Shock Waves (2006) 15(1): 31–41DOI 10.1007/s00193-005-0003-0

ORIGINAL ARTICLE

R. C. Mehta

Numerical simulation of supersonic flow past reentry capsules

Received: 24 June 2005 / Accepted: 31 August 2005 / Published online: 1 February 2006C© Springer-Verlag 2006

Abstract The flow fields over ARD (ESA’s atmosphericreentry demonstrator), OREX (orbital reentry experiments)and spherically blunted cone-flare reentry configurationsare numerically obtained by solving time-dependent, ax-isymmetric, compressible Navier–Stokes equations forfreestream Mach numbers range of 1.2–6.0. The fluid dy-namics are discretized in spatial coordinates employing afinite volume approach which reduces the governing equa-tions to semi discretized ordinary differential equations.Temporal integration is performed using the multistageRunge–Kutta time-stepping scheme. A local time step isused to achieve steady-state solution. The numerical simu-lation is carried out on a structured grid. The flow-field fea-tures around the reentry capsule, such as bow shock wave,sonic line, expansion fan and recirculating flow in the baseregion are obtained. A good agreement is found betweenthe calculated value of aerodynamic drag coefficient of thespherically blunted cone/fare reentry configuration with theexperimental data. The effects of geometrical parameters,such as radius of the spherical cap, half cone angle, withsharp shoulder edge and with smooth shoulder edge on theflow-field have been numerically investigated for variousreentry configuration which will be useful for optimizationof the reentry capsule.

Keywords Supersonic flows · Computational fluiddynamics · Reentry capsules

PACS 47.11.Df, 47.40.Ki

1 Introduction

A high-speed flow-past a blunt-body generates a bow shockwave which causes a rather high surface pressure and as

Communicated by K. Takayama

R. C. Mehta (B)Aerodynamics Division, Vikram Sarabhai Space Centre,Trivandrum 695022, IndiaE-mail: atulm@md4.vsnl.net.in

a result the development of high aerodynamic drag whichis needed for aerobraking. Highly blunt configurations aregenerally used to decelerate spacecraft for safe returning onthe Earth after performing the experiments. The bow shockwave is detached from the blunt forebody and is having amixed subsonic–supersonic region between them. The sur-face pressure distribution, the location of the sonic line andthe shock stand-off distance on the spherical cap have beenanalytically calculated at very high speeds with an adiabaticindex near to unity which gives a singular point at 60◦ fromthe stagnation point [1, 2]. The analytical approach for thehigh-speed flow-past the blunt-body is considerably difficultand complex [3]. The flow-field over the reentry module be-comes further complicated due to the presence of corner atthe shoulder and the base shell of the capsule.

Aeroassist flight experiment (AFE) configurations havebeen analyzed using two different Navier–Stokes flowsolvers by Venkatapathy et al. [4]. Aerodynamic heatingand pressure along with the forebody and wake-flow struc-ture during atmospheric entry of the Mars Pathfinder space-craft have been computed by Haas and Venkatapathy [5] us-ing the commerically available general atmospheric simula-tion program (GASP 2.2). An aerodynamic analysis of thecommercial experiment transport (COMET) reentry capsulehas been carried out by Wood et al. [6] solving the lami-nar thin layer Navier–Stokes LAURA code for low super-sonic to hypersonic speeds. The LAURA code is an upwind,point implicit, second-order-accurate fluid dynamics solverbased on an extension of the Roe flux-difference-splittingscheme. Yamomoto et al. [7] carried out flow-field com-putation over the OREX (orbital reentry experiments) us-ing computational fluid dynamics approach coupled with thethermal response of the heat shield material using finite el-ement method in conjunction with the aerodynamic flightdata. Tam [8] has computed flow field at hypersonic Machnumbers over Viking, Biconic and AFE reentry capsules us-ing IEC3D (inviscid equilibrium computation in three di-mension). Menne [9] has computed flow field over Viking(Apollo type) and Biconic cases for freestream Mach num-bers 2.0 and 3.0 by solving Euler equations. The inviscid

