A methodology is presented to model non-compact thermoacoustic phenomena using Reduced Order Models (ROM) based on the Linearized Navier-Stokes Equations (LNSE). The method is applicable to geometries with a complex flow field as in a gas turbine combustion chamber. The LNSE, and thus the resulting ROM, include coupling effects between acoustics and mean fluid flow, and are hence capable of describing propagation and (e.g. vortical) damping of the acoustic fluctuations within the considered volume. Such a ROM then constitutes the main building block for a novel thermoacoustic stability analysis method via a low-order hybrid approach. This method presents an expansion to state-of-the-art low-order stability tools, and is conceptually based on three core features: Firstly, the multi-dimensional and volumetric nature of the ROM establishes access to account spatial variability and non-compact effects on heat release fluctuations. As a result, it is particularly useful for high frequency phenomena such as screech. Secondly, the LNSE basis grants the ROM the capability to reconstruct complex acoustic performances physically accurate. Thirdly, the formulation of the ROM in state-space allows convenient access to the frequency and time domain. In the time domain, non-linear saturation mechanisms can be included, which reproduce the non-linear stochastic limit cycle behavior of thermoacoustic oscillations.
In order to demonstrate and verify the ROM’s underlying methodology, a test case using an orifice-tube geometry as the acoustic volume is performed. The generation of the ROM of the orifice-tube is conducted in a two-step procedure. As the first step, the geometrical domain is aeroacoustically characterized through the LNSE in frequency domain, and discretized via the Finite Element Method (FEM). The second step concerns the actual derivation of the ROM. The high-order dynamical system from the LNSE discretization is subjected to a modal reduction as order reduction technique. Mathematically, this modal reduction is the projection of the high-order (N ∼200,000) system into its truncated left eigenspace. An order reduction of several magnitudes (ROM order: Nr ∼100) is achieved. The resulting ROM contains all essential information about propagation and damping of the acoustic variables, and efficiently reproduces the aeroacoustic performance of the orifice-tube. Validation is achieved by comparing ROM results against numerical and experimental benchmarks from LNSE-FEM simulations and test rig measurements, respectively. Excellent agreement is found, which grants the ROM modeling approach full eligibility for further usage in the context of thermoacoustic stability modeling. This work is concluded by a methodological demonstration of performing stability analyses of non-compact thermoacoustic systems using the herein presented ROMs.
Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws