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

Chapter 2
δr
δr
dA t d d
δν δη
() = ( ν )( η )
(2.6)
* *
Now let another surface that is close to the flame surface be represented by r ( νη , , n )
such that:
,
* *
∂r ∂r ∂a ∂r ∂r ∂a
= + , = + , = +
∂ν ∂ν ∂ν ∂η ∂η ∂η
*
r r a
(2.7)
Where a is a small magnitude displacement vector; then:
tel-00623090, version 1 - 13 Sep 2011
∂
Where ∇ t
= eν
+ e
∂ν
* * * *
*
⎛δr δr ⎞ *
δr δr
dA t ⎜ ⎟ n d d d d
⎝ δν δη ⎠
δν δη
δr
δr
≅ ⎡1+∇ t. a⎤ d d = ⎡1+∇t.
a⎤dA t
δν δη
⎣ ⎦ ⎣ ⎦
() = × ⋅ ( ν)( η) ( ν )( η )
η
( ν)( η) ( )
(2.8)
∂
, which represents the gradient operator along the tangential
∂η
plane of the flame surface (Chung and Law, 1988).
It is useful to note that we can always decompose any arbitrary velocity into two
components: a tangential to the flame surface and another normal to the flame surface
as follows:
V = V + V = V n n+
V
( . )
n t t
(2.9)
Now considering a curved flame surface A(t) moving in the space with local velocity W
(note that each spatial point has its own velocity). Then, the rate of change of the flux
of a vector G across the flame surface is given by the following expression based on
the Reynolds’ transport theorem.
d
dt
∫
. ⎡∂G
GndA W .
G G .
W G . ⎤
= ∫ ⎢ + ∇ − ∇ + ∇W⎥
.
ndA
⎢⎣
∂t
⎥⎦
At ( ) At ( )
(2.10)
By specifying G = n , Eq. (2.10) yields:
d
dt
∫
⎡∂n
⎤
dA = ∫ ⎢ + W. ∇n −n. ∇ W + n∇. W . n dA
∂t
⎥
⎣
⎦
At ( ) At ( )
(2.11)
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