**4. Geometrical parameters of reentry vehicles**

A high-speed flow past a reentry capsule forms a bow shock wave which causes a high surface pressure. It yields high aerodynamic drag (ballistic coefficient) force, which is needed for aero-braking purposes. Therefore, the primary design consideration of the reentry capsules requires large spherical nose radius *RN* and fore-body diameter *D* as shown in **Figure 1(a)**. Reentry capsule configurations significantly differ from each other due to entry conditions and mission requirements. The sphere space capsule (Sputnik) permits the highest possible volumetric efficiency but does not give good maneuvering ability. Therefore, the reentry space vehicle requires a back-shell with an inclination in order to generate lift to reduce '*g*' forces on the crew tolerance levels. Bedin et al. [52] have illustrated sixteen types of space vehicles in which the frontal diameter *D* of the capsule is kept constant for all configurations

and varying geometrical parameters *αN*, *RC*, *αB*, and *L* in three groups. Experimental investigation of various combination of cone-segment bodies and spheres of Russian reentry capsules are carried out by Bedin et al. [52] in a pressure-tight ballistic range for ratio of specific heats of 1.14–1.67, Mach number varied from 0.5 to 10, and Reynolds number based on the base diameter varied from 2.5 × 105 to 5.0 × 106 .

In the first group, five capsules are having variation in the back-shell angle *α*<sup>B</sup> in the range of 0–30°. In the second group, five capsules are having variation of the overall length varied from 1.0 *D* to 0.375 *D*. In the last group the back-shell angle *αB*, overall length *L*, and shoulder radius *RC*, alignment with frontal cap are varied to evaluate the ballistic performance. Recently Minenkol et al. [53] have studied the effect of geometrical parameters on aerodynamic performance of the space vehicles such as the Apollo and the Soyuz.

The reentry capsules can be classified as a head-light shape as in the case of Soyuz, or bell shape as in the case of Apollo and ARD, or a saucer type as in the case of OREX. **Table 1** depicts the dimension of the Apollo, the OREX and the Soyuz capsules to emphasis the classification of the capsules based on *L*/*D* ratio. The nominal OCM geometry, based on the Apollo configuration, consists of a spherical fore-body transitioning to a conical back-shell section with a truncated base to accommodate docking hardware. The aerodynamic characteristic of the Orion is analyzed numerically and experimentally by Stremel et al. [54]. The OCM is similar in shape to the Apollo Command Module but is approximately 29% larger by length. The ARD resembles a 70% scaled version of Apollo capsule as mentioned by Walpot [6].

The schematic sketches of flowfield feature of the Apollo, the Soyuz, the OREX capsules are displayed in **Figure 3(a)**–**(c)**. The Apollo and the Soyuz configurations are having spherical-blunt nose segment. The fore-body of the OREX consists of spherical cap with a cone section. The bow shock wave is detached on the blunt fore-body in the case of SRE as delineated in **Figure 3(d)**. The fore-body of the SRE is having a mixed subsonic-supersonic region as seen in the figures. The flowfield in the wake region is affected due to the presence of the truncated cylinder. **Figure 3(e)** shows schematic flowfield features at high speed on a sharp-tipped double-cone configuration. The double cone capsule shows formation of an attached conical shock wave on the tip of the cone. The flowfield in the wake region of a reentry capsule is again found to be complex in nature and is attributed to the expansion fan at corner of the shoulder.

**Figure 4** shows the nomenclature of the geometrical parameters of the ARD, the Soyuz, the OREX, the SRE and the double cone reentry capsules. The Soyuz, the Apollo and the OREX capsules are having back-shell inclination angle *α*B of 9, 33 and 15° relative to the vehicle's axis of symmetry respectively. **Figure 5** depicts the geometrical details of the CARINA [13] and Beagle-2 [14] capsules. **Table 2** depicts the geometrical detail of Viking, MPF, MER, Phoenix and MSL which are having a 70° sphere-cone shaped (Mars space vehicles) with a back-shell needed for highspeed entry phase and a disk-gap band (DGB) type of supersonic parachute during the descent portion of the entry sequence. **Table 3** presents the dimensional details of the ARD, the Apollo, the OREX, the CARINA, the MUSES-C and the Beagle-2. **Table 4** depicts the dimensional details of the SRE capsule.


