Untitled - Aerobib - Universidad Politécnica de Madrid

112 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
Now, the system of equations (5.22), (5.23) and (5.24) can be written in the
following way, which evidences the influence of α and M,
(
p 1 + γM 2 − 4 3 α 1 )
dv
= i, (5.25)
v dε
(
c p T 1 + γ − 1 M 2 −
q
2 c p T ε − 1 α 1 dT
γP r M 2 T dε − 4 γ − 1
α 1 )
dv
= e, (5.26)
3 γ v dε
1 α dY
γS c M 2 − Y + ε = 0, (5.27)
dε
where P r = µc p /λ and S c = µ/ (ρD) are the Prandtl and Schmidt numbers, respectively,
for the mixture.
Since α ≪ 1, the last term of the left hand side of equation (5.25) can be
neglected when compared to unity, thus obtaining, instead of equation (5.25)
p ( 1 + γM 2) = i. (5.28)
Similarly, in equation (5.26) one can neglect the last term of the left hand side. Since
the Prandtl P r and the Schmidt S c numbers are of order unity, 5 the order of magnitude
of terms 1 α 1 dT
P r M 2 T dε of equation (5.26), and 1 α dY
S c M 2 of equation (5.27) depends
dε
on the values of the ratio α/M 2 . Here, there are two possible cases:
a) If the Mach number M of the flow is of the order of magnitude of unity or larger,
then α/M 2 ≪ 1, and the said terms can also be neglected.
b) Whereas if α/M 2 are of the order one, then said terms must be preserved. In
this case, however, M 2 ≪ 1, and γM 2 can be neglected in equation (5.28)
and γ − 1 M 2 can be neglected in equation (5.26). Therefore the two following
2
cases are obtained:
1) α ≪ 1, M 2 ∼ 1.
Will be designated case A. For this case, Eq. (5.28) is valid and will be
written in the form
p + ρv 2 = i.
(5.28.a)
In Eq. (5.26) the fourth and fifth terms of the left hand side disappear,
obtaining the following simplified equation
(
c p T 1 + γ − 1 M 2 −
q )
2 c p T ε = e, (5.29)
that can be written
c p T + 1 2 v2 − qε = e.
(5.29.a)
5 See Hirschfelder, Curtiss & Bird: Molecular Theory of Gases and Liquids, John Wiley & Sons Inc.,
New York, 1954, p. 16.