Untitled - Aerobib - Universidad Politécnica de Madrid

10.5. CHAMBER WITH SLOWLY VARYING CROSS-SECTION 259
0.5
0.4
0.3
U 0
= 0.100
U
0.2
U 0
= 0.087
0.1
U 0
= 0.050
0
0.2 0.4 0.6 0.8 1.0
ψ
Figure 10.10: Velocity profiles as a function of the burnt mass fraction for λ = 6 and three
values of the initial velocity (subcritical, critical and supercritical).
function, must be substituted by
ρu = 1 2 ρ 0u 0
R 2
r
∂ψ
∂r , (10.39)
where R is the radius of the chamber and r is the distance to its axis. Coefficient 1/2
is introduced so that stream function ψ changes from value zero at the axis to value 1
at the chamber wall.
The change of variable
( r
) 2 y =
R h , (10.40)
transforms (10.39) into (10.18) keeping conditions ψ = 0 at y = 0 and ψ = 1 at
y = 1. The remaining calculations are identical in both cases. Hence, the problem
reduces to the two-dimensional case. Fabri, Siestrunck and Fouré have also studied the
case where stabilizer is at the chamber wall. The annular stabilizer has been studied
by Ernst [9]. All these cases reduce to the two-dimensional problem by an adequate
change of variables.
10.5 Chamber with slowly varying cross-section
The preceding method can be applied to an approximate study by substituting equations
(10.18) or (10.39) for an equation that includes the variation of the cross-section