The sphere-cone and double-cone capsule geometries were modeled in ANSYS Fluent. Two nose configurations were tested for the double-cone capsule: 15°-55° and 55°-25°; Model A and B, respectively. In which the first angle refers to the first frontal cone and ectara; the angle is taken with respect to the horizontal. 

A rectangular control surface was used to capture the double-cone capsule's geometry. To create a mapped mesh, the control surface was divided into multiple zones (Figure 2). Each zone was refined by applying "number of divisions" to each line segment. To achieve a particular global cell size each segment was given a designation, and the appropriate "number of divisions" was calculated (Figure 3). Although this method is computationally fast, a sphere of influence was applied for further mesh refinement; doing so captures the boundary layer and shockwaves near the capsule. This process leads to the mesh being generated in Figure 4.

A C-shaped control surface was used to capture the frontal surface of the heat-shield. The previous process of creating a mapped mesh was reapplied (Figure 5). After applying a sphere of influence, the final mesh was generated (Figure 6).

• A double-point precision density-based solver was used to capture compressibility effects with the supersonic (M = 3) and hypersonic (M = 20) flight regimes at steady-state conditions. 

• Gravity effects were not considered. 

• 2D planar space was selected rather than axisymmetric to avoid divergence issues. 

• The two-temperature model more accurately captures thermal effects. This is because the fluid elements do not stay within a particular cell long enough for the thermal environment to reach equilibrium in supersonic and hypersonic flows.

• The ideal gas model for air was used for simplification purposes; literature review suggested that selecting a real gas would yield limited results.

• The Sutherland viscosity model was used to capture the viscous effects of air.

• The gamma-transport equation model was used to simulate the air transition from laminar to turbulent flow.

• The Kato-Launder setting was enabled to prevent the SST k-omega solver from overestimating turbulent behavior in areas with normal strain.

• The transition SST model allowed all simulations to achieve neutral if not total convergence over the standard SST k-omega model.

• Pressure far-field inlets were applied to the front, top, and bottom edges of the rectangular and C-shaped control surfaces.

  • The top and bottom surfaces were not classified as walls to prevent potential shockwave propagation and airflow interactions. Furthermore, classifying the top and bottom surfaces as walls would skew the airflow contours and peak magnitude values.

  • Although there are various inlet boundary conditions, the pressure far-field type most accurately models compressibility effects based on the freestream Mach number and static airflow parameters. 

• The right edge of the control surfaces were classified as pressure outlets.

• The capsule edges were classified as walls.

• Simulation method: Implicit Roe-FDS

• Spatial discretization: Least squares cell based

• The flow, intermittency, turbulent kinetic energy, and specific dissipation rate were all set to second order upwind.

• High speed numerics was enabled.

• FMG initialization was used, and the first-to-high-order blending was set to 50%.

• Initial courant number: 0.1.

• Maximum courant number: 10.

• An adaptive courant number was used throughout the simulation. In which, the courant number adjusted after every 10 iterations.

Aerodynamic theory specifies that for blunt bodies traveling at M > 1, an attached shock wave would be generated. Conversely, blunt bodies will generate a bow-shock for the same flight regime. There are three potential shock-wave cases for model A:

1. Due to the sharp geometry of the initial cone, an attached shock wave could be present throughout the entire capsule.

2. The first cone angle would generate an attached shock wave, but due to the blunt geometry of the second cone angle, the detached shockwave could recombine with a bow shock generated by the second cone.

3. Since the capsule is subjected to supersonic flight and not hypersonic flight, the overall bluntness of the second cone could generate a bow shock for the entire vehicle.

Of the three possibilities, the contour plots of model A showed that the capsule generated a bow shock. A potential reason of why this was the case is that because the second cone's overall size is triple that of the first cone, the frontal cone had minimal impact on the airflow. Thus, the second cone's geometry dominated the airflow characteristics, which led to the formation of the bow shock.

Both geometries shows the formation of a bow shock, the shear layer, wake, and recompression shock. The separation shock is present at the rightmost edge of the second cone. Aside from the shock waves, two airflow recirculation points are present behind the capsule. These recirculation points and wake have subsonic and supersonic airflow, respectively. 

Although the peak heating of both geometries are the same, model A experienced greater temperature concentrations around the stagnation points of the first cone. Model B experienced less heating due to the expansion waves present at the second cone.

