Thesis Ermina

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ARISTOTLE UNIVERSITY OF THESSALONIKI AND

E´ COLE POLYTECHNIQUE FE´ DE´ RALE DE

LAUSANNE

MASTER THESIS

NUMERICAL

SIMULATIONS OF THE

APOLLO 4 REENTRY

TRAJECTORY

A thesis submitted in fulfillment of the requirements

for the degree of Master of Mechanical engineering

Author’s name:

Ermioni Papaopo!"o!

Supervisors:

Dr. Penelope Leylandi!hil "aner#i

$lise %ay

La!#ann$% S!i##$% Mar&' ()*+


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Acknowledgment

&

During this pro#ect there have been numerous people that contributed time and thoughts

in order the present 'or! to be fulfilled. My than!s are addressed to:

Dr. Penelope Leyland( 'ho allo'ed me to 'or! on this pro#ect as a member of the )nterdis*

ciplinary Aerodynamics +roup ,)A+- and her dedication and help throughout the pro#ect.

My colleagues and friends( i!hil "aner#i( for his supervision and moral support( i! +e*

uns( /eremy Mora*Monteros( $lise %ay( 0leg 1otsur( +ennady Plyushchev( for the every

day support and the good times 'e spent in and out of the office and Angelo 2asagrande

for his support and patience.

Special than!s should be addressed to my supervisor in +reece( Prof. Anestis ).1alfas( for

giving me this opportunity to do my Master’s thesis in $P%L and for his constant control

and support for this pro#ect.


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Abstra

3

Sample return capsules( as the Apollo 2ommand Module have been 'idely used to ad*

vance the !no'ledge and planning of manned lunar and planetary return missions. Suchreentry vehicles undergo e4treme thermal conditions( caused by shoc!*heated air during

their super*orbital atmospheric re*entry. Such e4treme conditions can result in failure of

the aeroshell structure and loss of important payload. 5his technological challenge is ad*

dressed by the use of ablative thermal protection systems ,5PS-( 'hich dissipate the heat

a'ay from the vehicles front 'all via ablative products release into the boundary layer.

Additionally( such velocity and temperature magnitudes during reentry conditions intro*

duce significant radiative heat loads( filling the shoc! layer 'ith radiators that react 'ith

the ablative species in#ected by the capsule 'all.

5herefore( accurate numerical modeling techniques are required( so that the thermophys*

ical( thermochemical environment of a reentry capsule can be successfully reproduced

and predicted. 5he present 'or! aims to numerically rebuild certain significant tra#ectory

points( containing the pea! heating points of the Apollo 6 terrestrial re*entry. 5his re*

quires the coupling of the resolved flo'*field 'ith radiative and ablative effects in order

to accurately predict the convective and radiative heat flu4 for each tra#ectory point. 5he

results 'ill be compared to previous calculations and e4isting flight data.

5he numerical simulations are performed in 7D thermal non*equilibrium 'ith a compress*ible e4plicit avier*Sto!es solver( coupled to a radiation database and a thermal material

response code to implement the ablative effects. 5he calculations are performed also in &D(

using a commercial implicit avier*Sto!es solver. 5he results 'ill be used to reproduce

the capsules tra#ectory and verify the accuracy of the associated Modeling 5ools.


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Conten

8

A&,no-"$.m$n/# iii

A0#/ra&/ 1

Con/$n/# 1ii

Li#/ o2 Fi.!r$# 3i

Li#/ o2 Ta0"$# 31

1 In/ro!&/ion *

9.9 5he Arc Pro#ect........................................................................................................... 99.7 eentry flight missions............................................................................................... 9

9.& Applied 5ools........................................................................................................................7

9.6 Scope and 0vervie' of the present dissertation............................................................7

I T'$or 5

2 T'$or$/i&a" 6a&,.ro!n 7

7.9 ypersonic %lo'*fields........................................................................................................8

7.9.9 Stagnation flo'*field properties...........................................................................8

7.9.7 5he shoc!*standoff distance.................................................................................;

7.7 Aerothermodynamics of hypersonic flights.....................................................................;

7.7.9 Stagnation region flo'*field and thermodynamic state..................................<

7.& )nvestigation of eentry Shoc! Layers...........................................................................97

7.6 5he Apollo 2ommand Module Mission.........................................................................9&

3 E8!a/ion# an M$/'o# *7

&.9 2ompressible avier*Sto!es equations...........................................................................98

&.7 )mplemented equations.....................................................................................................

9


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Con/$n/# Con/$n/#

viii

&.7.9 A4isymmetric 7D flo's* 5'o temperature thermodynamic model . . 9


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Conten

<

6.7.7.7 eat flu4 calculations.........................................................................39

6.7.7.& adiation 2alculations.......................................................................3&

6.7.& 5ra#ectory points comparison.............................................................................3<

6.7.6 @alidation of the results......................................................................................>76.7.6.9 Semi*empirical correlations....................................................................>7

6.7.6.7 "ertin.....................................................................................................>&

6.7.6.& %ay*iddel.............................................................................................>&

6.7.6.6 Martin....................................................................................................>6

6.7.6.3 5auber*Sutton......................................................................................>6

6.7.6.> esults Discussion..............................................................................>3

Con/$n/# Con/$n/#

IIIN!m$ri&a" Sim!"a/ion# in +D 97

5 Sim!"a/ion# in +D 9:

3.9 +eneral Procedure.............................................................................................................><

3.9.9 Protocol...................................................................................................................8?

3.7 Pre*processing........................................................................................................... 89

3.7.9 +eometry 2onfiguration.............................................................................. 89

3.7.7 Mesh +eneration........................................................................................... .......87

3.7.7.9 %ore*"ody ..............................................................................................87

3.7.7.7 hole "ody ..........................................................................................8;

3.& %ore*"ody Simulations......................................................................................................;3

3.&.9 2%DBB Simulations..................................................................................... ;>

3.&.9.9 @iscous %lo'* Perfect gas model.....................................................;>

3.&.9.7 eal 3*species equilibrium gas model.............................................;;

3.&.9.& eal 3*species non*equilibrium gas model....................................


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Con/$n/# Con/$n/#

viii

3.&.7.9 $quilibrium gas model.......................................................................

3.6 hole "ody Simulations................................................................................................9?<

3.6.9 $quilibrium gas model..................................................................................... .9?<

3.6.7 5hermal on*$quilibrium gas model............................................................99&

IV Con&"!#ion **5

6 Con&"!#ion# an R$&omm$na/ion# **7

>.9 2onclusions.............................................................................................................. 998>.7 ecommendations.................................................................................................. 99;

6i0"io.rap' *(*


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99

A Con/o!r# +D Sim!"a/ion# *


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99

List of Figures

7.9 )llustration of the aerothermodynamic processes occurring in the shoc! layer

and on the surface of a re*entry capsule at pea! heating conditions( ta!en from

C&6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9?

7.7 )llustration of the chemical !inetic processes along the stagnation streamline

of an atmospheric reentry vehicle( ta!en from C&6...........................................................99

7.& 5he Apollo 2ommand Module geometry( caption ta!en from C7;................................96

7.6 5he distinguished flo' regions observed in front of the reentry blunt face of

Apollo 6( ta!en from C&&.......................................................................................................96

7.3 5he distinguished flo' regions observed in front of the reentry blunt face of

Apollo 6( ta!en from C79........................................................................................................93

&.9 5he control volume method scheme. %or the present case it is assumed that

δy n Eδy s E Fy/7 and δ x e Eδ x w E Fx/7....................................................................... 76

&.7 5he schematic representation of the definition of adiative intensity .......................... .&>

&.& Schematic of tangent*slab calculation domain along lines of cells on a multi*

bloc! grid...................................................................................................................................&;

&.6 $4ample of a radiation sub*grid on a simple 7D a4isymmetric grid..............................&<

&.3 $4ample of mapping of the radiation sub*grid onto the 2%D grid.................................69

6.9 "loc!ing of the domain of the Apollo 6 fore*body for a viscous case( 'ith and

'ithout ablation............................................................................................................ ...........68

6.7 Meshing of the domain for an inviscid and a viscous flo' calculation..........................6;

6.& @elocity profile along the stagnation line( 57( investigation of ablation effects

and diffusion model.......................................................................................................................6<

6.6 5emperature profile along the stagnation line( 57( investigation of ablation

effects and diffusion model....................................................................................................3?

6.3 Species mass fractions along the stagnation streamline( 57( ablative viscous flo' 39

6.> 2onductive and Diffusive heat flu4 for the viscous non*ablation case( second

tra#ectory point.........................................................................................................................37


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Li#/ o2 Fi.!r$# Li#/ o2 Fi.!r$#

6.8 2onvective heat flu4 for the viscous non*ablation case( second tra#ectory point.

2omparison of diffusion models......................................................................................... ..37

6.; Spectral and integrated heat flu4( pea! radiative heating point( tangent slab

radiation transport model......................................................................................................36

6.< Spectral heat flu4( pea! radiative heating point( tangent slab radiation trans*

port model. Divided spectra into specific important regions of the 'avelength

range...........................................................................................................................................33

6.9? Spectral heat flu4( pea! radiative heating point( tangent slab radiation trans*

port model( ablation included...............................................................................................3>

6.99 Spectral heat flu4 in 'avelength range 7??*97?? nm( pea! radiative heating

point( comparison of ablation and no*ablation cases.......................................................38

6.97 adiative divergence along stagnation streamline( Pea! radiative heating point(

5angent slab radiation transport model..............................................................................3;

6.9& 2onductive and adiative heat flu4es for 'avelength range ,?.?3*9.7 µm- along

'all surface( Pea! radiative heating point...........................................................................3;

6.96 @elocity profiles for the t'o tra#ectory points( viscous flo' 'ith and 'ithout

ablation............................................................................................................................ >?

6.93 5emperature profiles for the t'o tra#ectory points( viscous flo' 'ith and 'ith*

out ablation...............................................................................................................................>?

6.9> 2onvective eat flu4 along 'all surface( 2onstant Le'is umber diffusion

model( viscous flo' 'ithout ablation...................................................................................>9

6.98 2onductive eat flu4 along 'all surface( amsha'*2hang diffusion model( vis*

cous flo': )nvestigation of tra#ectory points heating and ablation effects.....................>9

6.9; Ma4imum 2onductive and radiative heat flu4es ,for 'avelength range C?.7*

9.7 µm- for the three tra#ectory points( amsha'*2hang model( viscous flo'

'ithout ablation.......................................................................................................................>7

3.9 5he geometry profile of the Apollo 2ommand Module .....................................................89

3.7 5he geometry of the Apollo 6 front*body and the flo' domain around it .....................87

3.& 5he geometry of the Apollo 6 entire body and the flo' domain around it . . . 8&

3.6 5he geometry of the Apollo 6 flo' domain and the individual parts as divided

in according to the boundary surfaces.................................................................................8&3.3 5he bloc!ing of the Apollo 6 fluid domain.........................................................................86

3.> 5he mesh generation of the Apollo 6 fluid domain...........................................................83

3.8 5he &4&4& determinant quality chec! results for the above bloc!ing.............................8>

3.; 5he mesh of the front*body as used for the SM" solver...............................................88

3.< 5he mesh of the front*body as used for the 2%DBB solver.............................................88

3.9? )nvestigation of convergence independence of mesh.........................................................8;

3.99 5he geometry of the Apollo 6 flo' domain as configured in )2$M ...............................8<

3.97 %irst split of bloc!ing...............................................................................................................8<

3.9& 0*grid splits of the bloc!ing..................................................................................................;?


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9&

Li#/ o2 Fi.!r$# Li#/ o2 Fi.!r$#

3.96 Deletion of unnecessary bloc!s..............................................................................................;9

3.93 2reation of ne' he4a bloc!s for the bac!*flo' part of the domain from e4isting

surfaces......................................................................................................................................;7

3.9> %inal vie' of the bloc!ing......................................................................................................;&

3.98 5he mesh of the 'hole*body case........................................................................................;6

3.9; 5he &4&4& determinant quality chec! results for the bloc!ing for the 'hole*

body case...................................................................................................................................;6

3.9< 2onvergence criteria for perfect gas model.........................................................................;>

3.7? Pressure and temperature field for perfect gas model......................................................;8

3.79 Mach number and velocity field for perfect gas model.....................................................;8

3.77 2onvergence criteria for 3*species real gas model in equilibrium..................................;;

3.7& Pressure and velocity contour levels for real 3*species gas model in equilibrium ;<

3.76 Mach number contours and chec! of the grid’s suitability for this case ........................;<

3.73 5emperature fieldG 3 species equilibrium gas model.........................................................


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Li#/ o2 Fi.!r$# Li#/ o2 Fi.!r$#

3.3? Pressure and density field( 3*species thermal non*equilibrium gas model...................9?3

3.39 5emperature field( 3*species gas model in thermal and chemical non*equilibrium 9?3

3.37 2omparison of the residuals graphs for t'o SM" cases 'ith different Mach

numbers....................................................................................................................................9?8

3.3& Pressure field( 3*species gas model in equilibrium for different Mach*number

cases..........................................................................................................................................9?;

3.36 esiduals for t'o time integration modes for the non*equilibrium cases in 2%DBB9?<

3.33 Pressure( density and Mach*number field( real equilibrium gas model.........................99?

3.3> @elocity field( real equilibrium gas model..........................................................................99?

3.38 5emperature field( 3*species equilibrium gas model........................................................999

3.3; Species distributions( 3*species equilibrium gas model case..........................................997

3.3< Pressure( density and Mach*number field( real equilibrium gas model.........................99&

A.9 Pressure and temperature field for perfect gas model.......................................................9 A.7 Mach number and velocity field for perfect gas model ......................................................7

A.& IB for the perfect gas model case................................................................................. 7

A.6 Pressure and temperature field............................................................................................. ..&

A.3 Mach number and velocity field..............................................................................................&

A.> Surface parameters.....................................................................................................................6

A.8 Species distributions for the 3 species equilibrium case....................................................3

A.; Pressure and temperature field...............................................................................................>

A.< Mach number and velocity field....................................................................................... .......8

A.9? Surface parameters.....................................................................................................................8 A.99 Species distributions for the 3 species non*equilibrium case............................................;

A.97 Pressure and temperature field...............................................................................................<

A.9& Mach number and velocity field..............................................................................................<

A.96 Surface parameters....................................................................................................... ...........9?

A.93 Species distributions for the 99 species equilibrium case..................................................99

A.9> Pressure and temperature field.............................................................................................97

A.98 @elocity field.................................................................................................................... 9&

A.9; Surface parameters...................................................................................................................9&

A.9< Species distributions for the 99 species non*equilibrium case.........................................

96


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List of Tables

7.9 5he Apollo 2ommand Module tra#ectory points to be numerically rebuilt . . . 9>

&.9 Symbols e4planation of avier*Sto!es equations...............................................................9<

&.7 2hemical reactions considered in Par!’s 7??9 model.......................................................&9

6.9 2omparison results for the convective and radiative heat flu4es for the Apollo

6 front body ...............................................................................................................................>3

3.9 5he results for the quality chec! concerning the &4&4& determinant.............................8>

3.7 5he results for the quality chec! concerning the &4&4& determinant............................;6

3.& Summary of the basic results from the 2%DBB simulations............................................


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9

Chater !

Introduction

1.1 T'$ Ar& Pro;$&/

5he present 'or! describes the contribution of )A+ ,)nterdisciplinary Aerodynamics

+roup- at $P%L to the $uropean pro#ect called A2: JAblation*adiation couplingJ.

5his study is led by a consortium composed of the )nterdisciplinary Aerodynamics

+roup of the $cole Polytechnique %ederale de Lausanne( the $uropean Aerospace

esearch $stablish* ment 2)A( the )S )nstitute of the Hniversity of Stuttgart( an

e4perienced Aerospace SM$( %+$( together 'ith subcontractors: the Hniversity of

Kueensland ,Australia-( a 'ell !no'n e4pert on ablation modelling( Dr. +. Duffa as a

consultant( and AS5)HM*St.

5his pro#ect addresses the ablation*radiation coupling for high speed reentry and the phys*

ical phenomena that are induced by hypersonic flight conditions: ablation and radiation

over a thermal protection system ,5PS-. 5herefore( both testing pro#ects in ma#or $uro*

pean plasma 'ind tunnels and coupling simulations are led simultaneously( based on and

compared 'ith data obtained from past $arth return missions. 5his particular contribu*

tion of )A+ deals 'ith the radiation*ablation*flo'*field numerical coupled simulation of

the reentry of the Apollo 2ommand Module.

