Hypersonic Turbulent Flow Simulation of FIRE IIReentry Vehicle Afterbody

D. Siva K. Reddy∗ and Krishnendu Sinha†

Indian Institute of Technology Bombay, Mumbai 400 076, India

DOI: 10.2514/1.41380

This paper presents a numerical investigation of the hypersonic reacting flow around the FIRE II reentry capsule.

At the chosen freestream conditions, the forebody boundary layer and the separated flow on the afterbody are

turbulent. The Reynolds-averaged Navier–Stokes method along with two commonly used turbulence models are

used to compute theflowfield.Accurate prediction of turbulent separatedflowat hypersonic conditions is challenging

due to the limitations of the underlying turbulence models. The presence of turbulent eddy viscosity in the flow

simulation results in a smaller separation bubble than the laminar solution at identical conditions. Also, the two

turbulence models predict different levels of eddy viscosity in the neck region. This has a dominant effect on the

separation bubble size and the surface pressure. On the other hand, the eddy viscosity values in the near-wall region

determine the heat transfer rate to the body. The two models predict comparable heating rates on the conical

frustum, and the results match in-flight measurement well. By comparison, surface pressure predictions are

appreciably higher than the data.

Nomenclature

D = diameter of the vehicle, mk = turbulent kinetic energy, J=kgM = Mach numberp = pressure, Paq = heat transfer rate,W=cm2

ReD = Reynolds number based on freestream conditions andbody diameter

s = arc length from the nose stagnation point, mT = translational-rotational temperature, KTv = vibrational temperature, KU = velocity, m=s� = molecular viscosity, Pa � s�T = turbulent eddy viscosity, Pa � s�T = turbulent kinematic viscosity, m2=s� = density, kg=m3

! = specific turbulent dissipation rate, 1=s

Subscripts

T = turbulentw = wall1 = freestream

I. Introduction

ATMOSPHERIC reentry vehicles are subjected to largeaerothermal loads, which play an important role in their design

and operation. During the later part of their descent, the flow aroundthe vehicle, especially in the wake region, transitions to turbulence.This can significantly increase the heat transfer to the vehiclewalls. Predicting turbulent heating rate is challenging due to theuncertainties in the turbulence models. Flow separation on the

afterbody typically accentuates these uncertainties. As a result, alarge safety factor is generally used in the design of afterbody heatshields [1].

Wright et al. [1] present a comprehensive compilation of availableaeroheating measurements on reentry capsule configurations, withthe focus on afterbody flowfield. Flight data show that there isenhanced heating on portions of the afterbody in several cases such asGemini, Mercury, and Apollo, which may be attributed to transitionto turbulence. In another recent review, Schneider [2] highlightsthe effects of transition on reentry capsule flowfields using datacollected from flight and ground tests. Several afterbody flowfeatures, including separation and reattachment locations, and theassociated heating rates show a strong dependence on whether theflow is laminar or turbulent.

Of all the reentry flight and ground tests listed in [1,2], only a fewhave been investigated using modern computational fluid dynamics(CFD) methods. Most notably, the FIRE II and Apollo AS-202flowfields are computed at a series of time points spanning theirrespective heat pulses. The axisymmetric simulation of the zero-angle-of-attack FIRE II configuration matches the experimental datawell [3]. Similarly, computation of the three-dimensional flowfieldaround the AS-202 vehicle yields heat transfer rates that are withinthe scatter of in-flight measurements [4]. However, the majority ofthe trajectory points considered in [3,4] have Reynolds numbers lowenough for the flow to be laminar. The only exception is the AS-202data during the second heat pulse, in which laminar calculationsunderpredict the heating measurements. Fully turbulent predictionsare higher than data, which indicate possible transitional flow at theseconditions. Thus, CFD validation for turbulent flow around reentryvehicles is very sparse in literature, and further work is required tounderstand and accurately predict turbulent reentry wake flows [1].

The ground-based tests of the Crew Exploration Vehicle (CEV)configuration [5] achieved significantly higher Reynolds numbersthan flight conditions. Transition to turbulence occurred on theforebody surface, and computational predictions are found toreproduce the extensive heat transfer rate measurements. Much ofthe experimental and computational effort is directed toward theforebody analysis, with limited data available on the afterbody. Inanother investigation, Brown [6] computed the flow around a sphere-cone geometry with heat transfer measurements on the modelafterbody and sting. The flowwas found to transition at reattachmenton the sting, and several turbulence models were evaluated againstthe experimental data. Although the findings are relevant to reentrywakeflowprediction, it is not clear how the presence of a sting affectsthe recirculating flow and the resulting afterbody pressure and heat

Presented as Paper 805 at the 45th Aerospace Sciences Meeting andExhibit, Reno, NV, 8–11 January 2007; received 6 October 2008; revisionreceived 31March 2009; accepted for publication 2 April 2009. Copyright ©2009 by the American Institute of Aeronautics and Astronautics, Inc. Allrights reserved. Copies of this paper may bemade for personal or internal use,on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0022-4650/09 and $10.00 in correspondence with the CCC.

∗Doctoral Student, Department of Aerospace Engineering.†Assistant Professor, Department of Aerospace Engineering. Member

AIAA.

JOURNAL OF SPACECRAFT AND ROCKETS

Vol. 46, No. 4, July–August 2009

745

transfer rate. Therefore, it is desirable to validate turbulence modelpredictions against in-flight afterbody measurements, which do nothave any sting-interference effect. The current work is a step in thisdirection.

We study the transitional/turbulentflow around the FIRE II reentryvehicle. The Reynolds number at the conditions considered here ismore than twice of the highest value considered for AS-202 [4].Transition to turbulence is therefore expected to occur earlier on thevehicle than in the Apollo case, possibly even on the forebody. Theobjective of the current work is to compute the reacting turbulenthypersonic flow around the FIRE II reentry module using Reynolds-averaged Navier–Stokes (RANS) equations and to evaluate theaccuracy of the aerothermal predictions against in-flight measure-ments. The effects of turbulence on the gas properties are studied bycomparing the RANS solution with the results of a laminar simu-lation at identical conditions. Furthermore, variation between com-monly used turbulence models is presented in terms of theireddy viscosity distribution in the flowfield. The resulting effect onthe mean flow structure and the corresponding difference in theaerothermal predictions are discussed in detail.

