Universit`a degli studi della Basilicata

Transcript

Università degli studi della Basilicata
Dottorato di ricerca in
“Ingegneria Industriale e dell’Innovazione”
TITOLO DELLA TESI:
Numerical Simulation of Multi-dimensional
Hypersonic Plasma Flows
Settore Scientifico-Disciplinare
“Macchine a fluido (ING-IND/08) e
Fisica dei reattori nucleari (ING-IND/18)”
Coordinatore:
Prof. Vinicio Magi
Dottorando:
Ra↵aele Pepe
..........................................
Tutor:
Prof. Aldo Bonfiglioli
..........................................
Dr. Ing. Antonio D’Angola
..........................................
A.A. 2013/2014 Ciclo XXVII
....................................
Giudizio del collegio dei docenti
Il dott. Raffaele PEPE espone il proprio lavoro concernente “Numerical Simulation Of
Multi-Dimensional Hypersonic Plasma Flows”.
Al termine della presentazione il Collegio si riunisce per decidere sull’ammissione del
dottorando all’esame finale. Il relatore, Prof. Aldo Bonfiglioli illustra l’attività svolta dal
dottorando nel triennio.
Durante tutto lo svolgimento del dottorato di ricerca, sei mesi del quale trascorsi presso
il von Karman Institute for Fluid Dynamics, il dott. Raffaele PEPE ha dimostrato acume
e dedizione al lavoro di ricerca, predisposizione al confronto costruttivo, anche in ambiti
multidisciplinari, rigore metodologico ed una notevole capacità di approfondimento dei
problemi.
L’attività di ricerca, rapidamente inserita in un contesto internazionale di notevole
spessore, è stata correttamente impostata grazie ad un assiduo lavoro di ricerca
bibliografica ed è stata sorretta costantemente da un grande impegno ed una buona
capacità organizzativa.
Tale attività ha portato alla sottomissione di 3 articoli su rivista internazionale, di cui
uno attualmente in stampa, 4 articoli ed 1 poster presentati a conferenze internazionali
ed un rapporto tecnico.
I risultati sono interessanti ed analizzati con buon senso critico. Nell’esposizione il
dottorando dimostra di avere ottima conoscenza delle problematiche trattate.
Il Collegio dei Docenti, sentito il parere del tutor, valuta più che positivamente il lavoro
svolto e delibera l’ammissione del dottorando Raffaele PEPE all’esame finale.
"Our best ideas are often those that bridge between two different worlds"
Marvin Minsky
Acknowledgments
I wish to thank various people for their contribution to this thesis work.
I would like to thank my supervisors, Prof. Aldo Bonfiglioli and Dr.
Antonio D’Angola for their constant support and guidance in the development of this work. I am very grateful to them for encouraging me to
pursue my passion for research. During last three years, their dedication
and rigor in research work has been a constant example for me.
A special thank goes to Dr. Gianpiero Colonna, from IMIP-CNR and
Prof. Renato Paciorri, from University of Rome for providing me the
necessary insight on nonequilibrium plasma models and on the subject
of hypersonics. I would like to thank both of them for their uncountable
theoretical suggestions and the practical help.
Thanks to Dr. Andrea Lani, from the Von Karman Institute. He was enthusiastic about the opportunity to collaborate from the first email that
I sent to him. It was a real honor and pleasure for me to work with him
at the VKI.
I would like to thank Jesuś Garicano Mena, from VKI, for all the useful
discussions and the practical help on the use of COOLFluiD. I wish him
all the best for his Ph.D.
I acknowledge Prof. Herman Deconinck, from VKI, for having accepted
my request to spend a research period at the VKI as visiting Ph.D. student.
i
ii
Acknowledgment
Thanks to Renato, Pasquale, Marianna and Michele who shared the office
with me during these years. I wish them all the best.
Thanks to my bandmates Antonio, Claudio, Daniele and Donato for all
the good vibes that music has given to me.
Thanks to all my friends: Francesco, Antonio S. senior, Claudio, Lina,
Cosimo, Antonio G., Dino, Donatella, Stefano, Anna, Angelica, Pasquale,
Rocco, Alessandro, Antonio S. junior, Giunio, Maurizio...etc.
Last but not least I would like to thank my family: my father Giuseppe,
my mother Agnese, my brother Donato and my sister Mariateresa. Thank
you for trusting in me.
Abstract
In the present thesis, the capabilities of eulfs, an unstructured 2D/3D
solver developed for thermally and calorically perfect gas, have been extended making it capable to deal with chemical nonequilibrium plasma
flows. Preliminary tests have been carried out for an ionized argon mixture flowing in a converging-diverging nozzle. 2D and 3D results obtained
by using the extendend version of the eulfs code have been compared
with those obtained with a well established quasi-unidimensional code
developed at the IMIP-CNR of Bari, showing a good agreement. The
CFD code can be coupled with a newly developed, unstructured, shockfitting algorithm which treats the discontinuities as moving boundaries
that border regions of the flow-field were a smooth solution to the governing PDEs exists. The unstructured shock-fitting algorithm has been
extended to deal with an ionized argon mixture to model shock waves
in chemical reacting flows. Promising results have been obtained using
the shock-fitting approach for a 2D hypersonic flow past the fore-body
of a circular cylinder. The unstructured shock-fitting algorithm has been
extended to deal with thermochemical nonequilibrium flows and thanks
to its modurality, has been coupled with COOLFluiD, an in-house shockcapturing CFD solver developed at the Von Karman Institute. Results
obtained in the computation of hypersonic flows past circular cylinders
have been obtained for both ideal gas and dissociated Nitrogen in thermochemical nonequilibrium.
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Contents
Acknowledgment
i
Abstract
iii
List of Figures
ix
List of Tables
xiii
List of Symbols
xv
1 Introduction
1
2 Physical model
5
2.1
2.2
2.3
Governing equations . . . . . . . . . . . . . . . . . . . . .
6
2.1.1
Mixture parameters . . . . . . . . . . . . . . . . . .
6
2.1.2
Equation of state . . . . . . . . . . . . . . . . . . .
7
2.1.3
Thermodynamic model . . . . . . . . . . . . . . . .
8
2.1.4
Conservation equations . . . . . . . . . . . . . . . . 13
Chemical models . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1
Kinetic model for an ionized argon mixture . . . . . 17
2.2.2
Kinetic model for a dissociated nitrogen mixture . . 21
Electrohydrodynamic model . . . . . . . . . . . . . . . . . 22
3 Computational tools
27
v
vi
Table of contents
3.1
Shock-capturing solver . . . . . . . . . . . . . . . . . . . . 28
3.1.1
Fluctuation and conservative linearization . . . . . 29
3.1.2
Signals or Residual Distribution . . . . . . . . . . . 32
3.1.3
Solution of the discretised equations . . . . . . . . . 34
3.2
Shock-fitting algorithm . . . . . . . . . . . . . . . . . . . . 36
3.3
Details on the implementation . . . . . . . . . . . . . . . . 40
4 Numerical results
4.1
4.2
4.3
43
eulfs results . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.1
Ionized argon, inviscid flow in a nozzle . . . . . . . 43
4.1.2
Ionized argon, inviscid flow over a circular cylinder
48
COOLFluiD results . . . . . . . . . . . . . . . . . . . . . . 51
4.2.1
Ideal gas, inviscid flow over a circular cylinder . . . 51
4.2.2
Ideal gas, viscous flow over a circular cylinder . . . 54
4.2.3
Dissociated nitrogen, inviscid flow over a circular
cylinder . . . . . . . . . . . . . . . . . . . . . . . . 60
EHD results . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1
Parallel electrodes . . . . . . . . . . . . . . . . . . . 62
4.3.2
Ionized argon, inviscid flow in a nozzle with two
opposite electrodes . . . . . . . . . . . . . . . . . . 65
5 Conclusions and future work
67
A Appendix: Split Jacobian matrix
71
A.1 Jacobian transformations . . . . . . . . . . . . . . . . . . . 75
A.2 Transformations between U and Z . . . . . . . . . . . . . 77
A.2.1 U ! Z . . . . . . . . . . . . . . . . . . . . . . . . 77
A.2.2 Z ! U . . . . . . . . . . . . . . . . . . . . . . . . 77
B Appendix: Nondimensionalization
79
B.1 TCneq, external flows . . . . . . . . . . . . . . . . . . . . . 79
Table of contents
vii
B.2 TCneq, internal flows . . . . . . . . . . . . . . . . . . . . . 81
B.3 EHD, external flows . . . . . . . . . . . . . . . . . . . . . . 82
B.4 EHD, internal flows . . . . . . . . . . . . . . . . . . . . . . 83
B.5 Nondimensional parameter vector . . . . . . . . . . . . . . 84
B.6 Nondimensional pressure derivatives . . . . . . . . . . . . . 85
C Appendix: Rate coefficients fits
87
C.1 Rate coefficients fit . . . . . . . . . . . . . . . . . . . . . . 87
References
91
viii
List of Figures
1.1
Schematic of the flowfield surrounding a space vehicle during the reentry phase (reproduced from [60]). . . . . . . . .
2
2.1
Ionization and recombination rates from the ground level
as a function of the electron temperature (courtesy of G.
Colonna) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2
Electrical conductivity of argon (p = 0.013atm), reprinted
from [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3
Scheme of the coupled circuit . . . . . . . . . . . . . . . . 26
3.1
Residual distribution concept. . . . . . . . . . . . . . . . . 28
3.2
Inward scaled normal. . . . . . . . . . . . . . . . . . . . . 33
3.3
Starting point: a) background mesh and b) shock boundary (right). . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4
a) Cell removal. b) Local remeshing. . . . . . . . . . . . . 38
3.5
a) Normal and tangential unit vector computation. b) Interpolation of the phantom nodes. . . . . . . . . . . . . . . 39
3.6
Interpolation of mesh point jumped by the shock boundary. 39
3.7
Modularity of the shock-fitting code. . . . . . . . . . . . . 41
4.1
Flow in a converging-diverging nozzle: 2D and 3D geometries flooded by Mach number. . . . . . . . . . . . . . . . . 44
4.2
Test A: distributions along the nozzle axis. . . . . . . . . . 45
4.3
Test B: distributions along the nozzle axis. . . . . . . . . . 46
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Table of contents
4.4
Flow in a converging-diverging nozzle: convergence histories for test B. . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5
Baseline mesh and “fitted” mesh. . . . . . . . . . . . . . . 48
4.6
2D flow past a circular cylinder: comparison between the
shock-capturing (S-C) and shock-fitting (S-F) solutions.
Both S-C and S-F solutions are first order accurate in space
(N scheme). . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.7
2D flow past a circular cylinder: comparison of shockcapturing (S-C) and shock-fitting (S-F) solutions along y = 0. 50
4.8
Grids used for the inviscid flow over a circular cylinder. . . 52
4.9
Pressure contour plot: comparison between the shock-fitting
(S-F) and shock-capturing (S-C) second-order-accurate solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.10 Pressure contour plot for the shock fitting solutions: comparison between COOLFluiD and eulfs solutions. . . . . . 54
4.11 Grid used for the viscous flow over a 1 m radius circular
cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.12 Pressure and temperature contour plot: comparison between the shock-fitting and shock-capturing second-orderaccurate accurate solutions. . . . . . . . . . . . . . . . . . 57
4.13 Wall distributions. . . . . . . . . . . . . . . . . . . . . . . 57
4.14 Heat flux distribution on the wall. . . . . . . . . . . . . . . 58
4.15 Distributions along the stagnation streamline. . . . . . . . 58
4.16 Distributions along the stagnation streamline (enlargement
of the boundary layer). . . . . . . . . . . . . . . . . . . . . 59
4.17 Adimensional total enthalpy. . . . . . . . . . . . . . . . . . 59
4.18 Grid used for the Nitrogen flow over a circular cylinder.
. 60
4.19 COOLFluiD + SF vs. Hornung’s experimental measurements: a) Non-dimensional shock-wall distance and b) finite interference fringe patterns. ✓ is the azimuthal angle
which takes value 0 at the stagnation point, /R is the
shock-wall distance divided by the cilinder’s radius. . . . . 61
Table of contents
4.20 Pressure contour plot: qualitative comparison between the
fitted solution obtained with COOLFluiDand the solution
obtained by Wang and Zhong [76]. Hornung’s experimental measurements of the shock stando↵ distance are represented by the red circles. . . . . . . . . . . . . . . . . . . 62
4.21 Electric potential contours for the electrode channel with a
constant electrical conductivity ( = 1 ⌦ 1 m 1 ): a) eulfs
results and b) reprinted from [32]. . . . . . . . . . . . . . . 63
4.22 Parallel electrodes with a non constant electrical conductivity ( = ): a) mesh b) boundary conditions. . . . . . . 64
4.23 Parallel electrodes with a non constant electrical conductivity ( = ): a) electric potential contour plot (numerical
vs. analytitcal solution) b) distribution of the electric potential along the y-axis (numerical vs. analytitcal solutions). 65
4.24 Flow in a converging nozzle with two parallel electrodes. . 66
4.25 Temperature distributions along the nozzle axis for di↵erent values of the generator potential. . . . . . . . . . . . . 66
C.1 Relative error associated to the e A forward rate coefficient vs. the translationa l temperature . . . . . . . . . . 89
xi
xii
List of Tables
2.1
Chemical species in the ionized argon mixture . . . . . . . 18
2.2
Chemical processes in the ionized argon mixture . . . . . . 18
2.3
Equilibrium constants for all processes accounted for in the
present model . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4
Chemical species in the dissociated nitrogen mixture . . . 22
2.5
Chemical processes in the dissociated nitrogen mixture . . 22
4.1
Area distribution along the nozzle. . . . . . . . . . . . . . 45
4.2
Inlet flow conditions. . . . . . . . . . . . . . . . . . . . . . 45
4.3
Freestream conditions for an ionized argon flow over a
0.05m radius cylinder. . . . . . . . . . . . . . . . . . . . . 49
4.4
Characteristics of the grids used for the inviscid hypersonic
flow over a circular cylinder. . . . . . . . . . . . . . . . . . 52
4.5
Freestream conditions for a viscous flow over a 1m radius
cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6
Freestream conditions for a nitrogen flow around a 1 inch
radius cylinder. . . . . . . . . . . . . . . . . . . . . . . . . 60
4.7
Boundary conditions for the EHD solver.
4.8
Inlet flow conditions. . . . . . . . . . . . . . . . . . . . . . 65
. . . . . . . . . 63
C.1 Coefficients of function f1 = log KeqI (cm 3 ) for the Ar
ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
C.2 Coefficients of function f2 = log kf1 (cm 3 ) for the reaction
1 (e A ionization from Ar0 ) . . . . . . . . . . . . . . . . 88
xiii
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Table of contents
2 (e A excitation) . . . . . . . . . . . . . . . . . . . . . . 88
1 (e A ionization from Ar⇤ ) . . . . . . . . . . . . . . . . 89
C.5 Rate coefficients kf (cm3 /s) for A-A processes . . . . . . . 89
List of Symbols
Acronyms
B
Blended scheme
CDT
Constrained Delaunay Triangulation
CFD
Computational Fluid Dynamics
Cneq
Chemical nonequilibrium
CRD
Contour Residual Distribution
EEDF Electron Energy Distribution Function
EHD
ElectroHydroDynamics
EOS
Equation Of State
FE
Finite Element
FS
Fluctuation Splitting
FV
Finite Volume
I/O
Input/Output
LDA
Low Di↵usion Advection scheme
LRD
Linear Residual Distribution
MHD MagnetoHydroDynamics
N
Narrow scheme
ODE
Ordinary Di↵erential Equation
PDE
Partial Di↵erential Equation
RD
Residual Distribution
R-H
Rankine-Hugoniot relations
S-C
Shock-Capturing
S-F
Shock-Fitting
TCneq Thermochemical nonequilibrium
VKI
von Karman Institute
xv
xvi
List of Symbols
Roman symbols
a
speed of sound
B
magnetic field vector
Cp specific heat at constant pressure
Cv specific heat at constant volume
Da Damköler number
e
internal energy per unit mass
ee
internal energy per unit mass of free electrons
i
e
internal contribution to internal energy per unit mass
e
e
electronic contribution to internal energy per unit mass
r
e
rotational contribution to internal energy per unit mass
t
e
translational contribution to internal energy per unit mass
v
e
vibrational contribution to internal energy per unit mass
E
total energy per unit mass
E
electric field vector
fe
electron energy distribution function
FB magnetohydrodynamic force
g
statistical weight
h
enthalpy per unit mass
f
h
formation enthalpy per unit mass
H
total enthalpy per unit mass
kb
backward rate constant
kf
forward rate constant
Keq equilibrium constant
j
current density vector
Js di↵usion flux of the species s
I
current intensity
L
reference lenght
mr reduced mass
M Mach number
M molar mass
n
normal unit vector
n
molar density
N
particles number density
m/s
T
J/(kg K)
J/(kg K)
J/kg
J/kg
J/kg
J/kg
J/kg
J/kg
J/kg
J/kg
V/m
N
J/kg
J/kg
J/kg
A/m2
kg/(m2 s)
A
m
kg
kg/mol
mol/m3
m 3
List of Symbols
Nr
Ns
p
pe
q
qv
Qs
QJ
r
Rc
Ṡev
Ṡs
t
t
T
Te
TE
TR
TV
Tv
u
x
Vdis
Vq
Wsh
number of chemical reactions
number of chemical species
gas pressure
electronic pressure
global heat flux
vibrational heat flux
partition function of species s
energy production rate due to the Joule e↵ect
radius
circuit resistance
rate of production of the vibrational energy
rate of production of the sth species
tangential unit vector
time
traslational temperature
electron temperature
electronic temperature
rotational temperature
vibrational temperature
vibro-electronic temperature
fluid velocity vector (u, v, w)
position vector (x, y, z)
potential between the electrodes
potential of the generator
shock wave velocity
xvii
Pa
Pa
W/m2
W/m2
W/m3
m
⌦
W/m3
kg/(m3 s)
s
K
K
K
K
K
K
m/s
m
V
V
m/s
xviii
List of Symbols
Greek symbols
↵
mass concentration
↵ion inization degree
✏
internal energy per unit volume
"
electron energy
isentropic coefficient
shock-wall distance
electric potential
residual
roto-translational thermal conductivity
v
vibrational thermal conductivity
µ
viscosity
0
⌫
stochiometric coefficients for the products
00
⌫
stochiometric coefficients for the reagents
⌦
element area
⇠˙
chemical reaction velocity
⇢
density
electric conductivity
e
electric conductivity tensor
&
cross section
⌧
stress tensor
⌧s
relaxation time of the sth species
⇥
azimuthal angle
J/m3
eV
m
V
W/(kg K)
W/(kg K)
kg/(m s)
m2
mol/(m3 s)
kg/m3
⌦/m
⌦/m
m2
Pa
s
List of Symbols
Subc
d
down
e
i
s
r
up
sh
w
1
⇤
xix
and Superscripts
convective fluxes
di↵usive fluxes
downstream states
eth element
ith node
sth chemical species
rth chemical reaction
upstream states
shock wave
wall
freestream conditions
adimensional variables
Constants
ec elementar charge
hP Planck constant
kB Boltzmann constant
me electron mass
NA Avogadro’s number
Rg universal gas constant
µ0 permeability of the free space
Species
Ar0 argon in the ground state
Ar⇤ argon in the metastable state
Ar+ argon ion
e
electron
N2 molecular nitrogen
N
atomic nitrogen
1.6022 ⇥ 10 19 C
1.3807 ⇥ 10 23 J/K
1.3807 ⇥ 10 23 J/K
9.11 ⇥ 10 31 kg
6.022 ⇥ 10 23 mol 1
8.314 J/(mol K)
4⇡ ⇥ 10 7 N/A2
xx
Chapter 1
Introduction
The accurate simulation of hypersonic flows past blunt bodies is still a
challenge, despite more than 20 years of algorithmic developments on
CFD solvers. In hypersonic conditions, the flowfield surrounding the