1.2 R$$n/r 2"i.'/ mi##ion#

5he e4ploration of space and the ma#or aerospace industry advancements have been

achieved through numerous pro#ects of un*manned flights( 'ith the use of command mod*

ules( aeroshell capsules. 5hese capsules( as the Apollo 2ommand Module( are basically

test*probes for the actual manned aircrafts( such as the case of the Apollo program( or

they are merely used for flight*testing of ne'ly designed equipment and technologies and


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*


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*


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Par/ I

T'$or


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8

Chater "

Theoretical#ackground

"his #hapter starts with the literat$re re%iew &or 'ypersoni# Fli(hts in (eneral) re&ers to

some testin( &a#ilities &or hypersoni# &lows and explains thoro$(hly the physi#al

phenomena o##$rrin( d$rin( an atmospheri# re*entry+ It with the important in&ormation

&or the ,pollo test*&li(ht+

2.1 Hp$r#oni& F"o-=2i$"#

A flo'*field 'here the Mach number is far greater than 9 ,conventionally the threshold isconsidered to be M ≥ 3-:

M EU∞

∞α∞

,7.9-

is defined as a ypersonic %lo'*field and is characteri=ed by certain physical phenomena

occurring in the shoc! and boundary layer of the flo' in front of the reentering bodies.

igh speed reentry velocities lead to a strong shoc! 'ave upstream of the vehicle 'hich

strongly modifies the state of the fluid behind it. 5he internal thermodynamic energy of

the free*stream fluid particles is small 'hen compared 'ith the !inetic energy of the free

stream C6. As the Mach number is increased( the flo' density increases progressively( so

the shoc! layer created by the strong bo' shoc! over the blunt nose becomes thinnerG the

detachment distance bet'een the 'all boundary and the shoc! 'ave is really small.

2.1.1 S/a.na/ion 2"o-=2i$" prop$r/i$#

)n the stagnation region in front of the capsule’s nose the free stream passes through the

normal portion of the bo' shoc! 'ave and decelerates isentropically to the point near the

surface of the body( 'hich constitutes the edge condition for the thermal boundary layer


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;

at the stagnation point. the pressure and the heating rate at the stagnation point are

useful reference values for characteri=ing hypersonic flo's.


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(< T'$or$/i&a" 6a&,.ro!n (-( 'hich for a

sphere* cone model( the ratio ∆ ( 'here F is the shoc!*standoff distance and is the nose

radius( is given: FE?.96&exp

R

&.76

M 2,7.7-

2.2 A$ro/'$rmonami o2 'p$r#oni& 2"i.'/#

)n hypersonic flights( the directional velocity of the fluid particles is much larger than the

fluctuation velocities of the molecules of the flo'. 5hus( the flo' !inetic energy before the

bo' shoc! is much larger than the internal thermodynamic energy of fluid particles. )n

atmospheric reentry conditions( i.e. high altitudes and hypersonic speeds three physical

phenomena affect the aerothermodynamic state of the flo' in the shoc! layer C7:

❼ @iscous interactions:

5he viscous interactions bet'een the viscous boundary layer and the inviscid unaf*

fected flo'*field are caused by the large amount of !inetic energy in the boundary

layer. 5hese viscous interactions affect the thic!ness of boundary layer( δ:

2

δ≈ √ ∞ ,7.&-Re

( 'hich in hypersonic speeds( 'here M∞ is e4tremely high( can be so thic! that it

practically merges 'ith the shoc! 'ave( forming a merged shoc! layer( and( thus

affecting the non*viscous part of the flo' ,outside of the boundary layer-.

❼ Lo' density flo':

At the atmospheric reentry altitudes the flo's density is so lo' that the distance

bet'een particles is too big to appro4imate the gas as a continuum. 5hus( imple*

mentation of the !inetic theory is required to describe the fluid state.

❼ igh* temperature flo':

1inetic energy of the high speed flo' is dissipated by the influence of friction 'ithin

the boundary layer. 5he loss of !inetic energy through the shoc! is transformed

into internal thermodynamic energy and creates a pea! of temperature behind the

shoc!. 5his viscous dissipation results in temperatures high enough to e4cite the

vibrational energy of the atmospheric molecules and cause dissociation( and even

ioni=ation 'ithin the gas( both in the boundary and the shoc! layer. 5hus( the

shoc! and boundary layers of hypersonic flo's are chemically reacting. )f 'e con*

sider the atmospheric air a mi4ture 'ith a composition of N2, O2, NO, O and N (

-

M


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at certain levels of temperature the follo'ing changes in the state of the mi4ture

can appear: 5hese so*called igh*temperature effects ,or real*gas effects- induce

T 7???K Dissociation ofO2 O 2 → 7 O

T >6???K N2 → 7N N2BO2 → 7N ONBO→N O+ B

e− T >


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absorbers( so that the coupling bet'een ablation and radiation cannot be ignored C9;.

%or a better understanding( Fi($re 21 describes the overall processes occurring during

reentry and Fi($re 22 describes the chemical !inetic processes occurring in the shoc! and

boundary layer of an reentering vehicle. 5PS design is mainly based on numerical tools.

A coupled approach bet'een the solid domain and the fluid one has to be lead as they

have both a strong thermal( physical( and chemical impact on each other.

Fi.!r$ ( )llustration of the aerothermodynamic processes occurring in the shoc! layerand on the surface of a re*entry capsule at pea! heating conditions( ta!en from C&6

2onvective heating consists of the effects of conduction and diffusion in the shoc! layer.

)t is considered to be the sum of the conductive heat flu4 in'ards and out'ards of the

surface 'all( 'hich is purely the result of temperature difference in the flo'*field and dif*

fusive heating( 'hich results from the enthalpy in#ected in the flo'*field by the pyrolysis

and char ablative products.

0n the other hand( radiative heating is highly dependent on the composition of the at*

mospheric gas. )t mainly consists of the radiative heat flu4 created from the ioni=ation

and vibrational e4citation of the atmospheric molecules( 'hich is transferred to'ards the


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Fi.!r$ ( )llustration of the chemical !inetic processes along the stagnation streamlineof an atmospheric reentry vehicle( ta!en from C&6

surface 'all( and the re*radiation( the radiative emission e4erted from the reentry body.

Sutton C&


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(< T'$or$/i&a" 6a&,.ro!n (


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layers during reentry. 5he operation of a shoc! tube includes a shoc! being driven di*

rectly through the test gas at pressures and velocities encountered in super*orbital flight.

Spectral measurements can be made of the radiation emitted in the shoc! layerG from the

shoc! frame*of*reference( the post*shoc! rela4ation phenomena encountered in flight along

the stagnation streamline are e4actly replicated C7>.

5he computational modelling of radiating shoc! layers requires the numerical solution of

an appropriate set of governing equations. ypersonic flo'*fields in the continuum regime

are 'ell described by the compressible avier*Sto!es equations( 'hereas in the flo' regimes

that the mean free path of the molecules becomes comparable to the characteristic scale of

the flo'( statistical methods such as the Monte*2arlo method should be applied. )n order

the numerical simulations to be accurate enough( a lot of 'or! has put on thermochemical

non*equilibrium modelling( 'ith the latest advances in this area to propose the use of acombination of "olt=mann distributions and !inetic modelling. 0n the other hand( the

radiation modelling is affected by: ,9- the internal state populations of chemical species(

,7- the spectral distribution of the electromagnetic energy and ,&- the transport of the

electromagnetic energy through the plasma. More details about the numerical modelling

tools applied are discussed in chapter &. N$meri#al re$ild and sim$lation of 'ea!ly ion*

i=ed gases over reentry vehicles has been successfully performed for numerous missions(

including FIRE C&8( C77( Apollo Command Module C7


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(< T'$or$/i&a" 6a&,.ro!n (


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orbital speed for a test case 'ith an ablative heat shield. 5he obtained measurements(

including pressure distribution and convective and radiative heating 'ere conducted 'ith

the use of pressure transducers( radiometers and surface*mounted calorimeters located at

the pitch lane and at various rays around the command module( as it is depicted in Fi($re

2! .

Fi.!r$ ( 5he distinguished flo' regions observed in front of the reentry blunt face of

Apollo 6( ta!en from C79

5he radiometer used for the measurements 'as embedded 'ithin the ablating 5PS( and

the interaction of the ablation products 'ith the optics 'as a concern. o'ever( post flight

inspection revealed an absence of carbon deposits on the radiometer 'indo'. 5herefore( it

is unli!ely that there 'as a strong concentration of carbon bearing species in the radiome*

ter cavity. 5he gas 'ithin the radiometer cavity 'as most li!ely dominated by molecular

hydrogen. 5he absorption properties of molecular hydrogen over the 'avelength rangetransmitted by the quart= 'indo' are lo'. As such( absorption in the radiometer cavity

is neglected for the purposed of this rebuild. 5he radiometer 'as mounted on the fore*

body( at a distance of 9.&( Par! C7


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5his equivalent sphere is also used for the 7D calculations of the present pro#ect( in order

for the comparison of the results 'ith the results achieved by Par! and other correlations

to be made successfully.

5he radiative flu4 'as obtained from flight data and 'as numerically rebuilt at three alti*

tudes of the tra#ectory in previous 'or!s C&> and C76.33> 3.?38Density ,kg/m3-

@elocity ,km/s-

9.8& ·

9?−4&.69 · 9?−4

9?.7373.?9 · 9?−4

all 5emperature ,1- 7&?? 73?? 733?

%light ecorded adiative

)ntensity ,?.7 − 6µm- ,W cm−2sr−1- 9?.3 73.? 9>.?