The paper is organized as follows. A brief outline of the FIRE IIvehicle geometry and the associated physical flow phenomena arepresented subsequently. The simulation methodology and the gridgeneration technique are discussed next. The grid is carefully refinedand adapted to the prominent flow features to ensure accurateprediction of the flowfield. Results include a description of turbulentflowfield solution and surface properties such as pressure and heattransfer. This is followed by comparison of turbulent vs laminarcomputations. Finally, variation between commonly used turbulencemodels and the effect of compressibility corrections are presented,followed by their comparison with flight data.

II. FIRE II Configuration

Project FIRE (Flight Investigation Reentry Experiment) flights [7]were conducted to investigate the heating environment on a blunt-nosed Apollo-shaped vehicle entering the Earth’s atmosphere at avelocity in excess of the escape velocity. Figures 1a and 1b, takenfrom [7], show a schematic of the FIRE II vehicle. The sphericalforebody consisted of a multilayer configuration made of threeprotective phenolic asbestos heat shields sandwiched between threeberyllium calorimeters. The first two calorimeters and their asso-ciated heat shields were jettisoned during the flight at predetermineddeceleration loads. The third heat shield was not ejected, and thisgives the shape of the vehicle through the later part of the reentry thatis of interest in this work. The outer mold line used in the currentsimulations is shown in Fig. 1c. The vehicle diameter is 58.79 cmwith a nose radius of 70.2 cm. The shoulder/corner radius is 0.61 cmand the base radius is 4.2 cm. The afterbody is conical with a 66 degincluded angle, and the C-band antenna at the base is replaced by aflat base, for simplicity. Also, geometrical details at the heat-shield/aftshell junction and the ringlike structure at the base are not con-sidered in the outer mold line presented in Fig. 1c. Their effect on theflow structure is studied separately in Sec. IV.D.

The FIRE II vehicle entered Earth atmosphere at 11:2 km=s. Itdecelerated to a velocity of 4950 m=s (Mach 16) at 35 km altitude.Slocumb [7] presents the time history of altitude and velocity over theentire descent. We simulate the conditions (listed in Table 1) at the35 km trajectory point, which corresponds to one of the lowestaltitudes and highest Reynolds numbers for whichmeasurements areavailable. The Reynolds number based on freestream conditions andbody diameter is 1:76 � 106. Blunt-body separation shear layer andinner wake transition correlations are presented by Lees [8]. Atransition criteria is defined in terms of the Reynolds number com-puted using local flow properties outside the wake. The distancemeasured along the shear layer, starting from the separation point, isused as the relevant length scale to identify the physical location atwhich theReynolds number approaches the transitional range. Basedon these correlations and data from preliminary simulations of theFIRE II configuration, the wake is expected to transition upstream ofthe neck.

Fig. 1 FIRE II reentry vehicle with afterbody instrumentation [7]:

a) side view b) rear view (�, △, and � represent gold calorimeters,pressure sensor, and radiometer, respectively) and c) outer mold line

used for the current simulations. All dimensions are in centimeters.

Critical geometrical points are 1) fore stagnation point s=D� 0, 2) fore

shoulder s=D� 0:51, 3) heat-shield/backshell interface s=D� 0:64, 4) aftshoulder s=D� 1:42, and 5) aft stagnation point s=D� 1:49.

746 REDDYAND SINHA

The vehicle afterbody was instrumented for pressure and surfacetemperature measurements. Pressure was measured at a singlelocation and 12 gold calorimeters were placed at four axial locationsalong three circumferential rays (�� 0, 120, and 240 deg). Here, �denotes the azimuthal angle around the vehicle, as shown in Fig. 1b.The system accuracy, including that of sensors and telemetry system,was approximately�20% for the pressure data and about�28 K forthe temperature measurements. The corresponding error in heattransfer measurement is estimated to be �15% [3]. The surfacetemperature measured by the calorimeters is presented as a functionof reentry elapsed time. From this temperature-time history, an aver-agewall temperature of 553.3K is taken at 35 km altitude. In the laterpart of the trajectory the vehicle has periodic motion about its axis,resulting in oscillations in the effective angle of attack. However,there is negligible effect of angle-of-attack variation on the afterbodymeasurements, with no appreciable difference between the heatingrates along the three circumferential rays [7]. The angle of attack istherefore assumed to be zero in the current simulations.

III. Simulation Methodology

The chemically reacting turbulent hypersonic flow around theFIRE II reentry configuration is simulated by solving the Reynolds-averaged Navier–Stokes (RANS) equations along with the speciesconservation equations and a thermal nonequilibrium model. Thefreestream total enthalpy at the current condition (12:73 MJ=kg) issimilar to that of Apollo AS-202 computations in [4], in whichionization is found to be negligible.Hence, air ismodeled as a neutralmixture of five perfect species (N2, O2, NO, N, and O) with threedissociation and two exchange reactions. Rotational modes of themolecules are assumed to equilibriate with the translational modesand are described by a translational-rotational temperature T.Vibrational modes are represented by a separate vibrational tempera-ture Tv [9]. An additional conservation equation is solved for thevibrational energy of the mixture to account for the thermalnonequilibrium. The vibrational relaxation source term is calculatedusing Landau–Teller formulation based on a harmonic oscillatormodel for diatomic molecules [10]. The Arrhenius rate constants forthe chemical reactions are evaluated using curve fits to experimentaldata by Park [9].

