Computational Investigations of Boundary Condition Effects on Simulations of  Thermoacoustic Instabilities

This dissertation presents a formulation of the Continuous Sensitivity Equation Method (CSEM) applied to the Computational Fluid Dynamics (CFD) simulation of thermoacoustic instability problems. The proposed sensitivity analysis approach only requires a single run of the CFD simulation. Moreover, the sensitivities of field variables, pressure, velocity and temperature to boundary-condition parameters are directly obtained from the solution to sensitivity equations. Thermoacoustic instability is predicted by the Rayleigh criterion. The sensitivity of the Rayleigh index is computed utilizing the sensitivities of field variables.

The application of the CSEM to thermoacoustic instability problems is demonstrated by two classic examples. The first example explores the effects of the heated wall temperature on the one-dimensional thermoacoustic convection. The sensitivity of the Rayleigh index, which is the indicator of thermoacoustic instabilities, is computed by the sensitivity of field variables. As the heat wall temperature increases, the sensitivity of the Rayleigh index decreases. The evolution from positive to negative sensitivity values suggests the transition from a destabilizing trend to stabilizing trend of the thermoacoustic system.

Thermoacoustic instabilities in a self-excited Rijke tube are investigated following the relatively simple thermoacoustic convection problem. The complexity of simulating the Rijke tube increases in both dimensions and mechanisms which incorporate the species transport process and chemical reactions. As a representative model of the large lean premixed combustor, Rijke tube has been extensively studied. Quantitative sensitivity analysis sets the present work apart from previous research on the prediction and control of thermoacoustic instabilities. The effects of two boundary-condition parameters, i.e. the inlet mass flow rate and the equivalence ratio, are tested respectively. Small variations in both parameters predict a rapid change in sensitivities of field variables in the early stage of the total time length of 1.2s. The sensitivity of the Rayleigh index "blows up" at a specific time point of the early stage. In addition, variations in the inlet mass flow rate and the equivalence ratio lead to opposite effects on the sensitivity of the Rayleigh index.

There exist some common findings on the application of the CSEM. For both thermoacoustic problems, the sensitivities of field variables and the Rayleigh index exhibit oscillatory nature, confirming that thermoacoustic instability is an overall effect of the coupling process between fluctuations of pressure and heat release rate. All the sensitivities of the Rayleigh index show rapid changes and "blow up" in the early stage. Although the numerical errors could influence the fidelity of computational results, it is believed that the rapid changes reflect the susceptibility to thermoacoustic instabilities in the studied systems. It should also be noted that the sensitivities are obtained for small variations in influential parameters. Therefore, the resulting sensitivities do not predict the occurrence of thermoacoustic instabilities under a condition that is far from the reference state determined by either CFD simulation results (employed in this dissertation) or experimental data.

The sensitivity solver developed for the present research has the feature of flexibility. Additional mechanisms and more complicated instability criteria could be easily incorporated into the solver. Moreover, the sensitivity equations formulated in this dissertation are derived from the full set of nonlinear governing equations. Therefore, it is possible to extend the use of the sensitivity solver to other CFD problems. The developed sensitivity solver needs to be optimized to gain better performance, which is considered to be the primary future work of this research.