**3.5 Shock stand-off distance**

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 layer.

Computations of Flowfield over Reentry Modules at High Speed 361

vector and pressure contour plot and they are 0.19 and 0.11 which show good agreement with the analytical values. The spherical spike shows the greatest change in velocity gradient as compared to the flat disk. The flow is compressed to subsonic condition, which appears as inviscid and incompressible (Truit, 1959; Liepmann & Roshko 2007) . The shock wave stands in front of the blunt body and forms a region of subsonic flow

(a) Hemispherical forebody (b) Flat-disc forebody

The inviscid flow field in the vicinity of the stagnation point is described in a fluid dynamics sense as the conversion of a unidirectional high velocity stream by a normal shock wave into a high temperature subsonic layer, which is taken to be inviscid and incompressible (Truit, 1959). The stagnation point of a blunt body, the incoming hypersonic flow brought to rest by a normal shock and adiabatic compression. The heat transfer rate is directly proportional to the enthalpy gradient at the wall and square root of the tangential velocity gradient at the edge of the boundary layer. The inviscid flow field in the vicinity of the stagnation point is described as the conversion of the uni-directional, high velocity stream by a normal shock wave into a high temperature subsonic layer. The enthalpy gradient is determined by the shape of the velocity profile in the boundary layer and by the variation of the air properties with temperature. Heat flux at the stagnation point can be calculated using following

0.6 0.763Pr *w w w w q Kh <sup>h</sup>*

where the subscripts ∞ and w are freestream and wall, respectively. Pr is Prandtl number, K stagnation point velocity gradient and h enthalpy of air. The value of stagnation point velocity gradient K is taken as 0.3 (White, 1991). The hemispherical spike shows the greatest changes in the velocity gradient as compared to the flat disk spike. The magnitude of the stagnation-velocity gradient indicates the maximum heat transfer over the hemispherical

 

(4)

Fig. 6. Close-up view of Velocity vector plot over fore body of the capsules.

around the stagnation region.

**3.6 Wall heat flux** 

expression of (Fay & Riddle, 1958)


Table 4. Fore-body aerodynamic drag.

For the case of flat-nosed capsule flying at hypersonic speeds, a detached bow wave is formed in front of the nose which is practically normal at the body axis (Truit, 1959; Liepmann & Roshko 2007). Since the flow behind the normal shock is always subsonic, simple continuity considerations show that the shock-detachment distance and stagnationvelocity gradient are essentially a function of the density ratio across the shock. The flow behind the shock wave is subsonic, the shock is no longer independent of the fardownstream conditions. A change of the spike shape (geometry) in the subsonic region affects the complete flow field up to the shock. The shock-detachment distance becomes smaller with increasing density ratio. (Hayer & Probstein 1959) gives expression for the shock detachment distance ∆F with diameter of the flat-disk DS ratio as

$$\frac{\Delta\_F}{D\_S} = 2.8 \sqrt{\frac{\rho\_w}{\rho\_0}} \tag{1}$$

where the density ratio across the normal shock (Ames Research Staff, 1953) is

$$\varepsilon = \frac{\rho\_{\text{ev}}}{\rho\_0} = \frac{\left(\gamma - 1\right)M\_{\text{ev}}^2 + 2}{\left(\gamma + 1\right)M\_{\text{ev}}^2} \tag{2}$$

The ratio of shock stand-off distance ∆S with hemispherical cap of diameter, DS is

$$\frac{\Delta\_S}{D\_S} = \frac{2\varepsilon}{1 + \sqrt{\frac{8\varepsilon}{3}}}\tag{3}$$

The values of ∆F/DS and ∆S/DS are found 0.1898 and 0.1109, respectively. The numerical values of the ratio of shock stand-off to cap diameter are calculated from the velocity vector and pressure contour plot and they are 0.19 and 0.11 which show good agreement with the analytical values. The spherical spike shows the greatest change in velocity gradient as compared to the flat disk. The flow is compressed to subsonic condition, which appears as inviscid and incompressible (Truit, 1959; Liepmann & Roshko 2007) . The shock wave stands in front of the blunt body and forms a region of subsonic flow around the stagnation region.

