Simulation and analysis of the over-expanded flow field in asymmetric nozzles with lateral expansion

The study presented a numerical simulation of three-dimensional asymmetric nozzles featuring lateral expansion to investigate the flow field characteristics under over-expanded conditions and assess the influence of the lateral expansion angle. The research findings elucidate the distinct shape of the separation zones within the nozzle resulting from varying lateral expansion angles. In the restricted shock separation with separation bubbles forming on the flap pattern, the central separation bubble and corner separation bubbles exhibit independent characteristics. With an increasing lateral expansion angle under the same nozzle pressure ratio, the corner separation shock wave significantly impacts the shape of the central separation bubble.


Introduction
The over-expanded flow separation in asymmetric nozzles can affect the stability of the exhaust system [1].To ensure effective control and enhance the stability of exhaust systems, it is imperative to have a comprehensive insight into this phenomenon.Yu et al. [2] examined the flow separation patterns in the two-dimensional asymmetric nozzles.They discovered various flow separation patterns, including Restricted Shock Separation with separation bubble forming on the flap pattern (RSS(flap)), Free Shock Separation pattern (FSS) and Restricted Shock Separation with separation bubble forming on the ramp pattern (RSS (ramp)).In the RSS pattern, the flow downstream re-attaches to the nozzle wall after the separation point, forming a separation bubble.While in the FSS pattern, reattachment does not occur after boundary layer separation, forming an open recirculation region.
However, studies have indicated that significant three-dimensional effects exist in the separated flow field within the rectangle channels with clear coupling between the corner separation and the main airflow [3][4][5].Additionally, with the advancement of hypersonic technology, three-dimensional nozzles incorporating lateral expansion and geometric constraints exhibit greater developmental potential than two-dimensional nozzles [6].However, the research in this area is relatively limited with a weak understanding of the flow characteristics.Therefore, this paper utilized numerical simulation methods to explore the three-dimensional separation flow field and assess how the lateral expansion angle affects the over-expanded separated flow field in asymmetric nozzles, which provides some reference for further research of three-dimensional asymmetric nozzle flow fields.

Numerical validation and grid independence verification
The experimental model is shown in Figure 1 (a).The model is designed with a design nozzle pressure ratio (NPR) of 20, flat side walls and equal width before and after.The Fluent software was employed to solve the flow field inside the nozzle.The computational domain was divided by using a structured grid with grid refinement near the walls to ensure the y+ value was below 5.A density-based solver, SST k-ω turbulence model and second-order upwind differencing scheme were utilized for an implicit solution.The medium was assumed to be an ideal gas and the viscosity term was calculated by using the Sutherland formula.For the SST k-ω turbulence model, optimization led to the selection of the Bradshaw constant a 1 and the constant a ∞ * : a 1 = 0.35756 and a ∞ * = 1.7375727 [7].The boundary conditions of numerical calculation are 93243.66Pa of the inlet total pressure, 31073.02Pa of the outlet static pressure, 31073.02Pa of the far-field static pressure, 300 K of the static temperature and no-slip conditions of the walls, which mirror the actual experimental conditions with the nozzle in an over-expanded state. .Experimental asymmetric nozzle model, grid and computational domain division.The sensitivity of the grid was also verified.Numerical simulations are performed by using three different grid densities: 1.1 million cells, 1.8 million cells and 3.2 million cells for coarse grid, medium grid and fine grid respectively.The pressure distribution along the centerline of the ramp and flap is normalized by the total pressure P i .The nozzle length X is normalized by the throat height H t .The results across the three grid densities from numerical simulations align well with experimental findings, displayed in Figure 2. Therefore, this numerical simulation method can be used for the study of asymmetric nozzles in over-expanded states.Considering the computational cost and other factors, it is decided to use the medium-density grid for the next research phase.

Physical model
A three-dimensional asymmetric nozzle model featuring lateral expansion is constructed in Figure 3, derived from the experimental model depicted in Figure 1 (a).In the z-direction, an angle β characterizes the lateral expansion, while the side wall retains a planar configuration.