The k-! model of Wilcox [11] and the Spalart–Allmaras (SA)model [12] are used for turbulence closure. The standard forms of theturbulence models, without compressibility corrections, are used inthe majority of the work. The effect of compressibility corrections isdiscussed separately. Chemical and vibrational source terms areevaluated using mean flow properties, such that effect of turbulence-chemistry interaction is neglected. The molecular viscosity of eachspecies is determined using curve-fit data of Blottner et al. [13].Thermal conductivities for the translational and rotational modesof the species are then determined using the Eucken relation [14].Viscosity and thermal conductivity of the mixture are computedusing Wilke’s mixing rule [15]. Palmer and Wright [16] show thatWilke’s mixing rule is a good approximation for computing themixture viscosity if the shock layer temperature is below ionizationlimits. Multicomponent mass diffusion due to concentration gradientis modeled using Fick’s law with a Lewis number of 1.4. Turbulenttransport of heat and mass are computed using a turbulent Prandtlnumber of 0.9 and a turbulent Lewis number of 1.0, respectively.

The axisymmetric form of the governing equations are discretizedusing the finite volume approach. Inviscid fluxes are computed usinga modified (low-dissipation) form of the Steger–Warming flux-splitting approach [17], and the turbulence model equations are fullycoupled to the mean flow equations [18]. The method is second-order-accurate in both streamwise and wall-normal directions. Theviscous fluxes and the turbulent source terms are evaluated usingsecond-order-accurate central differencing, and the implicit data-parallel line relaxation method [19] is used to obtain steady-statesolutions.

No-slip, noncatalytic, and isothermal boundary conditions arespecified at thewall. The boundary conditions for the k-! turbulencemodel are as per Menter’s recommendations [20]. In the freestream,!1 � 10U1=D and k1 � 0:01�1!1, where D is chosen as thecharacteristic length dimension for the current configuration. At thewall, kw � 0 and !w � 6�w=�y

22, where y2 is the distance between

first grid point and wall, and �� 3=40 is a model constant. For theSA turbulence model, boundary conditions are �T1 � 0:1�1=�1and �T;w � 0, as given in [12]. Freestream conditions are applied atthe outer boundary, and the extrapolation boundary condition isimposed at the exit station. The simulation results are found to beinsensitive to the specific values of k, !, and �T prescribed in thefreestream.

A typical grid used in the simulations is shown in Fig. 2. The outerboundary follows the shape of the bow shock and extends up to2.5 diameters downstream of the vehicle base. The flow at this pointis found to be supersonic, and therefore the location of the exitboundary is not expected to alter the flow solution in the vicinity ofthe vehicle. A typical grid consists of about 160 points in the wall-parallel direction i and 150 points in the wall-normal direction j.Grids with a larger number of points are also used in some cases.

The forebody grid in the i direction is designed such that points areclustered in the nose stagnation region and at the shoulder expansioncorner. The grid in the j direction is refined both in the vicinity of theshock wave and in the near-wall region. In addition, the outerboundary is carefully tailored to the bow shock wave. The forebodygrid refinement follows the same procedure as delineated in [21], anddetails are omitted for the sake of brevity. The current paper isfocused on the afterbody solution, and a careful grid refinement in theafterbody region is presented subsequently.

A. Afterbody Grid Refinement Study

A careful grid refinement study is undertaken for the afterbodyflowfield by identifying the critical regions, in which the flowsolution is most sensitive to the computational mesh. The grid issuccessively refined with a focus on the critical zones, and the gridlines are adapted to the prominent flow features such as the free shearlayer enclosing the recirculation bubble. The effect of grid changeson the afterbody surface pressure and heating rate is studied, and atolerance of 2% of the base stagnation-point value is taken as the

Table 1 Freestream conditions for the current

FIRE II simulations

Freestream conditions Values

Altitude, km 35Time from launch, s 1652.75�1, kg=m

3 0.0082U1, m=s 4950T1, K 237M1 16ReD 1:76 � 106

Fig. 2 Computational domain and a typical grid used in the simula-

tions. The inset shows amagnified viewof the grid near the vehicle, where

every second point is shown in each direction.

REDDYAND SINHA 747

grid-independence criterion. Figures 3 and 4 show the results fromthe grid refinement study using the SA turbulence model, in whichpressure, normalized by its freestream value, and heat transfer rateare plotted as a function of the normalized arc length s=Dmeasuredfrom the nose of the vehicle. The extent of conical frustum and flatbase are marked by vertical lines.

Afterbody flowfield is sensitive to the distribution of grid pointsand the quality of the computational mesh in regions in which theflow gradients are high. In the wall-parallel direction, for example,these correspond to the geometric corners, stagnation point at thebase, and the separation point on the conical frustum. Zonal gridrefinement in these regions is done to identify the most criticalrequirements. It is found that a fine grid near the separation pointalong with a low stretching factor on the conical frustum is necessaryfor grid convergence. The results are, however, not so sensitive to thegrid spacing at the rear stagnation point and the expansion corner atthe base. Three grids with successive refinement along the conicalfrustum, especially near the separation point, are used to compute theflow, and the results are presented in Fig. 3. The two finer grids(240 � 171 and 280 � 171) yield practically identical (within 1%)pressure and heat transfer results on the afterbody. A total of 240points in the wall-parallel direction is therefore taken to be adequatefor a grid-independent solution.

In the wall-normal direction, special attention is given to theaccurate resolution of the free shear layer enclosing the recirculationbubble. Three grids (160 � 150, 160 � 171, and 160 � 220) withsuccessive refinement in the j direction are used to compute the flow.

The majority of the refinement is localized in the free shear layer andthe recirculation bubble. The velocity distribution across the shearlayer (see Fig. 5) shows almost overlapping results from the two finergrids. The pressure and heating rates obtained on the three grids arealso comparable, with negligible difference between the two finergrids (Fig. 4). A total of 171 points is therefore taken to be adequate inthe wall-normal direction.

Further, the computational mesh in the near wake is modifiedsuch that the grid lines are parallel to the shear layer. This is doneby identifying a separate grid block enclosing the recirculationbubble and the shear layer (see Fig. 6) and carefully controlling thedistribution of points along the block boundaries. The shear-layer-adapted grid yields a practically identical size of separation bubbleand shear layer location comparedwith the baseline grid (bottomhalfof Fig. 6). The surface pressure and heating rate show negligibledifference between the results computed on the two grids (see Fig. 7).

