Untitled - Aerobib - Universidad Politécnica de Madrid

10.3. METHOD OF FABRI-SIESTRUNCK-FOURÉ 255
10.3 Method of Fabri-Siestrunck-Fouré
The flow in the combustion chamber is two dimensional and stationary. Therefore, a
stream function ψ exists. This function is defined by the following relations
ρu =ρ 0 u 0 h ∂ψ
∂y , (10.18)
ρv =ρ 0 u 0 h ∂ψ
∂x . (10.19)
Thus defined, ψ is non-dimensional.
The x axis is a streamline to which the value ψ = 0 is assigned. The upper
wall of the chamber, y = h, is also a streamline for which ψ = 1, as results from the
integration of (10.18). Therefore, in the upper half of the chamber, ψ varies from the
value zero, at the axis, to the value one, at the wall.
A linearization of the problem, by assuming that all velocities are small compared
to u, leads to the conclusion that pressure is constant at each cross section of
the chamber. Therefore, since the flow of unburnt gases is isentropic, their density,
temperature and velocities are equally constant at each cross section. Moreover, the
value of either one of these magnitudes determines the values for the others.
ψ
ψ’
D u’
ρ
1
B
C
ρ’’
E
u
u’’
ψ
ψ’
0
x
A
Figure 10.9: Notation for the method of Fabri-Siestrunck-Fouré.
Let us consider a cross section AB, at a distance x from the stabilizer, as shown
in Fig. 10.9. Here, evidently
∫
1 h
dy = 1, (10.20)
h 0
which, by virtue of (10.18), can be written in the following form
∫ 1
0
ρ 0 u 0
ρu
dψ = 1. (10.21)