Untitled - Aerobib - Universidad Politécnica de Madrid

212 CHAPTER 7. TURBULENT FLAMES
B
A
BURNED
UNBURNED
GASES
GASES
B’
A’
Figure 7.3: Structure of the turbulent flame in the case l ≫ d l and v ′ ≫ u l , according to
Shelkin.
zone shows islands of unburnt gas which have been dragged by the strong turbulent oscillations
towards the region of combustion products without time to burn, see Fig. 7.3.
In this case, the zone between AA ′ and BB ′ can be considered as a combustion wave
which advances with a propagation velocity u t . The propagation velocity of a combustion
wave is determined by an equilibrium between transport processes, which are
characterized by combustion time τ. Independently from the mechanism which determines
the values for ε and τ, a dimensionless analysis shows that u t must be of the
form
u t ∼
√ ε
τ . (7.9)
For the case shown in Fig. 7.3, ε is clearly the turbulent diffusivity. Combustion
velocity is determined by the mixing rapidity and therefore τ is proportional to the
time of turbulent mixing
τ ∼ l v ′ . (7.10)
When the expressions for ε and τ given by (7.2) and (7.10) are taken into (7.9)
u t ∼ v ′ , (7.11)
Hence, the propagation velocity is proportional to the intensity and independent from
the scale of turbulence. Furthermore, u t results to be independent from laminar velocity
u l . Such conclusion contradicts experimental evidence.
3.b) The intensity of turbulence is of the same order of magnitude or smaller
than the propagation velocity of the laminar flame
v ′
u l
1. (7.12)
In this case the islands disappear and the action of turbulence reduces to deforming