blunt bodies is characterized by several complex physico-chemical phenomena such as chemical reactions, thermal relaxation, ablation of the
surface, radiation, strong bow shocks, shock-shock and shock-boundary
layer interaction. Fig. 1.1 illustartes a schematic of the flowfield surrounding a capsule during the reentry phase in atmosphere. A fundamental problem is the description of the ionization and dissociation kinetics,
and, more in general, the chemical processes occurring in the flowfield past
the bow shock. A primary role is played by excited states of atoms and
molecules [19]. Excited levels, reduce the activation energy of exothermal
processes speeding up the reaction processes. An important role is played
by metastable states, that, because cannot decay by radiation emission,
survive for long time, with a large influence on the flow properties.
Another critical aspect is the accurate description of discontinuities, and
in particular the descripition of strong shock waves, like the bow shock
ahead of the blunt body. To simulate hypersonic flows, widespread “traditional” shock-capturing solvers often exhibit severe drawbacks, expecially when used on unstructured grids: stagnation point anomalies [39],
carbuncle phenomenon and spurious oscillations [45] and a reduction of
the order of accuracy within the entire shock-downstream region [18].
These drawbacks, defined as ”shock anomalies” [45] seem to be caused
by numerical details of the capturing process, since numerically captured
1
2
Introduction
Figure 1.1: Schematic of the flowfield surrounding a space vehicle during the reentry
phase (reproduced from [60]).
shock usually contains at least one computational cell forming numerical internal shock-structure, which is a pure numerical ”artefact” and
is not related to the real physical internal structure of the shock wave
[45, 69]. Despite unstructured codes are less e↵ective and accurate of
structured ones in the simulation of hypersonic regimes, various unstructured CFD tools used in the aero-thermodynamic design and analysis of
space vehicles entering planetary atmospheres have been developed, since
unstructured grids o↵er greater flexibility than structured ones in tackling complex geometries allowing to automatically adapt the mesh to the
local flow features. NASA’s FUN3D [3, 4], DLR’s TAU [33, 52] and LeMANS [66, 67] codes are three such examples of unstructured codes used
in hypersonic reentry applications.
The shock-fitting approach, which has been made popular since the mid
60s by Moretti and collaborators [55], has already proved to be immune
to the shock-capturing drawbacks. Shock-fitting consists in using the
Rankine-Hugoniot jump relations to explicitely track the motion of the
discontinuities in a Lagrangian manner. Thanks to its ability of accurately simulating shocks on coarse grids, shock-fitting was very popular
in the early computer era. With increasing computer power and because
of some algorithmic difficulties that plagued the shock-fitting approach on
structured grids, shock-capturing took over and is nowadays the method
Introduction
of choice for virtually all CFD simulations. Shock-fitting discretizations
based on the so-called “boundary” variant have been in use until the mid
90s [53, 54] to simulate supersonic and hypersonic flows: only the strong
bow shock was fitted and made to coincide with the upstream boundary of
a structured mesh; all other shocks were captured. The “floating” variant
of the shock-fitting technique, although more versatile since it allows to
fit also the embedded shocks, is algorithmically complex, so that only few
three-dimensional calculations have been reported in the literature [77].
Recent advances in unstructured grid generation and discretization techniques has allowed to develop an unstructured shock-fitting algorithm
[57] which is algorithmically simpler than the shock-fitting algorithms traditionally used in the structured-grid setting. This unstructured version
of the shock-fitting technique combines features of both the “boundary”
and “floating” variants proposed in the structured grid setting: it therefore allows not only to fit the bow shock, but also the embedded shocks.
Moreover, the geometrical flexibility o↵ered by the use of unstructured
triangular and tetrahedral meshes allows to deal much more properly
with interacting shock [42, 58] than it was possible in the structured-grid
context. Shock-fitting algorithms have been used in the past to simulate chemically reacting nonequilibrium flows on structured grids, see e.g.
the work by Pfitzner [63] and Paciorri et. al. [59] in the 90s. More recently, Prakash et al. [65] have developed an high order finite di↵erence
shock-fitting algorithm to simulate thermochemical nonequilibrium flows
on structured grids.
In the present thesis, the capabilities of eulfs [10, 13], an unstructured
2D/3D solver developed for thermally and calorically perfect gas, have
been extended making it capable to deal with chemical nonequilibrium
plasma flows. Tests have been carried out for an ionized argon mixture, including also the argon metastable state, flowing in a converging-diverging
nozzle. 2D and 3D results obtained by using the extendend version of
the eulfs code have been compared with those obtained with a quasiunidimensional code developed at the IMIP-CNR of Bari [20, 21, 24]. In
the present work the unstructured shock-fitting algorithm developed by
Paciorri and Bonfiglioli [57] has been extended to deal with an ionized
argon mixture to model shock waves in chemically reacting flows. Promising results have been obtained using the shock-fitting approach for a 2D
hypersonic flow past the fore-body of a circular cylinder.
3
4
Introduction
Thanks to its modularity, the unstructured shock fitting algorithm has
been coupled with COOLFluiD [50, 51], an in-house CFD solver developed at the Von Karman Institute (VKI), to investigate the causes of
anomalous heat flux distributions on the wall of blunt bodies immersed in
hypersonic flows. Moreover the shock-fitting algorithm has been extended
to deal with two temperature, multispecies thermochemical nonequilibrium flows and results have been obtained for an hypersonic dissociating
nitrogen flow past the forebody of a circular cylinder.
Finally a simple electrohydrodynamic model have been implemented within
eulfs. In this model the ionized gas flow is coupled with an electric
field controlled by considering a power supply and an external circuit resistence. Poisson’s equation and the Ohm’s law for the external circuit
have been added to the set of multispecies Euler equations. Preliminary
results an ionized argon mixture flowing in a subsonic nozzle with two
opposite electrodes, showing the e↵ect on the flowfield variables of the
ohmic heating due to the electric field.
The physical models used to model both viscous and inviscid flows in thermochemical nonequilibrium are described in Chapter 2. The computational tools used in the dissertation are described in Chapter 3. The main
features of the capturing solver used in conjunction with the shock-fitting
technique are provided in Section 3.1, while the unstructured shock-fitting
algorithm is described in details in the Section 3.2. Numerical results obtained using both codes, eulfs and COOLFluiD, are shown and discussed
in the Chapter 4. Finally conclusions on the present work and future work
are discussed in Chapter 5
Chapter 2
Physical model
When dealing with hypersonic flows, the perfect gas model can no longer
be considered, because of the very high temperatures reached after the
strong shock wave. The flowfield in the shock layer is characterized by
complex physiscal phenomena such as: chemical reactions, rotational,
vibrational and electronic excitation, ablation, radiative heat-flux and
viscous interactions. Under those conditions, thermodynamic properties
as the specific heat, and transport properties, as the viscosity and thermal
condictivity are not constant and vary with temperature, pressure and
chemical composition.
Concerning chemical reactions and internal energy relaxation, it is possible to define di↵erent regimes: frozen flow, equilibrium flow and nonequilibrium flow. The bounds of those regimes can be defined considering an
important nondimensional parameter, the Damköler number. The chemical Damköler number is defined as the ratio between the characteristic
time for the macroscopic processes occurring in the flow ⌧f , such as convective or di↵usive phenomena, and the chemical characteristic time ⌧c
for chemical processes:
Dac =
⌧f
⌧c
Then the three regimes are defined as follows:
5
(2.1)
6
Chapter 1. Physical model
Dac >> 1
Dac ⇡ 1
Dac << 1
Equilibriumf low
N on equilibriumf low
F rozenf low
Defining a characteristic time ⌧t for the energy relaxation, it is possible to
⌧
define another Damköler number Dav = ⌧ft , which can be used to define
regime of thermal equilibrium, nonequilibrium and freezing.
In the following sections of this chapter we will present the set of governing
equations used in this work to model nonequilibrium flows.
2.1
2.1.1
Governing equations
Mixture parameters
In this section we introduce the physical quantities which characterize
the composition of a mixture. There are di↵erent ways to define the
density of the sth chemical species and the global density of the mixture.
Considering the species mass density ⇢s , the mixture mass density ⇢ is
given by:
⇢=
Ns
X
⇢s
(2.2)
s=1
where Ns is the number of chemical species present in the mixture. The
number of the moles of the species s, ns is given by the ratio of the species
mass density and the molar mass Ms
ns =
⇢s
Ms
(2.3)
then the total number of moles n is given by the sum of the species
number moles:
Ns
X
n=
ns
(2.4)
s=1
7
The particles number density of the species s, Ns is easly obtained as
Ns = ns NA where NA is the Avogadro’s number. Total number density
is trivially given by Eq. (2.5):
N =
Ns
X
s=1
Ns
(2.5)
In Eq. (2.6) we define respectively the mass fraction ↵s and the molar
fraction s of the species s:
↵s =
⇢s
,
⇢
s
=
ns
Ns
=
n
N
(2.6)
Considering Eq. (2.2), (2.4) and (2.6) we obtain the following constraint
for the mass and the molar fraction:
Ns
X
↵s = 1,
s=1
2.1.2
Ns
X
s
=1
(2.7)
s=1
Equation of state
At high temperature the e↵ects of the intermolecular forces can be neglected, so that it is possible to consider a mixture of thermally perfect
gases [2]. The equation of state (EOS ) for the chemical species s is given
by Eq. (2.8)
p s = ⇢s
Rg
Ts
Ms
(2.8)
where ps and Ts are respectively the pressure and the kinetic temperature
of the species s. If the fluid is assumed to be sufficiently collisional, the
species rapidly thermalize with each other and it is possible to define
a single temperature T for all the particles, Ts = T . On the contrary,
in those circumstances in which ionization is high, the energy exchange
between electrons and heavy particles is too slow, so that the free electrons
are characterized by an electron temperature Te 6= T . The partial pressure
of the electrons is given by the following equation:
8
p e = ⇢e
Rg
Te
Me
(2.9)
The EOS of the mixture can be easly obtained from the Dalton’s law
which allows us to compute the pressure of the mixture p
p=
Ns
X
ps =
s=1
X
s6=e
⇢s
Rg
Rg
T + ⇢e
Te
Ms
Me
(2.10)
Finally if the electron are in thermal equilibrium with heavy particles the
pressure of the mixture is given by Eq. (2.11)
p=
Ns
X
s=1
where R =
2.1.3
P Ns
Rg
s=1 ↵s Ms .
⇢s
Rg
T = ⇢RT
Ms
(2.11)
Thermodynamic model
For high temperature flows it is not possible to neglect the electronic,
vibrational and rotational energy excitation, which implies that it is not
any longer possible to assume that the gas is calorically perfect. The
internal energy of each species can be expressed as the sum of a translational contribution ets , a contribution due to the excitation of the internal
energy modes eis and the formation enthalpy hfs [73]:
es = ets + eis + hfs .
(2.12)
If we use the rigid rotator and harmonic oscillator model to describe
rotational and vibrational motion of the molecules we can completely
separate the internal energy modes into three contributions: rotational
(r), electronic (e) and vibrational (v):
eis = ers + ees + evs .
(2.13)
Concerning the atomic species, the rotational and vibrational energy is
zero, so that the internal energy is only due to the electronic energy, which
9
is related to the electronic excitation. Electrons do not have an internal
structure, so their electronic energy is zero. We obtain the following
expressions for the internal energy per unit mass:
es = ets + ees + evs + ers + hfs
es = ets + ees + hfs Atoms
ee = ete + hfe Electrons
Molecules
(2.14a)
(2.14b)
(2.14c)
The expressions for the rotational, vibrational and electronic energy contributions can be obtained using the definition of the partition function
as described in the quasi-classical statistical mechanics [14, 46, 73].
Assuming that each internal mode follows a Boltzmann-Maxwell distribution, it is possible to define a single temperature for each degree of
freedom. These kind of models are known as multitemperature models
[37, 61] For completeness, we provide the expressions of vibrational, rotational and electronic energy e and enthalpy h as given by the multitemperature models.
Translational energy and enthalpy
Since the translational energy is assumed to be completely excited the
translational energy and enthalpy per unit mass are:
3
ets = Rs T,
2
(2.15)
hts = ets + Rs T,
(2.16)
If the free electrons are not in thermal equilibrium with the heavy particles, the translational energy and enthalpy of the electrons depends upon
the electrons temperature Te :
3
ete = Re Te ,
2
(2.17)
hte = ete + Re Te ,
(2.18)
10
t
The translational specific heat at constant volume Cv,s
and the specific
t
heat at constant pressure Cp,s are given by Eq. (2.19):
3
t
Cv,s
= Rs ,
2
5
t
Cp,s
= Rs
2
(2.19)
Rotational energy and enthalpy
Considering the molecules as a rigid rotor, the rotational energy and
enthalpy per unit mass are:
ers = hrs = Rs TR
(2.20)
where TR is the rotational temperature. The rotational specific heat at
r
r
constant volume Cv,s
and the specific heat at constant pressure Cp,s
are
given by Eq. (2.21):
r
r
Cv,s
= Cp,s
= Rs
(2.21)
Vibrational energy and enthalpy
Considering the molecules as harmonic oscillators, the vibrational energy
and enthalpy per unit mass are:
evs
=
hvs
=
exp
Rs ⇥vs
⇣ v⌘
(2.22)
⇥s
TV
1
where TV is the vibrational temperature and ⇥vs is the characteristic vibrational temperature of the species s. The rotational specific heat at
v
v
constant volume Cv,s
and the specific heat at constant pressure Cp,s
are
given by Eq. (2.23)
(⇥vs /TV )2 exp
v
v
Cv,s
= Cp,s
= Rs h
exp
⇣
⇥vs
TV
⌘
⇣
⌘
⇥vs
TV
i2
1
(2.23)
11
Electronic energy and enthalpy
The electronic energy and enthalpy per unit mass are given by Eq. (2.24):
ees
=
hes
= Rs
P1
e
l=1 gs,l ⇥s,l
P1
l=1 gs,l
exp
⇣
exp
⇣
⇥es,l
TE
⇥es,l
TE
⌘
⌘
(2.24)
where TE is the electronic temperature, gs,l is the degeneracy of the electronic level l of the species s and ⇥es,l is its characteristic temperature.
e
The electronic specific heat at constant volume Cv,s
and the specific heat
e
at constant pressure Cp,s are given by Eq. (2.25)
e
e
Cv,s
= Cp,s
=
where:
@ees
@TE
(2.25)
P1
2
l=1 gs,l ⌧s,l exp ( ⌧s,l )
= Rs P 1
@TE
g exp ( ⌧s,l )
⇥P1l=1 s,l e
⇤ P1
l=1 gs,l ⇥s,l exp ( ⌧s,l ) [
l=1 gs,l ⌧s,l exp ( ⌧s,l )]
Rs
P
2
TE [ 1
l=1 gs,l exp ( ⌧s,l )]
@ees
where ⌧s,l = ⇥es,l /TE
Considering the rotational energy fully excited, the rotational temperature can be set equal to the translational temperature TR = T . Following
[61] we can assume that the electronic temperature is in thermal equilibrium with the electron temperature TE = Te .
A 3-temperature model is then obtained:
5
Rs T + ees (Te ) + evs (TV ) + hfs
2
3
= Rs T + ees (Te ) + hfs Atoms
2
3
= Rs Te + hfe Electrons
2
es =
es
es
Molecules
(2.26a)
(2.26b)
(2.26c)
12
If all the species of the mixture are neutrals, further simplifications can
be introduced considering the 2-temperature model [37, 61, 62, 64], in
which the electronic temperature is equal to the vibrational temperature
TE = TV = Tv :
5
Rs T + ees (Tv ) + evs (Tv ) + hfs
2
3
= Rs T + ees (Tv ) + hfs Atoms
2
es =
es
Molecules
(2.27a)
(2.27b)
Finally if the flow is in thermal equilibrium it is possible to consider a
single temperature model Tve = T .
The mixture energy and enthalpy are given by:
e=
Ns
X
↵s es ,
h=
s=1
Ns
X
↵s hs
(2.28)
s=1
State-to-state model
In a state-to-state model, each excited internal state is convected as a
single chemical species, which implies that there are no electronic and vibrational contributions to the internal energy of each individual chemical
species [26, 43, 44]. For the k th excited level of the sth chemical species,
the internal energy is given only by the sum of the roto-translational and
formation energies:
5
Rs T + hfs,k
2
3
= Rs T + hfs,k
2
es,k =
Molecules
(2.29a)
es,k
Atoms
(2.29b)
There are no electronic or vibrational contributions to internal energy
of the single excited level. Therefore in each point in space, the total
electronic and vibrational energy for the individual chemical species can
be obtained by summing up the formation enthalpy of each excited level,
multiplied by the corresponding mass concentration:
ees
=
Ls,e
X
13
↵s,k hfs,k ,
evs
=
k=1
Ls,v
X
↵s,k hfs,k ,
(2.30)
k=1
where Ls,e and Ls,v are the electronic, resp. vibrational number of levels
of the sth chemical species.
2.1.4
Conservation equations
In this work we have considered multidimensional reacting flows under
the conditions of thermal equilbrium, with a single temperature, and
in thermal nonequilibrium with two di↵erent temperatures. Hereafter
we denote with thermochemical nonequilibrium model (TCneq) the 2temperature model and with chemical nonequilibrium model (Cneq) the
1-temperature model .
Conservation equation for the thermochemical nonequilbrium model are
given by the continuity equations for each chemical species, the momentum equation, the total energy equation and the vibrational energy equation [37].
Species continuity equations
The species continuity equations reads:
@⇢s
+ r · (⇢s u) = S⇢s
@t
r · Js
(2.31)
where Js is the di↵usion flux associated to the chemical species s and S⇢s
is the mass production term given by the Law of Mass Action Eq. (2.32):
S ⇢s = M s
Nr
X
00
⌫sr
0
⌫sr ⇠˙r ,
(2.32)
r=1
where ⇠˙r is the velocity for the rth chemical reaction:
◆⌫sr
Ns ✓
Y
⇢
s
⇠˙r = kf r
Ms
s=1
0
◆⌫sr
Ns ✓
Y
⇢s
kbr
.
M
s
s=1
00
(2.33)
14
In Eqs. (2.32 - 2.33) Ns is the number of chemical species, Nr the number
of chemical reactions, kf r and kbr resp. the forward and backward reaction
0
00
rates and ⌫sr and ⌫sr resp. the stoichiometric coefficients of the products
and the reagents.
The global continuity equation is retrieved summing up all the species
continuity equations:
@⇢
+ r · (⇢u) = 0
@t
(2.34)
since:
Ns
X
S⇢s = 0,
s=1
Ns
X
Js = 0.
(2.35)
s=1
Momentum equations
The momentum conservation equation is given by the Navier-Stokes equation:
@⇢u
+ r · (⇢uu) + rp = r · ⌧
(2.36)
@t
where ⌧ is the viscous stress tensor. Under the hypothesis of negligible
bulk viscosity e↵ects the viscous stress tensor is given by:

