CFD Simulations of Superorbital Reentry for a Phobos Sample

Return Capsule

Marta Fernandes Pereira Bruxelasmarta.bruxelas@tecnico.ulisboa.pt

Instituto Superior Tecnico, Universidade de Lisboa, Portugal

November 2017

Abstract

A simulation of the reentry flow for a Phobos Sample Return Mission is presented. The initialreentry velocity considered is 11.6 km/s and the capsule has the same forward body geometry asthe Hayabusa capsule. The simulations are performed using CFD code SPARK, developed andmaintained at IPFN. The heat fluxes along the capsule surface are analysed for several trajectorypoints, centered in the convective heating peak trajectory point, since the assessment of the totalheat fluxes is paramount for the proper design of a TPS. A mesh generation process and meshconvergence study are presented. Two methods for calculating the transport properties, Wilke andGupta-Yos, are applied and compared. The obtained results are also compared against a previoussimulation performed using the same software considering the RAM-C II capsule reentry at 7.65 km/s.For the highest reentry speed, both models diverge more than for the lowest speed. An analysis ofthermal nonequilibrium is performed applying Park’s two-temperature model to account for thermalnonequilibrium and compared against thermal equilibrium conditions. Two trajectories for the capsuledescent are considered in the scope of project DIVER, a steep and a shallow entry. The stagnationpoint convective heat flux is analysed at six key points for each trajectory and compared to theSutton-Graves semi-empirical correlation used in the preliminary design.Keywords: Reentry, Hypersonic, Aerothermodynamics, Convective Heat Fluxes

1. Introduction

Robotic planetary exploration missions rank amongthe most challenging endeavours in Space engineer-ing. The Phobos Sample Return Mission was de-signed to collect samples from the Martian moonPhobos and returning them to Earth. One ofthis mission’s most critical technologies is the onebehind the superorbital Earth reentry originatingfrom Mars/Phobos. During Earth reentry, thespacecraft is moving at velocities ranging from 13to 6 km/s, crossing several hypersonic flow condi-tions. These velocities lead to the formation of astrong and high-temperature bow shock around thecapsule. Because of the high temperature nature ofthis shock, the heat fluxes that reach the surface ofthe capsule will exceed what the surface materialcan withstand. Consequently, the proper selectionof a Thermal Protection System (TPS) is very im-portant when designing a reentry capsule. To thiseffect, an accurate assessment of wall heat fluxesduring the high speed, hyperbolic reentry is there-fore critical. These heat fluxes can be divided intotwo types, convective heating and radiative heating.Convective heating refers to the combined effect of

the convective heat flux and the conduction heatflux. The radiative heat flux is sourced in the radi-ation emitted by the ionized flow.

DIVER (Desenho Integrado de VEıculo de Reen-trada) is a Portuguese consortium project compris-ing Spin.works, INEGI, IST-DEM and IPFN (Insti-tuto de Plasmas e Fusao Nuclear), funded by Por-tugal2020. It gathers all the concurrent engineeringrequired in the preliminary design of a reentry vehi-cle in the Phobos Sample Return mission and aimsat demonstrating the national expertise on inter-planetary exploration missions design. Two trajec-tories for the capsule descent are considered, onewith a flight path angle (fpa) of -15.9, Trajectory15.9, and the other with fpa = -24.2, Trajectory24.2. In both trajectories, the maximum value ofthe reentry velocity is 11.6 km/s.

The purpose of this work is to 1) identify the keytrajectory points for the proper design of the cap-sule; 2) calculate the convective heat fluxes for eachtrajectory key points; 3) compare the stagnationheat fluxes against correlation values; and 4) pro-vide a first estimate of the heat fluxes for the nextphase of the design. In order to accomplish these

1

objectives, CFD simulations will be performed us-ing the in-house IPFN CFD code SPARK (SoftwarePackage for Aerodynamics, Radiation and Kinetics)[1]. It is a multidimensional, second-order finite vol-ume method discretization solver for Navier-Stokesand reactive flow governing equations in structuredmeshes. The simulations will consider two dif-ferent transport models (Wilke and Gupta-Yos).These transport models have been previously im-plemented and discussed in the scope of a previ-ous Master’s thesis [2] and will be briefly summa-rized here. A one-temperature model and a multi-temperature model will be considered to account forthermal equilibrium and nonequilibrium conditions.

There are three approaches to study the reentryflowfield around a capsule: flight testing, groundtesting and numerical simulations. The latter isthe most economical and viable way of simulatingreentry conditions. Nevertheless, this kind of simu-lations are challenging, especially the full capture ofthe multi-physics phenomena. As explained before,the gas is highly reactive in the shock layer owing tothe large post-shock temperatures. To accuratelymodel a hypersonic reentry flow, a large numberof complex gas dynamics processes must be takeninto account, such as molecular collisions leadingto dissociation/recombination and/or ionization ofmolecules and/or atoms, charged particle interac-tions, and radiation [3]. The physical models thathave been developed based on theory and experi-mental data are implemented in a computationalsolver that will perform numerical simulations withpurpose of validating those models. Many numeri-cal simulations have been performed for missions atsuperorbital velocities1, such as the Stardust mis-sion [4] [5], the Hayabusa mission [6], the FIRE IImission [1] [7] and the Apollo missions [8]. Notall numerical data obtained agreed well with ei-ther experimental data or flight data. This leadsto the conclusion that despite the proved success ofsome models and correlations in properly definingthe processes at reentry, there are still opportunitiesfor further development in this area.

2. Governing Equations and Physical Models

During a superorbital reentry, the flow is charac-terized as a thermal nonequilibrium reacting flow.Consequently, it has to be defined by a multi-component model, which means the gas is treatedas a mixture of individual chemical species, mixedin a single phase. Since the gas is not in chemicalequilibrium, a chemical-kinetic model needs to beapplied to model the gas composition.

During reentry the large amount of coherent ki-netic energy in the hypersonic freestream is con-

1These correspond to velocities above the escape velocityfrom Earth’s gravity.

verted into translational energy of the gas across thestrong bow shock wave. This energy is then trans-ferred to the molecules’ internal modes through fur-ther collision processes. There are four main ther-mal energy modes2 translational energy, rotationalenergy, vibrational energy and electronic excitationenergy [9]. In thermal equilibrium, every speciesthermal energy is associated to the same character-istic temperature T = Ttra = Trot = Tvib = Texc.However, usually species thermal energy modes areexcited differently and thus, multiple temperaturesmust be considered. This thermal nonequilibriumis taken into account through the use of a multi-temperature model.

Equations (1 - 4) present the conservative equa-tions for a thermal nonequilibrium reacting flow.

∂(ρci)

∂t+ ~∇ · (ρci~u) = ~∇ · ~Ji + ωi (1)

∂(ρ~u)

∂t+ ~∇ · (ρ~u⊗ ~u) = ~∇ · [τ ]− ~∇p (2)

∂(ρE)∂t + ~∇ · (ρE~u) = ~∇ ·

(∑k ~qCk

+∑i~Jihi + ~u · [τ ]− p~u

)(3)

∂∂t (ρεk) + ~∇ · (ρ~uhk) = ~∇ ·

(~qCk

+∑i~Jihi,k

)+ Ωk (4)

where ρ is the density, ~u the mean velocity in vec-torial form, ci the species i mass fraction, ~Ji thespecies i mass diffusion flux, ωi the species i sourceterm, [τ ] the viscous stress tensor, p the pressure,E the total energy, ~qCk

the conduction heat flux,h the enthalpy of the flow, εk the thermal energyassociated to each thermal energy mode and Ωk thethermal energy source term.

2.1. Kinetic ModelIn the above stated equation for the species con-tinuity (eq. 1), ωi is the species source termthat accounts for the destruction and productionof species. It is determined by the chemical-kineticmodel. Depending on the kinetic scheme consid-ered, the number of chemical reactions and theirrespective rates are different. The kinetic schemeconsidered in the present work is the 11-species airmodel developed by Park [10]. It considers thespecies O2, N2, NO, N, O, N+

2 , O+2 , NO+, N+, O+,

e−.

2.2. Multi-Temperature ModelThe two-temperature model by Park[11] is imple-mented in SPARK. It assumes that the transla-tional energy of the atoms and molecules and therotational energy of the molecules are characterizedby the same temperature Ttra and that Tvib char-acterizes the vibrational energy of the molecules,translational energy of the electrons and electronicexcitation energy of atoms and molecules.

2Assuming the Born-Oppenheimer approximation.

