THESE de DOCTORAT - cerfacs

6.6 Transmitted and Reflected Waves through an ideal Compressor: the enthalpy jump case 113
π c = p t,2 /p t,1 where the indices [1] and [2] represent the flow upstream and downstream of the
compressor respectively. It is then useful to rewrite Eq. (6.13) as function of the parameter π c .
F = (ρu 2 )
∣
2
1
+ p
∣
2
1
∫ x2
+
x 1
∫ x2
F = ∆p t + ∆e k +
x 1
∂
(ρu) dx (6.49)
∂t
∂
(ρu) dx (6.50)
∂t
where p t = p + ρu 2 /2 and e k = ρu 2 /2. From Eq. (6.50) it is observed that the compressor F
acts as an element that exerts a change in the total pressure p t of the flow as well as in its kinetic
energy e k . The parameter π c is now inserted into the expression leading to
∫ x2
F = p t1 (π c − 1) + ∆e k +
x 1
∂
(ρu) dx (6.51)
∂t
Equation (6.51) is now linearized and expressed in the frequency domain. It yields
∫ x2
̂F = ¯p t ˆπ c + ˆp t,1 ( ¯π c − 1) + ∆ê k − iω ( ¯ρû + û ˆρ) dx (6.52)
x 1
where ê k = ˆρū 2 /2 + ¯ρūû and ˆp t = ˆp + ˆρū 2 /2 + ¯ρūû. For a compact system the integral can be
neglected and Eq. (6.52) results in
ˆ F = ¯p t ˆπ c + ˆp t,1 ( ¯π c − 1) + ∆ê k (6.53)
In Appendix A, it is shown that π ′ c is a cumbersome term that depends mainly on the mean and
fluctuation value of the enthalpy both upstream and downstream of the compressor. Neglecting
this term might be an strong assumption. The conditions of a compressor such that π ′ c ≈ 0
are discussed in Appendix A. Nevertheless, it is practical as a first approximation and it is useful
to see the influence of the mean value of the total pressure ratio ¯π c . After this assumption,
the momentum equation (Eq. 6.22) becomes
ˆp t,1 ¯π c = ˆp t,2 (6.54)
Finally, the system of equations to resolve is expressed as function of the Mach Number
The continuity equation (Eq. 6.31) and the momentum equation (Eq. 6.54) read
¯M.