Untitled - Aerobib - Universidad Politécnica de Madrid

236 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
The ratio of velocities is
obtained from Eqs. (9.23) and (9.17).
v 2
v 1
= n, (9.24)
Therefore, once the density and temperature of the unburnt gases and the value
of n are known, the temperature of the burnt gases is determined by equation (9.22),
its density by equation (9.23), and the pressure drop across the flame by
p 2 − p 1
p 1
= γ 1 M 2 1 (1 − n), (9.25)
where γ 1 is the ratio of the heat capacity at constant pressure to the heat capacity at
constant volume of the unburnt gases, and
M 1 = v 1
a 1
(9.26)
is the flow Mach number for the unburnt gases, where
√
p 1
a 1 = γ 1 (9.27)
ρ 1
is the sound speed in these same gases.
9.4 Inclined flame front
Here, v n1 and v n2 are still small compared to the sound speed of the unburnt and
burnt gases respectively. Therefore, the pressure drop across the flame front is still
small, see Eq. (9.2), and the ratio between densities on each side of the flame front is
determined by the ratio of temperature T 2 to temperature T 1 . Let λ be this ratio, there
results
Let
λ = T 2
T 1
= ρ 1
ρ 2
= v n2
v n1
. (9.28)
δ = α 2 − α 1 (9.29)
be the velocity deviation across the flame front (see Fig. 9.2) . A simple calculation
gives for δ
tan δ = (λ − 1) tan α 1
1 + λ tan 2 α 1
. (9.30)
This ratio has been taken into Fig. 9.4 for different values of λ. As it can easily be
seen, the maximum deviation δ max corresponds to
tan α 1 = 1 √
λ
. (9.31)