2

2.3. Transport ModelsThe phenomena that take place during reentry leadto larger changes in the gas mixture transport prop-erties. Consequently, an adequate model needs tobe implemented. In the equations previously stated(eqs. 1 - 4), ~Ji, [τ ] and ~qCk

define the dissipativefluxes. In the present work, the mass diffusion flux~Ji is modelled by Fick’s Law of diffusion:

~Ji = ρDi~∇(ci) (5)

where Di represents the ith species mass diffusioncoefficient. Moreover, it is considered that the vis-cous stress tensor [τ ] assumes a Newtonian fluid andthe Stokes hypothesis for the normal stresses:

[τ ] = µ(~∇~u+ (~∇~u)T

)− 3

2µ(~∇ · ~u)[I] (6)

where µ is the viscosity coefficient. Furthermore,the conduction heat flux in each energy mode, ~qCk

,is assumed to be given by Fourier’s Law of heatconduction:

~qCk= λk ~∇Tk (7)

The most effective way of determining these prop-erties is by applying approximate mixing rules,which are simplified forms of the Chapman-Enskogsolution [12]. In this work, two different mix-ing rules are applied, the Wilke/Blottner/Euckenmodel and the Gupta-Yos/Collision Cross-Sectionmodel.

2.3.1 Wilke/Blottner/Eucken Model

Wilke’s Model [13] for gas mixture viscosities wasdeveloped through the application of kinetic the-ory to the first order Chapman-Enskog relation. Itassumes that all interactions between any particlespresent the same (hard sphere) cross-section [14].The gas mixture viscosity µ and the thermal con-ductivity λk for each gas temperature are deter-mined as follows:

µm =

NS∑i=1

xiµiφi

λk =

NS∑i

xiλk,iφi

(8)

where xi is the species molar fraction and µi repre-sents the individual viscosities. φi is a scale factorgiven by:

φi =NS∑j=1

[1 +

õi

µj

(Mj

Mi

) 14]2/√

8

(1 +

Mj

Mi

)(9)

where M∗ represents each species’ (i’s or f ’s) molarmass.

For the species viscosities, curve fits determinedby Blottner [15] are considered:

µi(Ttra,i) = 0.1exp((Ai lnTtra,i +Bi) lnTtra,i + Ci) (10)

where Ai, Bi and Ci are curve fitted coefficients foreach species.

Eucken’s relation [16] is used to determine eachspecies thermal conductivity:

λtra,i =5

2µiCVtra,i

λk 6=tra,i = µiCVk,i(11)

where CVk,irepresents the specific heat at a con-

stant volume of the i species in each energy mode.In thermal nonequilibrium, the contributions ofeach species should be accounted for differently inthe mixing rule, according to the multi-temperaturemodel considered.

The species mass diffusion coefficient is given bya single binary coefficient D assuming a constantLewis number, Le = 1.2:

Di = D =Leλ

ρCP(12)

where CP is the gas mixture total specific heat ata constant pressure and λ represents the total ther-mal conductivity of the gas mixture. The Lewisnumber Le corresponds to the ratio of the energytransport due to mass diffusion relative to the onedue to thermal conduction.

2.3.2 Gupta-Yos/Collision Cross-SectionModel

Another method to calculate the transport prop-erties of a gas mixture is by using the Gupta-Yosmodel [17], a simplification of the Chapman-Enskogsolution. The main difference between the Wilkeand the Gupta-Yos models is that the latter con-siders each collision’s corresponding cross-section.Consequently, it is assumed to be more accuratebecause the true nature of the viscosity collision in-tegrals is taken into account. The collision integralsare defined as follows:

∆ij(1)(Tc) = 8

3

[2MiMj

πRT (Mi+Mj)

]1/2πΩij

(1,1)(13)

∆ij(2)(Tc) = 16

5

[2MiMj

πRT (Mi+Mj)

]1/2πΩij

(2,2)(14)

where Ωij(1,1)

and Ωij(2,2)

represent weighted av-erages of the cross-sections. These collision in-tegrals are a function of the controlling temper-ature Tc that varies depending on the type ofparticles colliding. For collisions between heavy-particles, the controlling temperature is the trans-lational/rotational temperature (Ttra), whereas theone for collisions involving electrons is the vibra-tional/electron/electronic excitation temperature(Tvib) [5].