Based on the foregoing study, the 240 � 171 baseline grid wasfinalized, and all results using SA model presented subsequentlyare computed on this grid. A similar grid refinement study wasundertaken for the k-! model, but is not presented here.

All of the grids presented previously for the SAmodel have awall-normal spacing of 10�6 m on the entire vehicle. This is equivalent toy� of 0.9 or less along the forebody attached boundary layer and 0.02or less on afterbody. Further refinement is found to have negligible(less than 0.2%) effect on the heat transfer rate obtained using the SAmodel. The k-! turbulencemodel requires finer grid at thewall and isdiscussed later.

Fig. 3 Plots of a) normalized pressure and b) heat transfer rate on the afterbody computed using successively refined grids in thewall-parallel direction.

Fig. 4 Plots of a) normalized pressure and b) heat transfer rate on the afterbody computed using successively refined grids in wall-normal direction.

748 REDDYAND SINHA

B. Iterative Convergence

The flowfield is initialized to freestream values, except for a smallregion next to the conical afterbody and flat base, in which thevelocity is set to zero. The advancement of the solution is studied interms of flow feature development and the corresponding residualhistory. The rms density residual, normalized by its initial value, isplotted in Fig. 8 as a function of the number of time steps. Thecorresponding physical time, normalized by �f, is discussed subse-quently. The characteristic flow time �f is defined as �f �D=U1,and is 1 � 10�4 s for the current configuration.

There is a rapid drop in the residual for the first 20,000 time steps.This corresponds to the development of the bow shock and theboundary layer on the forebody and to the expansion fans at theshoulder. The flow on the forebody reaches steady state during thisperiod, which corresponds to about 12 �f. The separation bubblestarts forming at this time and it grows to its full size by about 46 �f .The recirculating flow in the near wake shows a very slow conver-gence to the steady state and it takes up to 2 � 105 iterations at aCourant–Friedrichs–Levy number of 100 to arrive at the finalsolution.

Beyond the initial drop, the rms residual continues to fluctuatebetween 2 � 10�5 and 10�4. These fluctuations correspond to a gridpoint located along the bow shock near the first expansion corner onthe shoulder. This may be caused by a slight misalignment of theshock with the grid lines. There is, however, no noticeable change inthe shock location or shape at this point, and the large value of thelocal residual does not appear to affect the rest of the flowfield.

The high local residual value at the bow shockmisalignment pointmasks the iterative convergence of the afterbody flowfield after thefirst 50,000 time steps. A second rms residual is therefore computedin the region downstream of the shoulder and is referred to as theafterbody residual in Fig. 8. Comparedwith the full-domain residual,the afterbody residual drops to a lower value, but oscillates around2:5 � 10�7. The origin of these oscillations is traced to the separationpoint on the conical frustum. In thevicinity of separation point, all theflow variables are found to fluctuate with very small amplitude (lessthan 0.05% of the mean). The location of the separation point is notaltered, however, and therefore these oscillations have negligibleeffect on the remainder of the flowfield. This can be seen from theresidual computed only in the region downstream of the separationpoint, denoted by the wake residual in Fig. 8. This includes themajority of the recirculation bubble, the neck region, and the farwake. Thewake residual shows a steady drop to about 3 � 10�9 after2:5 � 105 time steps. The iterative convergence of the afterbodyheating rate to its steady-state value is shown in Fig. 9. There is nonoticeable difference between the heat transfer rates obtained after2 � 105 and 2:5 � 105 time steps, and this is taken as an indicator forterminating the time-iteration process.

x, m

U,m

/s

0.4 0.45 0.50

500

1000

1500

2000

2500

3000

3500

4000

160x150

160x171

160x220

Vortex

Shearlayer

Outerinviscidregion

Fig. 5 Effect of wall-normal grid refinement on velocity distribution

across the free shear layer. Data are plotted along the line shown in

the inset.

Shear layeraligned grid

Baselinegrid

Fig. 6 Grid-point distribution in the baseline grid (bottom half) and a

modified grid that is adapted to the shear layer (top half). White lineidentifies the boundary of the grid block enclosing the shear layer and

separation bubble.

Fig. 7 Plots of a) normalized pressure and b) heat transfer rate on the afterbody computed using the baseline grid and amodified grid that is adapted to

the shear layer.

REDDYAND SINHA 749

Overall, it typically requires computing over 220 �f to reach thefinal steady-state solution for the current FIRE II flowfield. For a240 � 171 grid, this takes about 230 CPU hours on a 2.6 GHz Intel-processor-based machine.

IV. Results

Figure 10 shows the RANS solution computed using the SAturbulence model in terms of the Mach number contours. Theprominent features such as bow shock and flow expansion at thecorners can be easily identified. The flow on the entire forebody issubsonic, with Mach number approaching unity close to the firstexpansion corner. The boundary layer remains attached through thetwo successive expansions and separates downstream of the secondcorner. Flow separation on the conical frustum is found to generatea mild lip shock. The recirculation bubble on the afterbody ischaracterized by a single toroidal vortex that extends up to about0.36 diameters downstream of the base. The shear layer enclosing therecirculation region coalesces at the neck and a recompression shockwave is generated. The flow is subsonic in the recirculation bubble,and it expands to supersonicMach numbers downstream of the neck.

The postshock temperature in the fore stagnation region is about5700K and it varies less than 5%on the forebody.At this temperatureoxygen is completely dissociated, whereas nitrogen is mostly in itsmolecular form. There is a small amount of NO formed in this region.Isothermal condition drops the temperature to 553.3 K at the wall,causing recombination of oxygen atoms in the boundary layer. Therapid drop in fluid density associated with the expansion processleads to a frozen thermochemical state of the gas. The gas compo-sition on the afterbody is approximately the same as that in theforebody inviscid region.

Figure 11 plots the translational-rotational and vibrational tem-peratures of the gas mixture along a stream line (identified in Fig. 10)passing through the bow shock and the inviscid regions on theforebody, shoulder, and afterbody. The two temperatures areidentical in the postshock flow on the forebody. As the gas expandsaround the shoulder, the vibrational temperature freezes at about5000 K, whereas the translational-rotational temperature drops tovalues lower than 2000 K. In the recirculation bubble, the trans-lational temperature gets elevated due to viscous and turbulentdissipation, with values exceeding Tv by about 350 K. Thus, the gasgoes from a thermal equilibrium state on the forebody through anonequilibrium region on the shoulder to a thermally frozen state onthe afterbody.