32 R. C. Mehta

Fig. 1 Representation of flow features on blunted body at supersonic speeds

analysis is unable to predict the flow separation in the backregion of the reentry module. The aerodynamic characteri-zation of the CARINA reentry module in the low supersonicMach regimes has been performed employing numerical andexperimental approach [10]. The flow-field simulation pastthe ARD (ESA’s atmospheric reentry demonstrator) modulehas been computed numerically by Walpot [11] at hyper-sonic speeds. Developments relating to base pressure andbase heating prediction for wide range of cone and wedgehave been reviewed for supersonic and hypersonic Machnumbers by Lamb and Oberkampf [12]. Recently, flow-field solutions past Beagle-2 spacecraft have been obtainedby Liever et al. [13] using commerically available CFD-FASTRAN code for low supersonic to hypersonic speeds.A numerical simulation code has been used for super or-bital reentry flow and has been applied to the flow-field pre-diction around the MUSES-C reentry capsule [14]. Super-sonic and hypersonic flow over a slender cone [15] has beennumerically obtained by solving Navier–Stokes equationsusing an explicit multi-stage Roe’s scheme. The flow fieldaround blunt reentry capsules [16] were numerically stud-ied in order to understand the mechanism of dynamic in-stability of the capsule at freestream Mach number of 1.3.The blunt and short reentry capsule tends to be dynami-cally unstable at low supersonic Mach number attributedprimarily to the delay in base pressure [17]. Numericalstudies have been carried out by Ottens [18] using a lam-inar Navier–Stokes flow solver for two different types ofdelft aerospace reentry test (DART) demonstrators reentrymodules.

The above literature survey reveals that the forebodyshape of reentry capsules can be classified either using asa spherical cap as in the case of Apollo and ARD, or com-bination of the spherical nose with cone as in the case ofOREX and Beagle-2, or a spherical blunt-cone/flare config-uration as proposed by DART. The flow-field features overthe reentry capsule can be delineated through the experimen-tal and theoretical investigations at high speeds. The signif-icant flow features are described by the following: In the

forebody region, the fluid decelerates through the bow shockwave depending upon the cruise speed and altitude. At theshoulder of the capsule, the flow turns and expands rapidly,and the boundary layer detaches, forming a free shear layerthat separates the inner recirculating flow region behind thebase from the outer flow field. The latter is recompressedand turned back to freestream direction, first by the so-calledlip shock, and further downstream by recompression shock.At the end of the recirculating flow past the neck, the shearlayer develops in the wake trail. A complex inviscid wavestructure often includes a lip shock wave (associates withthe corner expansion) and wake trail (adjacent to the shear-layer confluence). The corner expansion process is a modi-fied Prandtl–Mayer pattern distorted by the presence of theapproaching boundary layer. Figure 1 shows schematic fea-tures of the flow field over OREX and DART.

The sonic line is located on the OREX-type configura-tion on the shoulder whereas in the case of DART’s reentrymodule, the location of the sonic line is at the junction of thespherical blunt cone. Thus, it is seen that the flow field overthe reentry capsule needs a high drag with good static stabil-ity margin which leads to the selection of an axisymmetricshape of large angle sphere-cone combination.

The forebody geometry of the ARD configuration is hav-ing a spherical cap; the OREX module is having combina-tion of spherical cap with cone. The other capsule consistsof spherically blunted cone/fare. These capsules cover mostof the existing reentry modules. The main aim of the presentpaper is to numerically analyze three types of reentry mod-ules such as ARD, OREX and DART configurations in orderto understand flow-field behavior and its influence on pro-files of the surface pressure and skin friction coefficients andalso aerodynamic drag for freestream Mach numbers rangeof 1.2–6.0. The numerical solution of solve-axisymmetriclaminar-compressible time-dependent Navier–Stokes equa-tions is carried out employing a multi-stage Runge–Kuttatime stepping scheme. The numerical scheme is second-order accurate in space and time. A local time step is usedto obtain a steady-state solution. The computation is carried

Numerical simulation of supersonic flow past reentry capsules 33

out on a structured grid. The effects of geometrical parame-ters of the reentry capsules on the wall quantities and aero-dynamic drag coefficient are analyzed using the numericallyobtained flow-field data.