Ta0"$ ( 5he Apollo 2ommand Module tra#ectory points to be numerically rebuilt


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98

Chater %

E&uations and Methods

"his #hapter re&ers to the (o%ernin( e$ations o& the &low*&ield n$meri#al modellin( &or

'ypersoni# &ields and real (ases and the n$meri#al methods applied &or the

dis#reti4ation and sol$tion o& the set o& e$ations+ Finally) the #hemi#al models and

heatin( transport models applied in the present work are shortly dis#$ssed+ Emphasis

is (i%en on the models implemented in order to set $p a #ase in the &low*&ield sol%er

$sed &or the 2D #al#$lations) Eilmer3 and &or the 3D #al#$lations) $sin( the 2

#ommer#ial does) CFD++ and NSMB+ "he (eneral pro#ed$re to r$n a sim$lation on

Eilmer3 is analy4ed in Chapter ) whereas the pro#ed$re &or the 3D #ases is dis#$ssed

in Chapter !+

3.1 Compr$##i0"$ Na1i$r=S/o,$# $8!a/ion#

During atmospheric reentry( the fluid flo' is compressible( as the fluid density varies due

to compressible phenomena induced in the flo'*field by the strong bo' shoc! 'ave in

front of the capsule’s face. 5he evolution of a compressible hypersonic flo' is described

'ith the compressible avier*Sto!es equations. 5his set of partial differential equations is

derived by applying the continuity equation to conserved quantities of mass( momentum

and energy of the mi4ture. According to the thermodynamic model implemented( therecan be more than one energy and mass continuity equations. 5he generic differential form

of the continuity equation is e4pressed as:

∂φB F E N ,&.9-

∂t

( 'hereφis the conserved quantity( Fis a vector describing the flu4 of φand N is a source

term describing the creation and destruction of φ.

∇ ·


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9;

5he conserved quantities to be applied to $quation &.9 'hen deriving the compressible

avier Sto!es equations are dependent on the degree of thermochemical non*equilibrium

to be considered. %or a gas in thermochemical equilibrium( the conserved quantities are


38/194

+< E8!a/ion# an M$/'o# +


39/194

+


40/194

3.2 Imp"$m$n/$ $8!a/ion#

%or an a4isymmetric flo'field case( the avier*Sto!es equations can be summari=ed in a

compact form as follo's:

∂→− U

→− →− −→ →−

here:

→−

ρ

∂t

B ∇ .,Fi −F% - E Q ,&.;-

❼ U E ρ→− v

: the conservative variables

ρE

ρ

→− v

→−

❼ Fi E

ρ→−

v∧→−

vB pI

: the non*viscous flu4 vector ρ→− v H

−

→−

J

−→ ❼ F% E

ττ .→ − v

ωO

: the viscous flu4 vector

❼

→− QE

ρ→

−

f

: the source term

ρ→− f .→− v

('here the symbols used are summari=ed in &.9

v @elocity vector vE ,u, v,

w- p Pressure

ρ Density

f $4ternal force vector

τ @iscous stress tensor:τEµr · , ρ v⊗v- B · , ρv⊗v-"

l Bη · vI

µ Dynamic @iscosity

ν 1inematic viscosity

H 5otal enthalpy

E 5otal energy

Ta0"$ + Symbols e4planation of avier*Sto!es equations

"he sym1ol ⊗ de&ines a tensor prod$#t) i+e+ ,u⊗v-i5 Euiv 5

∇ ∇ ∇


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3.2.1 A3i#mm$/ri& (D 2"o-#= T-o /$mp$ra/!r$ /'$rmonami& mo$" "he

&ollowin( se#tion re&ers to the &orm$lation o& the #ode &or ,xisymmetri# "wo*Dimensional

&lows $sin( &inite*rate #hemistry and m$lti*spe#ies (as) in the way that the e$ations are

&ormed &or the Eilmer3 #ode+


42/194

+< E8!a/ion# an M$/'o# +


43/194

+


44/194

5he viscous contribution F% for a t'o*temperature model is as follo's:

?

τ xx

τ

?

τ

xy

τF% E

i + P j ,&.97-

yx yy τ xx u x Bτyx uy Bq x

τ xyu x Bτyyuy Bqy

q x)%e

J x)s

qy)%e

Jy)s

ere( τrefers to the a4isymmetric viscous stresses components( qdenotes the heat*flu4

and Jis the diffusion flu4. )n order to apply the operator*splitting integration approach(

'hich is used in the $ilmer& code( the vector of source terms is separated into geometric(

chemical !inetic( thermal energy e4change and radiation contributions:

QEQ(eom+ BQ#hem+ BQtherm+ BQrad+ ,&.9&-

Q(eom is the source term for a4isymmetric geometries:

, p−τ(eom+

?

77 - A xy/V

?

,&.96-

ere( A xy is the pro#ected area of the cell in the ,4(y- plane.

Q#hem is the chemistry source term:

N mol

?

M ion

,&.93-

),

i

N6 C ),

EC 5

5

MsωO s

Q E

?

?

?

#hem+

?

?

?

i B N


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ere( N6 C is the vibration*chemistry energy e4change source term( NEC is the electron*

chemistry energy e4change source term( Ms is the molecular 'eight and ωO s is the

mass production source term. Qtherm is the thermal energy e4change source term:


46/194

N mol

?

M spe#ies

,&.9>-

),

i

6 " ),

E" 5

5

?

ere( N6 " is the vibration*translation energy e4change source term( NE" is the electron*

translation energy e4change source term. Qrad is the radiation source term:

?

,&.98-

−∇ .qrad

−∇ .q rad

?

+


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5he inviscid component of the flu4es(Fi ( becomes:


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+< E8!a/ion# an M$/'o# +


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+


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control volume scheme for a structured grid is presented in figure &.9. )n &D( finite*volume

cells are he4ahedral 'ith > quadrilateral surfaces interfacing the neighboring cells( called

orth ,-( $ast ,$-( South ,S-( est ,-( 5op ,5- and "ottom ,"-. %lu4 values are es*

timated at midpoints of the cell. 5he advantage of the fact that the computational nodes

are assigned to the 2ontrol @olume center is that the nodal values represent the mean over

the control volume at higher accuracy( since it is a second order discreti=ation scheme ,O2-.

Fi.!r$ + 5he control volume method scheme. %or the present case it is assumed that

δy n Eδy s E Fy/7 and δ x e Eδ x w E Fx/7

5he net flu4 through the control volume is calculated as the sum of the integrals over the

control volume surfaces ,in 7D the surfaces are 6: ( S( ( $- and the integral conservation

equation is appro4imated 'ith the follo'ing algebraic e4pression:

dU 9

dtE −V

'

,Fi −F% -.nPdAB Q ,&.77-N ES0

( 'here U and Qrepresent cell*averages values.

3.3.2 Tim$ #/$ppin. pro&$!r$

An operator*splitting approach ,as described by 0ran and "oris C8- is used to perform

the time*integration of this ordinary differential equation ,&.77-. 5he physical phenomena

are handled in a decoupled manner( creating different time differentiations for the inviscid(

viscous( chemical and thermal phenomena and the boundary conditions are imposed on the

edges of the control volumes. 5herefore( the time differentiation variable d8 is decomposed

into:

d


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dU

dt

dU

dt

dU

in%+

%is#

+

E'

,Fi -.nPdAB Q(eom+B Qrad+ ,&.7&-V

N ES0

E

'

,− F% -.nPdA ,&.76-N ES0

dt

dU

dt

#hem+

therm+

EQ#hem+ ,&.73-

EQtherm+ ,&.7>-

3.3.3 Tim$ i#&r$/i?a/ion m$/'o

3.3.3.1 E3p"i&i/ M$/'o

As aforementioned( in the 7D solver( $ilmer&( the set of 0rdinary Differential $quations

,0D$- is obtained by an operator*splitting approach. 5hus( each physical process can be

handled by a more efficient integration scheme( 'hich is useful for large chemical !inetic

models ,see C&9 for the chemical model-.

%or a better understanding(

→−

U is the vector containing all the conserved quantitiesprevi* ously described. )t is obvious that→− U is time and space*dependent and can be 'ritten as→−

U,→− x , t- for a one*dimensional problem. 0nce discreti=ed in space and time( 'e

end up

'ith:

i EU,xExi , tEtn-.

5he ne4t value of→− U( can be computed via the second order up'ind scheme as:

Un+1 n n n

i E f,Ui, Ui −1, Ui −2-.

5herefore( the value of the considered conservative variable in a cell at the ne4t time step

,tn+1- can only be computed if the past values ,tn- in the current and the t'o previous

cells are !no'n.