The thermal nonequilibrium in the flow is computed using a two-temperaturemodel. If a thermal equilibrium state is assumed, a singletemperature (T � Tv) can be used to characterize the translational,rotational, and vibrational energy modes of the molecules. Thevalue of T thus computed using thermal equilibrium is found to be

Time step

log

(Res

idua

l)

0 50 100 150 20010-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Full domainAfterbodyWake

X 103

Fig. 8 Residual history for a typical simulation. Full-domain indicatesthe rms residual in the entire computation domain, afterbody refers to

the rms residual in the afterbodydomain, andwake residual corresponds

to the afterbody region of the grid downstream of the separation point.

s/D

q,W

/cm

2

0.6 0.8 1 1.2 1.4

10

20

30

40

100

150

200

250

Iteration X103

Fig. 9 Afterbody heat transfer rate after different time steps.

Fig. 10 Mach number distribution in the turbulent (SA model)

flowfield around the FIRE II reentry capsule at Mach 16 and 35 km

altitude. Representative streamlines are shown to identify the recir-culation bubble.

s (m)

T,T

v(K

)

0 0.2 0.4 0.6 0.8

2000

4000

6000

8000

T

Tv

Foreb

ody

Should

er

Afterb

ody

Fig. 11 Variation of the translational-rotational and vibrationaltemperatures along a streamline (identified as streamline AB in Fig. 10)

that passes through the bow shock wave, forebody and afterbody

expansion regions.

750 REDDYAND SINHA

comparable with that obtained using the two-temperature model in amajority of the flowfield. The only exception is the inviscid expan-sion region around the shoulder, in which there is a large differencebetween the two temperatures. In the separation bubble, assumingthermal equilibrium does not alter the temperature level bymore than5% of the values computed using the two-temperature model. Theresulting effect on the afterbody heat transfer rate is therefore ex-pected to be minimal.

The dissociation reaction rates are a function of���������

TTvp

and maytherefore be dependent on the thermal nonequilibrium state (T ≠ Tv)on the afterbody. However, the large difference between T and Tv(see Fig. 11) is expected to have negligible effect on the chemicalcomposition of the gas. This is because the species concentrations inthe afterbody flowfield are frozen at levels corresponding to theforebody region. The low density that prevails downstream of theshoulder expansion inhibits any chemical reactions, irrespective ofthe thermal state of the gas. Thus, thermal nonequilibrium has mini-mal effect on the gas composition and therefore on the afterbody flowproperties at the current conditions.

Figure 12 shows the distribution of eddy viscosity normalized bythe local molecular viscosity. This ratio quantifies the effect ofturbulence at each point in the domain. Eddy viscosity is negligiblecompared with molecular viscosity in the nose stagnation region.The boundary layer is initially laminar, and it becomes transitionalwith �T=� ’ 5 just upstream of the first corner. Strong flow expan-sion at the shoulder relaminarizes the boundary layer. Subsequently,there is a buildup of turbulence in the shear layer. In the recirculationregion, �T � 20� and its value increases through the neck up toabout 360� in the wake. The effect of turbulence on the flow predic-tions is discussed subsequently.

Figure 13 presents the computed surface properties on forebody ofthe vehicle. The wall pressure follows a parabolic profile withmaximum at the nose stagnation point (s=D� 0) and a rapid dropdue to expansion at the shoulder (s=D� 0:52). Heat transferrate varies gradually on the forebody with a maximum value of219 W=cm2 at the nose. The stagnation-point heating matches thetheoretical estimate of Fay and Riddell [22] (shown by the symbol inFig. 13b). At the expansion corner the heating rate has a local peak

followed by a sharp decrease as the gas temperature drops due to flowexpansion.

The pressure and heating rate on the afterbody surface arepresented in Fig. 14. The vertical scales are magnified as comparedwith Fig. 13 to highlight the low afterbody levels. The extent ofthe conical frustum and flat base are identified in the figure. Flowexpansion on to the conical afterbody results in a sharp decrease inpressure at s=D ’ 0:65. A similar pressure drop due to the expansionof the reversed flow at the base corner is observed at s=D ’ 1:42.The pressure at the base stagnation point is about 7.23 p1, which isabout 2% of the nose stagnation-point value. The expansion cornersare also marked by local peaks in the heat transfer rate. Between thetwo expansion corners, the afterbody pressure and heating rate showa monotonic increase from the separation point at the shoulder to thebase corner. The base stagnation-point heating rate is 22:48 W=cm2,which is 10.3% of the corresponding nose stagnation-point value.

The afterbody heating rate predicted by the SA model is found toexhibit small waviness along the conical frustum. This wavinessoccurs when the value of the damping function fw in the eddyviscosity equation drops below zero. The model equation with thedamping function limited to nonnegative values results in a smoothvariation of the afterbody heating rate, which is shown in Fig. 14b.This modification to the SA turbulence model does not affect anyother part of the flow solution.

A. Effect of Turbulence

To study the effect of turbulence on the flowfield, the precedingRANS flowfield is compared with the laminar solution computed atidentical conditions. Figure 15 presents the temperature distributionand representative stream lines in the two cases. The forebodyflowfield and the expansion around the shoulder are almost identicalin the laminar and turbulent solutions. The main difference is in theafterbody flowfield. The laminar solution has a larger recirculationregion than the turbulent case, which extends to a diameter down-stream of the base. Also, in contrast to the turbulent solution, thereversed flow along the axis becomes supersonic in the laminar case.A normal shock is therefore formed at the base to slow the flow downto a subsonic Mach number before it reaches the rear stagnationpoint.