2 Problem definition and approach

2.1 Governing equations

The axisymmetric time-dependent compressible Navier–Stokes equations can be written in the following strong con-servative form:

∂W∂t

+ ∂F∂x

+ ∂G∂r

+ H =[∂R∂x

+ ∂S∂r

](1)

where

W = r

ρ

ρu

ρv

ρe

, F = r

ρu

ρu2 + p

ρuv

(ρe + p)u

,

G = r

ρv

ρuv

ρv2 + p

(ρe + p)v

are the state vector W and inviscid flux vectors F and G.The viscous flux vectors are R, S, and H is the source vectorterm.

R = r

0

σxx

τxr

uσxx + ντxr + qx

,

S = r

0

τxr

σrr

uτxr + νσrr + qr

, H =

0

0

σ+0

where σxx , σrr , τxr and σ+ are components of the stressvector, qx and qr are components of the flux vector, uand ν are axial- and radial-velocity components in x andr directions, respectively, e is the total energy. Thus, theviscous terms in the above equations become

σxx = −2

3µ∇ · U + 2µ

∂u

∂x

τrr = −2

3µ∇ · U + 2µ

∂ν

∂r

τxr = τr x = µ

(∂u

∂r+ ∂ν

∂x

)

σ+ = −p − 2

3µ∇ · U + 2µ

ν

r

∇ · U = ∂x

∂x+ ∂ν

∂r+ ν

r

qx = −Cpµ

Pr

∂T

∂x

qr = −Cpµ

Pr

∂T

∂r

where Cp is specific heat at constant pressure, U is the meanupstream velocity. Pr is the Prandtl number. The coefficientof molecular viscosity µ is calculated according to Suther-land’s law. The flow is assumed to be laminar, which is con-sistent with the numerical simulation of [6, 13, 15]. The tem-perature T is related to the pressure p and ρ by the perfectgas equation of state as

p

(γ − 1)=

[ρe − 1

2ρ(u2 + ν2)

](2)

The ratio of specific heats γ was assumed constant andequal to 1.4.

2.2 Numerical algorithm

2.2.1 Spatial discretization

To facilitate the spatial discretization in the numericalscheme, the governing fluid dynamics, Eq. (1), can be writ-ten in the integral form over a finite volume as

∂

∂t

∫�

Wd�+∫

�

(Fdr −Gdx) =∫

�

(Rdr −Sdx)−∫

�

Hd�

(3)

where � is the computational domain, � is the boundary ofthe domain. The contour integration around the boundary ofthe cell is taken in the anticlockwise sense.

The computational domain is divided into a finite num-ber of non-overlapping quadilateral cells. The conservationvariables within the computational cell are represented bytheir average values at the cell centre (i, j). When the inte-gral governing Eq. (3) is applied separately to each cell inthe computational domain, we obtain a set of coupled ordi-nary differential equations of the form

Ai, j∂Wi, j

∂t= Q(Wi, j ) − V(Wi, j ) + Ai, j Hi, j (4)

where Ai, j is the area of the computational cell, Q(Wi, j ),V (Wi, j ) and H(Wi, j ) are the inviscid and viscous fluxesand source term, respectively. These quantities are obtainedby a simple averaging of adjacent cell-centre values of de-pendent variables [19]. In viscous calculations, dissipatingproperties are present due to diffusive terms.

