PREFACE ix
LIST OF MAIN SYMBOLS xv
CHAPTER 1. GENERATION OF MULTIPHASE FLOWS 1
1.1. Creation of suspensions of solid particles in a gaseousphase 1
1.1.1. Creation of a homogeneous suspension of starch particles2
1.1.2. Soot formation 3
1.2. Creation of suspensions of bubbles in a liquid 5
1.2.1. Example of creation of a suspension of bubbles in aliquid 5
1.2.2. Influence of gravity on suspensions in pipes 7
1.2.3. Slug flows 8
1.3. Creation of suspensions of drops in a gas 9
1.3.1. Destabilization of fluid sheets and layers 10
1.3.2. Formation of droplets from filaments 22
1.3.3. Numerical simulation of primary atomization 32
1.3.4. Secondary atomization 42
CHAPTER 2. PROBLEMS AT THE SCALE OF A PARTICLE 47
2.1. Force exerted by a fluid on a spherical particle 48
2.1.1. Perfect incompressible fluid 48
2.1.2. Incompressible viscous fluid 51
2.2. Heat exchanges 59
2.3. Combustion of a drop of fuel in an oxidizing environment61
CHAPTER 3. SIMPLIFIED MODEL OF A NON-REACTIVE FLOW WITHPARTICLES 65
3.1. Variables characterizing the flow 66
3.2. Balance equations 68
3.2.1. Balances for the particles 69
3.2.2. Balances for the gaseous phase 71
3.2.3. Entropy balance and phenomenological relations 71
3.3. Application to the linearized study of sound propagation ina non-reactive dilute suspension 74
3.4. Two-phase dilute flows in nozzles 78
3.4.1. Flow with constant phase shifts 79
3.4.2. Numerical solutions 82
CHAPTER 4. SIMPLIFIED MODEL OF A REACTIVE FLOW WITH PARTICLES85
4.1. Balance equations for a reactive fog 85
4.1.1. Balances for the droplets 86
4.1.2. Balances of the mixture 88
4.1.3. Gaseous balances 90
4.1.4. Entropy balance of the spray and phenomenologicalrelations 90
4.1.5. Equations of the two-phase CEDRE solver 92
4.1.6. Modified equations to take account of an internaltemperature gradient of the drops: multi-layer model 93
4.2. Application to a spray flame 98
4.2.1. Application of a minimum model to the study of thethreshold of appearance of a pulsating flame 98
4.2.2. Application to the study of the resonant action of anacoustic wave on a spray flame 102
CHAPTER 5. RADIATIVE PHENOMENA 105
5.1. Basic values and fundamental relations in radiativetransfer 106
5.1.1. Definitions 106
5.1.2. Radiative Transfer Equation (RTE) 108
5.1.3. Radiative flux and power 110
5.1.4. Involvement of radiative heat transfer in the equationsof aerothermochemistry 113
5.1.5. Turbulence-radiation interaction (TRI) 118
5.1.6. Modeling of the radiative properties of gases 119
5.1.7. Modeling of the radiative properties of the particles124
5.2. Application to the hypersonic flow of atmospheric re-entry127
5.2.1. One-dimensional approximation for a re-entering body128
5.2.2. 3D calculations for a body experiencing re-entry 130
5.3. Application to the boundary layer above a flat plate withsoot formation and radiative transfer 132
5.3.1. Recap on a boundary layer with diffusion 133
5.3.2. Reminders about the Emmons problem 136
5.3.3. Influence of soot and radiative transfer 142
5.4. Application to combustion of aluminum-based solidpropellants 144
APPENDIX. CONCEPTS SURROUNDING THE HOPF BIFURCATION 149
BIBLIOGRAPHY 153
INDEX 173
List of Main Symbols
Latin characters
a, b partial derivatives of pressure with respect to
? and
A respectively
A chemical affinity; chemical species; monoatomic species
A2 diatomic species
A column matrix of chemical affinities in a multireactive medium
A, B Arrhenius coefficients
B Spalding parameter for mass transfer (
BM), or energy transfer (
BT), between a fluid mixture and an interface
c speed of sound; or speed of light in the void
c1,
cµ,
ce,
ck coefficients of the
k -
e method
C total number of moles per unit volume
Cj molar concentration per unit volume
Cp ,
Cv specific heat at constant pressure or at constant volume, respectively (
cp,
cv for the unit of mass)