The laminar recirculation bubble is marked by multiple vorticesoriginating from the primary, secondary, and tertiary separationpoints along the conical frustum. These vortices are found tooscillate, coalesce, and separate over time, resulting in an unsteadyflow in this region. The size of the recirculation bubble and thelocation of the base shock are also found to oscillate in time. Thelaminar data presented in Fig. 15 correspond to a single timerealization of the unsteady solution. The flow outside the wake is notaffected much by the unsteady vortices and is found to be mostly

360220

8020

Fig. 12 Normalized eddy viscosity distribution computed by turbulent(SAmodel) simulation. Representative streamlines are shown to identify

the recirculation bubble.

s/D

p w/p

∞

0 0.2 0.4 0.6

0

100

200

300

s/D

q,W

/cm

2

0 0.2 0.4 0.6

50

100

150

200

250

300ComputationalTheory (Fay & Riddell [22])

1

2

a) b)Fig. 13 Surface properties obtained using the SA turbulencemodel on the forebody: a) normalized pressure and b) heat transfer rate. Geometry points

1 and 2 are identified in Fig. 1c.

REDDYAND SINHA 751

steady. In the turbulent solution, the dominant effect of eddyviscosity (�T=� > 20) dissipates the smaller vortices and increasesthe gas temperature. The turbulent recirculation bubble is muchsmaller, with peak temperature in the core region significantly higherthan that in the laminar case.

Comparison of laminar and turbulent surface properties on theforebody is shown in Fig. 16. The turbulent heating rate deviates

from the laminar value at s=D ’ 0:15, marking the onset oftransition. However, the difference between the two solutions is lessthan 28% because of the relatively low levels of eddy viscosity(�T=� < 6) in the forebody boundary layer. The turbulence leveldecreases as the flow expands on to the shoulder, resulting inidentical heating rates in the laminar and turbulent solutionsimmediately downstream of the first corner (s=D� 0:52). Furthertransition on the shoulder results in a higher heating rate (by up to77% just before the second expansion corner) than in the laminarcase. As expected, surface pressure distribution is not affected bytransition to turbulence in the attached boundary layer on theforebody.

Afterbody surface pressure and heat transfer rate obtained from thelaminar simulation (Fig. 17) have multiple peaks and minima. Thelow values of pressure correspond to vortex cores and the high-pressure regions are local flow separation and reattachment points.On the other hand, a low heat transfer rate is observed at flowseparation points, and the reattachment points are marked by highlocal heating rates.

At the beginning of the separation bubble, pressure in the laminarsolution is higher than the turbulent value. This is because of asmaller bubble in the turbulent case leading to larger flow expansionaround the corner. Laminar pressure decreases along the frustum,whereas the turbulent pressure increases downstream such that theirvalues are comparable on the later part of the conical afterbody.At thebase, the presence of a normal shock in the laminar solution results inpressure level twice that in the turbulent case.

s/D

p w/p

∞

0.6 0.8 1 1.2 1.40

2

4

6

8

Conical frustum

Base

s/D

q,W

/cm

2

0.6 0.8 1 1.2 1.4

0

10

20

30

40

Conical frustum

Base

34

5

a) b)

Fig. 14 Surface properties obtained using the SA turbulencemodel on the afterbody: a) normalized pressure and b) heat transfer rate. Geometry points

3, 4 and 5 are identified in Fig. 1c.

2400

2400

2000

1800

4200 4400

32002200

Laminar

Turbulent(SA model)

Normalshock

Fig. 15 Comparison of temperature distribution and representativestream lines between laminar (top half) and turbulent (bottom half)

flowfields.

s/D

p w/p

∞

0 0.2 0.4 0.6

0

100

200

300

TurbulentLaminar

s/D

q,W

/cm

2

0 0.2 0.4 0.6

50

100

150

200

250

300

TurbulentLaminar

a) b)Fig. 16 Comparison of a) normalized pressure b) heat transfer rate in the laminar and turbulent solutions computed on the forebody.

752 REDDYAND SINHA

The turbulent heat transfer rate (see Fig. 17b) increases along theafterbody from values comparable with the laminar case near theseparation point to more than 5 times the laminar value at the end ofthe frustum. The only exceptions are the local flow-reattachmentregions in the laminar solution. The higher heat transfer rate in theturbulent case is mainly because of higher temperature in the core ofthe recirculation bubble. With identical wall temperature in the lami-nar and turbulent solutions, a higher core temperature in the turbulentcase yields a higher temperature gradient and therefore a higher heattransfer rate across the near-wall region. At the base, the laminar heattransfer rate increases drastically to 29:5 W=cm2, due to the presenceof the normal shock. It exceeds the turbulent value by 31.5%.

B. Comparison of Turbulence Models

Wilcox’s k-!model [11] is used to compute the FIRE II flowfield,and the solution is compared with the SA model results presentedpreviously. The grid-converged k-! solution is obtained on a 240 �191 computational grid with wall-normal spacing of 2:5 � 10�7 m.This is equivalent to y� � 0:25 or lower along the entire vehicle.Adaptation of the afterbody grid to the free shear layer, similar to thatof the SA model solution described earlier, is also carried out withidentical results.

Figure 18 compares eddy viscosity levels in the two cases, withfreestream molecular viscosity as a common reference for normali-zation. The k-!model shows a faster buildup of eddy viscosity alongthe free shear layer compared with the SA solution. An oppositeeffect is seen downstream of the base, with �T in the SA solutionexceeding the corresponding k-! value by a factor of 2 or more.Lower eddy viscosity in the neck region makes the k-! separationbubble larger by a factor of 2 than the SA solution. The separationpoint is also moved closer to the shoulder expansion corner, and theshear layer angle, measured with respect to the conical frustum, isincreased by about 2 deg more than SA. As discussed in the laminarcase, a larger separation bubble leads to lower flow expansion aroundthe shoulder, thereby resulting in a higher pressure level on theconical afterbody in the k-! solution than in SA (see Fig. 19).Forebody pressure computed using the two models is identical (seeFig. 20).