34 R. C. Mehta

2.2.2 Artificial dissipation

To suppress the tendency for odd- and even-point decouplingand to prevent the appearance of oscillations in regions con-taining severe pressure gradients near shock wave and stag-nation points, the finite volume scheme must be augmentedby the addition of artifical dissipation terms. Therefore,Eq. (4) is replaced by

Ai, j∂Wi, j

∂t+ Res(Wi, j ) − D(Wi, j ) = 0 (5)

where the residual Res(W ) is given by

Res(Wi, j ) = Q(Wi, j ) − V (Wi, j ) + Ai, j Hi, j (6)

where Di, j denotes the dissipative terms which are gener-ated by dissipative fluxes. The approach of Jameson et al.[20] is adopted to construct the dissipative terms consistingof a blend of second and fourth differences of the vectorconserved variables Wi, j with coefficients that depend onthe local pressure gradient. The dissipation term consists ofthe following operators in each direction.

D = (Dx + Dr )Wi, j (7)

The dissipative fluxes in each direction are

Dx Ui, j = di+ 12 , j − di− 1

2 , j

Dr Ui, j = di, j+ 12

− di, j− 12.

The dissipative flux di+ 12 , j is defined as

di+ 12 , j =

(A)i+ 12 , j

(t)i+ 12 , j

[d(2)

i+ 12 , j

− d(4)

i+ 12 , j

](8)

with

d(2)

i+ 12 , j

= ε(2)(Wi+1, j − Wi, j )

d(4)

i+ 12 , j

= ε(4)(Wi+2, j − 3Wi+1, j + 3Wi, j − Wi−1, j ).(9)

The terms di− 12 , j , di, j+ 1

2and di, j− 1

2are calculated in an

analogus manner. ε(2) and ε(4) are adaptive coefficients anddefined as

ε(2) = κ(2)max(νi+1, j , νi, j )

ε(4) = max(0, κ(4) − ε(2)

) (10)

are switched on or off by use of the shock wave sensor ν,with

νi, j =∣∣∣∣ pi+1, j − 2pi, j + pi−1, j

pi+1, j + 2pi, j + pi−1, j

∣∣∣∣ (11)

where κ(2) are κ(4) are constants, taken equal to 14 and 1

256 ,respectively, in the above calculations. The scaling quantity(A

t )i+ 12 , j in Eq. (8) confirms the inclusion of the cell vol-

ume in the dependent variable of Eq. (4). The blend of sec-ond and fourth differences provides third-order backgrounddissipation in smooth regions of the flow and first-order dis-sipation as shock waves. The dissipation terms in the r -direction are constructed in a similar way.

2.2.3 Time-marching scheme

The spatial discretization described above reduces the gov-erning flow equations to semidiscrete ordinary differentialequations. The integration is performed using an efficientmutistage three-stage Runge–Kutta time-stepping scheme[20]. The time-step advance of each variable W from time(t) to time (t + t) can be written as

W(0) = W(t)

W(1) = W(t) − 0.6

[t

A

] (Res(0) − D(0)

)

W(1) = W(t) − 0.6

[t

A

] (Res(1) − D(0)

)

W(1) = W(t) − 0.6

[t

A

] (Res(2) − D(0)

)

W(t + t) = W(3)

(12)

In order to minimize the computation time and increasethe stability margin for the dissipative terms, the expensiveevaluation of the artificial dissipation terms is carried outonly at the first intermediate stage (0) and then frozen for thesubsequent stages. A conservative choice of the Courant–Friedrichs–Lewy (CFL) number (1.4) is taken to achieve astable numerical solution. Local time steps are used to ac-clerate convergence to a steady-state solution by setting thetime step at each point to the maximum value allowed bythe local CFL condition. The present numerical algorithm isvalidated with many test cases [21, 22].

2.3 Initial and boundary conditions

The freestream conditions for each trajectory point are enu-merated in Table 1, which are used as the initial conditions.The subscript ∞ represents freestream value in Table 1.

Four types of boundary conditions are required for thecomputation of flow field, i.e. wall, inflow, outflow and sym-metric conditions. They are prescribed as follows:

At the solid wall, no-slip condition is enforced by setting

uw = vw = 0 (13)

together with an adiabatic wall condition where subscript wrefers to the wall condition.