Fig. 6. Close-up view of Velocity vector plot over fore body of the capsules.

#### **3.6 Wall heat flux**

360 Computational Simulations and Applications

Capsules M =1.2 M=1.4 M=2.0 M=3.0 M=5.0 M=6.0 ARD 1.41 – 1.43 1.40 1.38 – Apollo 1.30 – 1.50 1.43 1.40 – Apollo – II 1.32 – 1.47 1.43 1.05 –

shoulder) 1.79 1.77 1.50 1.30 1.16 – OREX (with sharp shoulder) 1.80 1.66 1.36 1.17 1.05 – CARINA 1.40 – 1.50 1.45 1.45 – MUSSES – C 1.27 – 1.24 1.36 1.38 – Beagle – 2 1.58 1.59 1.54 1.47 1.42 –

=25 deg 0.40 – 0.43 0.44 – 0.42

=30 deg 0.51 – 0.49 0.46 – 0.42

=35 deg 0.58 – 0.56 0.51 – 0.47

For the case of flat-nosed capsule flying at hypersonic speeds, a detached bow wave is formed in front of the nose which is practically normal at the body axis (Truit, 1959; Liepmann & Roshko 2007). Since the flow behind the normal shock is always subsonic, simple continuity considerations show that the shock-detachment distance and stagnationvelocity gradient are essentially a function of the density ratio across the shock. The flow behind the shock wave is subsonic, the shock is no longer independent of the fardownstream conditions. A change of the spike shape (geometry) in the subsonic region affects the complete flow field up to the shock. The shock-detachment distance becomes smaller with increasing density ratio. (Hayer & Probstein 1959) gives expression for the

0

2 2

1 2 1 *M M*

(1)

(2)

(3)

2.8 *<sup>F</sup> DS*

 

 

The values of ∆F/DS and ∆S/DS are found 0.1898 and 0.1109, respectively. The numerical values of the ratio of shock stand-off to cap diameter are calculated from the velocity

shock detachment distance ∆F with diameter of the flat-disk DS ratio as

where the density ratio across the normal shock (Ames Research Staff, 1953) is

0

The ratio of shock stand-off distance ∆S with hemispherical cap of diameter, DS is

*S DS*

OREX (with smooth

Spherically blunted cone,

Spherically blunted cone,

Spherically blunted cone,

Table 4. Fore-body aerodynamic drag.

The inviscid flow field in the vicinity of the stagnation point is described in a fluid dynamics sense as the conversion of a unidirectional high velocity stream by a normal shock wave into a high temperature subsonic layer, which is taken to be inviscid and incompressible (Truit, 1959). The stagnation point of a blunt body, the incoming hypersonic flow brought to rest by a normal shock and adiabatic compression. The heat transfer rate is directly proportional to the enthalpy gradient at the wall and square root of the tangential velocity gradient at the edge of the boundary layer. The inviscid flow field in the vicinity of the stagnation point is described as the conversion of the uni-directional, high velocity stream by a normal shock wave into a high temperature subsonic layer. The enthalpy gradient is determined by the shape of the velocity profile in the boundary layer and by the variation of the air properties with temperature. Heat flux at the stagnation point can be calculated using following expression of (Fay & Riddle, 1958)

$$q\_w = 0.763 \,\text{Pr}^{-0.6} \sqrt{\left(\rho\_w \mu\_w \mathcal{K}\right)} \sqrt{\frac{\rho\_w \mu\_w}{\rho\_w \mu\_w}} \left(h\_w - h\_w\right) \tag{4}$$

where the subscripts ∞ and w are freestream and wall, respectively. Pr is Prandtl number, K stagnation point velocity gradient and h enthalpy of air. The value of stagnation point velocity gradient K is taken as 0.3 (White, 1991). The hemispherical spike shows the greatest changes in the velocity gradient as compared to the flat disk spike. The magnitude of the stagnation-velocity gradient indicates the maximum heat transfer over the hemispherical

Computations of Flowfield over Reentry Modules at High Speed 363

Fig. 8. Instantaneous vector field, pressure contour plots and wall pressure distribution over

