<p dir="ltr">Thermo-acoustic interactions, characterized by the coupling between heat release and acoustic waves, are a phenomenon that can lead to combustion instability in high-speed propulsion devices. These interactions are highly undesirable as they can damage engine components and, in severe cases, cause catastrophic failure of the entire propulsion system. Mitigating these instabilities is crucial for ensuring reliable combustor operation. This work presents a computational investigation of combustion instability in Purdue's Continuously Variable Resonance Combustor (CVRC), focusing on the prediction of instability trend over the entire oxidizer-post length range. Computational fluid dynamics (CFD) studies in the past mainly focused on individual CVRC cases with specific oxidizer-post lengths. Those studies help understand the instability mechanism for individual CVRC cases but are limited in examining the applicability of model predictions over a wide range of instability conditions. No studies have been reported to assess the model predictivity over the entire oxidizer-post range in CVRC. </p><p dir="ltr">In this work, we first conduct a series of CFD simulations that cover the entire oxidizer-post length in CVRC to assess the models for a wide range of instability conditions. It is found that the CFD models generally fail to capture the instability trend over the entire oxidizer-post length although they can capture some individual cases. To understand the model failure, parametric studies are often deemed the first step of investigation. Such parameter studies, however, are expensive for CVRC since more than ten simulation cases to cover the entire oxidizer-post range are needed for each parametric study. Multiple parametric studies are typically needed to cover various uncertainties from numerics and physical models and those involved in the experimental conditions, making parametric studies for CVRC a computationally expensive task. Therefore, our focus next is on developing faster approaches.</p><p dir="ltr">The second part of this work is to develop a reduced-order model to quickly conduct the needed parametric studies. The developed reduced-order model leverages the instability mechanisms observed from the CFD simulations conducted in the first part. Monte Carlo approaches are employed to replace expensive CFD simulations by replicating the randomness in the combustor through statistical sampling. The developed reduced-order model is first validated by comparing its predictions with the CFD simulation results in a number of cases. The reduced-order model, despite its simplicity, reasonably reproduced the overall trend of instability from CFD simulations, making it an attractive alternative to the detailed model simulations for parametric studies. </p><p dir="ltr">The validated reduced-order model is then applied to parametric studies of CVRC to help identify the uncertainties of CFD predictions of CVRC. Four sets of parametric studies are conducted to provide a rapid examination of the effect of heat loss, the effect of oxidizer temperature, the effect of equivalence ratio, and the effect of turbulence on the instability predictions in CVRC. From the rapid reduced-order parametric studies, we found that the heat losses in upstream of the oxidizer inlet and the combustor wall are the two most contributing factors to the uncertainties of CFD model predictions. The turbulence level and the error involved in the equivalence ratio due to experimental uncertainties play an insignificant role in contributing to the CFD prediction uncertainties. </p><p dir="ltr">This work is a significant contribution to the combustion instability community by enabling an alternative rapid assessment of CFD model predictions. This capability facilitates the identification of major contributing factors of CFD modeling uncertainties with much less computational cost, thereby allowing for a more focused approach to CFD analysis and ultimately accelerating the improvement of CFD models for combustion instability studies. </p>

<p>Combustion instability, a complex phenomenon observed in combustion chambers is due to the coupling between heat release and other unsteady flow processes. Combustion instability has long been a topic of interest to rocket scientists and has been extensively investigated experimentally and computationally. However, to date, there is no computational tool that can accurately predict the combustion instabilities in full-size combustors because of the amount of computational power required to perform a high-fidelity simulation of a multi-element chamber. Hence, the focus is shifted to reduced fidelity computational tools which may accurately predict the instability by using the information available from the high-fidelity simulations or experiments of single or few-element combustors. One way of developing reduced fidelity computational tools involves using a reduced fidelity solver together with the flame transfer functions that carry important information about the flame behavior from a high-fidelity simulation or experiment to a reduced fidelity simulation.</p> <p> </p> <p>To date, research has been focused mainly on premixed flames and using acoustic solvers together with the global flame transfer functions that were obtained by integrating over a region. However, in the case of rockets, the flame is non-premixed and distributed in space and time. Further, the mixing of propellants is impacted by the level of flow fluctuations and can lead to non-uniform mean properties and hence, there is a need for reduced fidelity solver that can capture the gas dynamics, nonlinearities and steep-fronted waves accurately. Nonlinear Euler equations have all the required capabilities and are at the bottom of the list in terms of the computational cost among the solvers that can solve for mean flow and allow multi-dimensional modeling of combustion instabilities. Hence, in the current work, nonlinear Euler solver together with the spatially distributed local flame transfer functions that capture the coupling between flame, acoustics, and hydrodynamics is explored.</p> <p> </p> <p>In this thesis, the approach to extract flame transfer functions from high-fidelity simulations and their integration with nonlinear Euler solver is presented. The dynamic mode decomposition (DMD) was used to extract spatially distributed flame transfer function (FTF) from high fidelity simulation of a single element non-premixed flame. Once extracted, the FTF was integrated with nonlinear Euler equations as a fluctuating source term of the energy equation. The time-averaged species destruction rates from the high-fidelity simulation were used as the mean source terms of the species equations. Following a variable gain approach, the local species destruction rates were modified to account for local cell constituents and maintain correct mean conditions at every time step of the nonlinear Euler simulation. The proposed reduced fidelity model was verified using a Rijke tube test case and to further assess the capabilities of the proposed model it was applied to a single element model rocket combustor, the Continuously Variable Resonance Combustor (CVRC), that exhibited self-excited combustion instabilities that are on the order of 10% of the mean pressure. The results showed that the proposed model could reproduce the unsteady behavior of the CVRC predicted by the high-fidelity simulation reasonably well. The effects of control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size are also studied. The reduced fidelity model could reproduce the limit cycle amplitude within a few percent of the mean pressure. The successful constraints on the model include good spatial resolution and FTF with all modes up to at least one dominant frequency higher than the frequencies of interest. Furthermore, the reduced fidelity model reproduced consistent mode shapes and linear growth rates that reasonably matched the experimental observations, although the apparent ability to match growth rates needs to be better understood. However, the presence of significant heat release near a pressure node of a higher harmonic mode was found to be an issue. This issue was rectified by expanding the pressure node of the higher frequency mode. Analysis of two-dimensional effects and coupling between the local pressure and heat release fluctuations showed that it may be necessary to use two dimensional spatially distributed local FTFs for accurate prediction of combustion instabilities in high energy devices such as rocket combustors. Hybrid RANS/LES-FTF simulation of the CVRC revealed that it might be necessary to use Flame Describing Function (FDF) to capture the growth of pressure fluctuations to limit cycle when Navier-Stokes solver is used.</p> <p> </p> <p>The main objectives of this thesis are:</p> <p>1. Extraction of spatially distributed local flame transfer function from the high fidelity simulation using dynamic mode decomposition and its integration with nonlinear Euler solver</p> <p>2. Verification of the proposed approach and its application to the Continuously Variable Resonance Combustor (CVRC).</p> <p>3. Sensitivity analysis of the reduced fidelity model to control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size.</p> <p> </p> <p>The goal of this thesis is to contribute towards a reduced fidelity computational tool which can accurately predict the combustion instabilities in practical systems using flame transfer functions, by providing a path way for reduced fidelity multi-element simulation, and by defining the limitations associated with using flame transfer functions and nonlinear Euler equations for non-premixed flames.</p> <p> </p><br>