At the inflow, all flow variables are prescribed at thefreestream values as given in Table 1.

For the supersonic outflow case, all flow variables are ex-trapolated at the outer and wake regions of the computational

Table 1 Trajectory points and initial conditions

M∞ ν∞m/s p∞, (Pa) T∞, (K)

1.2 351 4519 2102.0 596 2891 2193.0 903 2073 2245.0 1532 1238 2326.0 1840 1064 234

Numerical simulation of supersonic flow past reentry capsules 35

Fig. 2 Geometrical parameters of reentry capsules

domain. At the centre line of the computational domain, thefollowing symmetric conditions are prescribed.

ρν = 0∂u∂r

= ∂T∂r

= ∂p∂r

= 0(14)

At the center line of the reentry capsule, the respectivecell faces are having zero surface area; therefore, it is simpleto implement the symmetric conditions.

3 Model and grid arrangement

3.1 Body configuration

The dimensional detail of the ARD capsule, shown inFig. 2a, is an axisymmetric design with a spherical bluntnose diameter, D = 2.8 m, spherical cap radius, RN =3.36 m and a shoulder radius, RC = 0.014 m. The back shellhas inclination angle αB = 33◦ relative to the vehicle’s axisof symmetry. A frustum of cone of radius 0.507 m with a12◦ half angle cone is attached to the base region. The over-all length of the module L = 2.04 m. The ARD resembles a70% scaled version of Apollo capsule [11].

The OREX geometry is depicted in Fig. 2b with the de-tailed dimensions. The forebody shape consists of RN =1.35 m, a half-angle cone of αN = 50◦, D = 3.4 m,L = 1.508 m and RC = 0.01 m. The OREX geometry in-corporates a rear cover with a small backward facing stepat the junction between back cover and heat shield. The aftbody is having a, αB = 15◦, half-angle cone relative to theplane of symmetry.

The spherically blunted-cone/flare configuration is illus-trated in Fig. 2c. The conical forebody has RN = 0.51 m,D = 2.03 m, L = 1.67 m and αN = 20◦. The flare has ahalf-angle cone of 25◦ and is terminated with a right circu-lar cylinder and is geometrically similar to the REV of theDART demonstrator [18].

3.2 Computational grid

One of the controlling factors for the numerical simulationis the proper grid arrangement. In order to initiate the nu-merical simulation of flow over the body, the physical shapeis discretized into nonuniform-spaced grid points. Thesebody-oriented grids are generated algebraically in conjunc-tion with homotopy scheme [23]. The typical computationalspace over the reentry capsule is defined by a number of gridpoints in cylindrical coordinate system. Using these surfacepoints as the reference nodes, the normal coordinate is thendescribed by exponentially structured field points, (xi, j , ri, j )extending outwards upto an outer computational boundary.The stretching of grid points in the normal direction is ob-tained using the following expression:

xi, j = xi,0

[e

( j−1)βnr −1 − 1

eβ − 1

]+ xi,w

[1 − e

( j−1)βnr −1 − 1

eβ − 1

]

ri, j = ri,0

[e

( j−1)βnr −1 − 1

eβ − 1

]+ ri,w

[1 − e

( j−1)βnr −1 − 1

eβ − 1

]

i = 1, 2, . . . , nxj = 1, 2, . . . , nr

(15)

