Impact of fuel supply impedance and fuel staging on gas turbine ...

3.2 Linear acoustic equations
p ′
ρ 0 c
= f (ξ)+ g (η)= f (x− c t )+ g (x+ c t ). (3.21)
The solution p ′ <strong>of</strong> the latter equation is, in general, normalized by the characteristic
<strong>impedance</strong> ρ 0 c. f <strong>and</strong> g are the so-called Riemann Invariants <strong>and</strong> are
determined by boundary or initial conditions. The Riemann Invariants represent
physically waves, which are traveling with speed <strong>of</strong> sound c in the right or
positive x-direction (f ) or rather in the left or negative direction (g ) as shown
in Fig. 3.1.
g
f
x
Figure 3.1: Riemann Invariants f <strong>and</strong> g traveling in positive <strong>and</strong> negative x-
direction
The acoustic velocity u ′ can be obtained from the acoustic pressure p ′
(Eqn. (3.21)) <strong>and</strong> the linearized momentum equation (3.17):
u ′ = f (x− c t )− g (x+ c t ). (3.22)
Using Eqn. (3.21) <strong>and</strong> (3.22), the Riemann Invariants can be conversely expressed
in terms <strong>of</strong> p ′ <strong>and</strong> u ′ :
f = 1 ( p
′
)
2 ρ 0 c + u′ , (3.23)
g = 1 ( p
′
)
2 ρ 0 c − u′ . (3.24)
49