Computed instantaneous Mach contour plots over the Apollo and the OREX capsules are also depicted in Figs. 7 - 8. It can be observed from the contour plots that the oscillatory motion of the flow field in the base region of the flow capsules. Vortices are generated at the capsules and are changing location as a function of time. The flow structure inside the recirculating region is influenced by the periodic motion of the separation bubble and the

The periodic phenomenon is analyzed using the surface pressure variation over the capsules. The figures also show the pressure coefficient Cp variation over the Apollo and the OREX modules. The x = 0 location is the stagnation point. The sonic point moves to the corner of the blunt bodies and affects the pressure distribution throughout the subsonic flow. The pressure coefficient shows over expanded flow in the case of the OREX module. These types of flow field features are also explained by (Bertin, 1994) in conjunction with the flow past reentry capsule. A sudden drop of Cp is noticed on the shoulder of the module followed by a negative pressure variation in the base region. A low pressure is formed immediately downstream of the base which is characterized by a low speed recirculation flow region which can be attributed to fill-up the growing space between the shock wave and body. In the base region, the Cp is increasing with time interval of the cycle. The effect of the shoulder radius of the capsule on the Cp can be seen in the figures. The Apollo capsule is having small conical angle at the base-shell as compared to the OREX base-shell geometry of semi-angle cone of 75 deg. The pressure over the fore body of the capsules is varied with time. Thus, the cyclic of the flow field in the base region is influenced the fore body flow. The different base-shell cone angles give significant effects in the time period. A wavy pattern is seen in the pressure distribution in the base region which may be attributed to complex geometry in the base region. The periodicity of the flow field in the base region are 0.12s (8.33 Hz) and 0.096s (1.04 Hz) for the Apollo and the OREX modules, respectively. It is important to say that the cycle of the flow field over the OREX is 8 times slower than the Apollo. Comparing the computed

base-pressure with a common engineering formula (Bonner et al., (1989).

OREX capsule.

vorticity of the flow.

spike. The computed values of the stagnation point heat flux are 0. 556107 W/m2 and 1.45x107 W/m2 for the flat-disk and the hemispherical disk spike, respectively. The stagnation point calculated using Eq. (4) is 0. 831x107 W/m2 and 1.55107 W/m2 for the flatdisk and the hemispherical disk spike, respectively. The discrepancy is due to value of the K. The value of K is difficult to calculate for the flat disk (Bertin, 1994). The results show reasonably good agreement between them. Close-up view of velocity vector plot over the blunt body gives a comparative velocity gradient. The velocity vector is turning in the stream line fashion on the hemispherical spike where as on the flat-disk of the reentry modules as appears that the flow is impinging.

## **3.7 Oscillation flow field characteristics**

The oscillation motion is noticed in the numerical simulations. The flow field can be visualized in the instantaneous velocity vector plots. One oscillation cycle is corresponding to approximately 0.12s for the Apollo and 0.96s for the OREX capsule. For the sake of brevity we are displaying one flow cycle in the six time intervals of 0.002s and 0.16s for the Apollo and the OREX capsules, respectively. Figures 6 and 7 show the close-up view of the computed instantaneous velocity vector field over the Apollo and the OREX modules at M = 1.2. It can be observed from the vector plots that all the significant flow field features such as bow shock wave ahead of the capsule, rapid expansion fans on the corner, flow recirculation region with converging free shear layer and formation of the vortex flow in the aft region of the capsule.

Fig. 7. Instantaneous vector field, pressure contour plots and wall pressure distribution over Apollo capsule.

The wake flow field immediately behind the capsule base exhibits complex flow characteristics. It is also seen that the formation of the bow shock wave on the fore-body of the capsules and which depends on the RN, RC and B. A gradual flow turning can be observed in Fig. 7 for the case of Apollo which is having smooth shoulder of radius RC = 0.01 m. The approaching supersonic boundary layer separates at the corner and free shear layer is formed in the wake region. The wake flow features also depicts vortex attached to the corner with a large recirculating flow behind the capsule adjacent to the axis of symmetry which depends on back shell cone angle. The separation point moves downstream zone and location of vortex depend on time as can be visualized from the vector and density contour plots.