3

For the calculation of the gas mixture viscosity,the mixing rule is used:

µ(1) =

NS∑i

mixiNS∑j

xj∆(2)ij

(15)

where mi is the ith species mass. The translationalmode of heavy species λtra reads:

λtra =15

4kB∑i

xi∑j αi,jxj∆

(2)ij

(16)

and the translational mode of electrons λe is:

λe =15

4kB

xe∑j

αe,jxj∆(2)ej

(17)

where kB is the Boltzmann constant and αi,j isgiven by:

αi,j = 1 +[1−Mi/Mj ][0.45− 2.54(Mi/Mj)]

[1 + (Mi/Mj)]2(18)

The global thermal conductivities associated withthe rest of the heavy species energy modes, λrot,λvib, λexc are:

λrot =∑i

ximiCVrot,i

NS∑j

xj∆(1)ij

(19)

λvib =∑i

ximiCVvib,i

NS∑j

xj∆(1)ij

(20)

λexc =∑i

ximiCVexc,i

NS∑j

xj∆(1)ij

(21)

In thermal equilibrium, the total thermal conduc-tivity λ is given by:

λ = λtra + λe + λrot + λvib + λexc (22)

In thermal nonequilibrium, the thermal conduc-tivity associated with each temperature mode is cal-culated by considering the individual contributionsof each species according to the multi-temperaturemodel considered, where the global temperature pa-rameter T is replaced by the appropriate mode tem-perature.

The mass diffusion coefficient Dij defines the dif-fusion velocity of each species relative to the otherspecies and reads:

Dij =kBTc

p∆(1)ij

(23)

An average of the diffusion coefficient relative tothe remaining gas mixture Di is determined by:

Di =1− xi∑j 6=i

xj

Dij

(24)

2.3.3 Ambipolar Diffusion

In an ionized gas, each charged particle induces anelectric field, which influences the other particlesmotion. This results in an increase in the diffusionof the ions, which are pulled by the more mobileelectrons, which in turn are slowed down by theheavier ions. The two species finally diffuse withthe same velocity, mainly imposed by the heav-ier species. Since the electro-static forces are notconsidered in the conservation equations, this phe-nomena can only be taken into account by artifi-cially introducing this effect in the calculation of thecharged particles diffusion. The ambipolar correc-tions account for this increase in diffusion velocityof ions and decrease in velocity of electrons due tothe electrostatic interactions between the two.

For ions, Chen [18] shows that the ambipolar dif-fusion coefficient is given by:

Daion =

(1 +

TeTion

)Dion (25)

where Dion is the ion free diffusion coefficient andTe and Tion are the translational temperatures ofthe electrons and the ions, respectively. In thermalequilibirum, this ionic factor is simply equal to 2(Da

ion = 2Dion), assuming that all heavy species arein equilibrium with a common translational tem-perature.

For electrons, the ambipolar diffusion coefficientis defined as [19]:

Dae = Me

∑i=ionD

ai xi∑

i=ionMixi(26)

This is only valid if all the ions have the same freediffusion velocity and electrical mobility.

3. Numerical SetupAfter correctly defining the problem, the conditionsin which the simulations will be performed need tobe established.

3.1. Selection of the Key Trajectory PointsThis works’ role in the DIVER project is to cal-culate the heat fluxes present at the wall so that asuitable TPS can be designed. Two separate trajec-tories were considered and for each of these trajec-tories, semi-empirical correlations were applied as afirst approach, such as it is customary when testinga large number of possible flight profiles and geome-tries, where full CFD simulations are impractical.

For the convective heating, the Sutton-Gravesequation [20] was applied. To analyse radiativeheating, the expression developed by Tauber andSutton[21] was considered. Both assume chemicalequilibrium flow. Figure 1 shows the results ob-tained by Spin.works for the convective (dark blueline) and radiative (light blue line) heat fluxes, the

4

Dra

g [

kN]

and V

eloc

ity

[km

/s]

Hea

t Fl

ux

[W/c

m2]

0

120

240

360

480

600 30

24

18

12

6

0

Time of Flight [s]739 743 747 751 755 759 763 767

ConvectionRadiationDragVelocityAnalysis Points

1 2 3 4 5 6

(a) Trajectory 15.9

ConvectionRadiationDragVelocityAnalysis Points

1 2 3 4 5 6

Hea

t Fl

ux

[W/c

m2]

Dra

g [

kN]

and V

eloc

ity

[km

/s]

Time of Flight [s]

60

48

36

24

12

0764 768 772 776 780 784 788

700

560

420

280

140

0

(b) Trajectory 24.2

Figure 1: Results for the semi-empirical correlationsfor both trajectories.

drag (dark red line) and velocity (light red line) dis-tributions during reentry and the points where theanalysis will be performed (green dashed line) for(a) Trajectory 15.9 and (b) Trajectory 24.2.