36 R. C. Mehta

Fig. 3 Enlarged view of computational grid

where ri,w and ri,0 are wall and outer surface points, respec-tively, β is the streching factor. nx and nr are total number ofgrid points in x and r directions, respectively. These strechedgrids are generated in an orderly manner. The typical com-putational space of the reentry module is defined by a num-ber of grid points in the cylindrical coordinate system. Usingthese surface points as the reference nodes, the normal co-ordinate is then described by exponentially streched struc-tured field points, extending up to an outer computationalboundary. The streched grid points in the direction is ob-tained using exponentially stretched relation. These grids aregenerated in an orderly manner. Grid independence tests arecarried out taking into consideration the effect of the com-putational domain, the streching factor to control the gridintensity near the wall, and the number of grid points in theaxial and normal directions. The outer boundary of the com-putational domain is varied from 5 to 12 times the maximumdiameter of the capsule. The grid streching factor in the ra-dial direction is varied from 1.5 to 5. The present numericalanalysis is carried out on 132 × 52 grid points. The gridstreching factor is selected as 5, and the outer boundary ofthe computational domain is kept about 4–7 times the maxi-mum diameter of the reentry module. In the downstream di-rection, the computational boundary is about 6–10 times themaximum diameter of the capsule. The minimum grid spac-ing at the wall is about 2 × 10−5 − 8 × 10−5 m, sufficientto resolve the boundary layer and complex flow field whichgives resulting Reynolds number after bow shock wave forthis minimum grid spacing as 33−41. The coarse grid helpsin reducing the computer time. A close-up view of the com-putational grid over different capsules can be seen in Fig. 3.

This grid arrangement is found to give a difference mea-sured and computed values of ±1.5% in the drag coefficient.The convergence criterion is based on the difference in den-sity values, ρ, at any grid point between two successive iter-ations, that is, |ρl+1 − ρl | ≤ 10−5, where l is the iterativeindex.

4 Results and discussion

The numerical procedure described in the previous sectionis applied here to compute flowfield over ARD (ESA’s at-

mospheric reentry demonstrator), OREX (orbital reentry ex-periments) with a smooth and a sharp shoulder edge anda spherically blunted cone-flare reentry modules and forfreestream Mach numbers range of 1.2–6.0.

4.1 Flow characteristics

Figures 4–7 show the enlarged view of the computed veloc-ity vector field over the above-mentioned vehicles at variousfreestream Mach numbers M∞. It can be visualized from thevector plots that all the significant flowfield features such asbow shock wave, rapid expansion fans on the corner, flow re-circulation region with converging free shear layer and for-mation of the vortex flow in the aft region of the capsule.The wake flow field, immediately behind the capsule base,exhibits complex flow characteristics. The formation of thebow shock wave on the forebody of the capsule is observed,which depends on RN and αN and the value of M∞. Thebow shock wave moves close to the forebody with the in-creasing M∞ and the stand off distance between bow shockwave and the forebody decreases with the increasing M∞.

Fig. 4 Enlarged view of velocity field over ARD

Numerical simulation of supersonic flow past reentry capsules 37

Fig. 5 Enlarged view of velocity field over OREX (with smooth shoul-der edge)

Fig. 6 Enlarged view of velocity field over OREX (with sharp shoulderedge)

In Fig. 7, the bow shock wave does not follow the body con-tour in the case of the spherically blunted cone-flare config-uration, which is attributed to small values of RN and αN ascompared to OREX. A gradual flow turning can be visual-ized in the case of OREX with smooth shoulder edge as seenin Fig. 5 whereas a sharp flow turning is found in the sharpshoulder edge of the OREX as noticed in Fig. 6. The ap-proaching supersonic boundary layer separates at the cornerand the free shear layer is formed in the wake region. Thewake flow feature also depicts vortex attached to the cornerwith a large recirculating flow behind the vehicle adjacent tothe axis of symmetry which depends on αB and M∞. Theseparation point moves downstream from the shoulder to-wards the base with the increase in M∞. Similar flow-fieldfeatures were observed in the analysis of the bulbous pay-load shroud of the heat shield of the launch vehicle [24].