⌧ = µ ru + (ru)t
2
r·u
3
(2.37)
where µ is the mixture coefficient of viscosity.
Total energy equation
The total energy conservation equation is given by:
@⇢E
+ r · (⇢Hu) = r · (⌧ · u)
@t
r·q
(2.38)
where E = e + u · u/2 is the total internal energy per unit mass and H
15
is the total enthalpy expressed as H = E + p/⇢.
In Eq. (2.38) q is the global heat flux, given by:
q=
where and
ductivity.
v
v
rT
Ns
X
rTv
Js hfs
(2.39)
s=1
are the roto-translational and vibrational thermal con-
Vibrational energy equation
The vibrational energy equation
PNs v express the conservation of the mixture
v
vibrational energy e = s=1 es .
@⇢ev
+ r · (⇢ev u) =
@t
r · q v + Se v
(2.40)
v
In Eq. (2.40) qv =
rTv is the vibrational heat flux while Sev is the
vibrational energy source term. In absence of ionzation the vibrational
energy source term accounts only for the energy exchange between the
roto-translational and the vibrational modes:
Se v =
Ns
X
s=1
⇢s
(evs (T )
evs (Tv ))
⌧s
,
(2.41)
where evs is the specific vibrational energy and ⌧s is the relaxation time of
the sth species.
Vectorial form of the conservation equations
Conservation equations (2.31),(2.36),(2.38),(2.40) can be rewritten in a
vectorial form as follows:
@U
+ r · Fc = r · F d + S
@t
(2.42)
16
where U stands for the vector of conserved variables:
U=
⇥
⇢i , ⇢u, ⇢E, ⇢ev
⇤t
.
(2.43)
The components of the convective flux tensor Fc are given by:
Fc =
⇥
⇢i u, ⇢uu + pId⇥d , ⇢uH, ⇢uev
⇤t
,
(2.44)
In Eq. (2.45) Id⇥d is the identity matrix of order d , where d is equal to
2 for two-dimensional (2D) flows and 3 for three-dimensional (3D) flows.
The di↵usive flux tensor Fd is given by:
Fd =
⇥
Js , ⌧, ⌧ · u
q,
qv
⇤t
,
(2.45)
S is a vector collecting the source terms due to chemistry and thermal
relaxation process.
S=
⇥
S⇢s , 0t , 0, Sev
⇤
.
(2.46)
Considering an inviscid flow, the di↵usive terms are neglected so the vectorial form of the conservation equations 2.47 is reduced to the following
expression:
@U
+ r · Fc = S
@t
(2.47)
When a chemical reacting mixture is in thermal equilibrium, the flow can
be characterized by only a single temperature and the vibrational energy
equation can be neglected. Finally for a single inert gas, governing equations are given only by the global continuity equation, the conservation
equation of momentum and the conservation equation of total energy.
Eq. (2.47), can be rewritten in a quasi-linear form as follows:
@U
+ A · rU = S
@t
(2.48)
where the Jacobian matrix of the inviscid fluxes is defined as A = @Fc /@U.
The expressions of the Fci fluxes and of the corresponding Jacobian matrices Ai are reported in Appendix A, see Eqs. (A.1) and (A.2).
2.2
Chemical models
In this work two di↵erent kind of reacting mixture have been considered:
• An ionized argon mixture containing only monoatomic species and
electrons.
• A dissociated nitrogen mixture containing only neutral atoms and
molecules.
In the following subsections the two mixtures are described in detail with
emphasis on the kinetic process considered.
2.2.1
Kinetic model for an ionized argon mixture
In this work we have considered a quasi-neutral argon plasma, i.e. the
molar density of the positive particles is assumed to be equal to the molar density of the negative particles. A precise description of an argon
plasma in non-equilibrium would require a detailed collisional-radiative
model. This implies that not only the chemical reactions between the
various species should be accounted for, but also all possible transitions
involving the atomic electronic excited levels [75]. Furthermore, the energy distribution of the free electrons may follow a non-Maxwellian distribution function [20, 22, 23, 27]. Due to the extremely high computational
cost, this type of modelling cannot be used in the context of a multidimensional CFD approach.
For an argon plasma, a good compromise is to consider a reduced number
of electronic excited levels for the atomic species and to use a Maxwellian
electronic energy distribution function (EEDF) for the electrons, which
amounts to define a single temperature for the electrons (Te ). Moreover,
we have made the hypothesis of thermal equilibrium between the electrons
17
18
and the heavy particles, which amounts to use a single temperature model
with Te = T .
In our model we take into account only three chemical species: the neutral atoms Ar, the positive ions Ar+ and the electrons e . Following [22],
we consider a two-levels system for the neutral atom, with the ground
state Ar0 and the 4s metastable state Ar⇤ , while we consider only the
ground state for the positive ion Ar+ . The chemical species are reported
in Tab. 2.1: we include metastable argon atoms since ionization of the
excited atoms is caused by collision with particles of lower energies, so
that the role of excitation and ionization from the metastable state cannot be neglected. Moreover, electron induced rate coefficients can be
deeply a↵ected by the presence of metastable states resulting in complex
structures in the EEDF [16].
Table 2.1: Chemical species in the ionized argon mixture
Chemical Species
Ground
Metastable
Positive Ion
Electron
Symbol
Ar0
Ar⇤
Ar+
e
Formation energy (eV)
0.0
11.55
15.76
0.0
Statistical weight
1
6
1
0
As shown in Tab. 2.2, the chemical processes we account for are: ionizationrecombination and electronic excitation-de-excitation induced by collision
with electrons (r = 1 3) and atoms (r = 4 6). Photoionization and
photo-recombination are not included in the model, since their contribution is negligible compared to electronic and atomic processes [78].
Table 2.2: Chemical processes in the ionized argon mixture
r
1
2
3
4
5
6
Reaction
Ar0 + e ⌦ Ar+ + e + e
Ar0 + e ⌦ Ar⇤ + e
⇤
Ar + e ⌦ Ar+ + e + e
Ar0 + Ar0 ⌦ Ar+ + e + Ar0
Ar0 + Ar0 ⌦ Ar⇤ + Ar0
⇤
Ar + Ar0 ⌦ Ar+ + e + Ar0
Description
e-A ionization
e-A excitation
e-A ionization
A-A ionization
A-A excitation
A-A ionization
The forward rate coefficients of the electron-atom (e A) processes have
been computed for di↵erent values of the electron temperature, integrating the cross section over the corresponding EEDF:
kr =
Z
19
1
&(")⌫e (")fe (")d".
(2.49)
0
For a Maxwellian EEDF, Eq. (2.49) becomes:
!1/2 Z
1
1
8kB T
kr =
(")"e
(kB T )2 ⇡mr
0
"/kB T
d".
(2.50)
These rate coefficients have been fitted to reduce the computational cost
and are reported in tables C.2, C.3 and C.4, in Appendix C. The fitted
forward and backward rate coefficients for the electron ionization from
the ground level are plotted in Fig C.1 versus the electron temperature.
The forward rate coefficients for the atom-atom (A
been taken from [5, 75]:
p
kf r = br T ("ij + 2kB T )e
"ij /kB T
A) processes have
(2.51)
where "ij is the di↵erence between the formation energies of the chemical
species involved in the reaction and br is a coefficient which depends
on the chemical process. Both "ij and br are reported in Tab. C.5, in
Appendix C.
As far as the atom impact processes are concerned, the backward rate
coefficients have been obtained using the detailed balance principle:
kbr =
kf r
.
Kr
(2.52)
The equilibrium constants Kr that appear in Eq. (2.52) are easy to compute once the global ionization equilibrium constant KeqI is known [14]:
+
KeqI
[Ar+ ][e ] QAr Qe
=
=
.
[Ar]
QAr
(2.53)
In Eq. (2.53) the concentrations are those at equilibrium and the equilibrium constant KeqI has been directly computed from the complete parti-
20
Figure 2.1: Ionization and recombination rates from the ground level as a function of
the electron temperature (courtesy of G. Colonna)
tion functions of Ar and Ar+ and then fitted as a function of the temperature; the fit function and the coefficients are reported in Appendix C, in
Tab. C.1 [15].
For the two-levels system described below, the Ar partition function is:
QAr = g0 + g⇤ e
"⇤ /kB T
(2.54)
where g0 and g⇤ are the statistical weights of the ground and metastable
species. The Ar+ partition function is equal to the contribution of the
ground state only:
+
QAr = g0+
(2.55)
21
and the electron partition function is:
Qe
✓
2⇡me kB Te
=2
h2P
◆ 32
.
(2.56)
To compute the backward coefficients rate of the atom-induced processes,
we need to know the expressions of the equilibrium constants for both the
ionization and the excitation processes. The equilibrium constants for the
ionization processes from the ith excited level can be computed using the
following equation:
QAr "i /kB T
Kr = KeqI
e
,
gi
(2.57)
while for the excitation processes between the ith and j th levels, the equilibrium constant reads:
Kr =
gj
e
gi
"ij /kB T
.
(2.58)
In Tab. 2.3 we report the expressions of the equilibrium constants of all
the processes accounted for in the present model.
Table 2.3: Equilibrium constants for all processes accounted for in the present model
2.2.2
r
1,4
2,5
Equilibrium constant
Ar
Kr = KeqI Qg0
Kr = gg⇤0 e "0⇤ /kB T
3,6
Kr = KeqI Qg⇤ e"⇤ /kB T
Ar
Kinetic model for a dissociated nitrogen mixture
For the dissociated nitrogen the two temperature model has been used,
in which the vibrational energy is not in equilibrium with the rototranslational energy. The dissociated nitrogen mixture is constituted by
22
only two neutrals species: the molecular nitrogen and the monoatomic
nitrogen. Details about the chemical species are shown in Tab. 2.4.
Table 2.4: Chemical species in the dissociated nitrogen mixture
Chemical Species
Molecule
Atom
Symbol
N2
N
Formation energy (kJ/mol)
0.0
470.818
Vibrational temperature (K)
3392.7
The set of chemical reactions includes the atom-molecule (A M ) dissociation and the molecule-molecule (M M ) dissociation (see Tab. 2.5).
Table 2.5: Chemical processes in the dissociated nitrogen mixture
r
1
2
Reaction
N2 + N ⌦ 3N
N2 + N2 ⌦ 2N + N2
Description
A-M dissociation
M-M dissociation
The Park model [61] has been considered to model the chemical reaction
rates. In this model the forward kf and backward kb rate constants are
expressed using the Arrhenius Law :
⌘r
kf r = Ar T e
✓r
T
(2.59)
where Ar and ⌘r are some semi-empirical coefficient and ✓r is the activation energy temperature for the reaction r. T in the Park model
[37, 40, 61] is an avereged temperature T = T q TV1 q , with q = 0.5. The
backward rates are computed using the detailed balance (see Eq.( 2.52)
and the equilibrium constants Kr are expressed using the curve fit proposed by Park [37, 61].
2.3
Electrohydrodynamic model
In this thesis the nonequilibrium model has been extended to the case in
which an external electromagnetic field is applied. We have investigated
the e↵ects of an external electromagnetic field on an inviscid chemical
reacting mixture in thermal equilibrium (Cneq). The conservative equations are given by the continuity equations for all the chemical species, the
23
momentum equation with the Lorentz force, FB as source term and the
total energy equation with the Joule heating, QJ as source term [8, 9, 34]:
@⇢i
+ r · (⇢i u) = Si
(2.60)
@t
@(⇢u)
+ r · (⇢uu) + rp = FB
(2.61)
@t
@(⇢E)
+ r · (⇢uH) = QJ
(2.62)
@t
The Lorentz force is given by the vectorial product of the current density
j and the magnetic field B.
FB = j ⇥ B
(2.63)
The Joule heating can be computed in each point of the domain as the
scalar product of the current density and the electric field E.
QJ = j · E
(2.64)
The electromagnetic variables, j, E and B can be computed coupling the
Maxwell equations to the conservation equations [34]. Some simplifications can be made in the cases of our interest since the magnetic Reynolds
number, Rem , is assumed to be small [9]:
Rem = uL µ0
(2.65)
In Eq. (2.65) u is the streamwise velocity of the flow, L is the reference
length, is the electrical conductivity and µ0 is the permeability of the
free space.
When Rem << 1, the induced magnetic field can be neglected and only
the external magnetic applied field is present in the flow, then the current
density can be determined solving the current continuity equation:
r · j = r · [e · (E + u ⇥ B)] = 0
(2.66)
where e is the electrical conductivity tensor. In Eq. (2.66) the current
density has been expressed using the generalized Ohm’s law, j = e ·
24
(E + u ⇥ B). The electric field can be expressed as the gradient of an
electrical potential as E = r .
Whithout an external magnetic field, the Hall e↵ect is absent, then the
electrical conductivity tensor reduces to a scalar and Eq. (2.66) reduces
to the following Laplace equation:
r · ( E) = r · ( r ) = 0
(2.67)
where is the scalar electric condutivity. The Ohm’s law is given by the
following equation:
j= E
(2.68)
The electrical conductivity of the mixture depends on thermodynamic
and electromagnetic variables. As pointed out by Bisek et al. [9], di↵erent electrical conductivity models can lead to very di↵erent results, in the
frame of MHD hypersonic flows on a blunt body. The experimental data
obtained for the electrical conductivity of an Argon plasma are plotted
versus temperature in Fig. 2.2. In the present work, the electrical con-
Figure 2.2: Electrical conductivity of argon (p = 0.013atm), reprinted from [8].
25
ductivity has been modeled using the semy-analytic model developed by
Chapman and Cowling for weakly ionized gas [9]:
= 3.34 ⇥ 10
12
↵ion
p ⌦
& T
1
cm
1
(2.69)
where ↵ion is the ionization degree defined by the ratio of the number of
ionized particles and heavy particles:
↵ion =
nion
ntot nion
(2.70)
and &[cm2 ] is the collision cross section of the gas. The collision cross
section is taken to be the total collision cross section for argon-argon
collisions using the hard sphere model, with a diameter of d = 4.04 ⇥
10 10 m [7], then & = 5 ⇥ 10 17 cm2 .
A more accurate model would require to include the e↵ect of chemical
nonequilibrium and the e↵ect of electric field.
External circuit coupling
The electrodes have been considered connected with an external electrical
circuit as shown in the Fig 2.3. The electric potential on the electrodes
is given by Ohms law for the external circuit:
Vdis (t) = Vg
Rc I (t)
(2.71)
where Vdis is the discharge voltage, Vg the power supply potential, Rc
the circuit resistence and I the electric current in the circuit. The electric current can be computed as the surface integral of scalar product of
current density vector and the surface vector over the elecrodes.
As the surface integral of the scalar product of the current
I=
Z
electrodes
j · dA
(2.72)
At each time step the discharge voltage changes according to the Eq. (2.71),
and the electric potential of one of the electrodes is set equal to ref +
26
Rc#
V g#
+"
#"
I#
electrodes"
Vdis#
Figure 2.3: Scheme of the coupled circuit
Vdis (t), where ref is a reference value for the electric potential (generally
ref is set equal to zero).
Chapter 3
Computational tools
In this chapter the numerical methods used to the discretize the governing
equations will be described. The chemical nonequilibrium model has
been implemented within eulfs [10, 13], an unstructured 2D/3D solver
developed for thermally and calorically perfect gas.
The CFD code can be coupled with a newly developed, unstructured,
shock-fitting algorithm which treats the discontinuities as moving boundaries that border regions of the flow-field were a smooth solution to the
governing PDEs exists. The unstructured shock-fitting algorithm has
been extended to deal with an ionised argon mixture to model shock
waves in chemical reacting flows and with
The shock-capturing discretization will be presented first, since it is also
used in the shock-fitting approach to solve the governing PDEs in the
smooth regions of the flow-field.
One of the key features of the unstructured shock-fitting algorithm is its
modularity: the shock-capturing code used to discretise the governing
PDEs (2.47) in the smooth regions of the flowfield is used as a “black
box” by the shock-fitting algorithm. The only constraint is that the
unstructured shock-capturing solver must feature a vertex-centred storage
of the unknowns.
In [12, 42, 57, 58] the unstructured shock-fitting algorithm had been coupled with the in-house eulfs shock-capturing code. In the present work,
the shock-capturing solver has been coupled, with a very limited coding
e↵ort, with the VKI-developed COOLFluiD [50, 51] solver. This was
27
28
Computational tools
done both for demonstrating the modularity of the unstructured shockfitting algorithm, but also because COOLFluiD features a wider range of
modelling capabilities than eulfs.
3.1
Shock-capturing solver
Both codes, eulfs and COOLFluiD, use Residual Distribution schemes
(RD) 1 for the spatial discretization, so we briefly introduce the main
characteristics of the RD discretization for the governing PDEs (2.47) [1,
28, 71]. The computational domain ⌦ ⇢ Rd is tessellated into triangles
in the 2D space, and tetrahedra in 3D. A dual tessellation is also defined,
which consists in the so-called median dual cells. Both the primal and
dual tessellations are shown in Fig. 3.1 for the 2D case: Ci is the median
dual cell centred about gridpoint i and ⌦e is the triangle e.
The dependent variables are stored at the vertices of the computational
mesh and are assumed to vary linearly and continuously in space, just as
in iso-P1 Finite Elements.
2
3
1
e
4
5
(a) The flux balance of cell ⌦e is scattered among
its vertices.
(b) Grid-point i gathers the fractions of cell residuals from the surrounding cells.
Figure 3.1: Residual distribution concept.
We use the term residual (or fluctuation) to refer to the sum of the
surface integral of the inviscid and di↵usive fluxes through the control
1
RD schemes are also known under the name of Fluctuation Splitting schemes (FS).
Computational tools
29
volume boundary and the volume integral of the source term. Observe
that the residual vanishes whenever U is a steady solution of Eq. (2.47).
At the discrete level, both cell and nodal residuals can be defined: for
cell residuals the control volume coincides with the simplicial element
(triangle/tetrahedron) whereas for nodal residuals the control volume is
the median dual cell. When using simplicial meshes, as we do with FS
schemes, the number of cells exceeds the number of gridpoints: by a factor close to 2 in two dimensions and ranging between 5 and 6 in three
dimensions; therefore, since the FS discretization is vertex-centred, at
steady-state it will only be possible to drive the nodal residuals to zero,
not the cell residuals.
With FS schemes, rather than calculating the nodal residual by numerical quadrature along the boundary of the median dual cell and over
its volume, as would be done with conventional Finite Volume schemes,
the nodal residual is obtained by collecting fractions of the cell residuals
of the triangles/tetrahedra that meet at mesh-point i. This is accomplished using a main loop over all triangular/tetrahedral elements in the
mesh: within each element the cell fluctuation (or residual) is computed
and then split into signals which are sent to the vertices of the cell (see
Fig. 3.1(a)). At the end of the loop, the nodal residual has been assembled
within each gridpoint by collecting the signals scattered by the elements
that surround that gridpoint (see Fig. 3.1(b)).
3.1.1
Fluctuation and conservative linearization
Given an element e with volume ⌦e and boundary surface @⌦e , the fluctuation (or residual ) e is defined as follows:
I
I
Z
e
c
d
e,d
e,s
=
F · ndS
F · ndS
SdV = e,c
(3.1)
@⌦e
@⌦e
⌦e
where e,c and e,d are resp. the net inviscid flux and the di↵usive flux
balance across @⌦e and e,s the volumetric integral of the source term.
The net inviscid flux over each triangular/tetrahedral element is evaluated by means of a conservative linearization that allows to compute the
inviscid flux balance over a cell using the quasi-linear form (2.48) of the
Euler system:
30
Computational tools
e,c
=
I
@⌦e
F · n dS =
Āi
✓
◆
@U
|⌦e |.
@xi
(3.2)
The problem stated by Eq. (3.2) amounts to define proper cell averages of
both the Jacobian matrices Āi and of the gradient of the conserved variables rU. Conservation is essential to guarantee correct shock-capturing,
whereas working with the quasi-linear form of the equations allows to give
an upwind flavour to the discretization.
When dealing with a single-component, calorically and thermally perfect
gas, this can be accomplished [29] by using Roe’s parameter vector:
Z=
p
⇢ (1, H, u)T
(3.3)
as the dependent variable featuring piece-wise linear variation over each
simplicial element. Indeed, since both the conserved variables U and the
components Fi of the inviscid flux vector are quadratic function of the
components of the vector Z, the cell fluctuation can be computed exactly
as follows:
e,c
I
Z
✓
=
F · n dS =
@⌦e
◆✓
◆ ⌦e
Z ✓
@Fi
@Z
dV =
@Z
@xi
⌦e
@Fi
@xi
◆
dV =
(3.4a)
✓
◆✓
◆
@Z
@Fi
|⌦e |
(3.4b)
@xi
@Z Z=Z
where we have taken advantage of the fact that rZ is cell-wise constant
and the value of @Fi /@Z, averaged over a cell, equals its analytical expression evaluated in the arithmetic average:
!
X
1
Z=
Zj
d + 1 j2e
since @Fi /@Z is also linear in the components of Z. The cell averaged
Jacobian matrices and the gradient of the conserved variables in Eq. (3.2)
can be written as in [29]:
Āi = Ai Z
rU =
✓
@U
@Z
◆
Z=Z
rZ.
Computational tools
31
The conservative linearization can be extended in a straightforward manner to a multi-species fluid in chemical non-equilibrium only in the restrictive case in which each species has the same number of degrees of
freedom (hence the same adiabatic index = Cp /Cv ) and behaves as a
calorically perfect gas. In this circumstance, it has been shown by Degrez
et al. [30] by using the following definition for Roe’s parameter vector:
p
Z = ⇢(↵i , H, u),
(3.5)
the vector of the conservative variables and of the inviscid fluxes are
homogeneous functions of second degree in Z, just as in the perfect gas
case. By contrast, when dealing with an arbitrary mixture of thermally
perfect gases or a single calorically imperfect gas, pressure is not any
longer a quadratic function of Z and the issue of finding an averaged
state becomes considerably more complicated. As far as FS schemes are
concerned, a Roe-type conservative linearization has only been obtained
in the case of arbitrary fluids in chemical non-equilibrium [30]. A di↵erent
approach, named Conservative Residual Distribution (CRD) [25], which
consists in calculating the flux integral by means of numerical quadrature,
thus avoiding the conservative linearization, is used when dealing with
thermochemical non-equilibrium flows; details can be found in Refs. [47–
49].
The availability of a conservative linearization is also a key ingredient to
generalise the scalar FS schemes to non-linear hyperbolic systems of equations. Although the individual Jacobian matrices Ai cannot be simultaneously made diagonal in more that one space dimension, hyperbolicity
of the Euler system implies that matrix Ān = Āi ni , n = ni ei being an
arbitrary unit vector, has real eigenvalues and a complete set of linearly
independent eigenvectors, so that it can be diagonalised as follows:
¯n
Ln Ān Rn = ⇤
(3.6)
1
where Ln and Rn = Ln
are the matrices of left and right eigenvectors
¯
and ⇤n the diagonal matrix of the corresponding eigenvalues. Detailed
expressions of the various matrices that appear in Eq. (3.6) are given in
Appendix A, see Eqs. (A.6), (A.7), (A.9) and (A.10).
The volumetric integral of the chemical source term is computed using
a quadrature rule [30]. The simplest possible choice is the one-point
32
Computational tools
quadrature rule:
e,s
=
Z
⌦e
S dV = S (xec ) ⌦e = S Z ⌦e .
(3.7)
where xec is the centroid of the element e.
3.1.2
Signals or Residual Distribution
Once the cell fluctuation has been computed according to Eq. (3.2), it is
e,d
scattered among the d + 1 vertices of the cell by sending signals e,c
i ,
i
and e,s
which
must
fulfil
the
following
conservation
constraints:
i
e,c
e,d
e,s
=
=
=
d+1
X
i=1
d+1
X
i=1
d+1
X
e,c
i ,
(3.8a)
e,d
i ,
(3.8b)
e,s
i .
(3.8c)
i=1
Eq. (3.8a) guarantees global conservation in space in the sense that the
inviscid flux through the boundary of the computational domain is exactly
recovered once the inviscid component of the fluctuations of all cells in
the mesh have been summed up.
Di↵erent criteria can be devised to split the cell fluctuation into signals.
Among them, there are schemes that depend linearly upon the solution
(when solving a linear PDE) and are either monotonicity preserving, but
limited to first order of accuracy, as the case of the N scheme, or, if second
order accurate, may lead to oscillatory behaviour in the neighbourhood
of a captured discontinuity, which is the case of the LDA scheme [72].
Within cell e, N and LDA schemes send the following signals:
e,N
i
=
K+
i (Ui
e,LDA
i
U ),
=
where:
U± =
d+1
X
j=1
±
j Uj
and
±
j
=
d+1
X
`=1
K±
`
!
+
i
e
(3.9)
1
K±
j .
(3.10)
Computational tools
33
and Ki is the nodal upwind parameter, defined as:
1
Ki = Āni Si
d
(3.11)
In Eq. (3.11) Āni = Ā · ni is the Jacobian matrix projected along the
direction of the unit inward normal to the face opposite to node i. Si is
the area of the face opposite to node i as it is shown in Fig. 3.2.
Si
ni
i
Figure 3.2: Inward scaled normal.
Using Eq. (3.6) we obtain:
1
¯ ± L n Si
K±
=
Rn i ⇤
ni
i
i
d
(3.12)
¯ ± is defined as:
where the generic element of the diagonal matrix ⇤
n
±
k
=
1
(
2
k
± | k |) .
is one of the m = d + Ns + 1 eigenvalues of matrix Ān . Non-linear
schemes that capture the discontinuities monotonically and preserve second order of accuracy in smooth regions of the flow-field can be constructed by using a solution-dependent weighting function which blends
the linear N and LDA schemes in such a way that the former scheme
is activated only in the neighbourhood of the captured discontinuities
whereas the latter is used elsewhere.
k
34
Computational tools
e,B
i
e,N
i
= ⇥e
+ (I
⇥e )
e,LDA
i
e
(3.13)
These kind of non-linear schemes (the matrix ⇥ depends upon the local
solution) are referred to as B (blended) schemes. The CRD variants of
the RD schemes are identified by a c character appended to the name of
the scheme; for example the CRD variant of the LDA scheme is named
LDAc.
The di↵usive contribution to the cell residual are distributed with the
Galerkin method. When a mesh made of iso-P1 (simplicial) elements is
used, the di↵usive cell residual sended to the node j reads:
d
i
=
Z
⌦e
Ni r · Fd dV
(3.14)
The fluctuation associated with the source term is split into signals using
the LDA weighting, i.e.
e,s
+ e,s
(3.15)
i
i =
The motivation for using an upstream weighted distribution of the source
terms lies in the analogy between FS and Petrov-Galerkin FE schemes
[17].
3.1.3
Solution of the discretised equations
Once the cell fluctuation has been computed and scattered among the
vertices of each cell, the nodal residual in gridpoint i is obtained by collecting all the signals scattered from its neighbouring elements.
Ri (U) =
=
I
c
@C
X⇣i
i3e
F · n dS +
e,c
i
+
I
d
F · n dS +
@Ci
⌘
e,d
e,s
.
i + i
Z
S dV
(3.16a)
Ci
(3.16b)
Equation (3.16) is the discretized version of the spatial terms in the governing conservation equations, where the control volume Ci is the median
dual cell.
Computational tools
Steady solutions of the governing conservation laws, have been calculated
by solving the following pseudo-unsteady system of equations:
dU
V = R (U)
(3.17)
d⌧
introducing the pseudo-time ⌧ . Since accuracy in pseudo-time is irrelevant, the mass matrix has been lumped into the diagonal matrix V whose
entries are the areas/volumes of the median dual cells and a first-order
accurate, two time levels Finite Di↵erence (FD) formula:
dU
U
=
+ O ( ⌧)
U = Un+1 Un
(3.18)
d⌧
⌧
is used to approximate the pseudo-time derivative in the l.h.s. of Eq. (3.17).
An explicit scheme is obtained by evaluating the nodal residual in Eq. (3.17)
at the known pseudo-time level n. Explicit time stepping is used when the
code is operating in shock-fitting mode (described in Sect. 3.2), whereas
implicit time stepping is used when the code works in shock-capturing
mode.
An implicit scheme is obtained if the residual R in Eq. (3.17) is evaluated
at the unknown pseudo-time level n + 1. expanding R about pseudotime level n and introducing the non uniform pseudo-time-step ⌧n , one
obtains the following sparse system of linear equations:

1
@R
V J (Un )
U = R (Un ) ,
J=
(3.19)
⌧n
@U
to be solved at each pseudo-time-step until the required convergence of
R is obtained.
Two distinct approaches are currently available for calculating the Jacobian matrix (J) of the residual: i) one in which some approximations are
introduced in convective terms and chemical source terms are neglected
in the calculation of the Jacobian and ii) an FD approximation of the “exact” Jacobian which accounts for all terms in the equations incuring only
into the truncation error of the one-sided FD formula. Even if the analytical approximation of the Jacobian is much cheaper to compute than its
FD approximation, it is less robust. Moreover the FD Jacobian approximation allows, in the limit of infinite ⌧n to recover Newton’s root-finding
35
36
Computational tools
algorithm, which is known to yield quadratic convergence when the initial
guess is close to the steady solution. However, unless a reasonably good
initial guess is available, the implicit scheme built upon the FD Jacobian
approximation may quickly diverge. A typical run is therefore split in
two steps. During the first step, the approximate, analytical Jacobian is
used to start the calculation from scratch until the residual has dropped
some orders of magnitude; in this first phase the pseudo-time-step length
is not allowed to increase beyond values corresponding to a CFL number of the order of one hundred. Then, the solution is re-started using
the FD Jacobian approximation and the pseudo-time-step is allowed to
grow un-boundedly so that quadratic convergence is eventually recovered
over the last pseudo-time-steps. At each inner iteration, The linear system (3.19) is solved using the suite of iterative, preconditioned Krylov
subspace algorithms available in the PETSc library [6]. Further details
regarding the Newton-Krylov algorithm can be found in Ref. [11].
3.2
Shock-fitting algorithm
The unstructured shock-fitting algorithm that has been recently developed by Paciorri and Bonfiglioli [12, 42, 57, 58] consists of two key ingredients: i) a local re-meshing technique that constructs a time-dependent
mesh in which the fitted discontinuities are internal boundaries of zero
thickness and ii) an algorithm for solving the Rankine-Hugoniot jump
relations that provides the Lagrangian velocity of the discontinuity and
an updated set of dependent variables within the downstream side of the
fitted shock. More precisely, in two space dimensions the fitted shock
fronts are made of polygonal curves, i.e. a connected series of line segments (which we call the shock edges) that join the shock points. The
downstream state and the shock speed are computed according to the
Rankine-Hugoniot jump relations and the fitted shock is allowed to move
throughout a background triangular mesh that covers the entire computational domain.
At each time level t the starting point is given by the solution on a
background mesh and a polyline describing the shock (see Fig. 3.3). The
Computational tools
37
(a)
(b)
Figure 3.3: Starting point: a) background mesh and b) shock boundary (right).
position of the shock nodes are totally independent by the location of
gridpoints on the background mesh. At each node of the shock two states
are associated: one for the upstream state and one for the downstream
state. The process of updating the mesh and solution to time level t + t
can be split into di↵erent steps that will be described in details:
1. Cell removal. The first step consists in finding and removing
the cells around the shock front. All cells of the background mesh
that are crossed by the shock boundary are removed with the gridpoints too close to the shock front. These mesh nodes are identified
as“phantom” nodes (dashed circles in Fig. 3.4(a)).
2. Local re-meshing. After the cell removal, the remaining part of
the background mesh is split into two separated regions. A constrained Delaunay triangulation (CDT) is performed to remesh the
hole (see Fig. 3.4(b)).
3. Computation of the normal and tangent vectors. The normal
n and tangent ⌧ unit vectors are computed for each shock node
(Fig. 3.5(a)). These vectors in a generic shock node are computed
by means of finite-di↵erence formulas which involve the coordinates
of the shock point itself and the neighbouring shock points.
38
Computational tools
(a)
(b)
Figure 3.4: a) Cell removal. b) Local remeshing.
4. Solution update using the S-C solver. The solution is updated
by the S-C code to time level t+ t on the modified mesh by treating
the shock front as an internal boundary. The shock downstream
states are not accurately updated except for the characteristic rising
from the downstream region (Rd ):
Rdt+
t
= at+
d
t
1
ut+
d
t
·n
(3.20)
2
where ad and ud are the values of the sound and flow velocity of the
downstream state of the shock nodes. The shock is treated as an
internal boundary by the S-C solver.
+
5. Shock calculation using the R-H relations. The upstream
state of each shock node have been correctly updated during the
S-C solver computation, whereas only the Rd quantity has been
correctly updated within the shock- downstream state. The exact
downstream state and shock speed Wsh of each shock point are
computed by solving a system of five algebraic non-linear equations
given by the four R-H relations, the Eq. (3.20), where the value Rd
has been computed by the S-C solver.
6. Interpolation of the phantom nodes. After the exact update
of the downstream state the solution is updated also on the phantom nodes which belong to the background mesh.This update is
Computational tools
(a)
39
(b)
Figure 3.5: a) Normal and tangential unit vector computation. b) Interpolation of the
phantom nodes.
performed finding the cell containing the phantom node and interpolating the phantom node state using the states of the cell vertices
(see Fig. 3.5(b))
7. Shock displacement. The position of the shock nodes at time
t + t is computed considering the following first order integration
formula:
t+ t
xt+ t = xt + Wsh
n t
(3.21)
Figure 3.6: Interpolation of mesh point jumped by the shock boundary.
The shock-fitting algorithm, originally developed for an ideal gas, has
been extended to Cneq and TCneq flows. The generalization to a reactive
40
Computational tools
mixture has required a minimum e↵ort, since modeling the bow shock as
a partly dispersed shock wave [73], the concentrations of chemical species
are kept frozen while passing through the bow shock. When dealing with
thermochemical non-equilibrium, the specific vibrational energy is kept
constant through the bow shock.
3.3
Details on the implementation
The shock-fitting algorithm has been implemented in a modular way, so
that the CFD solver, the remeshing tool are used as black boxes as shown
in Fig. 3.7. Comunication between the CFD solver and the remeshing tool
is performed by means of I/O files, then two converters are needed in order
to exchange the right data between the codes. Since the shock-fitting code
uses the adimensional Roe parameter vector Z⇤ (see Appendix A), a variable conversion is performed into the converters. Local re-meshing around
the shock is accomplished using public domain software: triangle [68]
in 2D and tetgen [70] and yams [31] in 3D.
Computational tools
Figure 3.7: Modularity of the shock-fitting code.
41
42
Chapter 4
Numerical results
In this chapter the numerical results will be presented. Results obtained
considering the ionized argon model implemented within the CFD code
eulfs are reported in Sect. 4.1. The results obtained using the CFD code
COOLFluiD for perfect gas and dissociating nitrogen flows are presented
in Sect. 4.2. Finally the results obtained considering the electrohydrodynamic interaction are shown in Sect. 4.3.
4.1
4.1.1
eulfs results
Ionized argon, inviscid flow in a nozzle
In order to validate the chemical model implemented within the eulfs
code, results obtained from the 2D and 3D simulation of the flow through
a converging-diverging nozzle have been compared with those obtained
using a Q1D code developed at IMIP-CNR [20, 21, 24]. In this reference code, the inviscid Q1D conservation equations are solved using a
space-marching algorithm and are coupled with state-to-state kinetics
and Boltzmann equation for free electron transport and the rates of electron collision processes are obtained by integrating the electron impact
cross section over the EEDF. Therefore, in order to make the code-to-code
validation meaningful, it has been necessary to downgrade the chemical
model available in the IMIP-CNR code to make it identical to the one,
described in Sect. 2.2.1, that has been implemented within the eulfs
43
44
Numerical results
CFD code. The analytical area distribution of the converging-diverging
nozzle is given in Tab. 4.1 whereas Tab. 4.2 shows the reservoir test conditions. In the 2D/3D simulations the total temperature, total pressure
and flow angles have been specified along the subsonic inlet section and
the metastable argon concentration has been set equal to that obtained
considering a Boltzmann distribution for the electronic levels. The flow
is supersonic at the outflow section and thus requires no boundary condition there. Figure 4.1 shows the 2D and 3D geometries with Mach
iso-contours superimposed: the 2D grid is made of 4439 grid-points and
8447 triangles and the 3D grid is made of 28581 grid-points and 155667
tetrahedral cells. The 2D/3D simulations use the second order accurate
LDA scheme for the spatial discretization.
Y
Z
X
M
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
Figure 4.1: Flow in a converging-diverging nozzle: 2D and 3D geometries flooded by
Mach number.
Comparison between eulfs and Q1D IMIP code have been made for two
di↵erent thermodynamic inlet conditions (see Tab. 4.2).
Figures 4.2 and 4.3 show a comparison of the area weighted distributions along the nozzle axis; data include: the molar fractions of ions and
metastable argon. The results obtained from the 2D and 3D simulations
have been area weighted in order to be compared with the Q1D result
Numerical results
45
Table 4.1: Area distribution along the nozzle.
x
0.3 < x < 0.0
0.0 < x < 0.5
area
A(x) = 0.010 + 0.6x2
A(x) = 0.010 + 0.39 x2 + 0.26 x3
Table 4.2: Inlet flow conditions.
Physical quantity
Total pressure
Total temperature
Ionisation degree
Test A
1 ⇥ 105 Pa
5000K
1 ⇥ 10 7
Test B
1 ⇥ 105 Pa
8000K
1 ⇥ 10 5
Mach number
Temperature
7
5000
Q1D IMIP
EulFS 3D
EulFS 2D
4000
Q1D IMIP
EulFS 3D
EulFS 2D
6
Mach number
5
T (K)
3000
2000
4
3
2
1000
1
0
-0.4
-0.2
0
0.2
0
-0.4
0.4
-0.2
0
0.2
(a) Temperature.
(b) Mach number.
Ar+
Ar*
1.5e+11
2e+07
Q1D IMIP
EulFS 3D
EulFS 2D
Q1D IMIP
EulFS 3D
EulFS 2D
particles number density, 1/m^3
N (1/m^3)
1e+11
5e+10
0
0.4
x (m)
x (m)
-0.2
0
0.2
x (m)
(c) Argon ions: Ar+ .
0.4
1.5e+07
1e+07
5e+06
0
-0.2
0
x, m
0.2
(d) Metastable argon: Ar⇤ .
Figure 4.2: Test A: distributions along the nozzle axis.
0.4
46
Numerical results
Mach number
7
8000
5
Mach number
6000
Temperature, K
Q1D IMIP
EulFS 3D
EulFS 2D
6
Q1D IMIP
EulFS 2D
EulFS 3D
4000
4
3
2
2000
1
0
-0.2
0
x, m
0.2
0
-0.4
0.4
-0.2
0
(a) Temperature.
0.4
(b) Mach number.
Ar+
Ar*
1e+12
3e+11
Q1D IMIP
EulFS 2D
EulFS 3D
6e+11
4e+11
Q1D IMIP
EulFS 2D
EulFS 3D
2.5e+11
Particles number density, 1/m^3
8e+11
Particles number density, 1/m^3
0.2
x (m)
2e+11
1.5e+11
1e+11
2e+11
5e+10
0
-0.2
0
x, m
0.2
(c) Argon ions: Ar+ .
0.4
0
-0.2
0
x, m
0.2
(d) Metastable argon: Ar⇤ .
Figure 4.3: Test B: distributions along the nozzle axis.
0.4
Numerical results
47
of the IMIP-CNR code. As shown in Fig. 4.2(a) and 4.3(a), comparisons
for temperature along the axis x, are satisfactory. Mach number distributions (Fig. 4.2(b) and 4.3(b)) show some di↵erences close to the outlet
boundary probably due to non-Q1D e↵ects in the diverging duct. The
2D and 3D results are superimposed, despite the fact that the 2D grid
is somewhat finer than the 3D one. The comparison between the 2D/3D
calculations and the reference Q1D calculation can also be considered satisfactory, given the di↵erent numerical models, inlet boundary conditions
and grid resolutions being used and confirms the correct implementation
of the chemical model. The flow in test A can be considered frozen, since
at low temperature, the characteristic time of chemistry is much greater
than the characteristic time of fluid dynamics. In test B the flow is in
non-equilibrium conditions. Sample convergence histories for the 2D calculation are shown in Fig. 4.4: the L2 -norm of the nodal residual is plotted
against the CPU seconds for each conservative variable. The convergence
history shown in Fig. 4.4(a) has been obtained using the analytical Jacobian approximation and a maximum CFL number equal to 100: it can
be seen that all residuals converge steadily towards machine zero. In
the convergence history shown in Fig. 4.4(b) the first 50 iterations were
run using the analytical Jacobian approximation, then the FD Jacobian
approximation was put in place and the maximum CFL number set to
infinity. By doing so, only 5 Newton steps (marked by ⇥ in Fig. 4.4(b))
are needed to drive the residuals to machine zero.
Picard
Newton
(a) Approximate, analytical Jacobian.
(b) Approximate, followed by FD Jacobian.
Figure 4.4: Flow in a converging-diverging nozzle: convergence histories for test B.
48
Numerical results
4.1.2
Ionized argon, inviscid flow over a circular cylinder
Hypersonic flows over blunt bodies are characterised by strong nonequilibrium conditions. For an hypersonic flow in argon, ionization and electronic excitation may significantly a↵ect the macroscopic physical quantities, such as temperature or Mach number, within the entire flow field
surrounding the body. In this work we have considered the 2D hypersonic flow past the fore-body of a circular cylinder in order to conduct
a comparative assessment of the predictive capabilities of the alternative
shock-capturing and shock-fitting options available in the unstructured
solver. The grid used for the shock-capturing calculation and also as the
background triangulation in the shock-fitting calculation is made of 38018
nodes and 19348 elements; the shock-fitting grid at steady-state, which
di↵ers from the background triangulation only in the neighbourhood of
the fitted shock (see Fig. 4.5), has 19613 gridpoints and 38190 triangles.
The grid has been refined near the wall and in particular in the stagnation region, since chemical activity is very important in this region. In all
0.15
0.1
0.02
0.05
y (m)
y (m)
0.01
0
0
-0.01
-0.05
-0.02
0.05
0.06
0.07
0.08
0.09
x (m)
-0.1
-0.15
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
x (m)
Figure 4.5: Baseline mesh and “fitted” mesh.
Numerical results
49
shock-capturing calculations presented herein, the spatial discretization
relies upon the first order accurate N scheme. Shock-fitting calculations
rely upon the first order N scheme and the second order non-monotone
LDA scheme. Flow conditions are given in Tab. 4.3: due to the low freestream temperature, the shock-upstream flow is in equilibrium conditions
and made only of neutral argon.
Table 4.3: Freestream conditions for an ionized argon flow over a 0.05m radius cylinder.
Physical quantity
M1
p1
T1
u1
Radius
Free-stream Value
11.2
543.95 Pa
298.7 K
3536.28 m/s
0.05 m
Figure 4.6 shows the static temperature (Fig. 4.6(a)) and Mach number (Fig. 4.6(b)) iso-contours computed by means of shock-capturing (upper half of the frames) and shock-fitting (lower half).
0.15
0.15
0.1
0.1
-0.05
0.02
0.05
10.5
9.5
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
0
0
-0.01
-0.05
-0.02
0.05
0.06
0.07
0.08
0.09
0.01
y (m)
0.01
y (m)
0
0.02
y (m)
y (m)
Mach
T (K)
12500
11000
9500
8000
6500
5000
3500
2000
500
0.05
0
-0.01
-0.02
0.05
x (m)
0.06
0.07
0.08
0.09
x (m)
-0.1
-0.1
-0.15
-0.15
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
0.05
0.1
0.15
0.2
0.25
x (m)
x (m)
(a) Temperature (K)
(b) Mach number.
0.3
0.35
Figure 4.6: 2D flow past a circular cylinder: comparison between the shock-capturing
(S-C) and shock-fitting (S-F) solutions. Both S-C and S-F solutions are first order
accurate in space (N scheme).
It is clear from the comparison between the two sets of calculations that
shock-fitting gives a much more realistic shock-thickness than does shockcapturing, without the need to adapt the mesh in the shock-normal di-
50
Numerical results
rection, as would be the case if mesh adaptation was used in conjunction with the shock-capturing solver. The remarkable thickness of the
X
12000
0.01
X X X X X X X X X X X
X X X X X X
X X X X X
X X X X X
X X X X
X X X X
X
X
10000
0
X X X X X X X X X X X X X X X X X X
X X
X X X X X X X X X X X X X X X X X
X X X X X X
X
-0.01
X
X
(H - Href)/Href
T (K)
8000
X
6000
X
-0.02
-0.03
-0.04
X
X
4000
-0.05
X
X
2000
Shock-Capturing (N)
Shock-Fitting (N)
Shock-Fitting (LDA)
X X X X X X
0
0.05
0.06
Shock-Capturing (N)
Shock-Fitting (N)
Shock-Fitting (LDA)
X
-0.06
0.07
0.08
-0.07
0.05
0.09
0.055
0.06
0.065
x (m)
10
-6
X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X
10-3
10-4
X
X
X
X
X
X
X
X
-7
X
X
X
10-8
X
X
X
10
-9
0.085
X
X
X
10
0.08
(b) Nondimensional enthalpy.
Molar fractions (Ar+)
Molar fractions (Ar*)
X X
-5
0.075
x (m)
(a) Temperature (K)
10
0.07
X
X
X
10
-5
10
-6
X
X
X
10-7
X
X
X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X X
X
X
X
X
X
X
X
10-8
X
Shock-Capturing (N)
Shock-Fitting (N)
Shock-Fitting (LDA)
X
10
-9
X
Shock-Capturing (N)
Shock-Fitting (N)
Shock-Fitting (LDA)
X
X
X
X
X
10
-10
0.05
X X X X X X
0.055
0.06
0.065
0.07
0.075
0.08
x (m)
(c) Ar⇤ molar fraction.
0.085
0.09
10
-10
0.05
X X X X X X
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09
x (m)
(d) Ar+ molar fraction.
Figure 4.7: 2D flow past a circular cylinder: comparison of shock-capturing (S-C) and
shock-fitting (S-F) solutions along y = 0.
captured bow-shock is particularly evident in Fig. 4.7(a), where symbols denote the intersections of the stagnation streamline (the x-axis)
with the triangular grid. As we can see in Fig. 4.7(b), shock-fitting allows to conserve total enthalpy across the captured shock. Even though
the shock-capturing solution recovers the “exact” post-shock state, all
those flow states that are inside the shock are a mere numerical artifact that also a↵ect the chemistry, as we shall describe later. Moreover,
Numerical results
the better description of the bow shock provided by shock-fitting has a
clear, beneficial impact also on the smooth flow within the entire shock
layer: both the temperature and Mach iso-contours within the shockdownstream region are smoother in the shock-fitting solution than they
are in the shock-capturing one, see Fig. 4.6. Finally, we observe that the
chemical activity in the flow-field is the consequence of the temperature
rise across the bow shock; this is evident from Fig. 4.7(c) and 4.7(d),
which show the molar concentrations of metastable argon and argon ions
along the stagnation streamline, for both sets of calculations. It is interesting to observe how the di↵erent modeling practices, capturing versus
fitting, have also an impact upon the non-equilibrium chemistry within
the shock-downstream region. Indeed, since the captured shock is remarkably thick, in the shock-capturing calculation chemical reactions are
activated already inside the shock, whereas they occur just behind the
fitted shock in the shock-fitting solution. Due to the memory e↵ect of the
non-equilibrium chemical model, the “artificial” chemical concentrations
that are created inside the captured shock are also felt downstream of the
shock-wave, even beyond the spatial location, see Fig. 4.7(c) and 4.7(d),
were both the shock-capturing and shock-fitting solutions have reached
the same post-shock temperature. Therefore, the molar concentrations
computed by means of shock-capturing and shock-fitting di↵er behind the
shock and the di↵erences will eventually vanish only further downstream.
Concerning the shock-fitting calculations, first order and second order accurate solutions di↵er slightly only in the post-shock region. Di↵erences
in molar concentrations between the shock-capturing and shock-fitting
solutions just downstream of the bow shock are due to the very di↵erent descriptions of the shock wave given by the two di↵erent modeling
options.
4.2
4.2.1
COOLFluiD results
Ideal gas, inviscid flow over a circular cylinder
An hypersonic inviscid flow past a circular cylinder at freestream Mach
number, M1 = 20, has been considered as a first test case. Results for
this same test case had already been obtained by Paciorri and Bonfiglioli
51
52
Numerical results
[57] using the eulfs solver. Two di↵erent grids with increasing spatial
resolution have been used to asses the improvements in accuracy of the
shock-fitting code. The two meshes shown in Fig. 4.8 while the number
of triangles and nodes are reported in Tab. 4.4. The shock-fitted grids
di↵er from the corresponding background ones because of the addition
of the shock points; the background grids have also been used for the
calculations in which the shock is captured. All the results obtained for
the ideal inviscid gas flows have been carried out by means of an explicit
backward Euler time integration scheme.
Table 4.4: Characteristics of the grids used for the inviscid hypersonic flow over a
circular cylinder.
coarse grid
Nodes Elements
351
610
411
654
Background grid
Shock-fitted grid
3
fine grid
Nodes Elements
5261
10151
5415
10229
3
2
2
0.6
0.6
0.4
0.4
1
1
0
y(m)
0
0.2
y(m)
y(m)
y(m)
0.2
0
-0.2
-1
-0.2
-1
-0.4
-0.6
1
1.2
1.4
1.6
1.8
0
1
2
3
4
-0.4
-0.6
2
x(m)
-2
-3
0
1
1.2
1.4
5
6
-3
1.6
1.8
2
x(m)
-2
0
1
2
3
4
x(m)
x(m)
(a) Coarse grid
(b) Fine grid
5
6
Figure 4.8: Grids used for the inviscid flow over a circular cylinder.
The comparison between the shock-capturing and shock-fitting results
is shown in Fig. 4.9, where pressure contour plots are shown for both
the coarse and the fine grid solutions. The non-linear B scheme has
been used to obtain a nominally second-order-accurate solution when
the shock is captured, whereas the linear LDA scheme has been used in
Numerical results
53
3
3
2
2
0.6
0.6
0
S-F
y(m)
y(m)
0.2
S-C
S-C
0
S-F
-0.2
-1
-0.4
-0.6
1
1.2
1.4
-3
1.6
1.8
0
1
2
3
4
0.2
S-C
0
S-F
S-C
0
S-F
-0.2
-1
-0.4
-0.6
2
x(m)
-2
1
1.2
1.4
6
-3
1.6
1.8
2
x(m)
-2
5
p(Pa)
5E+07
4.6E+07
4.2E+07
3.8E+07
3.4E+07
3E+07
2.6E+07
2.2E+07
1.8E+07
1.4E+07
1E+07
6E+06
2E+06
0.4
1
y(m)
0.4
1
y(m)
p(Pa)
5E+07
4.6E+07
4.2E+07
3.8E+07
3.4E+07
3E+07
2.6E+07
2.2E+07
1.8E+07
1.4E+07
1E+07
6E+06
2E+06
0
1
2
3
4
x(m)
x(m)
(a) Coarse grid
(b) Fine grid
5
6
Figure 4.9: Pressure contour plot: comparison between the shock-fitting (S-F) and
shock-capturing (S-C) second-order-accurate solutions.
the shock-fitting calculation. The use of a linear scheme in the shockfitting calculation is made possible by the fact that, when all shocks are
fitted, the LDA scheme is used only in smooth regions of the flow field.
Figure 4.9 clearly shows that, using grids of comparable resolution, shockfitting not only avoids smearing the shock, but it also allows to obtain
a much cleaner solution within the entire shock layer characterized by
a low numerical error also on the coarse mesh. It is worth to observe
that the shock-capturing solution computed on the coarse mesh features
a strong asymmetry with respect to the stagnation point, whereas the
asymmetry is completely absent in the shock-fitting solution computed
on the corresponding coarse mesh.
Finally, the results obtained for the this same testcase by coupling the
two di↵erent CFD solvers, COOLFluiD and eulfs, with the same shockfitting code, have been compared in Fig. 4.10, where pressure iso-contour
lines are displayed for both the fist-order-accurate N and second-orderaccurate LDA schemes. Not surprisingly, the results are superimposed
since the two codes implement the same FS discretization schemes.
54
Numerical results
0.4
y(m)
0.2
___ N - COOLFluiD
_ _ N - EulFS
___ LDA - COOLFluiD
_ _ LDA - EulFS
0
-0.2
-0.4
0.8
1
1.2
1.4
1.6
1.8
2
x(m)
Figure 4.10: Pressure contour plot for the shock fitting solutions: comparison between
COOLFluiD and eulfs solutions.
4.2.2
Ideal gas, viscous flow over a circular cylinder
The second configuration considered is an hypersonic flow over a 1 m radius circular cylinder. This problem has been proposed by NASA Langley
aerothermodynamic team [35, 36, 38] in order to asses the capabilities of
unstructured Finite Volume schemes to correctly predict the heat flux
wall distribution for hypersonic flows on blunt bodies. Tab. 4.5 summarizes the freestream conditions used for this testcase. The grid used
is a semi-unstructured mesh obtained by cutting into two triangles each
quadrilateral of a structured grid. The triangulation is symmetric along
the stagnation streamline and there is a slight mesh refinement around
the shock region, as shown in Fig. 4.11. The background mesh is made
of 6993 nodes and 13640 elements, while the shock-fitting grid has 7181
nodes and 13826 elements.
Pressure and temperature contour plots are shown in Fig. 4.12 for both
shock-capturing nad shock-fitting second order results. The B scheme
has been used for the fitted solution while the LDA scheme has been
used for the captured solution. As in the inviscid testcase shown before, shock-fitting provides a better accuracy in the shock-layer than the
shock-capturing. This is particular evident if we look at the temperature
Numerical results
55
Table 4.5: Freestream conditions for a viscous flow over a 1m radius cylinder.
Physical quantity
M1
p1
T1
Tw
Radius
Free-stream Value
17.5
57.65 Pa
200.0 K
500.0 K
1.0 m
2
0.4
1
0
y(m)
y(m)
0.2
0
-0.2
-1
-0.4
1
1.2
1.4
1.6
x(m)
-2
0
1
2
3
4
5
x(m)
Figure 4.11: Grid used for the viscous flow over a 1 m radius circular cylinder.
56
Numerical results
contour plot in the stagnation region (see Fig. 4.12(b)). The wall distribution of the pressure coefficient is displayed in Fig.4.13(a) while the
skin friction coefficient is displayed in Fig. 4.13(b). In all plots reporting
the distribution of the wall quantities, the abscissa is the azimuthal angle, which takes values zero at the stagnation point. The various curves
shown include the S-C solution obtained with COOLFluiD on the semiunstructured mesh showed in Fig. 4.11, the S-F solution and the reference solution obtained by Peter Gno↵o using FV NASA code LAURA
(FV) on a structured grid [38]. The comparison between the S-F and
the LAURA results is satisfactory as far as the pressure coefficient is
concerned, see Fig. 4.13(a). However, when looking at the skin friction
coecient, Fig. 4.13(b), the di↵erencies between S-C and S-F computation
are not significant. As it is possible see in Fig. 4.13(a), the comparison
between the S-F results and the LAURA results is satisfactory for the
pressure coefficient, however when looking at the skin friction coefficient
4.13(b) the di↵erencies between S-C and S-F computation are not significant. The heat flux distribution on the wall is displayed in Fig. 4.14; we
can see that even though the S-F results does not shown the spike present
in the S-C results, di↵erences of the order of 10
Temperature and pressure distributions along the stagnation streamline
are shown in Fig. 4.15. It is possible to notice that the S-F results are
in good agreement with the LAURA results in the shock-layer. The S-C
results are not monotone and show a peak in the post-shock region for
both pressure and temperature. In particular, the S-C pressure distribution over-predicts the reference one in all the shock layer region.
Pressure and temperature distributions within the boundary layer are
shown in Fig. 4.16. It is clear that post-shock conditions influence the
solution in the boundary layer. Finally taking a look to the adimensional
total enthalpy (see Fig. 4.17) it is possible to notice the capability of the
S-F algorithm to preserve total enthalpy through the shock.
Numerical results
57
p (Pa)
2
0.4
1
0.2
S-F
S-C
0
S-C
0
S-F
S-F
-0.2
-1
-0.4
S-F
-0.4
1
1.2
1.4
1.6
2
3
4
1
-2
x(m)
1
S-C
0
-0.2
-1
-2
0
0.4
0.2
S-C
0
y(m)
y(m)
1
T (K)
12500
11500
10500
9500
8500
7500
6500
5500
4500
3500
2500
1500
500
y(m)
2
y(m)
23000
21500
20000
18500
17000
15500
14000
12500
11000
9500
8000
6500
5000
3500
2000
500
5
1.2
1.4
1.6
x(m)
0
1
2
3
4
x(m)
x(m)
(a) Pressure
(b) Temperature
5
Figure 4.12: Pressure and temperature contour plot: comparison between the shockfitting and shock-capturing second-order-accurate accurate solutions.
2
0.006
COOLFluiD S-C
COOLFluiD S-F
LAURA
0.0055
0.005
1.5
0.0045
0.004
Cf
Cp
0.0035
1
0.003
0.0025
0.002
0.5
0.0015
0.001
COOLFluiD S-C
COOLFluiD S-F
LAURA
0
-50
0
0.0005
50
0
-50
0
50
θ(°)
θ(°)
(a) Pressure coefficient
(b) Skin friction coefficient
Figure 4.13: Wall distributions.
58
Numerical results
80
70
qW (W/cm2)
60
50
40
30
20
COOLFluiD S-C
COOLFluiD S-F
LAURA
10
-50
0
50
θ(°)
Figure 4.14: Heat flux distribution on the wall.
15000
22500
12500
20000
17500
10000
p (Pa)
T (K)
15000
COOLFluiD S-C
COOLFluiD S-F
LAURA
7500
COOLFluiD S-C
COOLFluiD S-F
LAURA
12500
10000
5000
7500
5000
2500
2500
0
0.2
0.4
0
0
0.2
0.4
x (m)
x (m)
(a) Temperature
(b) Pressure
Figure 4.15: Distributions along the stagnation streamline.
Numerical results
59
15000
25000
12500
24000
COOLFluiD S-C
COOLFluiD S-F
LAURA
23000
p (Pa)
T (K)
10000
7500
22000
5000
COOLFluiD S-C
COOLFluiD S-F
LAURA
21000
2500
0 -5
10
10
-4
10
-3
10
20000 -5
10
-2
10
-4
10
-3
x (m)
x (m)
(a) Temperature
(b) Pressure
10
-2
Figure 4.16: Distributions along the stagnation streamline (enlargement of the boundary layer).
1.1
H
2
0.9
0.2
S-C
0
S-F
y(m)
y(m)
1
1
0.4
S-C
0
S-F
0.8
COOLFluiD S-C
COOLFluiD S-F
0.7
-0.2
-1
H/Href
1
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
0.9
0.89
0.88
0.87
0.86
0.85
0.6
-0.4
1
-2
1.2
1.4
1.6
x(m)
0
1
2
3
4
5
0.5
0
0.2
0.4
x(m)
x (m)
(a) Contour plot
(b) Distribution along the stagnation line
Figure 4.17: Adimensional total enthalpy.
60
Numerical results
4.2.3
Dissociated nitrogen, inviscid flow over a circular cylinder
The nonequilibrium shock-fitting algorithm has been tested by reference
to the flow conditions of Hornung’s experiment [41], listed in Tab. 4.6.
The grid used is obtained by a 120⇥120 structured mesh cutting into two
triangles each quadrilateral. The triangulation is symmetric in the stagnation region, as shown in Fig. 4.18. The background mesh is made of
14884 nodes and 29282 elements, while the shock-fitting grid at steadystate, which di↵ers from the background triangulation only in the neighbourhood of the fitted shock, has 15039 nodes and 29371 elements.
0.02
0.002
y(m)
0.004
y(m)
0.04
0
0
-0.002
-0.02
-0.004
-0.018
-0.016
-0.04
-0.014
x(m)
-0.02
0
0.02
0.04
0.06
0.08
x(m)
Figure 4.18: Grid used for the Nitrogen flow over a circular cylinder.
Table 4.6: Freestream conditions for a nitrogen flow around a 1 inch radius cylinder.
Physical quantity
M1
p1
T1
Tv,1
↵N,1
Radius
Free-stream Value
6.13
2908.0 Pa
1833.0 K
1833.0 K
0.07
0.0254 m
The numerical simulations were carried out for inviscid flow, since the
Numerical results
61
