**3.3 Computational grid**

One of the controlling factors for the numerical simulation is the proper grid arrangement. The following procedure is used to generate grid in the computational region of the bluntedbody. The computational domain is divided into number of non-overlapping zone. The mesh points are generated in each zone using finite element method (Mehta, 2011) in conjunction with the homotopy scheme (Shang, 1984). The spiked blunt nosed body is defined by a number of grid points in the cylindrical coordinate system. Using these surface points as the reference nodes, the normal coordinate is then described by the exponentially stretched grid points extending outwards up to an outer computational boundary.

Grid independence tests (Mehta, 2006; and Mehta 2008) were carried out, taking into consideration the effect of the computational domain, the stretching factor to control the grid intensity near the wall, and the number of grid points in the axial and normal directions. The outer boundary of the computational domain is varied from 2.5 to 3.0 times the maximum diameter D and the grid-stretching factor in the radial direction is varied from 1.5 to 5. These stretched grids are generated in an orderly manner.

Fig. 4. Comparison between density contour and schlieren picture.

Computations of Flowfield over Reentry Modules at High Speed 359

Computed vector velocity plot around the various vehicles are depicted in Fig. 5. The Mach contour plots show the formation of vortices at the corner region of the capsule for M 3. Characteristics flow features around the blunt body at supersonic speeds is observed. The high surface pressure on the fore-body results the development of high aerodynamic drag which is required for the aerobraking application. Table 4 shows the fore body aerodynamic drag for various reentry modules. It is important to mention here that the aerodynamic drag

Fig. 5. Enlarged view of velocity vector field over Apollo, Apollo-II and ARD.

The computed velocity vector plots are shown in Fig. 6. Close-up views of the velocity vector plot over fore-body of the capsules and schematic shock location are depicted in the figure. The separated shear layer and the recompression shock from the reattachment point on the shoulder of the hemispherical cap are visible in the vector plot. The shock wave in front of the cap will increase the aerodynamic drag due to high pressure region. In the fore region of the spherical cap, the fluid decelerates through the bow shock wave. At the shoulder of the disk or hemispherical cap, the flow turns and expands rapidly, the boundary layer detaches, forming a free shear layer that separates the inner recirculating flow region behind the base from the outer flow field. The corner expansion over disk process is a modified Prandtl-Mayer pattern distorted by the presence of the approaching boundary

is highly influenced by the fore body shape.

**3.5 Shock stand-off distance** 

layer.

To verify the chosen grid delivers an accurate solution, the number of grid cells was increased until a steady-state solution occurred, that is, the resulting axial force on the investigated shape did not change anymore. Several test runs were made with a total doubled grid cell number. Therefore, the grid was highly refined in both directions. Grids are chosen with the number of grid points in the axial direction ranging from 187 for the shortest capsule to up to 220 for the longest reentry configuration, and the number in the radial direction ranging from 52 to 82. The present numerical analysis was performed on 187x62 grid points. The downstream boundary of the computational domain is maintained at 4 to 6 times the maximum diameter, D. This grid arrangement is found to give a relative difference of about ±1.5% for the drag coefficient. The convergence criterion less than the 10-5 is based on the difference in the density values at any grid point between two successive iterations. The minimum spacing for the fine mesh is dependent upon the Reynolds number. The finer mesh near the wall helps to resolve the viscous effects. The coarse-mesh helps reducing the computer time. It is important that the grid have sufficient points in the shoulder region of the reentry module in order to capture the rapid expansion and accurately predict the flow separation point and the angle of the resulting shear layer. A close-up view of the computational grid over the various reentry capsules is shown in Reference (Mehta, 2010). The structured grid generation and the mono block are suitable to accommodate spike shape. As seen in the figures of (Mehta, 2006; Mehta 2010), these types of grid use quadrilateral cells in 2-D in the computational array. The quadrilateral cells, which are very efficient at filling space, support a high amount of skew and stretching before the solution will be significantly affected. Additionally, the grid can be aligned with the flow, thereby yielding greater accuracy within the solver. Several grid arrangements are take check the grid independency check.

#### **3.4 Flow characteristics**

The flow field features over various capsules have been shown by (Mehta, 2010). Figure 4 shows the comparison between the density contour and schlieren picture for spherical blunted-cone at = 25 deg configuration at M = 6.0. All the essential flow field features are well captured and compared well with the schlieren picture. Figure 4 depicts the velocity vector plots for Apollo, Apollo-II and ARD reentry modules. It can be visualized from the vector plots that all the significant flow field behaviors such as bow shock wave, rapid expansion fans and the flow at the shoulder, flow recirculation region with converging free shear layer and formation of the vortex flow in the aft region of the capsules. The wake flow field immediately behind the vehicle base, exhibits complex flow characteristics. The formation of the bow shock wave on the fore body depends on RN and N and M. The bow shock wave moves close to the fore body with the increasing M and the stand off distance between the bow shock wave and the fore body decreases with the increasing M.