spike. The computed values of the stagnation point heat flux are 0. 556107 W/m2 and 1.45x107 W/m2 for the flat-disk and the hemispherical disk spike, respectively. The stagnation point calculated using Eq. (4) is 0. 831x107 W/m2 and 1.55107 W/m2 for the flatdisk and the hemispherical disk spike, respectively. The discrepancy is due to value of the K. The value of K is difficult to calculate for the flat disk (Bertin, 1994). The results show reasonably good agreement between them. Close-up view of velocity vector plot over the blunt body gives a comparative velocity gradient. The velocity vector is turning in the stream line fashion on the hemispherical spike where as on the flat-disk of the reentry

The oscillation motion is noticed in the numerical simulations. The flow field can be visualized in the instantaneous velocity vector plots. One oscillation cycle is corresponding to approximately 0.12s for the Apollo and 0.96s for the OREX capsule. For the sake of brevity we are displaying one flow cycle in the six time intervals of 0.002s and 0.16s for the Apollo and the OREX capsules, respectively. Figures 6 and 7 show the close-up view of the computed instantaneous velocity vector field over the Apollo and the OREX modules at M = 1.2. It can be observed from the vector plots that all the significant flow field features such as bow shock wave ahead of the capsule, rapid expansion fans on the corner, flow recirculation region with converging free shear layer and formation of the vortex flow in the

Fig. 7. Instantaneous vector field, pressure contour plots and wall pressure distribution over

The wake flow field immediately behind the capsule base exhibits complex flow characteristics. It is also seen that the formation of the bow shock wave on the fore-body of the capsules and which depends on the RN, RC and B. A gradual flow turning can be observed in Fig. 7 for the case of Apollo which is having smooth shoulder of radius RC = 0.01 m. The approaching supersonic boundary layer separates at the corner and free shear layer is formed in the wake region. The wake flow features also depicts vortex attached to the corner with a large recirculating flow behind the capsule adjacent to the axis of symmetry which depends on back shell cone angle. The separation point moves downstream zone and location of vortex depend on time as can be visualized from the vector and density contour

modules as appears that the flow is impinging.

**3.7 Oscillation flow field characteristics** 

aft region of the capsule.

Apollo capsule.

plots.

Fig. 8. Instantaneous vector field, pressure contour plots and wall pressure distribution over OREX capsule.

Computed instantaneous Mach contour plots over the Apollo and the OREX capsules are also depicted in Figs. 7 - 8. It can be observed from the contour plots that the oscillatory motion of the flow field in the base region of the flow capsules. Vortices are generated at the capsules and are changing location as a function of time. The flow structure inside the recirculating region is influenced by the periodic motion of the separation bubble and the vorticity of the flow.

The periodic phenomenon is analyzed using the surface pressure variation over the capsules. The figures also show the pressure coefficient Cp variation over the Apollo and the OREX modules. The x = 0 location is the stagnation point. The sonic point moves to the corner of the blunt bodies and affects the pressure distribution throughout the subsonic flow. The pressure coefficient shows over expanded flow in the case of the OREX module. These types of flow field features are also explained by (Bertin, 1994) in conjunction with the flow past reentry capsule. A sudden drop of Cp is noticed on the shoulder of the module followed by a negative pressure variation in the base region. A low pressure is formed immediately downstream of the base which is characterized by a low speed recirculation flow region which can be attributed to fill-up the growing space between the shock wave and body. In the base region, the Cp is increasing with time interval of the cycle. The effect of the shoulder radius of the capsule on the Cp can be seen in the figures. The Apollo capsule is having small conical angle at the base-shell as compared to the OREX base-shell geometry of semi-angle cone of 75 deg. The pressure over the fore body of the capsules is varied with time. Thus, the cyclic of the flow field in the base region is influenced the fore body flow. The different base-shell cone angles give significant effects in the time period. A wavy pattern is seen in the pressure distribution in the base region which may be attributed to complex geometry in the base region. The periodicity of the flow field in the base region are 0.12s (8.33 Hz) and 0.096s (1.04 Hz) for the Apollo and the OREX modules, respectively. It is important to say that the cycle of the flow field over the OREX is 8 times slower than the Apollo. Comparing the computed base-pressure with a common engineering formula (Bonner et al., (1989).