Based on aerodynamic theory for hypersonic flow, it is expected that an oblique shock wave will be attached to the frontal cone of the capsule. Considering that the frontal cone of model A is attached to a blunter secondary cone, it is expected that the shockwave will reattach past the sharper geometry of the fontal cone. The reattachment point is expected to occur near the base of the capsule. By examining the following contours, this expectation was met. It can be observed that a separation point is present within the frontal cone along with the subsequent oblique shock; an attached shock is present as well. After the initial oblique shock, a reattachment shock is formed at the base of the frontal cone. Due to the blunt geometry of the secondary cone, a strong bow shock is formed. Ultimately, these figures show that an oblique shock is formed by the frontal cone, and this shock reattaches with a bow shock due to the secondary cone.

Due to mesh limitations with ANSYS Fluent, the energy term residual maintained neutral convergence at 1e+00. Warnings involving turbulent viscosity were present, but because the number of affected cells was minimal and decreased throughout the simulation, these viscosity warnings were not of importance. To verify whether convergence limitations were limited to the SST k-omega model, other turbulent models were used. Although other turbulent models with corrective factors were used, there was no noticeable change in residual convergence.  

The velocity contour of model A shows that it is not symmetric above and below the frontal cone. A reason why this may be the case is that the sphere of influence cell size of 0.1 m is not sufficient to capture the boundary layer. Thus, using a sphere of influence with a cell size of 0.01 m could improve the shock profile. However, due to limitations of the student license and computation limitations, this cell size could not be generated. Aside from the contour profile, the boundary layer remains attached to the vehicle, the velocity magnitude at the walls is 0 m/s, and velocity at the stagnation points is 0 m/s. All of these observations are in line with theory expectations.

The pressure contours shows the shockwave more clearly. An oblique shockwave is attached to the frontal cone of model A. This shock reattaches to the flow before coming attached to the result bow shock from the rear cone. Past the recirculation region, the formation of the recompression shockwave can be observed.

Unlike the supersonic case, model A experienced greater peak heating. Model A experienced a peak heating of 2,060 K, whereas model B experienced a peak heating of 839 K. Model A experienced greater levels of heating at the frontal cone's point and at the points where the two cones meet. Model A experienced greater peak heating due to the stronger flow compression.

As previously discussed, the shockwave of the Orion capsule is expected to be a bow shock due to its blunt geometry. It is also expected that stagnation points will be present in front of the capsule, and that the recirculation points will occur at the top and bottom edges of the capsule. Next, flow separation is expected to be at the end of the leading edge of the capsule, which will lead to the formation of the free-shear layer. This free-shear layer should then eventually lead to the formation of a recompression shock wave downstream of the wake. The CFD simulation show that all of these expectations have been fulfilled.

The velocity and pressure contours shows that the stagnation point occurs along the leading edge of the capsule. Past this stagnation point, the airflow accelerates around the heatshield and detaches from the vehicle. The low velocity contour near the sides and rear of the capsule show the recirculating zones. This viscous area propagates downstream of the capsule and leads to the formation of the separation shock. The pressure plot shows the pressure values along the capsule's surface. 

The temperature contour shows that most of the heating is present in the airstream in front of the capsule rather than at the capsule's surface. Since the shockwave is not attached to the capsule, most of the heating is present within the air rather than the body. Aside from the frontal section of the capsule, most of the other heating occurs within the recirculating flow zones and within the wake.

Existing literature has stated that blunt bodies are expected to generate a bow shock when subjected to an environment where M > 1. Thus, CFD results should be similar to existing literature results. However, due to the extreme hypersonic environment, there is a possibility where the resulting shockwave is a strong bow shock due to the strong compressive effects. Aside from the bow shock, the flow separation point, free shear line, recirculation zones, recompression shocks, and a detached bow shock should be present. By examining the following contours, it is evident that the bow shock is not uniform. Instead, it appears that the airflow has become stagnant in front of the capsule. As previously stated, the quality of the mesh could be improved to reduce the coarse effects in the contours, but this is not possible due to student license limitations. At the top and bottom tips of the capsule, the flow separation point is present along with a faint free shear line. There does not appear to be any recirculation zones.

Initially, the SST k-omega model was used without considering the transition from laminar to transition flow to reduce computation time. Since the y-velocity term diverged, the transition model was reincorporated along with curvature correction and corner flow correction. Doing so allowed the x-velocity, y-velocity, continuity, and energy terms to achieve neutral convergence.