Fig. 7 Enlarged view of velocity field over spherically blunted cone-flare module

Computed Mach contour plots around the variouscapsules are shown in Figs. 8–11 for various freestreamM∞. The Mach contour plots show the formation ofvortices at the corner region of the capsule for M∞ ≤ 3.Characteristic features of the flow field around the bluntbody at supersonic speeds, such as bow shock wave ahead ofthe capsule, the wake, and the recompression shock wavesemanating from the neck point, are observed in the Machcontour plots. In Figs. 8–10, the bow shock wave followsthe body contour and the forebody is entirely subsonic uptothe corner tangency point of the ARD and the OREX wherethe sonic line is located. In the case of spherically blunted

Fig. 8 Mach contours over ARD

38 R. C. Mehta

Fig. 9 Mach contours over OREX (with smooth shoulder edge)

Fig. 10 Mach contours over OREX (with sharp shoulder edge)

Fig. 11 Mach contours over spherically blunted cone-flare module

cone-flare module, the sonic line is located at the junction ofthe sphere cone as seen in Fig. 11. The Mach contour plotsreveal many intresting flow features of the reentry capsule.The flow expands at the base corner and is followed by therecompression shock downstream of the base which realignsthe flow. The flow then develops in the trailing wake. Theflow ground of the capsule is divided into regions inside andoutside of the flow recirculating zone, and two regions areseparated by the shear layer. The wake structure includesone vortex attached to the conical after-body frustum anda large recirculating vortex behind the reentry module.As observed in Figs. 8–11, vortices are generated at thecapsule surface and are then moving and changing locationwith M∞. One can also see the strong vortex flow overthe shoulder of the capsule at Mach number 1.2 and 2.0.The flow may become highly unsteady at supersonic Machnumbers [16, 17] due to the formation of the vortices. Note,however, that use of a fixed CFL number in the presentnumerical flow simulation leads to a local time step sizewhich differs throughout the flow domain. The local timestepping scheme gives rapid convergence for steady-flowproblem but cannot compute time accurate behaviour. Rapidexpansion around the fore body corners produces highMach numbers in the outer inviscid region of the wake.

Figures 12–15 show the pressure coefficient [Cp =2(p/p∞) − 1/γ M2∞] variation along the surface for dif-ferent reentry capsules and freestream Mach numbers. Thes/D = 0 location is the stagnation point, where s is the dis-tance measured along the surface from the stagnation pointand D is the maximum diameter of the capsule. The pres-sure coefficient on the spherical cap of the capsule decreasesgradually for a given M∞. In Figs. 13–15, Cp falls on thesphere-cone junction and remains constant over the cone.In the case of the ARD and the OREX, the sonic pointmoves to the corner of the blunt bodies and affects the pres-sure distribution throughout the subsonic flow. In the case ofthe OREX with αN = 50◦, the pressure coefficient shows

Fig. 12 Variation of pressure coefficient along the surface (ARD)

Numerical simulation of supersonic flow past reentry capsules 39

Fig. 13 Variation of pressure coefficient along the surface (OREX withsmooth shoulder edge)

Fig. 14 Variation of pressure coefficient along the surface (OREX withsharp shoulder edge)

overexpanded flow. The spherically blunted cone-flare con-figuration (αN = 20◦) gives underexpanded flow as seen inFig. 15. These types of flow-field features are also explainedby Bertin [25] in conjunction with flow past reentry cap-sules. A sudden drop in Cp is observed on the shoulder ofthe module followed by a negative Cp variation in the baseregion. A low pressure is formed immediately downstreamof the base which is characterized by a low-speed recirculat-ing flow region which can be attributed to fill-up the growingspace between the shock wave and the body [3]. In the baseregion, Cp is decreasing with increasing M∞. The effect ofthe corner radius on Cp can be observed in Figs. 13 and 14.The value of Cp is higher on the corner as compared with thesharp shoulder edge of the OREX module. At M∞ = 1.2, awavy pattern is observed in the pressure distribution in the

Fig. 15 Variation of pressure coefficient along the surface (sphericallyblunted cone-flare module)

Fig. 16 Variation of skin friction coefficient along the surface (ARD)

base region which may be attributed due to complex geom-etry in the base region of the OREX.

