Untitled - Aerobib - Universidad Politécnica de Madrid

7.2. TURBULENT COMBUSTION THEORIES 215
valid for
which also contradicts (7.16).
v ′
u l
≪ 1,
Wohl has shown that (7.22) can also be deduced from (7.13) through purely
geometric considerations by assuming that the distortion of the flame front consists in
the formation of a system of prisms instead of the cones proposed by Shelkin.
Scurlock and Graver [17] calculate σ l by assuming with Shelkin that the laminar
surface is formed by a set of cones whose base is proportional to the square l 2
of the scale of turbulence and whose height is proportional to the root mean square
displacement ¯X of the turbulent oscillations suffered by a flame element. Therefore,
as in Eq. (7.14), u t is given by expression
√
u t
= 1 + k ¯X 2
u l l 2 . (7.25)
The problem lies in computing ¯X. Its magnitude, according to Scurlock, is
governed by three different mechanisms:
1) Turbulent diffusivity tends to increase indefinitely the value of ¯X as time elapses.
Figure 7.5 shows two consecutive positions of the flame front in Scurlock’s
model after the instant at which the flame was plane.
Initial flat flame
Flame after passage
of short time
Flame after passage
of longer time
Figure 7.5: Schematic diagram showing wrinkling with passage of time of an initially flat
flame element exposed to turbulence.
2) Laminar combustion tends to absorb oscillations reducing the value of ¯X. Karlovitz
and his collaborators [19] were the first to acknowledge this fact which is
represented in Fig. 7.6
3) The flame generates turbulence 2 and this tends to increase the value of ¯X.
2 See §4.