The CFD setup was verified to be correct based on [2, 3], which used ANSYS Fluent to conduct CFD simulations on the Orion capsule; both papers used double point precision and the SST k-omega turbulent model. [2] used a density-based solver with steady state conditions. However, rather than using a 2D planar space, 2D axisymmetric was used in [2]. A pressure far-field inlet, pressure outlet, and walls with a no slip condition was used; the global operating pressure was to 0 Pa [2]. Air was configuration as an ideal gas and Sutherland viscosity was used [2]. [2] Used the Boltzmann-kinetic theory equation, Eucken-relation, and Blotter curve fit, whereas [3] used Cp temperature coefficients to model thermal effects. This project initially used kinetic theory for Cp and thermal conductivity, but doing so resulted in divergence. [3] used a control surface with a height of 100D and  length of 150D. [3] used the SST omega turbulence model with compressibility effects and the production limiter enabled. [3] Specifies that these settings are necessary to capture the heat flux generated by viscous effects. The two temperature model was enabled to better capture the energy transfer compared to the one-temperature model [3]. The control surface specifications were not specified in [3], but it appears that the control surface was sized 20D based on the capsule dimensions.

Ultimately, the papers verify that the ANSYS CFD simulation setup was similar and configured correctly.

The pressure plot from the reference article [1] (left) and simulation results (right) show similar hypersonic flow conditions. Each plot features a sharp pressure rise near the cone transition point, which corresponds to flow compression and shock interaction. The peak pressures in both cases are followed by a gradual decrease downstream that is evidence of flow separation. Additionally, both plots display secondary pressure peaks downstream, which is due to the disturbance caused by complex flow surface flow interaction.

Comparing the velocity contour shows the formation of a shockwave and regions of separated flow downstream where the flow begins to reaccelerate [1]. Similarly, the results (right) show strong flow deceleration near the vehicle's surface follow by recovery further downstream. Although the shock features are more prominent in the paper, CFD results are similar.

CFD simulations in literature have simulated the Orion's crew exploration vehicle (CEV) and launch abort system within the supersonic and hypersonic flow environment. However, there is no existing literature that have simulated the Orion CEV at M = 3 and M = 20 at 0° AOA. Two CFD articles that were the most similar to the project's simulation configuration were selected from comparison purposes. [2] did not specify the initial temperature reference values, but it is assumed that the default value of 300 K in ANSYS was used. [2] simulated the Orion capsule within the supersonic (M = 1.4) and hypersonic (M = 5) flight regimes. 

Project results experienced a peak pressure of 1,209,000 Pa and peak temperature of 833.8 K. Previously it was discussed that project results fulfill theory expectations because all shockwave characteristics were observed. When compared to [2], similar results were obtained. Generally, as the Mach number increases for a blunt body, it is expected that the bow shock will become stronger; the bow shock becomes more curved. By comparing project results to the results obtained from [2], this trend is fulfilled. When comparing the temperature values, the project temperature result of 833.8 K, is between the peak heating values from [2]. This further validates the accuracy of project results for when M = 3. However, when examining the temperature contour for M = 5 from [2], the reattachment shock is weaker when compared to the project's reattachment shock. A reason why this may be the case is that [2] used a 3D axisymmetric simulation compared to the project's 2D planar simulation. Regardless of this difference, results from the project successfully captures the supersonic (M = 3) airflow environment. 

[3] simulated the Orion capsule within the supersonic (M = 3) and hypersonic (M = 7) flight regimes. Although [3] simulated the capsule at an AOA of 15°, project results could still be used to compare the overall behavior of the system. By comparing project results to [3], the bow shock has the same relative distance away from the capsule. The separation and stagnation points are also similar to [3]. In [3], the recirculation points are located at the bottom side and back side of the capsule. However, if [3] was at AOA of 0°, the recirculation points show occur at the sides and backside of the capsule as shown in [2]. Unlike [2], [3] does not show the formation of reattachment shock. It does, however, show the extended wake as shown by project results.

Ultimately, project results are similar to the results from [10] and [11], showing that the project successfully captured the supersonic environment. The M = 20 case cannot be directly compared against existing literature because there are no setups that are similar enough for comparison purposes.

[1] Youssefi, M. R., and Knight, D., “Assessment of CFD Capability for Hypersonic Shock Wave

Laminar Boundary Layer Interactions,” Aerospace, Vol. 4, No. 2, 2017.

https://doi.org/10.3390/aerospace4020025

[2] Al Qubeissi, M., Shaban, F., Almohammadi, K. M., and Shah, R. M. R. A., “Parametric

Optimisation of Hypersonic Re-Entry Capsules with Air-Duct Systems.”

https://doi.org/10.20944/preprints202309.0103.v1

[3] Kubeeran, M., Manikandan, K., Kumar, R. S., Tangudu, P. K., Vahini, M. S., and Mallapragada, S.,

“Computational Aero Thermodynamic Study on Orion and Undulated Reentry Capsules,” 2021.

https://doi.org/10.1007/978-981-19-6970-6_23