Six key points per trajectory were selected to per-form CFD simulations. Point 1, 2, 4 and 6 wereselected to properly compare the CFD results withthe correlations used. Point 3 represents the con-vective heating peak, it is the most critical pointin each trajectory. Point 5 corresponds to the peakin drag, it is the point where the mechanical loadsare the highest. Tables 1 and 2 show the selectedpoints data for Trajectory 15.9 and Trajectory 24.2,respectively. The data was calculated consideringthe US Standard Atmosphere 1976 to model Earth’sAtmospheric profile as a function of altitude.

3.2. Mesh GenerationA mesh generation study was performed consider-ing the initial conditions for the peak heating (point3) of Trajectory 15.9. Figure 2 shows the capsulegeometry where the forward body is highlighted inthe drawing. The body is axisymmetric and there-

Table 1: Points where the analysis will be per-formed for Trajectory 15.9.

Trajectory 15.9

Velocity [km/s] Density [g/m3] Temperature [K]

Point 1 11.58 0.024 202.03

Point 2 11.21 0.141 230.24

Point 3 10.07 0.486 257.18

Point 4 8.63 0.964 270.65

Point 5 6.89 1.726 266.79

Point 6 4.22 3.599 252.34

Table 2: Points where the analysis will be per-formed for Trajectory 24.2.

Trajectory 24.2

Velocity [km/s] Density [g/m3] Temperature [K]

Point 1 11.56 0.048 210.86

Point 2 11.29 0.178 235.07

Point 3 10.02 0.758 267.56

Point 4 8.73 1.451 270.32

Point 5 6.92 2.776 257.35

Point 6 4.18 5.888 243.10

Figure 2: DIVER capsule geometry.

fore, only half of the front body will be consideredin the computational domain, since a 0 angle ofattack is also assumed.

At the end of the mesh generation process, a fi-nal mesh configuration was selected. Afterwards,a mesh convergence exercise was performed, to ob-tain the best results at the least computational ef-fort. Figure 3 shows a comparative analysis of (a)the temperature profiles along the stagnation lineand (b) the wall heat fluxes for three mesh resolu-tions. Ni refers to the number of mesh cells alongthe stagnation line.

Analysing Fig. 3 it can be seen that Ni = 90(red line) and Ni = 120 (black line), present closerresults than Ni = 45 (blue line). Consequenty, itwas considered the latter mesh is too rough for thedesired simulations. Because the heat flux profilesobtained for Ni = 120 are still 16% higher than forNi = 90, it was concluded that the most suitable

5

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0

Stagnation Line x [m]

Tem

per

ature

[K]

14000

12000

10000

8000

6000

4000

2000

0

16000

Ni = 45Ni = 90Ni = 120

(a) Stagnation Line Temperature

0 0.1 0.2 0.3 0.4 0.5

Curvilinear Distance s [m]

700

600

500

400

300

200

100

0

Hea

t Fl

ux

[W/c

m2]

Ni = 45Ni = 90Ni = 120

(b) Wall Heat Flux

Figure 3: Stagnation Line Temperature (a) andWall Heat Flux (b) profiles for three different valuesof Ni, where Ni is the number of mesh cells alongthe stagnation line.

mesh for the simulations is the one where Ni = 120.More refined meshes were tested, however these didnot manage to converge to a realistic flow profile.Figure 4 shows the final mesh. The mesh is refinednear the wall to account for the boundary layer.

3.3. Initial and Boundary Conditions

The initial conditions for each trajectory point aredetailed in tables 1 and 2. Figure 4 shows the meshand the boundary conditions considered in the sim-ulations performed in this work.

In addition to the boundary conditions previouslystated, a condition will be considered to accountfor how the flow interacts with the surface. A fullycatalytic wall boundary condition is applied, assum-ing that all atoms/electron-ion pairs which collidewith the surface recombine into molecules/neutralspecies. It represents a pessimistic limit for theoverall heat flux (since these catalytic reactions areexothermal).