purpose of this study is to asses the capability of the unstructured shockfitting technique to correctly predict the shock position on blunt bodies
in the case of thermochemical nonequilibrium flows. Thermochemical
properties have been provided by the library MUTATION [60], while the
2-temperature model of Park [61, 62] has been used. Since the PDEs
governing dissociating nitrogen can be very sti↵, it has been necessary
to use the implicit backward Euler scheme to obtain the steady-state
solution. As shown in Fig. 4.19, the shock-wall distance and the interference fringe patterns agree reasonably well with the experimental results
obtained by Hornung [41].
1.2
1
COOLFluiD S-F (Nc)
COOLFluiD S-F (LDAc)
Hornung’s experiment
∆/R
0.8
0.6
0.4
0.2
0
0
20
40
60
80
θ(°)
(a)
(b)
Figure 4.19: COOLFluiD + SF vs. Hornung’s experimental measurements: a) Nondimensional shock-wall distance and b) finite interference fringe patterns. ✓ is the azimuthal angle which takes value 0 at the stagnation point, /R is the shock-wall distance
divided by the cilinder’s radius.
In Fig. 4.20 the pressure contour plot computed using the first-orderaccurate Nc scheme is compared with the one obtained by Wang and
Zhong [76] using a third-order-accurate finite di↵erence shock fitting scheme
on structured meshes. The good agreement between the two solutions obtained by the two di↵erent schemes with di↵erent order of accuracy proves
that both solutions are grid-independent, i.e. characterized by a very low
numerical error.
62
Numerical results
0.02
y(m)
p
140000
130000
120000
110000
100000
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
-0.02
-0.02
0
0.02
0.04
x(m)
(a) COOLFluiD + SF
(b) Reproduced from [76]
Figure 4.20: Pressure contour plot: qualitative comparison between the fitted solution obtained with COOLFluiDand the solution obtained by Wang and Zhong [76].
Hornung’s experimental measurements of the shock stando↵ distance are represented by
the red circles.
4.3
4.3.1
EHD results
Parallel electrodes
In order to validate the EHD solver, a 2D channel with two parallel
electrodes has been considered. The channel walls are 1 m apart and the
electrodes are 0.5 m wide, so that the domain simulated has a length of
1 m in both the x and y-directions. In this test the electrical conductivity
has been kept constant ( = 1 ⌦ 1 m 1 ) therefore Laplace’s equation
(r2 = 0) has been solved in spite of Eq. (4.3). The electrodes have
a specified potential: one volt on the upper electrode and zero on the
lower; Neumann boundary conditions are applied along the remaining
@
boundaries so that the normal component of the gradient is zero, @n
= 0.
Tab. 4.7 lists the boundary conditions used for the EHD solver.
For this testcase, the electric potential is symmetric about the center
of the electrode as seen in Fig. 4.21, where Fig. 4.21(a) has been obtained using a uniform grid made of 7931 nodes and 15606 elements and
Fig. 4.21(b) is obtained by Gaitonde [32]. The comparison between the
Numerical results
63
Table 4.7: Boundary conditions for the EHD solver.
CFD boundary conditions
Far field
Subsonic inlet
Supersonic inlet
Subsonic outlet
Supersonic outlet
Wall (insulated)
Wall (electrode)
EHD boundary conditions
Neumann
Neumann
Neumann
Neumann
Neumann
Neumann
Dirichlet
Condition
@ /@n = 0
@ /@n = 0
@ /@n = 0
@ /@n = 0
@ /@n = 0
@ /@n = 0
= specif ied
two solution shows a good agreement.
0.85
1
0.95
0.75
0.85
0.75
0.8
0.75
0.95
0.85
0.75
0.65
0.55
0.45
0.35
0.25
0.15
0.05
0.65
0.65
0.6
0.55
Y
0.55
0.45
0.45
0.4
0.35
0.35
0.25
0.2
0.2
5
0.15
0.05
0
-0.4
-0.2
0
0.2
0.4
X
(a)
(b)
Figure 4.21: Electric potential contours for the electrode channel with a constant
electrical conductivity ( = 1 ⌦ 1 m 1 ): a) eulfs results and b) reprinted from [32].
A second validation test has been performed, simulating two parallel electrodes separated by a distance of one meter along the y-axis with a nonuniform electrical conductivity. As in the previous test, the elecrodes
have a specified potential such that the top electrode plate is equal to
one volt and the bottom is set to zero. Fig. 4.22(a) illustrates the domain
with a nonuniform mesh used in the simulation, which is made of 22335
nodes and 44091 elements. Considering = it is possible to obtain an
analytical solution for the Eq. (2.67) as it follows.
r·( r )=0
(4.1)
64
Numerical results
r =
✓
k 1 k2
,
◆
(4.2)
@
@
k1
k2
dx +
dy = dx + dy
@x
@y
Z
Z
Z
d = k1 dx + k2 dy
d =
1
2
2
=
(4.3)
(4.4)
= k 1 x + k2 y + k 3
(4.5)
p
2 (k1 x + k2 y + k3 )
(4.6)
where k1 , k2 and k3 are trhee constants.
Considering the set of boundary conditions shown in Fig. 4.22(b) we
obtain the following analytical solution
p
= y
(4.7)
1
1
Dirichlet (φ = 1V)
0.8
0.8
0.6
Neumann
(∂φ/∂n = 0)
Y
Y
0.6
0.4
0.4
0.2
0.2
0
0
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-0.6
Dirichlet (φ = 0V)
-0.4
-0.2
0
0.2
X
X
(a)
(b)
0.4
0.6
Figure 4.22: Parallel electrodes with a non constant electrical conductivity ( = ):
a) mesh b) boundary conditions.
A comparison between the analytical and the numerical electrical potential contour plots is shown in In Fig. 4.23(a). In Fig. 4.23(b), the
Numerical results
65
theoretical and computed distributions of the electrical potential along
the y-axis are shown for a constant ( = 1) and a variable ( = )
electrical conductivity model.
Numerical
1
Analytical
1
Numerical (σ = φ )
Analytical (σ = φ )
Numerical (σ =1 )
Analytical (σ =1 )
φ
0.8
Y
0.6
0.4
0.2
0
-0.4
-0.2
0
0.2
0.8
0.6
Y (m)
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.4
0.2
0
0.4
0
0.2
0.4
X
φ
(a)
(b)
0.6
0.8
1
Figure 4.23: Parallel electrodes with a non constant electrical conductivity ( = ):
a) electric potential contour plot (numerical vs. analytitcal solution) b) distribution of
the electric potential along the y-axis (numerical vs. analytitcal solutions).
4.3.2
Ionized argon, inviscid flow in a nozzle with
two opposite electrodes
Preliminary results have been obtained for a subsonic nozzle with two
parallel electrodes on the surface. Tab. 4.8 shows the reservoir test conditions while the electrodes configuration and the geometry of the nozzle
are shown in Fig. 4.24(a). The mesh used is made of 26174 nodes and
51228 elements. Fig. 4.24(b) shows the contour plot of the nondimensional electrical potential.
Table 4.8: Inlet flow conditions.
Physical quantity
Total pressure
Total temperature
Ionisation degree
Value
1 ⇥ 105 Pa
5000 K
1 ⇥ 10 7
Fig. 4.25 shows the area weighted distributions along the nozzle axist of
the temperature, for di↵erent values of the generator potential, Vg . It is
66
Numerical results
0.15
0.04
0.1
0.02
0
Y
Y
0.05
1.92
1.82
1.72
1.62
1.52
1.42
1.32
1.22
1.12
1.02
0
-0.05
-0.02
-0.1
-0.04
-0.15
-0.3
-0.2
-0.1
0
-0.3
-0.28
-0.26
-0.24
-0.22
-0.2
X
X
(a) Nozzle geometry and electrodes configuration.
(b) Contour plot of the nondimensional electrical
potential.
Figure 4.24: Flow in a converging nozzle with two parallel electrodes.
possible to see the e↵ect on the flowfield of the ohmic heating due to the
electric potential imposed on the elctrodes.
5200
5000
4800
T (K)
4600
4400
4200
4000
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
x (m)
Figure 4.25: Temperature distributions along the nozzle axis for di↵erent values of
the generator potential.
Chapter 5
Conclusion and future
work
In this work, the investigation of a chemical non-equilibrium plasma has
been carried out by using eulfs, an in-house developed, shock-capturing
CFD code which uses compact-stencil, fluctuation splitting schemes on
unstructured grids made of triangular and tetrahedral elements. In the
code, the inviscid conservative equations have been succesfully coupled
with a kinetic model for argon plasma where argon metastable has been
considered as an independent species. In presence of shock waves, the
CFD code can either capture the shock or be coupled with a newly developed, unstructured, shock-fitting algorithm which treats the discontinuities as moving boundaries that border regions of the flow-field were a
smooth solution to the governing PDEs exists. When operating in shockfitting mode, the shock-capturing code is only used to solve the smooth
regions of the flow field and it is left to the shock-fitting algorithm to
enforce the Rankine-Hugoniot jump relations and preserve the species
concentrations through the fitted shocks.
The shock-fitting algorithm has been developed in a modular way then
it has been possible also to couple it with the VKI’s COOLFluiD code.
Thanks to the vaste choice of thermochemical models o↵ered by COOLFluiD,
it has been possible to extend the proposed shock-fitting technique to
thermochemical non-equilibrium flows. Results obtained in the computation of the flowfield over circular cylinders immersed in perfect gas,
chemical and thermochemical nonequilibrium hypersonic flow, show that
67
68
Conclusion and future work
shock-fitting not only allows a better modelling of the bow shock providing a good estimate of the position and the shape of the discontinuity,
but also improves the solution quality within the entire shock-layer.
Finally, the eulfs CFD solver has been successfully coupled with a Poisson
solver; this will pave the way for simulations of non equilibrium plasma
in which the electromagnetic interaction is also accounted for. Future
work may include the addition of an equation to calculate the electron
temperature, which is necessary to model discharges, overcoming a limit
of the present model that equalizes the electron gas temperature with that
of heavy species. Interesting future applications of this model could be
the investigation of plasma-flow interaction in DBDs, a device proposed
as actuator in high speed flow. Other possible applications can be the
expansion of the LIBS plasma in di↵erent environment, such as high
pressure gas or in liquid. In all these cases, because of the small size of
the plasma an accurate characterization of the plasma require accurate
determination of the shock wave shape.
Conclusions and future work
Publications related to the thesis
The results presented in this dissertation have led to the following publications in international conference and journals:
Journal articles
• R. Pepe, A. Bonfiglioli, A. D’Angola, G. Colonna, R. Paciorri,
Shock-Fitting Versus Shock-Capturing Modeling of Strong Shocks
in Nonequilibrium Plasmas. IEEE Transactions on Plasma Science,
Volume 42, Issue 8, Part 1, 2526-2527, 2014.
• J. Garicano Mena, R. Pepe, A. Lani, H. Deconinck, Assessment of
Heat Flux Prediction Capabilities of Residual Distribution Method:
Application to Atmospheric Entry Problems, under review in Communications in Computational Physics.
• R. Pepe, A. Bonfiglioli, A. D’Angola, G. Colonna, R. Paciorri,
An unstructured shock-fitting solver for hypersonic plasma flows in
chemical non-equilibrium, under review in Computer Physics Communications.
Conference Papers
• R. Pepe, A. Bonfiglioli, R. Paciorri, A. Lani, J. G. Mena, C. F.
Olliver-Gooch, Towards a modular approach for unstructured shockfitting. 11th World Congress on Computational Mechanics, ECCOMAS 2014. 20-25 July 2014, Barcelona, Spain.
• J. G. Mena, R. Pepe, A. Lani, H. Deconinck, Assessment of Residual Distribution Method Heat Flux Prediction Capabilities: Application to Atmospheric Entry Problems. 52nd Aerospace Sciences
Meeting, AIAA. 13-17 January 2014, National Harbor, U.S.A.
• R. Pepe, G. Colonna, A. Bonfiglioli, A. D’Angola, R. Paciorri, A
selfconsistent unstructured solver for weakly ionized gases. XXXI
69
70
Conclusion and future work
International conference on phenomena in ionized gases, 14-19 July
2013, Granada, Spain. (Poster)
• R. Pepe, A. Bonfiglioli, A. D’Angola, G. Colonna, R. Paciorri, An
unstructured solver for argon plasma flows with reduced state-tostate kinetics. 44th AIAA Plasmadynamics and Lasers Conference,
AIAA. 24-27 June 2013, San Diego, U.S.A.
• M. Onofri, R. Paciorri, R. Pepe, A. Bonfiglioli, A. D’Angola, G.
Colonna, A new computational technique for re-entry flow calculations based upon a shock-fitting technique for unstructured grids.
The 4th International ARA Days, The Atmospheric Reentry Association. 27-29 May 2013, Arcachon, France.
Technical reports
• R. Pepe, An unstructured shock-fitting solver for hypersonic flows.
Von Karman Institue for Fluid Dynamics Project Report, Chausse
de Waterloo, 72 B-1640, Rhode-Saint-Gense, Belgium.
Appendix A
Split Jacobian matrix
In this Appendix the derivation of the split Jacobian matrices introduced
in Chapter 3 and used to compute the generalised inflow parametershas
been described. The Cartesian x, y and z components of flux vector
Fc = Fci ei read:
0
B
B
Fc1 = B
B
@
⇢i u
⇢uH
⇢u2 + p
⇢uv
⇢uw
1
C
C
C,
C
A
0
B
B
Fc2 = B
B
@
⇢i v
⇢vH
⇢uv
⇢v 2 + p
⇢vw
1
C
C
C,
C
A
0
B
B
Fc3 = B
B
@
⇢i w
⇢wH
⇢uw
⇢vw
⇢w2 + p
1
C
C
C (A.1)
C
A
In Eq. (A.1) we have used u,v and w to denote the components of the
fluid velocity along the Cartesian axes and ei is the unit vector of the
Cartesian axis i.
The Jacobian matrices matrices Ai = @Fci /@U, already introduced in
Eq. (2.48), read:
0
B
B
A1 = B
B
@
( ij ↵i )u
0
↵i
0
0
u(⇧⇢j H) u(1 + ⇧⇢E ) H + u⇧⇢u u⇧⇢v u⇧⇢w
⇧⇢j u2
⇧⇢E
⇧⇢u + 2u ⇧⇢v ⇧⇢w
uv
0
v
u
0
uw
0
w
0
u
71
1
C
C
C
C
A
(A.2a)
72
0
( ij ↵i )v
0
0
↵i
0
B v(⇧⇢
H) v(1 + ⇧⇢E ) v⇧⇢u H + v⇧⇢v v⇧⇢w
j
B
uv
0
v
u
0
A2 = B
B
@ ⇧⇢j v 2
⇧⇢E
⇧⇢u ⇧⇢v + 2v ⇧⇢w
vw
0
0
w
v
0
B
B
A3 = B
B
@
1
C
C
C
C
A
( ij ↵i )w
0
0
0
↵i
w(⇧⇢j H) w(1 + ⇧⇢E ) w⇧⇢u w⇧⇢v H + w⇧⇢w
uw
0
w
0
u
vw
0
0
w
v
2
⇧⇢j w
⇧⇢E
⇧⇢u ⇧⇢v ⇧⇢w + 2w
(A.2b)
1
C
C
C (A.2c)
C
A
Partial derivatives ⇧⇢j , ⇧⇢E , etc. of the functional p = ⇧(⇢i , ⇢E, ⇢u) with
respect to the conservative variables have been introduced in the matrices
defined by Eq. (A.2). Observe that p = ⇧(⇢i , ⇢E, ⇢u) is not a thermodynamic relation [56] because pressure is a function of only Ns + 1 independent thermodynamic variables, for a gas in chemical non-equilibrium.
If we choose the densities ⇢i of the chemical species and the energy per unit
of volume ✏ = ⇢e as the set of independent thermodynamic variables [74]
and denote with i , resp. , the partial derivatives of the thermodynamic
pressure w.r.t. this set of independent variables:
=
i
✓
@p
@⇢i
◆
,
",⇢j ,j6=i
=
✓
@p
@"
◆
(A.3)
⇢i ,i=1...Ns
the following relations between the two sets of partial derivatives can be
easily established:
⇧ ⇢i =
@p
=
@⇢i
i
+
u·u
,
2
⇧⇢E =
@p
= ,
@⇢E
⇧⇢u =
@p
=
@⇢u
u.
(A.4)
In the case of a mixture of chemical species having the same number of
degrees of freedom, such as the argon plasma described in the Sect. 2.2.1,
the thermodynamic pressure derivatives are constant and, since all species
behave as a monoatomic atom, equal to:
i
5
= hfi ,
3
5
= .
3
(A.5)
73
Let us now compute the following matrix:
0
1
( ij ↵i )un
0
↵ i nt
A
Hnt + un ⇧⇢u t
An = Ai ni = @ un (⇧⇢j H) (1 + ⇧⇢E )un
⇧⇢j n un u
⇧⇢E n
⇧⇢u n + un + un Id⇥d
(A.6)
where n = ni ei is an arbitrary unit vector, un = u · n and terms like un
denote a dyadic tensor. Matrix An has the following eigenvalues:
0
B
B
B
B
⇤n = B
B
B
B
@
un
1
...
un
un
un + a
un
a
un
C
C
C
C
C
C
C
C
A
(A.7)
where a is the sound speed, given by:
a
2
=
Ns
X
u · u)
↵i ⇧⇢i + ⇧⇢E (H
i=1
= (1 + ⇧⇢E )
(A.8a)
p
⇢
(A.8b)
Each eigenvalue is associated with a right and a left eigenvector. Grouping
the right eigenvectors by column and the left eigenvectors by row, the
following matrices are obtained:
0
B
u·u
Rn = B
@
ij
u
0
⇧⇢ j
⇧⇢E
⇢u · s ⇢
⇢s
h
⇢ ↵ai
H
ah
+u·n
i
u
⇢ a +n
i
⇢
h
⇢ ↵ai
H
ah
⇢
u
a
u·n
i
n
i
0
1
C
⇢u · t C
A
⇢t
(A.9)
74
0
Ln = (Rn )
1
B
B
B
B
B
=B
B
B
B
@
↵ i ⇧⇢ j
a2
u·s
⇢
ij

