Untitled - Aerobib - Universidad Politécnica de Madrid

4.5. APPLICATIONS 101
By substituting these values and that of the temperature given by Eq. (4.37)
into Eq. (4.36), the following expression, for the Hugoniot curve, is obtained
xy = q + γ 2 − 1
(y − x) . (4.40)
γ 2 + 1
Therefore, the Hugoniot curve is an equilateral hyperbola, with an asymptote parallel
to the axis y for x 0 = (γ 2 − 1) / (γ 2 + 1). This curve cuts axis x at the point x 1 =
q (γ 2 + 1) / (γ 2 − 1).
Let us see how the Chapman-Jouguet points are determined. The equation of
an isentropic for the burnt gases is
yx γ2 = const. (4.41)
Therefore, the Chapman-Jouguet points are obtained by solving the system formed
by Eq. (4.40) and by the equations that express the tangency condition of the curves
Eq. (4.40) and Eq. (4.41), namely,
(γ 2 + 1) y + (γ 2 − 1)
(γ 2 + 1) x − (γ 2 − 1) = γ y
2
x . (4.42)
The solution of Eq. (4.40) and Eq. (4.42) gives for x and y the following equations
y 2 − ( (γ 2 + 1)q − (γ 2 − 1) ) y + q = 0, (4.43)
x 2 − 1 γ 2
(
(γ2 + 1)q + (γ 2 − 1) ) x + q = 0. (4.44)
When solving these equations the root x 1 < 1 must be assign to the root y 1 > 1 and
root x 2 > 1 to the root y 2 < 1.
In virtue of Eq. (4.8), the propagation velocity is
v 1
= 1
√
y − 1
√
a 1 γ1 1 − x , (4.45)
where a 1 is the sound velocity of the unburnt gases, and γ 1 the ratio of their heat
capacities.
Similarly, the velocity of the burnt gases with respect to the wave front is
v 2
=
x
√
y − 1
√
a 1 γ1 1 − x . (4.46)
The corresponding Mach number M 2 is
√
M 2 = √ 1 ( )
x y − 1
. (4.47)
γ2 y 1 − x