d molecular droplet diameter; distance; or differential
D diffusion coefficient; or diameter
Da;i diffusion coefficient of species
a at quantic level
i strain rate tensor
Da Damköhler number
DT thermal diffusion coefficient
ev, a vibrational energy of diatomic species
a orthonormal basis vector
E internal energy (
e per unit mass)
E(
k) energy spectrum of turbulence
Ea activation energy
f parameter; reduced chemical production rate; or Blasius function
f´,
f? Reynolds, Favre fluctuation, respectively
f force acting on each unit mass
fj force acting on the unitary mass of the species
j f?(
r,
s) volumetric distribution in particle radius, at abscissa
s of a light beam of
? wave number
F Helmholtz free energy (
f for the unit of mass); generalized force; or any extensive quantity (
f for the unit of mass);
F force of
Fx,
Fy, Fz components
F source of impulsion due to molecular collisions
G Gibbs free enthalpy (
g for the unit of mass)
g acceleration due to gravity (of modulus
g)
gj chemical potential per unit mass of the species
j in a mixture
H enthalpy (
h per unit mass)
I? (
M,
u) directional monochromatic radiative intensity at any
M point in the
u direction Planck function
1 unit tensor
j, k chemical species
J flux with respect to an unmoving reference frame
J flux with respect to the barycentric motion of the fluid
JDk diffusion flux of the species
k equal to
?k (
vk -
v)
k Boltzmann's constant; wave number; or kinetic energy
v2/2
k(
T) specific reaction rate
K kinetic energy (
k per unit mass); number of chemical reactions in a mixture; heat exchange coefficient; or wave number
l length; latent heat per unit mass; or mean free path
l transfer length; or integral scale of turbulence
lD diffusion thickness of a non premixed flame
lf,
ld respective thicknesses of preheating, of reaction of a premixed flame
lK length scales of Kolmogorov
L length; molar latent heat; number of chemical elements in a mixture; or phenomenological coefficient
Le Lewis number
Lij,
lii phenomenological coefficients matrix of phenomenological coefficients of chemical reactions
m total mass
M molecular mass; diluent; Mach number; or material point
M molar mass
mj mass of the species
j Mj molar mass of the species
j unit mass flow rate; mass flow rate; mass flow rate of a nozzle
n total number of moles
nj number of moles of the species
j N number of species; number of molecules per unit volume; coordinate normal to an interface; or number of elements in a statistical calculus
N number of Avogadro
n,
N unitary normal to an interface; or to a surface
p thermodynamic pressure
P probability density pressure tensor
PR (
M) radiative power at any point
M , equal to - (?·
qR)(
M) Pr Prandtl number
q parameter; or heat flux volume flow rate
q heat flux vector
qR (
M) radiative flux vector in
M qrad radiative flux
Q partition function; quantity of heat; total enthalpy released per unit mass of pyrolysed fuel at the surface; or efficiency (
Qabs for absorption,
Qsca for scattering) heat released per unit time at the walls of a chemical reactor; or lateral heat flux in a nozzle molar enthalpy of formation of the species
j ( per unit mass)
r perfect gas constant per unit mass; or radius
R molar universal gas constant; radius; or electrical resistance
R reference frame Re Reynolds number
S entropy (
s per unit mass); area; or area of the cross section of a nozzle (
s) Arrhenius exponent
s curvilinear abscissa
SL, ,
St combustion velocity respectively laminar, standard, and turbulent
S, S surface
S symmetrical part of the velocity gradient tensor
Sc Schmidt number
S? (
s´,
u) monochromatic source function
t time; or diffusion thickness
T absolute temperature
Tf flame temperature
Ta,
Tad activation temperature, adiabatic temperature of a reaction, respectively u, v, w velocity components in Cartesian coordinates (
vr, v?, vz in cylindrical...