**121**

**Figure 4.**

*space vehicles.*

**Figure 3.**

*(e) double-cone capsule.*

*Numerical Simulation of Base Pressure and Drag of Space Reentry Capsules at High Speed*

*Schematic sketches of flowfield over various reentry capsules (a) Apollo; (b) Soyuz; (c) OREX; (d) SRE; and* 

*Geometrical parameters of reentry capsules, (a) ARD; (b) Soyuz; (c) OREX; (d) SRE; and (e) double-cone* 

*DOI: http://dx.doi.org/10.5772/intechopen.83651*

#### **Table 1.**

*Dimension of the Apollo, the OREX and the Soyuz reentry capsules.*

*Numerical Simulation of Base Pressure and Drag of Space Reentry Capsules at High Speed DOI: http://dx.doi.org/10.5772/intechopen.83651*

#### **Figure 3.**

*Hypersonic Vehicles - Past, Present and Future Developments*

such as the Apollo and the Soyuz.

and varying geometrical parameters *αN*, *RC*, *αB*, and *L* in three groups. Experimental investigation of various combination of cone-segment bodies and spheres of Russian reentry capsules are carried out by Bedin et al. [52] in a pressure-tight ballistic range for ratio of specific heats of 1.14–1.67, Mach number varied from 0.5 to 10, and

In the first group, five capsules are having variation in the back-shell angle *α*<sup>B</sup> in the range of 0–30°. In the second group, five capsules are having variation of the overall length varied from 1.0 *D* to 0.375 *D*. In the last group the back-shell angle *αB*, overall length *L*, and shoulder radius *RC*, alignment with frontal cap are varied to evaluate the ballistic performance. Recently Minenkol et al. [53] have studied the effect of geometrical parameters on aerodynamic performance of the space vehicles

The reentry capsules can be classified as a head-light shape as in the case of Soyuz,

**Figure 4** shows the nomenclature of the geometrical parameters of the ARD, the Soyuz, the OREX, the SRE and the double cone reentry capsules. The Soyuz, the Apollo and the OREX capsules are having back-shell inclination angle *α*B of 9, 33 and 15° relative to the vehicle's axis of symmetry respectively. **Figure 5** depicts the geometrical details of the CARINA [13] and Beagle-2 [14] capsules. **Table 2** depicts the geometrical detail of Viking, MPF, MER, Phoenix and MSL which are having a 70° sphere-cone shaped (Mars space vehicles) with a back-shell needed for highspeed entry phase and a disk-gap band (DGB) type of supersonic parachute during the descent portion of the entry sequence. **Table 3** presents the dimensional details of the ARD, the Apollo, the OREX, the CARINA, the MUSES-C and the Beagle-2.

**Table 4** depicts the dimensional details of the SRE capsule.

*Dimension of the Apollo, the OREX and the Soyuz reentry capsules.*

**Capsule** *R<sup>N</sup> D R<sup>C</sup> L* **α***<sup>N</sup>*

Apollo 4.595 3.95 0.186 2.04 — 33.0 OREX 1.35 3.40 0.001 1.508 50.0 15.0 Soyuz 2.235 2.2 0.014 2.142 — 7.0

or bell shape as in the case of Apollo and ARD, or a saucer type as in the case of OREX. **Table 1** depicts the dimension of the Apollo, the OREX and the Soyuz capsules to emphasis the classification of the capsules based on *L*/*D* ratio. The nominal OCM geometry, based on the Apollo configuration, consists of a spherical fore-body transitioning to a conical back-shell section with a truncated base to accommodate docking hardware. The aerodynamic characteristic of the Orion is analyzed numerically and experimentally by Stremel et al. [54]. The OCM is similar in shape to the Apollo Command Module but is approximately 29% larger by length. The ARD resembles a 70% scaled version of Apollo capsule as mentioned by Walpot [6]. The schematic sketches of flowfield feature of the Apollo, the Soyuz, the OREX capsules are displayed in **Figure 3(a)**–**(c)**. The Apollo and the Soyuz configurations are having spherical-blunt nose segment. The fore-body of the OREX consists of spherical cap with a cone section. The bow shock wave is detached on the blunt fore-body in the case of SRE as delineated in **Figure 3(d)**. The fore-body of the SRE is having a mixed subsonic-supersonic region as seen in the figures. The flowfield in the wake region is affected due to the presence of the truncated cylinder. **Figure 3(e)** shows schematic flowfield features at high speed on a sharp-tipped double-cone configuration. The double cone capsule shows formation of an attached conical shock wave on the tip of the cone. The flowfield in the wake region of a reentry capsule is again found to be complex in nature and is attributed to the expansion fan at corner of the shoulder.

to 5.0 × 106

.