1 ⇧⇢ j
2⇢
a

1 ⇧⇢ j
2⇢
a
⇧
↵i a⇢E
2
0
u·n
⇧⇢E
2⇢a
+u·n
⇧⇢E
2⇢a
u·t
⇢
0
1
2⇢
1
⇧
t
↵i a⇢E
2 u
t
s
⇢

⇧⇢E t
a u
nt

1
t
2⇢ n +
⇧⇢E t
a u
tt
⇢
C
C
C
C
C
C
C
C
C
A
(A.10)
where n, s e t are a triad of orthogonal unit vectors.
Once the eigenvalues and the left and right eigenvectors are known, it is
possible to factorise the matrix An as follows:
A n = R n ⇤n L n
(A.11)
which, in turn, allow to compute the positive and negative parts of An :
±
A±
n = R n ⇤n L n
(A.12)
Lengthy algebra reveals that:
0
1
a±
a±
a±
i,j
i,N +1
i,N +2
±
±
A
@ a±
A±
N +1,j aN +1,N +1 aN +1,N +2
n =
±
±
a±
N +2,j aN +2,N +1 AN +2,N +2
with i, j = 1, ..., Ns and the individual entries read:

✓ ±
◆
±
±
±
⇧
+
un
⇢
j
+
+
±
±
±
ai,j = i,j 0 + ↵i 2
0
a
2
2
a
✓ ±
◆
±
↵
⇧
+
i ⇢E
+
±
a±
0
i,N +1 =
2
a
2