The pressure, density and Mach contour plots show the enlarged view of the computed vector field over the OREX, ARD, Apollo at various freestream Mach number M. It can be visualized from the vector plots that all the significant flow field features such as bow shock wave, rapid expansion fans on the shoulder of the capsule, flow recirculation region with converging free shear layer and formation of the vortex flow in the aft region of the capsules. The wake flow field, immediately behind the capsule base, exhibits complex flow characteristics. The formation of the bow shock wave on the fore body depends on RN and N and M. In the case of spherically blunted-cone/flare (SRE) capsule, the bow shock wave does not follow the body contour, which is attributed to small values of RN and N as compared to the OREX, the ARD and the Apollo.

To verify the chosen grid delivers an accurate solution, the number of grid cells was increased until a steady-state solution occurred, that is, the resulting axial force on the investigated shape did not change anymore. Several test runs were made with a total doubled grid cell number. Therefore, the grid was highly refined in both directions. Grids are chosen with the number of grid points in the axial direction ranging from 187 for the shortest capsule to up to 220 for the longest reentry configuration, and the number in the radial direction ranging from 52 to 82. The present numerical analysis was performed on 187x62 grid points. The downstream boundary of the computational domain is maintained at 4 to 6 times the maximum diameter, D. This grid arrangement is found to give a relative difference of about ±1.5% for the drag coefficient. The convergence criterion less than the 10-5 is based on the difference in the density values at any grid point between two successive iterations. The minimum spacing for the fine mesh is dependent upon the Reynolds number. The finer mesh near the wall helps to resolve the viscous effects. The coarse-mesh helps reducing the computer time. It is important that the grid have sufficient points in the shoulder region of the reentry module in order to capture the rapid expansion and accurately predict the flow separation point and the angle of the resulting shear layer. A close-up view of the computational grid over the various reentry capsules is shown in Reference (Mehta, 2010). The structured grid generation and the mono block are suitable to accommodate spike shape. As seen in the figures of (Mehta, 2006; Mehta 2010), these types of grid use quadrilateral cells in 2-D in the computational array. The quadrilateral cells, which are very efficient at filling space, support a high amount of skew and stretching before the solution will be significantly affected. Additionally, the grid can be aligned with the flow, thereby yielding greater accuracy within the solver. Several grid arrangements are

The flow field features over various capsules have been shown by (Mehta, 2010). Figure 4 shows the comparison between the density contour and schlieren picture for spherical blunted-cone at = 25 deg configuration at M = 6.0. All the essential flow field features are well captured and compared well with the schlieren picture. Figure 4 depicts the velocity vector plots for Apollo, Apollo-II and ARD reentry modules. It can be visualized from the vector plots that all the significant flow field behaviors such as bow shock wave, rapid expansion fans and the flow at the shoulder, flow recirculation region with converging free shear layer and formation of the vortex flow in the aft region of the capsules. The wake flow field immediately behind the vehicle base, exhibits complex flow characteristics. The formation of the bow shock wave on the fore body depends on RN and N and M. The bow shock wave moves close to the fore body with the increasing M and the stand off distance

between the bow shock wave and the fore body decreases with the increasing M.

The pressure, density and Mach contour plots show the enlarged view of the computed vector field over the OREX, ARD, Apollo at various freestream Mach number M. It can be visualized from the vector plots that all the significant flow field features such as bow shock wave, rapid expansion fans on the shoulder of the capsule, flow recirculation region with converging free shear layer and formation of the vortex flow in the aft region of the capsules. The wake flow field, immediately behind the capsule base, exhibits complex flow characteristics. The formation of the bow shock wave on the fore body depends on RN and N and M. In the case of spherically blunted-cone/flare (SRE) capsule, the bow shock wave does not follow the body contour, which is attributed to small values of RN and N as

take check the grid independency check.

compared to the OREX, the ARD and the Apollo.

**3.4 Flow characteristics** 

Computed vector velocity plot around the various vehicles are depicted in Fig. 5. The Mach contour plots show the formation of vortices at the corner region of the capsule for M 3. Characteristics flow features around the blunt body at supersonic speeds is observed. The high surface pressure on the fore-body results the development of high aerodynamic drag which is required for the aerobraking application. Table 4 shows the fore body aerodynamic drag for various reentry modules. It is important to mention here that the aerodynamic drag is highly influenced by the fore body shape.

Fig. 5. Enlarged view of velocity vector field over Apollo, Apollo-II and ARD.
