Untitled - Aerobib - Universidad Politécnica de Madrid

214 CHAPTER 7. TURBULENT FLAMES
time counted from the point at which measurements were initiated. Let t = l/u l be
the time taken by the laminar flame to cross a turbulent vortex. The total path travelled
by the flame in this time is ¯X + l and when this expression is divided by t one obtains
for u t
That is to say
u t = u l
l
The value for ¯X is given by Taylor’s formula
d ¯X 2
dt
¯X + u l , (7.16)
u t
= 1 + ¯X
u l l . (7.17)
= 2v ′ 2
∫ t
0
R t dt, (7.18)
where R t is the correlation coefficient between the velocities of the same particle at
two different instants. Karlovitz uses from R t the following expression
where
R t = e −tv′ l ′ , (7.19)
∫ ∞
l ′ = v ′ R t dt (7.20)
0
is the Lagrangian scale of turbulence. By substituting (7.19) into (7.18) and with
t = l/v ′ it results for ¯X
√
¯X = l
(
2a v′
1 − au [
l
u l v ′ 1 − exp
(
− 1 )])
v ′
, (7.21)
a u l
Here a is the ratio from the Lagrangian to the Eulerian scales of turbulence. Karlovitz
assigns to a a value 1 while Scurlock and Grover [17] and Wohl [18] choose 1/2.
Taking (7.21) into (7.17) one obtains for u t
√ (
u t
= 1 + 2a v′
1 − au [ (
l
u l u l v ′ 1 − exp − 1 a
)])
v ′
. (7.22)
u l
For very intense turbulence (7.22) reduces to
√
u t
≃ 1 + 2a v′
, (7.23)
u l u l
which is valid provided that
v ′
≫ 1.
u l
Eq. (7.23) is in contradiction with (7.12) and for weak turbulence is reduces to
u t
u l
≃ 1 + v′
u l
, (7.24)