Untitled - Aerobib - Universidad Politécnica de Madrid

238 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
By virtue of equation (9.15) and of the relation
the following expression is obtained for λ
T s1
= 1 + γ 1 − 1
M1 2 , (9.38)
T 1 2
λ = n + γ 1 − 1
2
(
n − c )
p1
M1 2 , (9.39)
c p2
which shows that for large values of M 1 , that is, when the flame front is very inclined
to the incident flow, λ can appreciably differ from n.
The values of λ/n as a function of M 1 , for c p2 = c p1 and different values of
χ = γ 1 − 1 n − 1
, have been represented in Fig. 9.5.
2 n
λ / n
1.20
1.18
1.16
1.14
1.12
1.10
1.06
1.06
1.04
1.02
χ=(γ 1
−1)(n−1)/2n
χ=0.20
χ=0.18
χ=0.16
χ=0.14
χ=0.12
χ=0.10
1.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M 1
Figure 9.5: Values of the ratio λ/n as a function of incident Mach number M 1.
Relation (9.39) is only valid for very inclined flame front, that is, for not too
small values of M 1 . For values of M 1 close to zero, it must be substituted by
λ ≃ n. (9.40)
The first case, that is to say, the case of slow flows (M 1 ≪ 1) with flame
fronts, has been examined in detail by Gross and Esch [4], reaching the following
conclusions:
1) If ϕ, λ and q are constant and the motion is irrotational before the flame, after
the flame it continues being irrotational.