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lecture_23 - 16.512 Rocket Propulsion Prof Manuel...

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16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 23: Liquid Motors: Stability (High Frequency); Acoustics Combustion Instability: High Frequency Methods of Analysis for High Frequency Instabilities Prior to the advent of large-scale 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 m m P t P t n t P m τ 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 Martinez-Sanchez Page 1 of 21
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Fig 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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