3.3.3.2 Imp"i&i/ M$/'o#

5he method used for implicit time integration for the steady state problems encountered

in this thesis( 'ith the use of the &D soft'are( is the Lo'er*Hpper Symmetric +auss*Seidel

,LH*S+S- method. 5he scheme is based on a lo'er*upper factori=ation and a symmet*

ric +auss*Seidel rela4ation. 5he governing equations are discreti=ed separately in space

and in time. 5his ensures that the steady state solution 'ill be independent of the time

discreti=ation procedure and therefore independent of the time step. 5he conservation

− 9

− 9

V


52/194

equations are transformed into a diagonally dominant form( in order to meet the stability

requirements of the rela4ation method.

%or the viscous flo' cases( the contribution of the viscous terms is ta!en into account


53/194

+


54/194


55/194

,&.&?-


56/194

+< E8!a/ion# an M$/'o# +


57/194

are smaller than that of chemistry( a flo' in chemical equilibrium is al'ays assumed

to be in translational*rotational and vibrational equilibrium.

%or flo's in chemical equilibrium( the governing equations are similar to those for a

caloric perfect gas e4cept for the equation of state( pE p, ρ, T-.

❼ on*$quilibrium gas: As aforementioned( collisions bet'een particles in 'hich the

collision energy is so high that a particle dissociates or ioni=es lead to dissociation

or ioni=ation of the gas respectively. )f the characteristic time for the read#ustment

by collisions is of the same order of magnitude as the characteristic time of the

flo'( the flo' is in non*equilibrium. 5he thermally perfect gas ,or #ust chemical

non*equilibrium- is the gas state( in 'hich the gas components have all a perfect

,collisional- behavior but each component has all internal energy modes e4cited to

an equilibrium described by a single temperature.%or flo's in chemical non*equilibrium( the governing equations are supplemented

'ith equations describing the mass conservation of the chemical species in the react*

ing mi4ture. As a result( the continuity equation has to be solved for each chemical

species.

❼ 5hermal on*$quilibrium gas:

As aforementioned( the collisions bet'een free stream particles and molecules of

the dense shoc! layer e4cite the rotational( vibrational and electronic modes of the

molecules( and give rise to a non*equilibrium bet'een the temperatures and energiescorresponding to these modes C96. +enerally( translational and rotational energy

modes require a fe' collisions to equilibrate( so they are usually considered in equi*

librium even for ypersonic flo' conditions( for the sa!e of time*saving. 0n the

other hand( chemical dissociation and vibrational e4citation require more collisions

to reach their equilibrium state and non*equilibrium effects may be important for

hypersonic flo' conditions( especially at high altitudes. ith the assumption of

thermal non*equilibrium( a multi*temperature model is applied( in 'hich every tem*

perature dictates a different energy mode C&7. 5his 'ay( the total internal energy of

any species i is formed by the contributions of translational( rotational( vibrationaland electronic energies. 5he same applies to the thermodynamic and transport

properties of the gas mi4ture C9&.

eEet Ber Be% Bee

Molecules are presented 'ith all energy modes( 'hile only translational and elec*

tronic modes are associated 'ith atoms and free electrons.

Modern non*equilibrium models ideally associate a modal energy ems depending on

temperature Tms for mode m for each species s. 5he internal energy of the gas

mi4ture of species is hence given by:


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eE'

ys'

ems,Tms- ,&.&9-

s m

('here ys is the mass fraction of each species s in the gas mi4ture.

5he model used in the present 'or! is a commonly*used thermal non*equilibrium(

t'o*temperature model( developed by Par! C&9. )n the specific model( the system is

described by t'o temperatures:Ttr ( 'hich stands for the translation*rotation energy

mode of the heavy particles and T%e( 'hich represents the distribution of vibrational(

electronic e4citation and electron translation energies.

)n the t'o*temperature model the total energy can be split into !inetic energy(

translation*rotation energy and vibration*electron*electronic energy:

EEetr Be%e B1u u

5he internal energy e for Nmodes thermal modes is given by the sum of the specific

energy of each mode:

eE),N modes

%or the t'o*temperature model( the above equation is transformed into: eE etr Be%e.

5his means that every energy and heat*flu4 is defined as the sum of the t'o thermal

modes. %or e4ample( the conduction heat*flu4 term is defined as:

q# ond Eq#o nd)tr Bq#on d)%e E − ktr∇ Ttr −k%e∇ T%e ,&.&7-

3.6.1 C'$mi&a" mo$""in.

Par!’s chemical model developed in 7??? C&9( used among simpler models in the present

'or! ,e.g. 8 species model of atmospheric air gas mi4ture for the &D cases-( is an atmo*

spheric chemical model( including ablative materials emerging from P)2A*li!e materials.

As can be seen in 5able ( it considers 7? species divided into three categories:

9. ,ir spe#ies: species that are present in the $arth’s atmosphere.

7. ,lati%e spe#ies: species that are emerging from the ablative 'all as ablative gases.

&. Bo$ndary layer spe#ies: species that are formed as the ablative species are bro!en

do'n as they pass through the boundary layer and the post*shoc! layer.

2

m


59/194

Ta0"$ + 2hemical reactions considered in Par!’s 7??9 model

R$a&/ion# M , " a[2 ] n R$a&/ion# M , " a[2 ] n

Di##o&ia/ion r$a&/ion#

N" + M −� −−***− N + N + M All 7+0"! 113200 −1+6 O" + M −� −−***− O + O+ M All 2+0"! 59360 −1+5 C 3+0"" 113200 −1+6

C 1+0"" 59360 −1+5O 3+0"" 113200 −1+6 O 1+0"" 59360 −1+5 N 3+0"" 113200 −1+6 N 1+0"" 59360 −1+5

H 3+0"" 113200 −1+6 H 1+0"" 59360 −1+5e' 3+0"$ 113200 −1+6 H" + M−� −−***− H + H + M All 2+2

!$

48300 0+0

C"

+ M −� −−***− C + C + M All 3+7!$ 69900 0+0 H" 5+5!$

48300 0+0

CN + M� −−−***− C + N + M All 2+5!$ 87740 0+0

N$!/ra" $3&'an.$

r$a&/ion#

N"

+ O −� −−***− NO + N − 5+7!" 42938 0+42 CN + O−� −−***− NO + C

− 1+6!%

14600 0+10 NO + O� −−−***− O" + N − 8+4

!" 19400 0+0 CN + C−� −−***− C" + N − 5+0!% 13000 0+0CO + C−� −−***− C" + O − 2+0

!( 58000 −1+0 CO + C"−� −−***− C% +

O − 1+0!" 41200 0+0CO + O −� −−***− O" + C − 3+9

!% 69200 −0+18 C% + N −� −−***− CN + C"− 1+0!" 34200 0+0

CO + N−� −−***− CN + O − 1+0!$ 38600 0+0 C% + C−� −−***− C" + C"− 1+0!" 16400 0+0

N" + C−� −−***− CN + N − 1+1!$ 23200 −0+11 C"H + H −� −−***− C" +

H" − 1+0!" 16770 0+0

E"$&/ron impa&/

ioni?a/ion r$a&/ion#

O + e' −� −−***− O) + e' + e' − 3+9%% 158500 −3+78 C + e'−� −−***− C) + e' + e'− 3+7%! 130720 −3+00 N + e' −� −−***− N) + e' + e'− 2+5%$

168200 −3+82 H + e'

−� −−***− H)

+ e'

+ e'

− 2+2%*

157800−2+80A##o&ia/i1$ ioni?a/ion

r$a&/ion#

N + O−� −−***− NO) + e' − 5+3!" 31900 0+0 N +

N−� −−***− N"+ e' − 4+4( 67500 1+5

('here A is the frequency factor inr m3/mole

1N −1s−1( Ta is the temperature that cor*

responds to the activation energy Ea in K and n is the temperature e4ponent of the

Arrhenius equation( that describes the temperature dependency of the reaction rates of

each reaction:

kE AexpIE

a

n1

— RT

,&.&&-

)t should be noted that some species as HCN ( CN O( N H and O+ are missing from

the model( even though their effects are particularly important in reentry conditions.

5heir concentrations are small( 'hile O+ presents no significant impact on chemical

reactions

and radiation.

)

2

2


60/194

3.6.2 C'$mi&a" Kin$/i

%or an accurate characteri=ation of the shoc! layer in atmospheric reentry( a good !no'l*

edge of the species mass fractions consumed and produced by chemical reactions in the

shoc! layer is required. 5herefore( apart from the species identified( the reactions and the

!inetics of the species should also be investigated. 5he 7? species identified in Par!’s model

form a 76*reaction scheme ,3 dissociation reactions( 97 e4change reactions( 6 electron im*

pact ioni=ation reactions( and 7 associative ioni=ation reactions-. All those reactions are

governed by the chemical !inetics theory 'hich deals 'ith the reaction rates corresponding


61/194

to the speed of reaction. 5hese reaction rates must be computed in order the evolution

of the gases mi4ture composition and thus the chemical behavior of the flo' field to be

described accurately.

