THESE de DOCTORAT - cerfacs

6.6 Transmitted and Reflected Waves through an ideal Compressor: the enthalpy jump case 117
ˆp t
ˆp t,1
=
¯p t
¯p t,1
thus
∂ ˆp t
∂x = ˆp t,1 ∂ ¯p t
¯p t,1 ∂x
(6.64)
The profile of ˆp t is given by a smooth hyperbolic tangent relation as seen in Fig. 6.14. From
Eq. (6.64), the (∂ ˆp t /∂x) term in Eq. (6.63) must be written as
x m
x ups
2
¯p t / ¯p t1
| ˆp t |/|ˆp t1 |
1.8
1.6
1.4
1.2
1
0 0.1 0.2 0.3 0.4
X (m)
Figure 6.14: Mean and Fluctuation profiles of total pressure
∂ ˆp t
∂x = ˆp t1 f (x, ¯π c ) with (6.65)
[ ( )] [ ]
f (x, π c ) = 0.5 ( ¯π c − 1) 1 − tanh 2 k(x − xm ) k
(6.66)
x m − x ups x m − x ups
6.6.4 Results
Two different cases are considered, corresponding to two different values of π c = ¯p t2 / ¯p t1 . The
value of the reflection coefficient R = w + 2 /w− 2
is computed by using both the analytical and the
numerical approach (SNozzle). In order to observe the influence of the forcing frequency on
the values of R, results are plotted as a function of a non-dimensional frequency.
Ω = f L/ ¯c 2 (6.67)
where L represents the length of the compressor and ¯c 2 the velocity of sound at the outlet. The