✓ ±
◆
±
±
±
t
↵
u
+
i
+
+
±
±
ai,N +2 =
⇧E
+
nt
0
a
a
2
2
✓
◆✓ ±
◆
⇣
⌘u ±
±
⇧ ⇢j H
n +
++
±
2
±
aN +1,j =
un
0 + ⇧⇢j H
2
a
2
a
2
 ✓ ±
◆
±
±
±
H
un
++
+
±
±
±
aN +1,N +1 = 0 + ⇧⇢E 2
+
0
a
2
2
a
(A.13)
(A.14a)
(A.14b)
(A.14c)
±
(A.14d)
(A.14e)
a±
N +1,N +2 =
a±
N +2,j
=
✓
✓
u n nt
⇧⇢E H t
u
a2
⇧⇢j
u
a2
◆✓
un n
✓
◆✓
±
+
+
2
±
+
+
2
±
+ ±
2
✓
◆✓ ±
⇧⇢E
++
= nn
uu
2
a
2
a±
N +2,N +1
A±
N +2,N +2
75
⇧⇢E
=
a
±
+
±
◆
◆
H
u
n
±
nt ⇧E ut
0 +
2
a
a
(A.14f)
◆
◆
±
± ✓⇧
u
⇢
n
j
±
+ +
n
u
0
2
a
a
(A.14g)
◆
±
±
± u
+ +
n
(A.14h)
0
a
2
◆ ✓
◆ ±
±
1
⇧⇢E
+
±
un
nu
+ ±
0 +
0 Id⇥d
a
a
2
(A.14i)
±
±✓
±
+
where:
0
A.1
= un ,
±
=
0
± a.
(A.15)
Jacobian transformations
In this section we report the Jacobian transformations between the conserved variables U and the parameter vector Z:
0
1
zi + ij z⇢
0
0t
@U @
@p
@p A
zH @z
z⇢ @z@pH
=
@zu
j
@Z
zu
0
z⇢ I
0
1
1 zi
0
0t
ij z⇢
2 z⇢2
@Z B
⇧⇢ j
zu
1 zH
1
=B
z⇢
2 z⇢2
z⇢ (1 + ⇧⇢E )
z⇢2 ⇧⇢E
@
@U
1 zu
1
0
2 z2
z⇢ I
⇢
Knowing the expression of p(Z):
p(Z) =
z⇢
1 + ⇧⇢E
Ns
X
j=1
⇧⇢j zj + ⇧⇢E zH + ⇧⇢u · zu
we can write the pressure deivatives z⇢ =
P Ns
s=1 zs
and
(A.16)
1
C
C
A
(A.17)
!
@p
@p
@zj , @zH
(A.18)
,
@p
@zu :
76
@p
1
=
⇧⇢s z⇢ +
@zs
1 + ⇧⇢E
Ns
X
j=1
⇧⇢j zj + ⇧⇢E zH + ⇧⇢u · zu
!
(A.19)
@p
⇧⇢E
=
z⇢
@zH
1 + ⇧⇢E
(A.20)
@p
⇧⇢u
=
z⇢
@zu
1 + ⇧⇢E
(A.21)
Here we report Jacobian transformations between the flux vectors F, G,
H and the parameter vector Z
0
zu ij
B 0
@F B
@p
=B
@zj
B
@Z @
0
0
0
zv ij
B 0
@G B
=B 0
@p
@Z B
@ @z
j
0
0
B
@H B
=B
@Z B
@
0
zu
0
0
zi
0
zH
0
@p
@p
2zu + @zu @zv
zv
zu
zw
0
0
zv
0
0
0
zv
@p
@zH
@p
@zH
@p
@zu
0
0
zw ij 0
0
zw
0
0
0
0
@p
@zj
@p
@zH
zi
zH
zu
@p
2zv + @z
v
zw
0 0
0 0
zw 0
0 zw
@p
@zu
@p
@zv
1
0
0 C
@p C
C
@zw C
0 A
(A.22)
1
0
0 C
C
0 C
@p C
A
@z
(A.23)
zu
w
zv
zi
zH
zu
zv
@p
2zw + @z
w
1
C
C
C
C
A
(A.24)
A.2
77
Transformations between U and Z
The transformations between the conservative variables U = (U⇢i , U⇢E , U⇢u )
and the parameter vector Z = (Z⇢i , ZH , Zu ) for a mixture of monoatomic
and thermally perfect gas, are reported in this section.
A.2.1
U!Z
ZH = p
U⇢

U⇢
Z ⇢i = q P i
Ns
U⇢
=p i
U⇢
i=1 U⇢i
U⇢E
✓
◆
Ns
1 U⇢u · U⇢u X
f
+
U⇢i hi
2U⇢
i=1
U⇢u
Zu = p
U⇢
A.2.2
(A.25)
(A.26)
(A.27)
Z!U
U⇢i = Z⇢i
Ns
X
j=1
Z ⇢j = Z ⇢i Z ⇢
(A.28)
78
U⇢E =
Z⇢
U⇢E = Z⇢ ZH

✓
ZH + (
Z⇢
1 + ⇧⇢E
N
s
Zu · Zu X
1)
+
Z⇢i hfi
2Z⇢
i=1
"
Ns
X
j=1
◆
⇧⇢j Z⇢j + ⇧⇢E ZH + ⇧⇢u · Zu
U⇢u = Zu Z⇢
(A.29)
#
(A.30)
(A.31)
Appendix B
Nondimensionalization
In this Appendix details on the nondimensional form of governing equations are provided. Two di↵erent kinds of nondimensionalization are considered for internal and external flows.
B.1
TCneq, external flows
The following choice of reference variables are considered:
Freestream density
Freestream velocity magnitude
Freestream temperature
Freestream viscosity
Reference length
⇢1 ,
q1 ,
T1 ,
µ1 ,
L.
(B.1)
Denoting the non-dimensional variables by an asterisk, one obtains:
x⇤ = x/L
2
p⇤ = p/ ⇢1 q1
2
e⇤ = e/q1
where:
⇤
⇢ =
Ns
X
s=1
t⇤ = t/ (L/q1 ) u⇤ = u/q1
⇢⇤s = ⇢s /⇢1 T ⇤ = T /T1
2
e⇤v = ev /q1
⇢⇤s
Ns
X
⇢
⇢s
=
=
⇢
⇢1
s=1 1
79
(B.2)
(B.3)
80
Replacing definitions B.2 in the in the conservation equations, one obtains:

⇢1 q1 @⇢⇤s
+ r⇤ · (⇢⇤s u⇤ ) = Ss
⇤
L
@t

2
⇢1 q 1
@(⇢⇤ u⇤ )
+ r⇤ · (⇢⇤ u⇤ u⇤ ) + rp⇤ = 0
⇤
L
@t

3
⇢1 q 1
@(⇢⇤ E ⇤ )
+ r⇤ · (⇢⇤ u⇤ H ⇤ ) = 0
⇤
L
@t

3
⇢1 q 1
@(⇢⇤ e⇤v )
+ r⇤ · (⇢⇤ u⇤ e⇤v ) = Sev
⇤
L
@t
(B.4)
(B.5)
(B.6)
(B.7)
Dividing Equations. (B.4) to (B.7) by the factor appearing in parenthesis
on their left hands side, their non-dimensional form is obtained:
@⇢⇤s
L
⇤
⇤ ⇤
+
r
·
(⇢
u
)
=
S
s
s
@t⇤
⇢1 q 1
(B.8)
@(⇢⇤ u⇤ )
+ r⇤ · (⇢⇤ u⇤ u⇤ ) + rp⇤ = 0
⇤
@t
(B.9)
@(⇢⇤ E ⇤ )
+ r⇤ · (⇢⇤ u⇤ H ⇤ ) = 0
⇤
@t
(B.10)
@(⇢⇤ e⇤v )
L
⇤
⇤ ⇤ ⇤
+
r
·
(⇢
u
e
)
=
S
e
v
v
3
@t⇤
⇢1 q 1
(B.11)
The non-dimensional equation of state reads:
N
N
s
s
X
T1 X
⇤
⇤
p = 2
⇢s R s T =
⇢⇤s Rs⇤ T ⇤
q1 s=1
s=1
⇤
(B.12)
81
where
Rs⇤ =
B.2
T1
Rs
2
q1
(B.13)
TCneq, internal flows
The following reference variables are considered:
Inlet total pressure
Inlet total temperature
Inlet total density
Inlet velocity magnitude
Inlet total viscosity
Reference length
p01 ,
T10 ,
0
⇢
p1 ,
R1 T10 ,
µ01 ,
C.
(B.14)
which gives rise to the following set of non-dimensional variables:
⇣p
⌘
p
⇤
⇤
0
x = x/C t = t
R1 T1 /C u⇤ = u/ R1 T10
(B.15)
p⇤ = p/p01
⇢⇤s = ⇢s /⇢01
T ⇤ = T /T10
e⇤ = e/R1 T10
e⇤v = ev /R1 T10
The adimensional conservation equations of the single chemical species
are
@⇢⇤s
C
⇤
⇤ ⇤
p
+
r
·
(⇢
u
)
=
S
(B.16)
s
s
@t⇤
⇢01 R1 T10
and the adimensional vibrational energy is:
@(⇢⇤ e⇤v )
C
⇤
⇤ ⇤ ⇤
+
r
·
(⇢
u
e
)
=
S
e
v
v
3/2
@t⇤
⇢0 (R1 T 0 )
1
(B.17)
1
The non-dimensional equation of state reads:
N
N
N
s
s
s
X
⇢01 T10 X
1 X
⇤
⇤
⇤
⇤
⇤
p = 0
⇢ Ri T =
⇢ Rs T =
⇢⇤s Rs⇤ T ⇤
p1 s=1 s
R1 s=1 s
s=1
where
Rs⇤
⇢01 T10
= 0 Rs
p1
(B.18)
(B.19)
82
B.3
EHD, external flows
The following choice of reference variables has been considered for EHD
external flows:
Freestream density
Freestream velocity magnitude
Freestream temperature
Reference length
Reference electric potential
⇢1 ,
q1 ,
T1 ,
L,
(B.20)
ref .
Denoting the non-dimensional variables by an asterisk, we obtain:
x⇤ = x/L
2
p⇤ = p/ ⇢1 q1
2
e⇤ = e/q1
t⇤ = t/ (L/q1 ) u⇤ = u/q1
⇢⇤s = ⇢s /⇢1
T ⇤ = T /T1
⇤
= / ref E⇤ = EL/ ref
(B.21)

⇢1 q1 @⇢⇤s
+ r⇤ · (⇢⇤s u⇤ ) = Ss
⇤
L
@t
(B.22)

2
⇢1 q 1
@(⇢⇤ u⇤ )
+ r⇤ · (⇢⇤ u⇤ u⇤ ) + rp⇤ = 0
⇤
L
@t
(B.23)

2
3
⇢1 q 1
@(⇢⇤ E ⇤ )
⇤
⇤ ⇤ ⇤
+ r · (⇢ u H ) = ref
E⇤ · E⇤
⇤
2
L
@t
L
(B.24)
Conservative equations in non-dimensional form are given by the following
epressions:
@⇢⇤s
L
+ r⇤ · (⇢⇤s u⇤ ) =
Ss
⇤
@t
⇢1 q 1
(B.25)
@(⇢⇤ u⇤ )
+ r⇤ · (⇢⇤ u⇤ u⇤ ) + rp⇤ = 0
⇤
@t
(B.26)
2
@(⇢⇤ E ⇤ )
ref E⇤ · E⇤
+ r⇤ · (⇢⇤ u⇤ H ⇤ ) =
⇤
3
@t
L⇢1 q1
(B.27)
83
In Eq. (B.27), it possible to identify a nondimensional expression for the
electrical conductivity:
2
⇤
=
ref
3
L⇢1 q1
(B.28)
Subsituting Eq. (B.28) in Eq. (B.27), one obtains:
@(⇢⇤ E ⇤ )
+ r⇤ · (⇢⇤ u⇤ H ⇤ ) =
⇤
@t
⇤
E⇤ · E⇤
(B.29)
Considering Eq. (B.28), the Poisson’s equation becomes:
3
⇢1 q 1
r⇤ · [ ⇤ r⇤ ⇤ ] = 0
ref L
(B.30)
r⇤ · [ ⇤ r⇤ ⇤ ] = 0
(B.31)
then:
B.4
EHD, internal flows
The following set of reference variables is introduced:
Inlet total pressure
Inlet total temperature
Inlet total density
Inlet velocity magnitude
Reference length
Reference electric potential
p01 ,
T10 ,
0
⇢
p1 ,
R1 T10 ,
C,
ref .
(B.32)
which gives rise to the following set of non-dimensional variables:
⇤
x = x/C
p⇤ = p/p01
e⇤ = e/R1 T10
⇣p
⌘
p
0
t =t
R1 T1 /C
u⇤ = u/ R1 T10
⇢⇤i = ⇢s /⇢01
T ⇤ = T /T10
⇤
= / ref
E⇤ = ECref / ref
⇤
(B.33)
84
Then the conservation equations of the single chemical species are
@⇢⇤s
C
⇤
⇤ ⇤
p
+
r
·
(⇢
u
)
=
S
s
s
@t⇤
⇢01 R1 T10
(B.34)
and the conservation of the energy equation is:
2
@(⇢⇤ E ⇤ )
ref
+ r⇤ · (⇢⇤ u⇤ H ⇤ ) =
E⇤ · E⇤
0
0
3/2
@t⇤
C⇢1 (R1 T1 )
(B.35)
Introducing the non-dimensional electrical conductivity, one obtains:
2
⇤
=
ref
C⇢01 (R1 T10 )3/2
(B.36)
@(⇢⇤ E ⇤ )
+ r⇤ · (⇢⇤ u⇤ H ⇤ ) = ⇤ E⇤ · E⇤
(B.37)
⇤
@t
Similarly to external flows the non-dimensional Laplace’s equation reads:
⇢01 (R1 T10 )3/2 ⇤
r · [ ⇤ r⇤ ⇤ ] = 0
ref C
(B.38)
r⇤ · [ ⇤ r⇤ ⇤ ] = 0
(B.39)
then:
B.5
Nondimensional parameter vector
The adimensional parameter vector Z⇤ is given by the following expressions:
• Cneq
• Tcneq
Z⇤ =
Z⇤ =
where
↵s⇤
p
p
⇢⇤ (↵s⇤ , u⇤ , H ⇤ )t
(B.40)
⇢⇤ (↵s⇤ , u⇤ , H ⇤ , e⇤v )t
(B.41)
⇢⇤s
⇢s /⇢ref
⇢s
= ⇤=
=
= ↵s
⇢
⇢/⇢ref
⇢
(B.42)
B.6
85
Nondimensional pressure derivatives
In this section, a procedure to determine the pressure derivatives in nondimensional form is given.
P Ns ⇤ ⇤
⇢s R s
⇤ = PNs=1
(B.43)
s
⇤C ⇤
⇢
s=1 s Vs
⇤
s
= Rs⇤ T ⇤
⇤ e⇤s
(B.44)
2
First the pressure derivative k ⇤ is expressed. Defining Rsref = q1
/T1 for
0 0 0
external flows and Rsref = p1 /⇢1 T1 = R1 for external flow, one obtains:
P Ns
⇤ Rs
s=1 ↵s Rsref
 ⇤ = P Ns
⇤ C Vs
s=1 ↵s Rs
ref
P Ns
= PNs=1
s
↵ s Rs
s=1 ↵s CVs
=
(B.45)
In the Eq. [B.45] the relation ↵i⇤ = ↵i is considered. The following
non-dimensional form is used for external flows:
⇤
s
= Rs
T1 T
2 T
q1
1
es
s
= 2
2
q1
q1
(B.46)
es
s
=
0
R1 T 1
R1 T10
(B.47)

and for internal flows:
⇤
s
=
Rs T
R1 T10

86
Appendix C
Rate coefficients fits
C.1
Rate coefficients fit
In this Appendix numerical fits used to approximate the rate coefficients
for the electron-atom processes are reported. The following fitting functions have been used:
f 2 = m1 + m2
f 2 = m1 + m2
1000
Te
! m3
1000
Te
+ m4 exp
!m3
✓
+ m4 exp
Te
m5
◆
✓
Te
m5
+ m6 exp
◆
" ✓
(C.1)
Te
m7
◆2 #
(C.2)
The coefficients mi , reported in the following Tabs. C.1-C.5, have been
obtained using the least square method. In the following tables the coefficients with the associated standard errors are reported for the equilibrium
constant and the forward rate coefficients. In table C.1 the coefficients
mi for the equilibrium constant are reported. In Tables C.2, C.3 and C.4
the coefficients used for the fit of kf of e A processes are reported.
In Tab. C.5 the coefficients of Eq. (2.51) for the computation of the forward rate coefficients of A A processes are reported. In Eq. (2.51) T is
in K while the products kB T are in eV.
87
88
Table C.1: Coefficients of function f1 = log KeqI (cm 3 ) for the Ar ionization
Coefficient
m1
m2
m3
m4
m5
Value
52.222
-188.2
0.99254
16.486
319.23
Error
0.0.035015
0.099176
0.00015501
0.49331
8.5891
Table C.2: Coefficients of function f2 = log kf1 (cm 3 ) for the reaction 1 (e
ionization from Ar0 )
Coefficient
m1
m2
m3
m4
m5
m6
m7
Value
-18.8
-176.7
1.018
-9.464
1545.0
-2.242
9690.0
Coefficient standard error
0.0225
0.2488
0.00079
0.61466
58.61
0.27119
114.67
excitation)
Coefficient
m1
m2
m3
m4
m5
m6
m7
Value
-18.98
-133.6
1.003
-2.114
8409.0
-0.6737
2047.0
A
0.01023
0.12937
0.00056
0.07367
96.81
0.07929
71.71
A
Rate coefficients fits
89
ionization from Ar⇤ )
Coefficient
m1
m2
m3
m4
m5
m6
m7
Value
-15.39
-49.61
0.994
-1.02
6865.0
0.2212
9272.0
A
0.00115
0.01293
0.000159
0.02424
45.19
0.00979
105.53
Table C.5: Rate coefficients kf (cm3 /s) for A-A processes
Reaction
4
5
6
✓
cm3
br
s K1/2 eV
3.7755755 ⇥ 10
1.1436529 ⇥ 10
3.3010664 ⇥ 10
◆
17
17
16
"ij (eV )
15.76
11.55
4.21
0.3
0.2
Relative error (%)
0.1
0
-0.1
-0.2
-0.3
kf1
kf2
kf3
-0.4
-0.5
10000
20000
30000
T(K)
Figure C.1: Relative error associated to the e
translationa l temperature
40000
50000
A forward rate coefficient vs. the
90
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