X [m]

Y [

m]

0

0.1

0.2

0.3

0.4

0.5

-0.2 -0.1 0 0.1 0.2 0.3 0.4

Inflow

Outflow

Symmetry

Isothermal +

Fully Catalytic Wall

Figure 4: Boundary conditions considered for theCFD domain.

4. Results

4.1. Impact of Transport Model and VelocityThe following analysis was performed on point 3 ofTrajectory 15.9. The results obtained were com-pared considering two transport models, Wilke andGupta-Yos. The impact of the velocity was studiedby comparing the results obtained in the presentwork to those obtained by Loureiro [2]. Both simu-lations were performed on SPARK and consideredthe same kinetic scheme and thermal equilibrium.In the simulations ran by Loureiro the RAM-C IIcapsule geometry and a reentry velocity of 7.65km/s were considered.

Figure 5 shows the stagnation line temperatureprofiles for both mixing rules, at reentry speeds of10 km/s (a) and 7.65 km/s (b).

Focusing on the results obtained in the presentwork (Fig. 5a), it can be seen that the greatest dis-crepancy in the results obtained for each transportmodel lies in the boundary layer. Around x = -0.0025 m, the Gupta-Yos model (black line) shows asteeper decrease in temperature when compared tothe Wilke model (red line). Moreover, the Gupta-Yos model predicts a peak temperature 7% higherthan the Wilke model.

Comparing the results obtained for each reentryvelocity, there are two main differences. 1) The re-sults agree better at lower velocities. This will beaddressed later on at the end of this section; and2) for the lowest velocity, there is no region wherea quasi steady state flow is achieved between theshock wave and the boundary layer. The reason forthe absence of this region is that the chemical re-actions, in the shock region, occur at a slower rate,when compared to the highest velocity. One wouldexpect the temperature peak value corresponding tothe highest velocity to be much higher. However,due to the roughness of the mesh in this area con-sidered in the present work, the shock peak is notfully captured and therefore, the peak temperature

6

16000

14000

12000

10000

8000

6000

4000

2000

0

Wilke Model

Gupta-Yos Model

Tem

per

ature

[K]

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0

Stagnation Line x [m]

DIVER Geometry

(a) 10 km/s

RAM-C II Geometry

Wilke Model

Gupta-Yos Model

Tem

per

ature

[K]

0

2000

4000

6000

8000

10000

12000

14000

16000

Stagnation Line x [m]-0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0

(b) 7.65 km/s

Figure 5: Stagnation line temperature profiles forboth mixing rules, for a reentry speed of 10 km/s(a) and 7.65 km/s (b).

appears similar in both cases.

Figure 6 shows the wall heat flux profiles for bothtransport models, for a reentry speed of 10 km/s (a)and 7.65 km/s (b).

Focusing on the results performed at 10 km/s(Fig. 6a), there is a great difference in the wallheat flux profiles depending on the selected trans-port model. The peak heat flux obtained is 640W/cm2 for the Wilke model, 19% higher than forthe Gupta-Yos model. It can be seen that for bothcurves, there is a slight disturbance around s = 0.04m, which is currently attributed to numerical errors.A possible cause for this is the carbuncle problem[22]. It is defined as a local displacement of the bowshock wave shape near the stagnation line, whichcompromises the accuracy of the heat transfer pre-dictions made by numerical simulations of hyper-sonic flows.

Comparing the results for both reentry velocities,the heat fluxes at the wall are about three timeshigher for 10 km/s, which agrees well with the in-creased temperature gradient at the surface.

Analysing Figs. 5 and 6, the results obtained for

650

600

550

500

450

400

350

300

250

200

Hea

tFl

ux

[W/c

m2]

Wilke Model

Gupta-Yos Model

0 0.1 0.2 0.3 0.4 0.5

Curvilinear Distance s [m]

DIVER Geometry

(a) 10 km/s

Wilke Model

Gupta-Yos Model

Curvilinear Distance s [m]

Hea

t Fl

ux

[W/c

m2]

250

200

150

100

50

00 0.2 0.4 0.6 0.8 1 1.2 1.4

RAM-C II Geometry

(b) 7.65 km/s

Figure 6: Wall Heat Flux profiles for both mixingrules, for a reentry speed of 10 km/s (a) and 7.65km/s (b).

both transport models agree better at the lowestvelocity. This is related to the increase in the degreeof ionization of the flow at higher velocities, becausethese models were formulated assuming a weaklyionized gas.

