` degli studi della Basilicata Universita

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 Jesu´s 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.

iii

iv

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 ix

x

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

xiv

Table of contents C.3 Coefficients of function f2 = log kf2 (cm 3 ) for the reaction 2 (e A excitation) . . . . . . . . . . . . . . . . . . . . . . 88 C.4 Coefficients of function f2 = log kf3 (cm 3 ) for the reaction 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 Damk¨oler 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 S˙ ev S˙ s 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 Damk¨oler number. The chemical Damk¨oler 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

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

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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 Jesu´s 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.

iii

iv

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 ix

x

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

xiv

Table of contents C.3 Coefficients of function f2 = log kf2 (cm 3 ) for the reaction 2 (e A excitation) . . . . . . . . . . . . . . . . . . . . . . 88 C.4 Coefficients of function f2 = log kf3 (cm 3 ) for the reaction 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 Damk¨oler 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 S˙ ev S˙ s 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 Damk¨oler number. The chemical Damk¨oler 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