Untitled - Aerobib - Universidad Politécnica de Madrid

7.2. TURBULENT COMBUSTION THEORIES 213
BURNED
σ
l
UNBURNED
σ
GASES
GASES
Figure 7.4: Structure of the turbulent flame in the case l ≫ d l and v ′ u l , according to
Shelkin.
the laminar flame front thus increasing its surface as shown in Fig. 7.4. Let σ l be the
effective surface of combustion and σ the surface of access to the flame of the unburnt
gases. Velocity u t is given in this case by expression
u t
u l
= σ l
σ . (7.13)
The problem lies then in estimating the value for σ l , for which several models have
been proposed.
Shelkin assumes that σ l is formed by a system of cones whose base is proportional
to the square of the scale of turbulence l 2 and whose height is proportional to
distance v ′ l/u l travelled by a mass of gas with a diameter l due to turbulent oscillations
during the time l/u l needed by the laminar flame to cross this mass. Here, σ l /σ
is the ratio of the lateral to the base area of the cones, and from (7.13), it results
√
( )
u t
v
′ 2
= 1 + k , (7.14)
u l u l
where k is a numerical coefficient of the order of magnitude unity. Therefore, u t is
also independent from the scale of turbulence. When turbulence is weak, Eq. (7.14)
shows that its effect is of the second order, whilst for very intense turbulence Eq. (7.14)
reduces to (7.11). Through a different analysis M. Tucker [15] obtains the following
expression for the case of weak turbulence
( )
u t
v
′ 2
= 1 + k(θ) , (7.15)
u l u l
which is valid if v ′ /u l ≪ l. In this formula, k(θ) is a coefficient depending on ratio
θ of temperature T f of the burnt gases to temperature T 0 of unburnt gases. It is seen
that when v ′ /u l ≪ l. Eqs. (7.14) and (7.15) agree.
Karlovitz [16] reasons as follows. Let ¯X =
√ ¯X2 be the root mean square
displacement of a particle due to turbulent oscillations. ¯X is a increasing function of