4.2. Impact of Multi-Temperature Model

As discussed in section 2.2, Park’s two-temperaturemodel is implemented in SPARK to account forthermal nonequilibrium. Figure 7 shows profiles for(a) the stagnation line temperatures and (b) thewall heat flux for thermal nonequilibrium and ther-mal equilibrium. This simulation was performedon the same peak heating point for trajectory 15.9,point 3. A fully catalytic boundary condition at thesurface and the Gupta-Yos transport model wereconsidered.

Focusing on Fig. 7a, the translational/rotationaltemperature (red line) profile follows the sameoverall trend as the temperature profile consid-ering thermal equilibrium (black line), althoughit does not stabilize. However, the peak intranslational temperature reaches 22000 K, 27%higher than for the thermal equilibrium case. The

7

TtraTvibThermal Eq.

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0

Stagnation Line x [m]

0

5000

10000

15000

20000

Tem

per

ature

[K]

25000

(a) Stagnation Line Temperatures

0 0.1 0.2 0.3 0.4 0.5

Curvilinear Distance s [m]

650

600

550

500

450

400

350

300

250

200

Hea

t Fl

ux

[W/c

m2]

Thermal Eq.

Thermal Noneq.

(b) Wall Heat Flux

Figure 7: Profiles for (a) the stagnation line tem-peratures (b) wall heat flux for thermal equilibriumand nonequilibrium.

vibrational/electron/electronic excitation temper-ature (blue line) follows a more distinct pro-file. The discrepancy in the profiles for thetranslational/rotational temperature and the vibra-tional/electron/electronic excitation temperatureleads to the conclusion that there is a very strongthermal nonequilibrium, as would be expected forthe flow initial conditions. Thermal equilibrium isonly reached in the boundary layer.

Figure 7b presents the wall heat flux results ob-tained for thermal equilibrium (red line) and ther-mal nonequilibrium (black line). Both present thesame overall trend after s = 0.09 m. Before thispoint, both heat fluxes present a disturbance at s =0.04 m, supporting the previous assumption of thecarbuncle problem. The stagnation point heat fluxpredicted for the thermal nonequilibrium is approx-imately 580 W/cm2, 9% lower than the one pre-dicted for the thermal equilibrium case. It wouldbe expected that the heat fluxes at the wall be thesame along the capsule since thermal equilibrium isreached at the boundary layer. However, the tem-perature gradients differ, leading to slightly differ-

Stagnation Line x [m]

Mol

e Fr

action

s

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0

100

10-1

10-2

10-3

10-4

10-5

10-6

10-7

N2 NO

E-N+

NO+NO N2+

O2

O+

O2+

(a) Species Mole Fractions

0

3200

6400

9600

12800

16000

Tem

per

ature

[K]

and V

eloc

ity

[m/s

]

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0

Stagnation Line x [m]

0

10

20

30

40

50

Pres

sure

[kP

a]

TemperaturePressureVelocity

(b) Temperature Pressure Velocity

Figure 8: Stagnation line profiles for (a) speciesmole fractions and (b) temperature, pressure andvelocity.

ent heat fluxes. Nonetheless, the differences remainlower than 10% in the critical region of the stagna-tion streamline, allowing for the use of a more sim-plified and time-efficient single-temperature ther-mochemical model in the rest of the analysis.

4.3. Peak Heating Point Analysis

This analysis was performed for point 3 of Trajec-tory 15.9, a key point in the trajectory for the de-sign of the TPS, since it corresponds to the peak inconvective heating.

Figure 8 shows the results for (a) species molefractions and (b) temperature, pressure and velocityprofiles along the stagnation line.

Figure 8a shows that the mole fractions of N2

and O2 decrease after the bow shock (x = -0.022m), whereas the rest of the species mole fractionsincrease after this point. Gas chemical compositionstabilizes from x = -0.020 m up to x = -0.003 m(the quasi steady state region). Afterwards, con-centrations of N2, O2 and NO increase due to therecombination reactions taking place near the cap-sule surface. This is balanced by the decrease in themolar fractions of species N+

2 , O+2 , NO+, e− and N.

8

Figure 9: Temperature 2D field.

One would expect the molar fraction of O to de-crease at the surface due to the recombination ofO2 (O + O + wall ⇒ O2), however that is not thecase, owing to the fact that this species is involvedin more recombination reactions.

