Untitled - Aerobib - Universidad Politécnica de Madrid

11.4. SCALING OF ROCKETS FOR NON-STEADY CONDITIONS 273
The similarity conditions that still need to be satisfied refer to the dynamic
characteristics of the feeding system which determine the value for τ i in Fig. 11.1.
For their study, we refer the reader to the papers by Crocco [8] and Penner-Fuhs [10]
previously mentioned.
Low-frequency oscillations may be analyzed by adopting the assumption that
the state of the gases in the chamber is constant, because their period is very long compared
to the propagation time of a wave through the chamber. To the contrary, when
both the period of the oscillations and the propagation time of the wave are of the
same order of magnitude, it is necessary to consider the differences in state between
different points of the chamber. Such is the situation in the case of high-frequency
oscillations, where, as shown by experimentation, the chamber oscillates like an organ
pipe with longitudinal and transversal oscillations. The existence of a time lag
makes possible the maintenance of these oscillations in a similar way as it happens
for low-frequency oscillations. High-frequency oscillations are particularly dangerous
because, in addition to pressure oscillations being very large, the transmision of
heat to the walls or injectors increases drastically and they are destroyed within a few
seconds. The study of these oscillations is less advanced than for the low-frequency
ones which are easier to control.
As for the scaling law, it appears evident that the residence time τ r should be
substituted by propagation time τ p of a pressure wave through the chamber. Therefore,
in order to mantain the same level of stability for high-frequency oscillation when
passing from the model to the rocket, the following condition must be satisfied
τ ch1
τ ch2
= τ p1
τ p2
. (11.53)
However, since the velocity of sound is the same for both rockets, between τ p1 and
τ p2 the following ratio exists
τ p1
τ p2
= K. (11.54)
Consequently we obtain
τ ch1
τ ch2
= K. (11.55)
It happens also as for low-frequency oscillations, that this condition can not always be
satisfied. In particular, it can be verified that for n = 1 and applying Crocco’s rule the
above condition may be attained.