Afterbody heat transfer rate in the k-! solution is higher than SAby 3–4 W=cm2 along the conical frustum. The base heating isenhanced by about 44%. The difference in base heating predictionsof the two models can be explained by studying the temperature and

s/D

0.6 0.8 1 1.2 1.4

0

2

4

6

8

10

12

TurbulentLaminar

s/D

q,W

/cm

2

0.6 0.8 1 1.2 1.4

0

10

20

30

40

TurbulentLaminar

p w/p

∞

a) b)

Fig. 17 Comparison of a) normalized pressure b) heat transfer rate in the laminar and turbulent solutions computed on the afterbody.

974

564

256410

615

462 615

2340290

411

170

1255

1315 1496

Spalart-Allmarassolution

k-ω solution

Fig. 18 Comparison of normalized eddy viscosity contours andseparation bubble computed using the SA (top half) and k-! (bottom

half) turbulence models.

Fig. 19 Surface properties obtained using the SA and k-! turbulence models on the afterbody: a) normalized pressure and b) heat transfer rate.

REDDYAND SINHA 753

eddy viscosity distribution along the base symmetry line (Fig. 21).The temperature at the neck is higher in the SA solution than k-!,owing to the higher eddy viscosity and the resulting turbulentdissipation. An opposite trend is seen in the near-wall region, withk-! eddy viscosity exceeding the corresponding SA value. Thetemperature gradient at the wall is determined by the eddy viscositylevel in the vicinity of the wall. Therefore, the k-! heating level isfound to be higher than the SA prediction.

There are large differences between the forebody heating ratepredictions by the two turbulence models (see Fig. 20). Unlike thegradual increase in heating rate predicted by the SA model, the k-!solution gives a distinct jump in heat transfer rate at abouts=D� 0:12. Compared with the SA results, the turbulent heatingpredicted by the k-! model is significantly higher, with a peakheating (420 W=cm2) about twice the stagnation-point value.Similar heating rate augmentation is experimentally observed on theCEV geometry by Hollis et al. [5].

Figure 20b also shows that Re� computed along the forebodyboundary layer grows up to a value of 350 immediately before theexpansion corner. The heating predictions of the two turbulencemodels deviate from the laminar value at s=D� 0:12, marking theonset of transition. This corresponds to Re� � 50, which issignificantly lower than the commonly used transition criterion of

200. Computation of transitional/turbulent flow on CEV forebodyusing an algebraic turbulence model exhibits a similar trend [5].

C. Compressibility Corrections

The standard forms of turbulence models used previously weredeveloped for incompressible flows. Several compressibility correc-tions are proposed in literature for their extension to high-speedflows. In two-equation models, compressibility corrections modelthe effect of dilatational dissipation [23] and pressure-dilatationcorrelation [24]. The net result is a reduction in the turbulent kineticenergy production and increase in the turbulent dissipation rate. Bothof these lead to a reduced turbulence level in compressible mixinglayers, thereby reducing the free-shear-layer growth rate comparedwith incompressible flows.

The compressibility correction proposed by Wilcox [25] isincorporated in the k-! model, and the computed results are com-pared with the baseline k-! solution. Normalized turbulent kineticenergy contours plotted in Fig. 22 show the reduction in turbulencelevel (by about 50%) due to the compressibility corrections.A similardecrease in the eddy viscosity levels is also observed between the twosolutions. The majority of the reduction is in the free shear layer andthe neck region, and it increases the separation bubble size in thecompressibility-corrected k-! solution. This in turn enhances thesurface pressure on a significant portion of the conical frustum,compared with the baseline k-! prediction (see Fig. 23a).

The temperature level in the recirculation region is comparable inthe two solutions. The resulting heat transfer rates are almostidentical over the majority of the conical frustum (see Fig. 23b). Thedifference in the base heating value is caused by a higher reverse

Fig. 20 Surface properties obtained using the SA and k-! turbulencemodels on the forebody: a) normalized pressure and b) heat transfer rate andRe�.

Fig. 21 Comparison of normalized eddy viscosity and temperature

along wake symmetry axis between SA and k-! model. The location

x� 0:53 corresponds to the base stagnation point.

0.4

0.9

0.8

0.4

0.2

0.5

0.1

0.2

0.1 0.2 0.4 0.6 0.8 1.0

k-ω model withcompressibility corrections

Baselinek-ω model

k/U2∞ (%)

Fig. 22 Comparison of normalized turbulent kinetic energy contours

and separation bubble between k-! (bottom half) and compressibility-

corrected k-!model (top half).

754 REDDYAND SINHA

velocity along the wake symmetry line for the compressibility-corrected model. In this case, the flow approaching the base stag-nation point reaches up to Mach 0.91, as compared with a maximumMach number of 0.55 in the baseline k-! solution. This also resultsin a higher base pressure in the solution computed using the com-pressibility corrections.

For the SA turbulence model, Paciorri and Sabetta [26] propose aMach-number-based modification to the production of eddy visco-sity in the model equation. This decreases the eddy viscosity level incompressible mixing layers and thus reduces their spreading rate.

Shur et al. [27] achieve a similar reduction in spreading rate byadding a negative source term to the eddy viscosity equation. Thecompressibility corrections to the SAmodel have an effect similar tothe k-! results discussed previously. The surface pressure is elevatedcompared with the baseline model. In addition, the heat transfer rateis reduced compared with the standard SA model.

Based on the comparison with flight data presented subsequently,compressibility corrections are expected to have a detrimental effecton the turbulence model predictions, especially for the surfacepressure. This is in contrast to the findings in [28], in which thecompressibility corrections improved results for supersonic baseflows. This is probably because the Reynolds number in the currentsimulations is about an order of magnitude lower than that in the baseflow calculations by Forsythe et al. [28]. Compressibility correctionsmay not be required at these low Reynolds number conditions andtherefore are not pursued further.

D. Effect of Geometric Simplifications

The FIRE II vehicle geometry has ringlike structures at the heat-shield/aftshell interface and at the base corner. The simplifiedgeometry presented previously does not include these details. Anupdated outer mold line is simulated to study the effect of theseadditional geometrical features, if any, on the computed flowpredictions. The exact dimensions of these ringlike structures couldnot be found in the literature. Their sizes are approximated fromgeometry sketches given in [29] and they aremodeled as finite heightsteps at the respective locations (see Fig. 24).