Focusing on Fig. 8b, for the shock peak (x= -0.022 m) there is an increase in temperatureand pressure, whereas the velocity decreases. Af-ter reaching its peak, temperature decreases rapidlyuntil it stabilizes at 10000 K, from x = -0.02 m upto x = -0.003 m. It then decreases abruptly untilreaching the capsule surface at x = 0 m with theimposed wall temperature. Regarding the pressure,it rises until 47.7 kPa in the shock region, at x =-0.022 m, and remains stable until the capsule wall(x = 0 m). The velocity drops abruptly from 10000to ≈ 960 m/s in the shock region and afterwardsgradually decreases until V = 0 m/s at the capsulewall.

Figure 9 depicts the temperature 2D field. Itshows the shock wave position very clearly due tothe abrupt temperature rise. Towards the outflowboundary condition the temperatures in the shocklayer are gradually lower (above Y = 0.3 m). This isdue to the shock wave position relative to the flow.In the stagnation line, the shock wave is normal tothe flow and the temperature gradient is very high.Moving along the shock wave, it becomes an obliqueshock resulting in a lower temperature gradient.

4.4. Heat Fluxes for Project DIVER

For the key trajectory points detailed in section3.1, CFD simulations were performed consideringthe Gupta-Yos transport model, thermal equilib-rium and a fully catalytic wall, owing to the conclu-sions of previous sections. A comparison between

Velocity [km/s]0 2 4 6 8 10 12

Hea

t Fl

ux

[W/c

m2]

Sutton-GravesCFD Points

1

2

3

4

5

6

0

100

200

300

400

500

600

(a) Trajectory 15.9

Velocity [km/s]0 2 4 6 8 10 12

Hea

t Fl

ux

[W/c

m2]

1

2

3

4

5

6

Sutton-GravesCFD Points

0

100

200

300

400

500

600

700

800

(b) Trajectory 24.2

Figure 10: Comparison of the data obtained in thiswork and the Sutton-Graves correlation for the con-vective heat flux along (a) Trajectory 15.9 and (b)Trajectory 24.2.

the 6 selected points and the continuous correlateddata-points for the two reentry trajectories has beencarried out.

Figure 10 compares the data obtained in thiswork with the Sutton-Graves correlation for theconvective heat flux along (a) Trajectory 15.9 and(b) Trajectory 24.2.

For both trajectories, some CFD points predict aslightly higher convective stagnation heat flux thanthe Sutton-Graves correlation. The highest discrep-ancy (48%) was found for Trajectory 24.2 in thepoint with the highest velocity, point 1. The peakheating point, for both trajectories is point 4, as op-posed to number 3 in the correlation. The discrep-ancies found between the results are due to the factthat, as mentioned in section 3.1, the Sutton-Gravesequation assumes chemical equilibrium. OverallCFD results correlate well with the semi-empiricalresults, providing confidence to the thermal protec-tions system design process.

9

5. Conclusions

A successful comparison between the Wilke andthe Gupta-Yos transport models was performed foran hyperbolic Earth reentry. It was found thatfor higher reentry velocities, the selection of themethod to calculate transport properties influencesthe results more than for lower velocities, on ac-count of the increased degree of ionization of theflow.

An analysis of the impact of applying a multi-temperature model was carried out. Park’s two-temperature model was considered. It was con-cluded that the flow at these reentry conditions, at10 km/s, presents a very strong thermal nonequilib-rium, as would be expected. However, this does notsignificantly impact the overall aerothermodynam-ics of the flow due to the fact that the mole frac-tions of the molecules in the shock layer are verylow and thus constitute a very small percentage ofthe flow composition. In this region, the ions andatoms characterize the flow and hence the vibra-tional temperature loses its meaning.

The trajectory point corresponding to the peakin convective heating was thoroughly analysed. Thespecies molar fractions validated the assumption ofa fully catalytic wall boundary condition.

The stagnation point convective heat flux wasanalysed for all the key trajectory points and com-pared to the Sutton-Graves correlation. The resultsagreed reasonably with the correlation. Discrepan-cies may be attributed to the fact that the corre-lation assumes that the flow is in chemical equilib-rium.

Acknowledgements

I gratefully acknowledge the support of Prof. MarioLino da Silva, Dr. Maria Castela, Prof. Paulo Gil,Dr. Bruno Lopez and company Spin.works for thesuccess of the present work.

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