Rigorous Modeling of High-Frequency Instabilities

in Liquid Rocket Combustors

W.S. Yoon[1] and G.Y. Lee

Combustion instability has been a major design criterion and severe problem in reliable and stable rocket engine development, since it was discovered in the late 1930s. But combustion instability problem have been solved and studied mostly by trial and error or experimental method that is time and cost-consuming, analytical methods that starts from many assumptions, and computational methods that need heavy computational load, because of insufficient knowledge of the driving mechanisms and fundamental combustion-related physics. Therefore, more effective and reasonable analysis method of combustion instability has been needed to investigate the mechanisms and characteristics of instability phenomena and apply the methods to developing rocket engines.  This paper is concerned with the new analysis method of combustion instability to overcome the critical problems of analytical method and computational method as stated below. Conventional analytical methods suggested made assumptions with uncertainties in the derivation and application of governing equations. It assumes that mean flow variables are uniform, all flow variables can be represented by solutions of wave equation and the effects of source terms on perturbed solutions are sufficiently small. It has also severe problems and uncertainties in integrating source terms, applying the orthogonality of functions to governing equations, and calculating the amplitudes of each pressure perturbation mode. The critical problems of the computational method are insufficient calculation capability and excessively simplified modeling. To obtain more accurate output of numerical simulation, more complex modeling and heavier calculation are needed. Moreover, the accurate unsteady solution is not easy to obtain with economy.  In this study, a new prediction method of combustion instability is suggested based on replacement of the partial differential equations of conservation by an equivalent system of ordinary differential equations and application of steady and unsteady solution of numerical simulation. Governing instability equations are derived in terms of the time rate of change of amplification factor for each variable based on Navier-Stokes system of equations. All variables are decomposed into mean and fluctuation quantities. Each fluctuation quantity is assumed to be the sum of products of amplification factors and orthogonal modes for each variable. Using the set of steady state solution and unsteady solution at arbitrary instant obtained by numerical simulation, the assumption of flow variable distributions can be avoided, the calculation load is reduced to the utmost and the orthogonal modes for each variable and the spatial integration terms were computed. Initial value and its temporal variation of each variable are calculated based on amplification factors and orthogonal modes. FFT is used to identify the frequencies and amplitudes of eigenmodes of system with temporal variation data of each variable.  In this paper, the validity of suggested analysis method is proved with analytical solution or artificially generated wave solutions and the identification of eigenmodes of nonlinear acoustic waves in combustion chambers and prediction of their behaviors are accomplished.