The finite step at the beginning of the conical afterbody moves theseparation point upstream to the corner location. A small secondaryvortex (see Fig. 25a) is formed close to the separation point.

Fig. 23 Effect of compressibility corrections (denoted by cc) on k-! turbulence model afterbody predictions: a) surface pressure b) heat transfer rate.

400 1225 2050 2875 3700 4525 5350 6175 7000T (K)

Baseline geometry

Modified geometry

Basering

Heatshield-aftshellinterface

Fig. 24 Comparison of temperature contours and separation bubblebetween baseline geometry (top half) and modified geometry (bottom

half) using the k-!model.

Heatshield-aftshellinterface

Base ring

a) b)Fig. 25 Enlarged view of secondary flow separation regions in the flowfield over modified geometry model a) heat-shield/aftshell interface b) base ring.

REDDYAND SINHA 755

Secondary separation is also observed at the base ring (see Fig. 25b).In spite of these changes, the overall size of the recirculation bubbleand the shear layer angle are found to be comparable with thebaseline geometry (Fig. 24).

The surface pressure and heat transfer rates computed using themodified geometry are close to that of the baseline geometry overmajority of the conical frustum (Fig. 26). The only differences arelocalized in the regions of geometry variation. The finite step at theheat-shield/aftshell interface is found to increase the surface pressurein this region. On the other hand, the secondary vortex at the basecorner reduces the surface pressure and heating rate. There is anegligible effect of the geometry variations on the heat transferprediction in the range 0:7< s=D < 1:3, in which comparison ismade with the in-flight measurements.

E. Comparison with Flight Data

The pressure and heating ratemeasured duringflight are comparedwith the different computational results in Fig. 27. Pressure ismeasured at s=D� 1:27, and surface heating values are available ats=D� 0:77, 0.96, 1.13, and 1.27. For a fixed axial station, there islittle variation between the heating data at different circumferentiallocations. Therefore, the heat transfer rates along the threecircumferential rays (�� 0, 120, and 240 deg) are not distinguishedin Fig. 27b.

Experimental measurements in turbulent base flow yield uniformpressure in the separation bubble [30]. This may be because of the

unsteady turbulentmixing in the coreflow that helps in equalizing thepressure on the base and thus yields a uniform surface pressure in thetime-averaged solution. A similar trend is expected to be valid in theturbulent recirculation bubble at the current conditions, and thereforethe single pressure data point is taken as representative of the entireconical frustum. This is indicated by the dotted line in Fig. 27a. Thein-flight measurement is lower than all the different computations,with the SA turbulence model predictions being the closest overmajority of the conical frustum. The k-! results are appreciablyhigher. Laminar simulation yields the highest pressure near theseparation point and it predicts one of the lowest values in the laterpart of the conical frustum.

There is large scatter in the pressure measurement between therange 1651.75 and 1655.75 s, which brackets the current trajectorypoint. This is because the afterbody pressure levels during this periodare close to the upper operating limit of the pressure sensing system.The actual afterbody pressure may therefore be higher than the valuereported in [7]. A higher in-flight pressure level is expected toimprove the comparison between computation and measurement.

Comparing the heating rate data in Fig. 27b, we see that thelaminar results are far lower than flight data, except for the localpeaks in heat transfer. Even an average value will considerablyunderpredict the in-flight measurements. By comparison, RANSresults are closer to the data, with values lower than measurement atthe beginning of the recirculation bubble. Both SA and k-! modelpredictions monotonically increase to exceed the flight data in thelater part of the conical frustum.

s/D

0.6 0.8 1 1.2 1.4

2

4

6

8

10Baseline geometryModified geometry

s/D

q,W

/cm

2

0.6 0.8 1 1.2 1.4

10

20

30

40 Baseline geometryModified geometry

p w/p

∞

a) b)Fig. 26 Effect of heat-shield/aftshell interface and base ring on surface properties: a) normalized pressure b) heat transfer rate.

s/D0.6 0.8 1 1.2 1.4

0

2

4

6

8

10

SAk-ωLaminarFlight

s/D

q,W

/cm

2

0.6 0.8 1 1.2 1.4

0

10

20

30

40

SAk-ωLaminarFlight

p w/p

∞

a) b)Fig. 27 Comparison of computed afterbody a) surface pressure and b) heat transfer rate with in-flight measurements [7] for flow over the FIRE II

capsule at 35 km altitude.

756 REDDYAND SINHA

V. Conclusions

Hypersonic turbulent flow around the FIRE II reentry module iscomputed using the Reynolds-averaged Navier–Stokes method.Five-species air chemistry and a two-temperature vibrationalrelaxation model is used in the simulations. Turbulence closure isachieved using the one-equation Spalart–Allmaras (SA) and two-equation k-!models. The standard forms of the model equations areused in the computation. Compressibility corrections are not found toimprove prediction over the baseline models.

The boundary-layer flow on the forebody transitions to turbu-lence, with large discrepancy between the turbulent heating levelspredicted by the two turbulence models. Also, the computed tran-sition location is upstream of that given by usually accepted tran-sition criteria. On the afterbody, there are large differences in theeddy viscosity level in the neck region between the twomodels. Thishas a dominant influence on the size of separation bubble, shear layerangle, and surface pressure value. The SA solution has a much lowerafterbody pressure than the k-! model. Both turbulence models arefound to overpredict the in-flight pressure measurements signi-ficantly. Discrepancy in the afterbody heating rate is much lower,with the two turbulence models predicting comparable temperaturelevels in the recirculation-bubble core flow. Higher eddy viscosity inthe k-! solution in the near-wall region, however, yields a slightlyhigher heating rate on the frustum and base than the SA solution. Onthe whole, both models are found to predict the in-flight heatingmeasurements reasonably well.

Acknowledgments

The authors would like to thank the reviewer for his/her extensivecomments that have helped in improving the manuscript signifi-cantly. The authors also thank Indian Space Research Organiza-tion (ISRO) for supporting this research under the RESPONDprogramme.

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G. PalmerAssociate Editor

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