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16.512, Rocket Propulsion
Prof. Manuel MartinezSanchez
Lecture 23: Liquid Motors: Stability (High Frequency); Acoustics
Combustion Instability: High Frequency
Methods of Analysis for High Frequency Instabilities
Prior to the advent of largescale computations, the most successful
theoretical development in this area was the “sensitive time lag theory” of L. Crocco
[26]. More than a detailed physical theory, this was a model in which a few basic
parameters were introduced from intuitive considerations, and then used to correlate
experimental observations on stability thresholds. The principal parameter was the
sensitive time lag, during which the various rates which eventually resulted in
vaporization at the total time lag
τ
T
after injection were assumed to vary with
pressure, velocity, stoichiometry, etc. This variation was characterized by means of
other important parameter, the “sensitivity index”. For pressure sensitivity, this is
()
Rates
n
P
∂
=
∂
ln
ln
(
1
)
and the definition of
is such that the variations in gas generation rate due to this
sensitivity are given locally by
τ
mm
Pt Pt
n
t
P
m
τ
••
•
−∂
−
−
=−
=
∂
'( )
'(
)
(2)
Similar sensitivity indices can be introduced for velocity, etc. Once this parameterization
is accepted, it is only a matter of mathematical modeling to obtain the stability limits of
a given acoustic wave or cavity. This modeling could be linear or even allow for non
linearities in the gas dynamics. It is one of the strengths of this theory that the acoustic
part of the problem, namely, combustor geometry, steady state combustion and heat
release, etc. are separated from the unsteady combustion effects, which allows for
generalization of test results and accumulation of meaningful stability data.
The results of calculations using the linear sensitive time lag theory are
displayed as shown in Fig. 1 (Ref. 26).
16.512, Rocket Propulsion
Lecture 23
Prof. Manuel MartinezSanchez
Page 1 of 21
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View Full DocumentFig 1. An
n
τ
diagram, showing instability zones for various modes.
Here are shown the loci of marginal stability for severe modes of a combustor,
on a map of interaction index
n
vs. sensitive lag
τ
. Each point on one of the lines
corresponds to a particular oscillation frequency, and these frequencies are found to
be within
10% of the undisturbed acoustic frequency of the mode. The goal of the
designer is to manipulate the factors influencing
n
and
±
τ
in order to place the
operating point outside all the stable regions of the various modes.
The parameters
and
n
(into which is lumped the modifying effects of
velocity or other sensitivities) are basically empirical, and a large data base has been
laboriously accumulated on their dependencies upon many design factors. As an
example, Figs. 2(a) and 2(b) (Ref. 26) show data on
τ
τ
for coaxial injectors, for
which n
0.5 throughout.
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 Fall '05
 ManuelMartinezSanchez
 Propulsion

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