Computations of Flowfield over Reentry Modules at High Speed 365

blunt body capsules such as ARD (ESA's Atmospheric Reentry Demonstrator), Apollo II, MUSES-C, OREX (Orbital Reentry EXperiments) with and without shoulder curvature and spherically blunted cone with flare angle of 30 and 35 degree. The inviscid analysis takes into consideration centrifugal force and expansion fan at the shoulder of the reentry capsules. The effects of the capsule geometry on the flow field may be useful for optimization of the reentry capsule. The Newtonian flow assumptions are used to calculate forebody aerodynamic drag for various blunt-bodies in conjunction with the NISA software. A comparison between CFD and the Newtonian flow assumptions for various reentry modules are made, and comparison shows an agreement between them. The numerical

A finite volume cell is specified by eight corners, which are connected by straight lines. Figure 9 shows hexahedral cell in the computational domain. The discrete values of the flow quantities are calculated at the centre of the cell. The surface integral ) over the convective part of the flux density tensor is evaluated for each component cell using an arithmetic average of the flux quantities at the vertices to determine the values on each of the cell faces. Then, the resultant convective inflow of mass, momentum, energy associated with cell centre point is computed by summing the contributions of the component cells. Any opensurface element for a given boundary has a unique, effective surface vector S that is independent of the shape of the surface. This is because, by applying the divergence theorem to a constant vector, the integral of the outward-oriented surface normal over a closed surface vanishes. For instance, the surface vector S-5678 is independent of the choice of which partitioning surface diagonal is used to define the cell volume with vertices 1 to 8. Whether the four vertices are on a plane or not, the surface vector is equal to one-half the cross product of its diagonal line segment. Given eight arbitrary corner points prescribing a general hexahedral, a simple way to define a shape whose volume can be precisely calculated is to partition each face into two planar triangles. The volume is then dependent on which diagonal is used on each face. The volume is the dependent on which diagonal is based on each face, since the diagonal of four non-planar points do not intersect. (Kordulla

scheme is described in detail in References (Mehta & Tiwari, 2008; Mehta, 2007).

& Vinokur, 1983) had been used here to calculate cell volume.

Fig. 9. A hexahedral computational grid of point (i,j,k).

$$
\Delta C\_{D,base} = \frac{1}{M\_{\odot}^2} - \frac{0.57}{M\_{\odot}^4} \tag{5}
$$

The relation is useful to determine the base drag of the capsule as well as to validate the numerical results.

### **4. Three-dimensional Navier-Stokes equations**

The reentry capsule considered here is typified by the Apollo command module. This shape encompasses most of the problems that arises from both viscous and inviscid flow field considerations. The module is at an arbitrary angle of attack that achieved by an offset centre of gravity relative to the free-stream flow. Thus, though the body is axisymmetrical, the geometry of the flow field is not. The only plane of symmetry is the pitch plane, which contains the free-stream velocity vector and the body axis. The bow shock wave in front of the body is fully detached; a subsonic region exists behind the shock wave on the front side of the capsule. As the flow expands around the body, a sonic surface is created (Mach = 1); since the flow continues to increase in speed, it becomes supersonic downstream of this surface. At the downstream the boundary layer tends to separate. The details of the flow vary with geometry and freestream conditions, the general flow features remain the same. The fore body flow field is dominated by a strong bow shock wave. As the flow turns around the shoulder of the reentry module it rapidly expands and can separate. The leeward side flow separates just after the shoulder, whereas the windward side flow remains attached until the rear apex. A shear layer separates the outer flow from the recirculating inner core, which contains multiple counter rotating vortices. The separation shear layer eventually coalesces, forming the neck or narrowest point in the wake. A wake shock is formed at that point, which compresses the flow, leading to local maxima in pressure and temperature. Further away from the neck is far wake region, which extends for many body diameters downstream as the momentum deficit created by the passing module is slowly recovered. Emanating from the separation regions the dividing stream surface, that defines the boundary of the recirculation region – an area in which the flow is trapped. The flow field outside the dividing stream lines proceeds to a single flow stream at the neck of the wake. Oblique shocks are generated near the neck because of the flow external to the wake is turned supersonically. The surrounding inviscid flow and the viscous mixing layer flow build up in the wake much like a boundary layer and develop into the far wake downstream of the base flow field.