The skin friction coefficient Ct along the surface of thecapsule is computed using following relation.

C f = −µ∣∣ du

dr

∣∣wall

12ρu2∞

(16)

Figures 16–19 depict the variation of C f along the sur-face of the capsule with M∞ as a parameter. C f decreaseswith increasing M∞ on the forebody. In Fig. 19, the skinfriction increases in the spherical region then decreases onthe first cone generator and again starts increasing on theflare. A sudden drop in skin friction is found at the shoulderof the capsule. This may be attributed to sudden expansionof the flow on the corner. Negative skin friction can be seen

40 R. C. Mehta

Fig. 17 Variation of skin friction coefficient along the surface (OREXwith smooth shoulder edge)

on the base, which is due to the flow separation. The separa-tion zone is found to be function of M∞ and geometry of thebase region of the capsule. Pressure fore drag is calculatedby integrating the pressure distribution on the body surfaceexcluding the base of the capsule and can be expressed as

CD = 2πri Cp∫

i tan θ dx

Amax(17)

where r and θ are local radius and local inclination angle inthe x-direction station i respectively. Amax is the maximumarea of the capsule. Table 2 gives pressure-drag coeffi-cient of the ARD, the OREX- and the spherically bluntedcone-flare configurations at different M∞. CD increasedas the nose radius is increased; as expected, the increase isrelatively small. High aerodynamic drag is seen in the ARD-and the OREX- type module as compared with the sphericalblunt configurations. High CD is found for the OREXwith the smooth shoulder edge as compared to the cornerradius. A good agreement is found between the calculatedvalue of forebody aerodynamic drag coefficient of thespherically blunted cone-fare reentry configuration with theexperimental data.

Table 2 Forebody pressure drag coefficient

M Calculated value of CD Experimentalvalue of CD

ARD OREX with smooth OREX with sharp Spherically blunted Spherically bluntedshoulder edge shoulder edge cone flare cone flare

1.2 1.45 1.82 1.80 0.43 0.422 1.43 1.50 1.36 0.42 0.413 1.28 1.30 1.17 0.39 0.405 1.13 1.16 1.04 – –6 – – – 0.37 0.38

Fig. 18 Variation of skin friction coefficient along the surface (OREXwith sharp shoulder edge)

Fig. 19 Variation of skin friction coefficient along the surface (spheri-cally blunted cone-flare module)

Numerical simulation of supersonic flow past reentry capsules 41

5 Conclusion

The flow field over various reentry axisymmetric configu-rations are studied numerically by solving time-dependentcompressible Navier–Stokes equations. The governing fluid-flow equations are discretized in spatial coordinates employ-ing a finite volume approach which reduces the equationsto semi-discretized ordinary differential equations. Tempo-ral integration is performed using the two-stage Runge–Kutta time-stepping scheme. A local time step is used toachieved steady-state solution. Flow field around the cap-sules have been calculated in the freestream Mach numberrange of 1.2–6.0 for different configurations. The essentialflow-field features around the capsules are captured for vari-ous reentry capsules. The effects of geometrical parameters,such as radius of the spherical cap, half cone angle, withthe sharp shoulder edge and with the smooth shoulder edgeon the flow field have been numerically investigated for var-ious reentry configuration. The ARD, the OREX capsuleshave the sonic line over the forebody shoulder whereas thespherically blunted cone-flare module is having sonic lineover the spherical cap region. The flow field behind the bowshock wave is either subsonic or mixed subsonic–supersonicregion depending upon the geometrical parameters of thereentry configuration. Thus, the reentry configurations canbe distinguished by the location of the sonic line over thefore-body at supersonic Mach number which influences wallpressure and skin friction variations and also aerodynamicdrag coefficient. The shoulder edge affects the corner expan-sion wave. The pressure coefficient, the skin friction coeffi-cient variation along the surface and the integrated value ofpressure coefficient will be useful quantities for optimizationof the reentry capsule.

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