An usual set of nchemical reactions involving N sspecies needs to ta!e into account aparameter rto distinguish the different reaction. )t can be described by:

N s N s'

νi)r Cxi �'

νii Cxi ,&.&6-

i =1 i =1

herexi is the species mole fraction and νi)r and νii

are the stoechiometric coefficient.

$quation &.&6 can be split into t'o individual relations: one for the reaction going from

left to right and the other for that going from right to left. Let us denote by k&)r the

for'ard reaction rate coefficient and k)r the bac!'ard one. 5a!ing into account this

transformation and summing over all reactions 'e end up 'ith the La' of Mass action

e4pressed in terms of mass production for a particular species α:

n N s N s

ωOs EM:',ν:)r −ν

ii -.{ k&)rnCxi D

; i)r −k1)rnCxi D

; i)r } ,&.&3-

r =1 i =1 i =1

)n equation &.&3 the for'ard and bac!'ard reaction rate coefficients can be e4pressed

using the equilibrium reaction rate constant k# :

k Ek & )r

#)r k)r

E A.T<

e4p

− EART

,&.&>-

k# follo's the e4tended Arrhenius la' 'here E , is the activation energy ( ηis the temper*

ature e4ponent and is the gas constant. )n Par!’s model( the Arrhenius la' 'as defined

as:

k,T- EC Tn exp,T

a-, ,&.&8-

T

( 'here C( nand Ta ,activation temperature- are presented in for the Par! 7? specieschemical model.

3.6.3 Tran#por/ prop$r/i$#

5he choice of the chemistry model influences the transport properties required to solve

the avier*Sto!es equations( such as the gas mi4ture’s viscosity and thermal conductivity.

%or the purposes of this section( the modelling of the fluid viscosity is further discussed.

)n a given gas mi4ture( the collisions bet'een the different particles ,atoms( molecules(

ions( electrons- create an induced momentum transfer. 5he amount of momentum transfer

depends on the electromagnetic forces and( thus( on the shape and si=e of the particles

i)

i)

:)

−


62/194

involved. 5he !inetic gas theory proposes the particles to be modeled as hard*spheres and

that is ho' Ma4'ell defined the follo'ing proportionality relation for the viscosity C6?:


63/194

+


64/194

+< E8!a/ion# an M$/'o# +


65/194

entry. )n presence of a moving fluid heat is transported mainly by convective heating. )t

is usually described by e'ton’s la' of cooling:

q#oOn% EhA,T∞ −Ts- ,&.66-

('here Ts is the temperature on the body surface( T∞ is the free*stream temperature and

his the heat transfer coefficient( 'hich is dependent on the type of transfer media( gas or

liquid( the flo' properties( such as velocity and viscosity and other flo' and temperature

dependent properties.

2onvection is usually used to describe the combined effects of heat conduction 'ithin the

fluid 'ith diffusion mechanisms. 5herefore( it is crucial to model diffusion in such a 'ay(

so that the results obtained are accurate.

Mo"$&!"ar Di22!#ion

%or the case of an ablating boundary surface( the mass balance equation must be ta!en

into consideration for the flu4 of chemical species entering or leaving the ablating surface.

ith the assumption that no solid material is removed in a condensed phase( the mass

balance of species i at the surface is 'ritten as:

ρDi5

∂YiI

I

Iw

BmO #)i BmO ()i E

, ρi vn-w B

N r ea#tions'

r

ωO i)r ,&.63-

( 'hereDi5 is the binary diffusion coefficient( for t'o interacting components(Yi the mass

fraction of species i( ρthe density( vthe velocity and

mOthe mass blo'ing for either gas or

char for the g and c subscripts respectively. 5he term ωO i)r is the creation or

destruction of chemical species i during surface reaction r.

5he diffusive species mass flu4 at the 'all:

Ji w E ρDi5

∂ YiI

I

Iw

,&.6>-

e4presses the relative motion of chemical species 'ith respect to the motion of their mov*

ing center of mass. Di is the effective binary diffusion coefficient of each species i into the

gas mi4ture( i.e. the mass diffusivity. 5he higher the diffusivity of one substance relative

to another( the faster they diffuse into each other.

T'$rma" i22!#ion

I

∂

I

∂


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5hermal diffusion affects the diffusion mass flu4 if there is a temperature gradient in

the flo'*field. 5he diffusion mass flu4 e4erted from thermal diffusion is equal to: J" E

nD"1

∇ T( 'here D" is the thermal diffusion coefficient and n is the total number of "


67/194

molecules per unit volume. 5he induced diffusive heat flu4 is calculated:

∂ci

q E ρ D h∂y

,&.68-

i w

All the calculations presented in this 'or! have been solved for t'o diffusion models:

❼ 2onstant Le'is umber: 5he Le'is number in reacting flo's( i.e. flo's 'here there

is simultaneous heat and mass transfer by convection( is a dimensionless number de*

fined as the ratio of thermal diffusivity to mass diffusivity: LeE : ( 'here αis the

thermal diffusivity and D the mass diffusivity. 5he thermal diffusivity is defined as:=

/C p

( 'here κis the thermal conductivity( ρis the density and C p is the specific

heat capacity and it relates the ability of a material to conduct thermal energy to

its ability to store thermal energy.

A simplified 'ay to obtain effective individual species diffusion coefficients consists

in deriving them from an assumed constant Le'is number. 5his simplification is

based on the concept of individual species Le'is number changing little through the

reactive area of the hypersonic flo'*field. o'ever( such a simplification can hardly

be considered accurate( as the Le'is number is highly dependent on the temperature

of the mi4ture( 'hich radically changes during reentry conditions.

❼ Ambipolar Diffusion in 5'o*5emperature Multicomponent Plasmas ,amsha'*2hang

Model-

5his approach C&3 calculates the diffusional mass flu4es of the individual compo*

nents of species relative to the mass*averaged velocity of the fluid are calculated after

e4tracting the diffusion coefficients ,binary molecular and thermal- from relations

containing collision integrals( such as the follo'ing:

Di5 E3k B " i " 5

16 p>i5 Ω+!,

(1)" i5 )

('here kB is "olt=mann’s constant( µi5 Emi m 5/,mi Bm 5 - is the reduced mass for

the pair ,i(#-( mi is the mass of a single particle of species i( N(1)

,9, Ti5 - is astandard

collisional integral andTi5 Emi " 5 +m 5 " i .

3.7.2 Raia/ion Tran#por/ Mo$""in.

Processes such as dissociation and ioni=ation ma!e the gas particles radiate electromag*

netic energy in the flo' ,radiative heating- and also transfer energy to the surface 'hen

(as)di& & − ' "

D

α

"

i

i

mi +m 5


68/194

they collide 'ith it ,convective heating-. 5he radiating gas can be either absorptive and

gain energy or transparent and emit energy in the flo'*field.


69/194

adiation is a non*local phenomenon( i.e. electromagnetic energy emitted at one location

in a plasma field may be absorbed at any location 'ithin a line*of*sight of the origin. A

first order appro4imation of radiation transport is to assume an optically thin or thic!

medium( representing the cases 'here 9?? and ? local re*absorption occurs. %or atmo*

spheric entry shoc! layers( ho'ever( the actual local re*absorption is generally some'here

bet'een these t'o limiting cases. 5he tangent*slab model ma!es various simplifications

to the integro*differential equations governing radiative transport such that they can be

more easily solved.

o'ever( the tangent*slab model has been sho'n to not accurately predict radiative heat*

flu4 due to shoc! layer curvature C;. 5he accurate solution of the radiation transport

equations requires the discreti=ation of the computational domain via ray*tracing. Such

an approach has been implemented by considering the radiation field as a discrete quantity

in Monte*2arlo models C9>.

+


70/194

%or applications to computational grids it is convenient to e4press equation &.3? as the


71/194

difference bet'een local emission and absorption:

− ∇ .− q− r →

ad Er ∞ r

0

4

r ∞κ;I;dωdν− 6π

0

j;dν ,&.3?-

hereκ; is the spectral absorption coefficient and j; is the spectral emission coefficient.

3.7.3 F"o-=2i$" Raia/ion &o!p"in.

A useful parameter for estimating the degree of radiation flo'*field coupling is the +oulard

number:

7 q O r ad R E

1∞

,&.39-

( 'hereqOrad is the radiative heat flu4 incident at the stagnation point( ρ∞ the free*

stream density and u∞ the free*stream velocity. 5he +oulard number is a measure of

the conversion of energy flu4 in the free*stream to radiative energy flu4 incident on the

vehicle. hen the +oulard number becomes large ,R > ?.?9- radiation flo'*field

coupling should be ta!en into consideration due to significant levels of radiative flu4

through the shoc! layer.

