel on the mid-plane of these two models is demonstrated in Fig. 3. When the gap of these two injectors is low, the performance of the fuel jet is almost near the single expanded jet. While the mechanism of mixing zone differs in the model with high gap. In the model with high gap of injector, the circulation is produced and accordingly, the fuel mixing is boosted. Besides, the fuel penetration is also extended and mixing zone fully covers back of the strut. The fuel concentration of the jet decreases by the formation of the circulation in the model with high gap distance.

Fig. 3
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Comparison of the fuel jet mixing behind the strut.

Figure 4 displays the Mach contour on the mid-plane in these two models. As the gap of these two jets is high, the effects of the free stream and the bottom wall on the shock waves (i.e. barrel shock, ) that are produced near the injector. As demonstrated in the image, the deformation of the barrel shock near the injector confirm this effect on model with high jet gap. The subsonic region is produced in the distance of these injector and consequently, this region is potential for the creation of the circulation. In the Case 1, the jet interactions are important while free stream flow has little impacts on the interaction behind the strut.

Fig. 4
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Comparison of Mach contour.

The three-dimensional feature of jet flow behind the strut in these two jet configuration is verified in Fig. 5. This image shows the main difference of the jet flow behind the strut and the interactions of the free stream with fuel jet flow are demonstrated to disclose the real physics associated with the jet gap on the mixing of the multi fuel jet behind the strut. The free stream enters to the wake behind the strut in two passages: side and top. As the free stream flow enters behind the strut from side of the strut, its velocity is less than the velocity enters from the top, this velocity difference is the source of the circulation formation in the strut. The use of the two jets for the fuel injection behind the strut also changes the circulations size behind the strut as depicted in Fig. 5.

Fig. 5
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Comparison of jet flow structure behind the strut in these two configurations.

Figure 6 illustrates the changes of the mixing zone and Mach contour on two planes located 5 mm and 10 mm behind the strut. As mentioned earlier, the velocity between the side flow and top flow is the primary source for the formation of the circulation behind the strut. The jet space is also results in the circulation as illustrated in Fig. 7a. When the mixing zone of these two cases with different gaps are examined in Fig. 7a, the main vortex is produced in different locations in these two models. The model with high gap expand the fuel jet in larger domain and the circulation (shown as A in Fig. 7a) is produced near the bottom close to the strut on plane 5 mm behind the strut.

Fig. 6
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Assessment of (a) zone of Mixing and (b) Mach change behind these two jet configurations.

However, the equivalent Mach contour (Fig. 6b) on the same plane (5 mm behind the strut) confirms that the source of this vortex is low subsonic region surrounded by supersonic flow. However, the main vortex is produced in the center of the domain for the model with high jet space (case 2). The Mach contour on the Fig. 6b at plane located 5 mm behind the strut shows that the induced vortex is the results of the high velocity gradient in the gap of the two injectors. As jet flow move to downstream (10 mm downstream of the fuel jet), the center of these vortex move to the boundary of the fuel mixing zone where the subsonic region behind the strut contact with freestream supersonic flow. Comparison of the fuel mixing region on far plane located 10 mm behind the strut also show that the mixing region expand more in downstream when the jets are close to each other’s. In fact, the higher space of the fuel jet would limit the fuel mixing due to low interaction of the fuel jet in the back of strut.

Fig. 7
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Comparison of circulation distribution downstream of these two configurations.

The comparison of the jet strength behind the strut for these two configuration (Fig. 7) shows that increasing the space of fuel injectors would enhance the circulation strength behind the strut. Indeed, the circulation in the gap of the jets and interaction of the free stream with upper jet are the main reason for this phenomenon. Although this difference in the strength of the circulation will be reduced in the far distance, there are still expressive alteration in the strength of the circulation near the strut injection.

Fig. 8
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Comparison of mixing efficiency of these two introduced models.

Figure 8 displays the changes of the fuel mixing of these two jet configuration behind the strut with equivalent single jet. Unlike circulation strength, the configuration with lower jet space has more efficient fuel mixing behind the strut. Besides, the change of the single injector with two injectors enhance the fuel mixing efficiency about 80% behind the strut. Although the difference of the fuel mixing of two injector configurations is about 10%, the higher efficiency of the model with low jet space confirm that the injection within the region with lower velocity is more efficient for the fuel diffusion.

Conclusion

In conclusion, the analysis of fuel jet mixing behind the strut demonstrates that the spacing between injectors plays a noteworthy role in determining the mixing efficiency and flow characteristics. For a model with a low gap between injectors, the fuel jet behavior closely resembles that of a single expanded jet, with less circulation but higher overall mixing efficiency. Conversely, in the model with a high injector gap, circulation is produced, enhancing fuel mixing and extending penetration. However, this increased circulation comes with a reduction in fuel concentration due to the mixing zone extending to cover the back of the strut.

The comparison of Mach contours further confirms that the injector gap influences shock wave behavior and the formation of low-velocity subsonic regions, which serve as the primary source of circulation. Larger injector gaps result in stronger circulations but limit fuel mixing due to lower interactions between the jets in the back of the strut, particularly at farther downstream locations.

The findings suggest that while increasing injector spacing strengthens the circulation behind the strut, a lower injector gap leads to more efficient fuel diffusion due to its placement in regions of lower velocity. The use of dual injectors, as opposed to a single injector, significantly improves fuel mixing, with a notable 80% increase in mixing efficiency behind the strut, particularly in the configuration with a smaller gap.