**<sup>0</sup> α***<sup>B</sup>*

**0**

Reynolds number based on the base diameter varied from 2.5 × 105

**120**

**Table 1.**

*Schematic sketches of flowfield over various reentry capsules (a) Apollo; (b) Soyuz; (c) OREX; (d) SRE; and (e) double-cone capsule.*

#### **Figure 4.**

*Geometrical parameters of reentry capsules, (a) ARD; (b) Soyuz; (c) OREX; (d) SRE; and (e) double-cone space vehicles.*

#### **Figure 5.**

*Geometrical parameters of reentry capsules (a) CARINA; (b) Beagle-2.*


#### **Table 2.**

*Geometrical parameters of Viking, MPF, MER, Phoenix and MSL.*


#### **Table 3.**

*Dimension of the reentry capsules.*


#### **Table 4.**

*Dimension of the blunted-spherical cone (SRE) reentry module.*

The effects of the module geometrical parameters, such as radius of the spherical cap radius, shoulder radius, semi-cone angle and back-shell inclination angle on

**123**

*Numerical Simulation of Base Pressure and Drag of Space Reentry Capsules at High Speed*

provide a useful input for the optimization of the reentry module.

the flowfield characteristics hence the base drag coefficient are analyzed which will

As discussed above the base pressure measurements in the wind-tunnel testing are affected by presence of sting attached to model. The free-flight data depend on quality of the transmitted telemetry data. The fluid dynamic equations describing the flowfield around a space vehicle include equations of continuity, momentum, and total energy. A numerical simulation of unsteady, compressible, axisymmetric laminar Navier-Stokes equations is an alternative to the expensive experimental testing of the reentry vehicles. The governing fluid dynamics equations can be written in the following conservation form in order to capture shocks and disconti-

Temperature *T* is related to pressure and density by the perfect gas equation of state. The ratio of the specific heats *γ* is assumed constant and is equal to 1.4. The coefficient of molecular viscosity is evaluated in the flow solver employing Sutherland's formula. The flow is assumed to be laminar, which is consistent with

To simplify the spatial discretization in numerical technique, Eq. (3) can be written in the integral form over a finite computational domain *Ω* with the bound-

The contour integration around the boundary of the cell is performed in anticlockwise sense in order to keep flux vectors normal to boundary of the cell. The computational domain *Ω* is having a finite number of non-overlapping quadrilateral cells. The conservation variables within the computational cell are represented

The inviscid fluxes are computed at the centre of the cell resulting in flux balance. The summation is carried out over the four edges of the cell. The derivatives of primitive variables in the viscous flux are evaluated by using the method of lines. A system of ordinary differential equations in time is obtained after integrating Eq. (4) over a computational cell. In the cell-centered spatial discretization scheme is non-dissipative, therefore, artificial dissipation terms [55] are added by blending of second and fourth differences of the vector conserved variables. The blend of second and fourth differences provides third order back ground dissipation in

The spatial discretization described above reduces the integral equations to semi-discrete ordinary differential equations (ODE). The ODE is solved using multi-stage Runge-Kutta time stepping scheme of Jameson et al. [55]. The numerical algorithm is second-order accurate in space discretization and time integration.

smooth region of the flow and first-order dissipation in shock waves.

*dt* <sup>∫</sup><sup>Ω</sup> *<sup>U</sup>d*<sup>Ω</sup> <sup>+</sup> ∫Γ(*Fdr* <sup>−</sup> *<sup>G</sup>dx*) <sup>+</sup> <sup>∫</sup><sup>Ω</sup> *<sup>H</sup>d*<sup>Ω</sup> <sup>=</sup> <sup>0</sup> (4)

experimental results of Cassanto [37] and Bulmer [42].

(3)

*DOI: http://dx.doi.org/10.5772/intechopen.83651*

**5. Numerical algorithm**

**5.2 Numerical technique**

ary of the domain *Γ* as

\_\_*<sup>d</sup>*

by their average values at the cell centre.

nuities as

**5.1 Governing fluid equations**

*Numerical Simulation of Base Pressure and Drag of Space Reentry Capsules at High Speed DOI: http://dx.doi.org/10.5772/intechopen.83651*

the flowfield characteristics hence the base drag coefficient are analyzed which will provide a useful input for the optimization of the reentry module.