3.7.3.1 Tran#por/ mo$"#

A variety of models are implemented in the %lo'*field Solver for the radiative source term

to be calculated:

1. Op/i&a""=/'in mo$"

5he optically*thin model neglects re*absorption( transforming &.3? into:

−∇ .− q− r →

ad E − 6π{∞ j%dv.

%or the radiating shoc! layers of hypersonic reentry cases( the ma#ority of the radia*

tive emission is located in the vacuum*ultraviolet spectral region 'here re*absorption

is significant. 5he optically*thin model 'ill therefore substantially overestimate the

radiative divergence( and is not an appropriate model for this 'or!.

2. Tan.$n/=#"a0 mo$"

5he tangent*slab model allo's the effect of re*absorption to be modeled 'hile

avoid* ing a complete directional integration of the local intensity field. A one*

dimensional variation of properties is considered along each line*of*sight normal

to the vehicle surface. 2omputationally( a line of cells is used to represent the

normal line*of*sight as demonstrated in figure &.&. 5his is a good appro4imation

0


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to the body normal direction for the shoc!*aligned grids ,such as the ones built

for blunt reentry bodies-. )f a single column of bloc!s is used to define the

computational domain bet'een the


73/194

inflo' and vehicle surface boundaries( the tangent*slab model is realisable( as all the

information required for the calculation is contained in the local bloc!.

Fi.!r$ + Schematic of tangent*slab calculation domain along lines of cells on a multi*

bloc! grid

+iven that the infinite*slab arrangement 'ill result in =ero net radiative flu4 in the

transverse directions( the definition of the radiative divergence for slab i reduces to:

− ,∇ qrad -i E −

. 3 r ad

.s

i

−(3r ad +i )!,−3r

ad +i ,)

∆si

('hereq(i )

is the radiative flu4 at theith cell interface ,i.e. preceding the cell from

right*to*left- and Fsi is the 'idth of the cell in the ,appro4imately- body normal di*

rection. 5he solution for the radiative flu4 in a gaseous medium bet'een t'o parallel(

infinite*slabs is a function of the spectral optical thic!ness τ% . )f the computational

domain is a collection of Nslas isothermal slabs 'ith the spectral range discreti=ed

into N% frequency intervals( the radiative flu4 at interface i can be e4pressed as:

q(i )

N %

τ (i )

B7π),N sla1s( 5 )

r

E3

I(i ) ( 5 )

I

I (i ) ( 5 −1)

I l

rad E),

k =1 7πI% k )wall E3

% k

5 =1

S% k

I

% k

— τ% kI

—

E3

I

% k

— τ%

k

IFv

I

('hereS( 5 )

is the source function for the i hisothermal cell atv

% k t

k frequency(I% k)wall

is the intensity emitted by the 'all and the optical thic!ness τ(i )

is calculated as:

E

ra

k

I I I

%


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τ(i ) i (l )

% k E ),l =1κ% kFsl

5he En term is the nt horder e4ponential integral 'ith form:


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&<

En,x- E{∞ω−nexp,− ωx-dω

5he $& curve fit that is implemented in the tangent*slab model is a semi*empirical

formula:

E3 E ?.?


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5he total radiative divergence for a finite*volume cell is calculated as the difference

bet'een the total emissive po'er Eems and absorptive po'er Eas divided by the

cell volumeV: , E ems+ E a1s+ -

− ∇ .− q− r

→

adE

− − ,&.37-

('here:

r r 4 r % max N ems+rays N % Eems+E

6 0% mi n

j%dvdωdVE

' '

Er)n ,&.3&-r n

N a1s+rays N %

Ea1s+ E' '

,− FEr)n- ,&.36-

r n

('here Na1s+rays is the total number of ray segments traversing the current cell(

Nems+rays is the total number of rays emitted by this cell and the frequency domain

has been divided intoNn intervals bet'eennmin andnmax . Er)n is the po'er carried

by a photon pac!etn 'ith frequency interval F%n from rayr 'ith solid angle Fr :

Er)n E j% FvnFrV ,&.33-

5he radiative po'er lost by a photon pac!et n 'hile traversing from point si to s&

along a ray is calculated from:

− FEr)n E − 9 −e−= % n

(s)∆s

Er)n,si - ,&.3>-

( 'here FsEs& −si . Similarly( the radiative heat flu4 incident on any 'all element

qradis calculated as the sum of the remaining energy from all incident rays Nin#rays

divided by the 'all element area A:Eas ),N in#+rays ),N %

qrad E A

E nEr)n

,&.38- A

Ra=/ra&in. an raia/ion #!0=.ri

5he ray*tracing method for planar and a4isymmetric grids of quadrilateral cells is

based on the creation of a radiation sub*grid and the connection of this to the main

V

r


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xEx0 BLcos,φ-cos,θ- ,&.3;-

yEy0 BLsin,φ- ,&.3?-

,&.>9-

5he corresponding 2%D grid coordinates are then calculated from the follo'ing

transformation ,in 7D application-:

xEx ,&.>7-

yE/yS2 BzS2 ,&.>&-

5his transformation has the effect of reflecting rays intersecting the symmetry a4is

at yE ?( as required. %or planar geometries the radiation sub*grid is formed in the

xyplane as the 2%D domain is symmetrical along the za4is.

5he core of the ray*tracing method is a cell searching algorithm that allo's the

radiation sub*grid to be mapped onto the 2%D grid( such as the one sho'n in

%igure &.3

Fi.!r$ + $4ample of mapping of the radiation sub*grid onto the 2%D grid


79/194


80/194

6&

Par/ II

N!m$ri&a" Sim!"a/ion# in (D


81/194


82/194

63

Chater $

Simulations in "-

"his #hapter presents the o%erall pro#ed$re to set $p and r$n a sim$lation on Eilmer3)

the two*dimensional sol%er) de%eloped y the 8ni%ersity o& $eensland) ,$stralia) and the

main res$lts o& the present #al#$lations

4


83/194

4


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stagnation point of a blunt body is governed primarily by the shoc! standoff( it becomes

possible to define a 7D equivalent sphere. %rom Schlieren studies carried out by ied et

al ,ref.ied et al.-( Par! ,ref.Par!stagnation- chose a sphere of 7.;3m radius( 'hich gave

the same shoc! standoff as seen on the Apollo 6 geometry at 73.3 deg angle of attac!.)n this study( the same equivalent sphere radius is used in order to be able to verify the

results from $ilmer& 'ith Par!’s results. )ndeed( the final geometry implemented in the

7D simulations is the one depicted in figure 6.9G it consists of the equivalent sphere of

Apollo 6( 'hile the case is built as a4isymmetric. 5he boundary conditions are the same

for each case for the inflo'( outflo' and symmetry boundaries( 'hereas it changes on

the 'all surface( according to the case simulated each time. More specifically( the east

boundary of the domain changes from a slip*'all for an inviscid calculation to an energy*

balance surface for a viscous calculation and to an ablative boundary condition for an

ablation calculation.

h h

kg

kg

f f

jd

jd

!"!#$%&A'A"(! )*#+A(!

y,

-./

i -.0a x

A&'A123! 4A''

y,

-./

i -.0a x

)'2P

4A''

- -.5

-.6)'2P

4A''

- -.5 -.6

+a, .iscousFlow

+b, .iscous flow/ ablation included

Fi.!r$ 4 "loc!ing of the domain of the Apollo 6 fore*body for a viscous case( 'ithand 'ithout ablation

4.2.1 Pr$=pro&$##in.

5he viscous simulations are initiali=ed 'ith the converged inviscid case( 'hich is set up

'ith a coarser mesh( preventing the solution to diverge 'hen inserting the viscous effects.

0nce the flo' is 'ell established from the inviscid solution( the viscous effect is added

incrementally. 5he transport properties ,e.g. viscosity- are incremented at each time stepfor stability reasons as stated in the relation: µn+1 Eµn ×Fin# ( 'here Fin# is the incre*

i i

mental factor( 'hich for the present calculations 'as set to 9 · 9?−5 .

7 7


85/194

5he generated mesh for the inviscid and viscous cases for the 7nd tra#ectory point( 'hich

is also the pea! radiative heating point( is presented in figure 6.7. )t is 'orth mentioning

that for the viscous cases the mesh 'as not only refined at the boundary layer region(


86/194


87/194


88/194

08-0-- ,8-

-- 8- -

all ?@@)

Fi.!r$ 4 @elocity profile along the stagnation line( 57( investigation of ablation effects and diffusion model


89/194

0.-,=5

,.8

in9isid

9is:;s ('

9is:;s#(

,.-

a7lati:n ('

a7lati:n #(

-.8

-.- 08- 0-- ,8- ,-- 8- -

,.0

,=5

,.-

-.6

-./

-.5

-.0

in9isid 9is:;s ('

9is:;s #(

a7lati:n ('

a7lati:n

#(

all ?@@)

-.- 08- 0-- ,8- ,-- 8- -

all ?@@)

Fi.!r$ 4 5emperature profile along the stagnation line( 57( investigation of ablationeffects and diffusion model

5he species mass fractions’ profiles in the shoc! layer are presented in figure 6.3 for the

ablation case. 5he graphs are illustrated in a logarithmic scale. 5he calculation 'as

performed 'ith the 77*species gas model( presented in section &.>.9. As ablative species

have been in#ected in the gas mi4ture the composition of the gas mi4ture of air and ablativespecies has been highly modified. )n the region close to the 'all the main ablative species

is carbon and its mass fraction on the 'all is more than ten times higher than the follo'ing

highest mass fractions. 5he ablative mass fractions never become significant compared

to the air species mass fractions. )t should also be noticed that the ablative species are

present in an e4tremely small region close to the 'all surface ,≤ 96mm-( compared 'ith

the shoc!*standoff distance ,≈ 93cm-.

