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

Chapter 2
Where, r c represents the local radius of the flame curvature. The first term of Eq. (2.21)
represents the effect of the strain; the second term represents the effect of the fluid
expansion of fluid (dilatation) and, the third term, the effect of the flame curvature. It is
evident that the flame can be stretched by the combined effect of strain, volume
expansion of the fluid and the curvature of the flame, which arises from the
nonuniformities of the flow and the normal propagation of the flame front.
For the present purposes, an appropriate unified tensor expression, in terms of strain
rate, κ s , and the stretch rate due to flame curvature, κ c , is that of Candel and Poinsot,
(1990).
κ = κ + κ
(2.22)
s
c
tel-00623090, version 1 - 13 Sep 2011
With
κ
s
=−nn: ∇ S+∇.
S
(2.23)
κ = S ∇.
n
(2.24)
c
u
In spherical expanding flames it is convenient to use spherical coordinates (r, θ, φ), the
components of n and S are written as (n r , n θ , n φ ) and (s r , s θ , s φ ), respectively. Then:
⎡
2⎛∂sr ⎞ 2⎛1 ∂s
2 1 s
θ
sr ⎞ ⎛ ∂
φ sr
sθ
cotθ
⎞
κs
=− ⎢nr
⎜ n n
r
⎟+ θ ⎜ +
φ
r θ r
⎟+ ⎜ + + ⎟
⎣ ⎝ ∂ ⎠ ⎝ ∂ ⎠ ⎝r sinφ ∂φ
r r ⎠
⎛∂s
1 s
s
r
s
φ 1 s s
θ
∂
θ ⎞ ⎛∂
∂
r φ ⎞
+ nn
r θ ⎜ + − nn
r φ
r r θ r
⎟+ ⎜ + − ⎟
⎝ ∂ ∂ ⎠ ⎝ ∂r r sinθ ∂φ
r ⎠
⎛1 ∂sφ
1 ∂sθ
cotθ
⎞⎤
+ nn
θ φ ⎜ + − sφ
⎟⎥
⎝r ∂θ r sinθ ∂φ
r ⎠⎦⎥
2
1 ∂ ( r sr
) 1 ∂ ( sθ
sinθ
) 1 ∂sφ
+ +
+
2
r ∂r
r sinθ
∂θ r sinθ ∂ φ
(2.25)
2
( r nr
) ∂ ( n sinθ
)
⎡ 1 ∂ 1 1 ∂n
⎤
θ
φ
κc
= un
+ ⎢ + + ⎥
2
⎢r
∂r r sinθ ∂θ r sinθ ∂φ
⎥
⎣
⎦
(2.26)
For an outward spherically propagating flame, the flame surface is identified by the cold
front of radius r u , n r =1, n θ =n φ =0, and s r =s g , s θ =s φ =0 (Bradley et al., 1996). The burning
velocity, s u , is associated with this surface and the gas velocity ahead of it is s g . The
flame speed, dr u /dt, is equal to s g + n r s u , and is indicated by S n . Applying this conditions
to Eqs. 2.25 and 2.26 gives
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