#### **4.1 Three-dimensional Euler equations**

The flow field analysis over various reentry configurations is studied numerically by solving time-dependent compressible Euler equations. The governing fluid flow equations are discretized in spatial coordinates employing a finite volume approach, which reduces the equations to semi-discretized ordinary differential equations. Temporal integration is performed using multi-stage Runge-Kutta time stepping scheme. A local time step is used to achieve steady state solution. The numerical computation is carried out for freestream Mach number of 10.0 and angle of attack of 10.0 degree. The flow features around the blunt body are characterized by a bow shock wave, expansion wave and base flow region. The numerical scheme captures all the flow field features well. Comparisons of the flow field and surface pressure distribution results are made between different configurations of the

, 2 4 1 0.57 *CD base <sup>M</sup> <sup>M</sup>*

The relation is useful to determine the base drag of the capsule as well as to validate the

The reentry capsule considered here is typified by the Apollo command module. This shape encompasses most of the problems that arises from both viscous and inviscid flow field considerations. The module is at an arbitrary angle of attack that achieved by an offset centre of gravity relative to the free-stream flow. Thus, though the body is axisymmetrical, the geometry of the flow field is not. The only plane of symmetry is the pitch plane, which contains the free-stream velocity vector and the body axis. The bow shock wave in front of the body is fully detached; a subsonic region exists behind the shock wave on the front side of the capsule. As the flow expands around the body, a sonic surface is created (Mach = 1); since the flow continues to increase in speed, it becomes supersonic downstream of this surface. At the downstream the boundary layer tends to separate. The details of the flow vary with geometry and freestream conditions, the general flow features remain the same. The fore body flow field is dominated by a strong bow shock wave. As the flow turns around the shoulder of the reentry module it rapidly expands and can separate. The leeward side flow separates just after the shoulder, whereas the windward side flow remains attached until the rear apex. A shear layer separates the outer flow from the recirculating inner core, which contains multiple counter rotating vortices. The separation shear layer eventually coalesces, forming the neck or narrowest point in the wake. A wake shock is formed at that point, which compresses the flow, leading to local maxima in pressure and temperature. Further away from the neck is far wake region, which extends for many body diameters downstream as the momentum deficit created by the passing module is slowly recovered. Emanating from the separation regions the dividing stream surface, that defines the boundary of the recirculation region – an area in which the flow is trapped. The flow field outside the dividing stream lines proceeds to a single flow stream at the neck of the wake. Oblique shocks are generated near the neck because of the flow external to the wake is turned supersonically. The surrounding inviscid flow and the viscous mixing layer flow build up in the wake much like a boundary layer and develop into the far wake

The flow field analysis over various reentry configurations is studied numerically by solving time-dependent compressible Euler equations. The governing fluid flow equations are discretized in spatial coordinates employing a finite volume approach, which reduces the equations to semi-discretized ordinary differential equations. Temporal integration is performed using multi-stage Runge-Kutta time stepping scheme. A local time step is used to achieve steady state solution. The numerical computation is carried out for freestream Mach number of 10.0 and angle of attack of 10.0 degree. The flow features around the blunt body are characterized by a bow shock wave, expansion wave and base flow region. The numerical scheme captures all the flow field features well. Comparisons of the flow field and surface pressure distribution results are made between different configurations of the

numerical results.

downstream of the base flow field.

**4.1 Three-dimensional Euler equations** 

**4. Three-dimensional Navier-Stokes equations** 

(5)

blunt body capsules such as ARD (ESA's Atmospheric Reentry Demonstrator), Apollo II, MUSES-C, OREX (Orbital Reentry EXperiments) with and without shoulder curvature and spherically blunted cone with flare angle of 30 and 35 degree. The inviscid analysis takes into consideration centrifugal force and expansion fan at the shoulder of the reentry capsules. The effects of the capsule geometry on the flow field may be useful for optimization of the reentry capsule. The Newtonian flow assumptions are used to calculate forebody aerodynamic drag for various blunt-bodies in conjunction with the NISA software. A comparison between CFD and the Newtonian flow assumptions for various reentry modules are made, and comparison shows an agreement between them. The numerical scheme is described in detail in References (Mehta & Tiwari, 2008; Mehta, 2007).