1 r a n s l a t i : n a l B # : t a t i : n a l

1 = @ C =

r a t ; r = ? D )

3 i 7 r a t i :

n a l B ! l = .

! l = t r : n i 1 = @ C = r a t ; r = ? D )


90/194

,--

,-B,

,-B0

,-BE

,-B5

,-B8

,-B/

,-BF

,-B6,6-

,/- ,5- ,0- ,-- 6- /- 5- 0-

G "

"0

"GG0

GHCl;s "HCl;s "GHCl;s

"0HCl;s =H@in;s

-

,--

,-B,

,-B0

,-BE

,-B5

,-B8

,-B/

,-BFB6

all?@@)

(I

(G(0

(" I0 (E (0I

(HCl;sIHCl;s

,-,5

,0 ,- 6 / 5 0 - all ?@@)

Fi.!r$ 4 Species mass fractions along the stagnation streamline( 57( ablative viscous

flo'

4.2.2.2 H$a/ 2"!3 &a"&!"a/ion#

%or each case( viscous 'ith or 'ithout the ablative boundary condition ta!en into consid*

eration( the heat flu4es 'ere calculated during the simulations. )n the follo'ing section

the calculated heat flu4es ,conductive and convective- are compared bet'een the cases

'ith or 'ithout ablation. 5'o diffusion models 'ere used and the results they gave are

also compared in this section.

%igure 6.> illustrates the conductive and diffusive heat flu4 participation in the heat flu4

field along the capsule’s 'all for a viscous case 'ithout ablation. As it is easily e4tracted(

the diffusive heat flu4 can be considered insignificant in this case.

%igure 6.8 illustrates the convective heat flu4 on the 'all of the Apollo 6 capsule( as

calculated 'ith the use of the amsha'*2hang and the 2onstant Le'is umber Diffusion

A t @ : s C h = r i s C = i = s

J a s s + r a t i : n s

A 7 l a t i 9 = s C = i =

s

J a s s + r a t i : n s


91/194

Models. )t is observed that the 2onstant Le'is umber diffusion model calculates a

conductive heat flu4 almost t'ice the value of the the one calculated 'ith the amsha'*

2hang model. As it 'ill be sho'n in section 6.7.6.9( the values of convective heating for


92/194

large nose radii are particularly small. 5hus( the amsha'*2hang Model is considered

more appropriate for this simulation case.

,-F

,-/

,-8

,-5

,-E

,-0

,-,

,--

-.- -.8 ,.- ,.8 0.- 0.8 E.- E.8 5.- 5.8 B(hang (:nstant '=>is

( : n 9 = t i 9 = I = a t + l ; x ? 4 @ 0 )


93/194

4.2.2.3 Raia/ion Ca"&!"a/ion#

5he viscous converged simulation is used as a starting point for the adiation calculations.

adiation calculations are also performed for the ablation cases( considering the 9? more

ablative species. 2alculations 'ere run assuming the KSS ,EKuasi Steady State- non*

equilibrium electronic level population model( 'hereas t'o different radiation transport

models 'ere investigated: the tangent slab and the Monte*2arlo ray*tracing method ,'ith

9> rays-. )t is 'orth noticing that the e&rad.e4e e4ecutable file that is used for the ay*

5racing Approach is not calculating the spectra in the case( #ust the radiative heat flu4

in the flo'*field. )n order for spectral data to be calculated for these cases( the solution

should be post*processed 'ith the tangent slab appro4imation. 5he radiation library used

for the radiation calculations is the photaura library implemented in $ilmer&.

5he spectral 'avelength range 'as considered bet'een >? ≤ λ≤ 6???nm( as most of the

e4istent data used the spectral range of ,?.?>*6 µm-. 5he radiators considered are the

follo'ing:

2( 0( ( ( 20( 22( 2( 2( 0( 02( 2( 0+( +( +( 0+( 2

+( e –

5he radiators considered in the KSS non*equilibrium electronic level population model are

the follo'ing:

2( 0( ( 20( 22( 0

2

)t is noticed that the species containing 'ere proved to be unstable for the KSS popu*

lation model and therefore 'ere not ta!en into consideration.

%igure 6.; illustrates the spectral flu4 and the integrated flu4 calculated 'ith the tangent

slab radiation transport method for the second tra#ectory point( 'hich is also the pea!

radiative point. )t is noticed that above 3? of the total integrated flu4 is reached in

the 'avelength range bet'een ,?.?>*?.7 µm-( 'hich is designated as the @H@ ,E@acuum*

Hltra*@iolet- range. o'ever( the spectral lines above 8?? nm have a valid contributionin the total spectra of the case. %igure 6.


94/194

6=5

0

F

/

8

5

E

0

-0-- 5-- /-- 6-- --- 0--

4a9=l=ngth ?n@)

+a, Sectra in wa0elength range 4*3$*** nm

E.8=/

0

E.-

0.8

0.-

.8

.-

-.8

-.- 0-- 5-- /-- 6-- --- 0-- 4a9=l=ngth ?n@)

+b, Integrated flu5 in wa0elength range 4*3$*** nm

Fi.!r$ 4 Spectral and integrated heat flu4( pea! radiative heating point( tangent slabradiation transport model

C = t r a l M ; x ? 4 @ 0 B @ )

2 n t = g r a t = d M ; x ? 4 @ 0 )


95/194

6,=,5

F

6 =E

0

F

/ /

8 8

5 5

E E

0 0

-/- 6- -- 0- 5- /- 6- 0--

00-

4a9=l=ngth ?n@)

-08- E-- E8- 5-- 58-

4a9=l=ngth ?n@)

+a, Sectra in wa0elength range4*3""* nm +b, Sectra in wa0elength range "!*3$6*

nm

6,=,E 0.8

=5

F

0.-

/

8.8

5

E.-

0

-.8

-58- 8-- 88- /-- /8-

F-- 4a9=l=ngth ?n@)

-.-F-- F0- F5- F/- F6- 6--

4a9=l=ngth ?n@)

+ , Sectra in wa0elength range$"*3(!* nm +d, Sectra in wa0elength range (**37!*

nm

0.8=5 .-

=5

0.--.6

.8-./

.--.5

-.8-.0

-.-6-- -.-

+ , Sectra in wa0elength range 7**3!"** nm

+f, ) Sectra in wa0elength range !***3$*** nm

Fi.!r$ 4 Spectral heat flu4( pea! radiative heating point( tangent slab radiation

transport model. Divided spectra into specific important regions of the 'avelength range

C = t r a l f l ; x ? 4 A

@ 0

)

C = t r a l f l ; x ? 4 A @

0

)

C = t r a l f l ; x ? 4

A

@ 0

)

C = t r a l f l ; x ? 4 A @

0 )

C = t r a l f l ; x ? 4 A @

0

)

C = t r a l f l ; x

? 4 A @

0

)

68- N-- N8- --- -8- -- 8- -- 8-- 0-- 08-- E--- E8-- 5--- 4a9=l=ngth ?n@) 4a9=l=ngth ?n@)


96/194

6=5

F

/

8

5

E

0

-0-- 5-- /-- 6-- --- 0--

4a9=l=ngth ?n@)

+a, Sectra in wa0elength range 4*3!"** nm

6=5 ,.0

,=,5

F.-

/

-.68

5 -./

E-.5

0

-.0

-/- 6- -- 0- 5- /- 6-

0--

4a9=l=ngth ?n@)

-.-0-- 5-- /-- 6-- --- 0--

4a9=l=ngth ?n@)

+b, Sectra in wa0elength range4*3"** nm + , Sectra in wa0elength range "**3

!"** nm

Fi.!r$ 4 Spectral heat flu4( pea! radiative heating point( tangent slab radiation transport model( ablation included

) C = t r a l f l ; x ? 4

A @

0

)

C = t r a l f l ; x ? 4 A @

0

)

C = t r a l f l ; x ? 4 A @

0

)


97/194

E.-=5 ,.0

,=,5

0

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