Separation flow field characteristics in the RSS (flap) pattern
Numerical simulations were conducted on asymmetric nozzles with lateral expansion angles β = 2 °, 6 °, 10 °, 15 ° and 20 ° to investigate the over-expanded flow conditions during the start-up processes.The initial NPR was set to 2.5.During the simulations, the NPR was continuously increased to simulate the start-up of the nozzle.In this paper, the RSS (flap) pattern is focused.To facilitate further discussion, the coordinate system is described as follows.The flow direction aligns with the x-axis, the normal direction with the y-axis and the spanwise direction with the z-axis.
Due to the pressure difference existing between the separation bubble/recirculation region and the mainstream, the backflow is formed with the streamwise velocity experiencing zero and negative values.Exploiting this characteristic, Figure 4 (a) displays the regions where the streamwise velocity component is equal or less than zero (excluding on the wall) to illustrate the separation regions near the wall of the asymmetric nozzle (β = 2 °) during the RSS (flap) pattern.It can be observed that a recirculation region forms downstream of the ramp separation point.For the separation regions near the flap, apart from the central separation bubble, additional separation bubbles are formed near the corners of the side wall and flap, but they are considerably smaller compared to the separation bubble along the centerline.It is also observed that the central separation bubble spans a considerable width in the spanwise direction compared to the flow direction, combined with the Mach contour slice shown in Figure 6 (a).The central separation bubble exhibits a slight bulge in the normal direction with flattened ends, forming a relatively flat bubble structure.The separation at the corners occurs earlier than in the central region, forming a narrow and elongated "spindle-shaped" structure.
Through depicting the reverse pressure gradient, Figure 4 (b) displays the separation shocks near the separation regions.The separation shock in the corner occurs early, propagating in the flow, normal and spanwise direction towards the downstream within the nozzle.It intersects with the λ shock originating from the symmetry and interacts with each other, leading to a conical shock structure near the corner region, with the side wall as the bottom, extending to the ramp and interior of the nozzle.The two intersections of the shear stress curve with y = 0 represent the starting point and reattachment point of the separation region and the starting point of the separation region is also reflected on the starting point of the λ shock at the center symmetry plane.Figure 5 (b) presents the shear stress curve at the corner region near the flap.Through Figure 5, it can be observed that at β ≤ 10 °, as the lateral expansion angle increases, the separation bubble gradually moves to the throat.However, the size of the central separation bubble on the flap does not change significantly in the flow direction, while the size of the corner separation bubble gradually decreases.After β > 10 °, the central separation bubble almost stops moving and the starting point of the λ shock remains relatively unchanged, but the shock foot continues to move to the throat, while the corner separation bubble continues moving.Overall, the displacement of the corner separation bubble exceeds that of the central region, resulting in subtle alterations in the shape of the separation bubble and the shock structure on the flap.Comparing the flow characteristics of the asymmetric nozzle under different lateral expansion angles, it can be observed that the overall structure of the separation region and separation shock is similar to the description mentioned earlier.However, due to the displacement difference caused by the movement of the separation bubble at the central region and the corner region with increasing lateral expansion angle, they exhibit slightly different structures at the same NPR.As the relative position of the corner separation bubbles and the central separation bubble changes, the corner separation gradually approaches the central separation bubble in the spanwise direction while moving away from it in the flow direction.Due to the corner separation occurring earlier than the central separation, as the proximity of the corner separation shock, the end of the central separation bubble is compressed.As previously mentioned, the corner separation bubble diminishes progressively with an increase in the lateral expansion angle, indicating a gradual attenuation of the corner separation shock.Consequently, as the corner separation bubble gradually moves away and the separation shock weakens, the central separation bubble slowly resumes its progression toward the side wall, pointing towards the direction of the corner separation bubble.Simultaneously, due to the continuous interaction between the corner weakened separation shock and the central separation shock, as the corner separation moves forward, the shock wave structure at the front of the central separation bubble transforms gradually into an arc shape.

Conclusion
In the RSS (flap) pattern of asymmetric nozzles with lateral expansion, the flow reattached after the separation point of the flap, forming a central separation bubble and two corner separation bubbles, which were independent of each other.The corner separation is earlier than the center separation and

Figure 1
Figure 1.Experimental asymmetric nozzle model, grid and computational domain division.The sensitivity of the grid was also verified.Numerical simulations are performed by using three different grid densities: 1.1 million cells, 1.8 million cells and 3.2 million cells for coarse grid, medium grid and fine grid respectively.The pressure distribution along the centerline of the ramp and flap is normalized by the total pressure P i .The nozzle length X is normalized by the throat height H t .The results across the three grid densities from numerical simulations align well with experimental findings, displayed in Figure2.Therefore, this numerical simulation method can be used for the study of asymmetric nozzles in over-expanded states.Considering the computational cost and other factors, it is decided to use the medium-density grid for the next research phase.

Figure 2 .
Figure 2. Verification of grid independence and a comparison between CFD and experimental for the pressure distribution on the ramp and flap.

Figure 3 .
Figure 3. Three-dimensional model of nozzle with lateral expansion.