A finite volume cell is specified by eight corners, which are connected by straight lines. Figure 9 shows hexahedral cell in the computational domain. The discrete values of the flow quantities are calculated at the centre of the cell. The surface integral ) over the convective part of the flux density tensor is evaluated for each component cell using an arithmetic average of the flux quantities at the vertices to determine the values on each of the cell faces. Then, the resultant convective inflow of mass, momentum, energy associated with cell centre point is computed by summing the contributions of the component cells. Any opensurface element for a given boundary has a unique, effective surface vector S that is independent of the shape of the surface. This is because, by applying the divergence theorem to a constant vector, the integral of the outward-oriented surface normal over a closed surface vanishes. For instance, the surface vector S-5678 is independent of the choice of which partitioning surface diagonal is used to define the cell volume with vertices 1 to 8. Whether the four vertices are on a plane or not, the surface vector is equal to one-half the cross product of its diagonal line segment. Given eight arbitrary corner points prescribing a general hexahedral, a simple way to define a shape whose volume can be precisely calculated is to partition each face into two planar triangles. The volume is then dependent on which diagonal is used on each face. The volume is the dependent on which diagonal is based on each face, since the diagonal of four non-planar points do not intersect. (Kordulla & Vinokur, 1983) had been used here to calculate cell volume.

Fig. 9. A hexahedral computational grid of point (i,j,k).

Computations of Flowfield over Reentry Modules at High Speed 367

Capsules M=1.5 M=2.0 M=3.0 M=5.0 M=7.0 COMET 1.56 1.55 1.49 – Beagle – 2 1.48 1.50 1.502 1.508 1.482

In the Newtonian flow concept the shock wave is assumed to coincide with the body surface. It is also assumed that the freestream maintain its speed and direction unchanged until it strike the surface exposed to the flow. The impact theory does not specify the pressure on surface that do not "visualize" the flow, that is, the surfaces on which gas dynamics would predict Prandtl-Mayer flow. When the flow is curved in the direction of freestream, the difference in pressure from the shock wave to the body surface equals the

A computer program is written to compute the aerodynamic coefficients using the modified Newtonian sine-squared law Cp = Cp,max sin2, where is the angle between the freestream velocity vector, and Cp, max maximum pressure coefficient, which is calculated at a stagnation point behind a normal shock using the Rayleigh pitot tube formula for frozen gas chemistry (constant specific ratio of heats). (Lees, 1955) proposed a modified Newtonian theory, which consists in scaling down so as to be exact at stagnation point, where the correct value is known. The forebody aerodynamic drag is also computed using Newtonian flow assumptions. The numerical simulation can be validated with the post flight data of the Gemini, Apollo and Star dust flights data (Griffith, 1967; Griffith & Boylan, 1968; Lockman,

The aerodynamic characteristics of the reentry capsules are studied experimentally employing the shock tube, the high enthalpy tunnel, wind-tunnel, and free-flight testing. The experimental set up has to design to meet the specific requirements and conditions for the reentry probes. The post flights test data are available for many reentry modules. The flow fields over many reentry modules are generated using computational fluid dynamics for various test conditions. The computational fluid dynamics can be used to analyze and understand the complex flow field features in the wake region of the modules. Numerical simulation can be used to get aerodynamic characteristics, wall heat flux, and stability

The author is indebted to his parents and Vikram Sarabhai Space Centre, Trivandrum, India for their valuable encouragement, help and contributions to build the research career.

Allen, J. S. and Cheng, S. I. (1970). Numerical solution of the compressible Navier-Stokes equations for the near wake, Physics of Fluids, Vol. 13, No. 1, pp. 37-52 Ames Research Staff (1953). Equations, tables and charts for compressible flow, NACA

Table 5. Numerically computed aerodynamic drag coefficient.

centrifugal force, due to the curvature of the flow.

margin for the optimization of the trajectory of the capsules.

**4.3 Newtonian impact theory** 

1970; & Wood, 1997).

**6. Acknowledgment** 

report 1135

**7. References** 

**5. Conclusion** 
