Etude de la combustion de gaz de synthèse issus d'un processus de ...

Chapter 2
taken into account here. In this relation the proportionality of the burning velocity to the
flame stretch is represented by a parameter that was considered as a characteristic
length scale of the flame. This parameter is now generally known as the Markstein
length, which denotes the sensitivity of a flame to flame stretch. Asymptotic analyses of
Clavin and Williams, (1982) and Matalon and Matkowsky, (1982) and detailed modeling
of Warnatz and Peters, (1984) show a linear relationship between stretch rate and
burning velocity in the low-stretch regime. Thus, it is assumed that,
S − S = L κ
(2.33)
0
n n b
Where
gases.
0
S
n
is the unstretched flame speed and L b is the Markstein length of burned
tel-00623090, version 1 - 13 Sep 2011
When the observation is limited to the initial part of the flame expansion where the
pressure does not vary yet, then a simple relationship links the unstretched flame
speed to the unstretched burning velocity.
S
S
0
n
0
u
ρ
u
= = (2.34)
ρ
b
σ
Where σ is the expansion factor and ρ u and ρ b are, respectively, the unburned and
burned densities.
The same behavior of the unstretched burning velocity regarding the stretch can be
observed (Lamoureux et al., 2003):
S − S = L κ
(2.35)
0
u u u
where L u represents the unburned Markstein length, which is obtained dividing the
burned Markstein length by the expansion factor.
The normalization of the laminar burning velocity by the unstretched one introduces
two numbers which characterize the stretch that is applied to the flame, the Karlovitz
number (Ka), and its response to it, the Markstein Number (Ma):
S
MaKa
S = − (2.36)
u
1
0
u
δ
Ka = κ
(2.37)
S
0
u
L u
Ma = (2.38)
δ
where δ is the flame thickness defined, in this work, using the thermal diffusivity, α:
α
δ = (2.39)
0
S u
47