The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Dynamics and stability of lean-premixed swirl-stabilized combustion

A review of oscillation mechanisms and the role of the precessing vortex core (PVC) in swirl combustion systems

Combustion dynamics constitutes one of the most challenging areas in combustion research. Many facets of this subject have been investigated over the past few decades for their fundamental and practical implications. Substantial progress has been accomplished in understanding analysis, modeling, and simulation. Detailed laboratory experiments and numerical computations have provided a wealth of information on elementary dynamical processes such as the response of flames to variable strain, vortex rollup, coupling between flames and acoustic modulations, and perturbed flame collisions with boundaries. Much recent work has concerned the mechanisms driving instabilities in premixed combustion and the coupling between pressure waves and combustion with application to the problem of instability in modern low NO x heavyduty gas turbine combustors. Progress in numerical modeling has allowed simulations of dynamical flames interacting with pressure waves. On this basis, it has been possible to devise predictive methods for instabilities. Important efforts have also been directed at the development of the related subject of combustion control. Research has focused on methods, sensors, actuators, control algorithms, and systems integration. In recent years, scaling from laboratory experiments to practical devices has been achieved with some successebut limitations have also been revealed. Active control of combustion has also evolved in various directions. A number of experiments on laboratory-scale combustors have shown that the amplitude of combustion instabilities could be reduced by applying control principles. Full-scale terrestrial application to gas turbine systems have allowed an increase of the stability margin of these machines. Feedback principles are also being explored to control the point of operation of combustors and engines. Operating point control has special importance in the gas turbine field since it can be used to avoid operation in unstable regions near the lean blowoff limits. More generally, closed loop feedback concepts are useful if one wishes to improve the combustion process as demonstrated by applications to automotive engines. Many future developments of combustion will use such concepts for tuning, optimization, and emissions reduction. This article proposes a broad survey of these fast-moving areas of research.

Combustion dynamics and control: Progress and challenges

Higher operating efficiencies, fewer pollutant emissions, and low capital investment have made gas turbines a dominant technology for a new power generating capacity in the US and worldwide. This book offers gas turbine users and manufacturers a valuable resource to help them sort through issues associated with combustion instabilities. In the last ten years, substantial efforts have been made in the industrial, governmental, and academic communities to understand the unique issues associated with combustion instabilities in low-emission gas turbines. The objective of this book is to compile these results into a series of chapters that address the various facets of the problem. The Case Studies section speaks to specific manufacturer and user experiences with combustion instabilities in the development stage and in fielded turbine engines. The book then goes on to examine The Fundamental Mechanisms, The Combustor Modeling, and Control Approaches.

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Combustion Instabilities In Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling

Lean blowoff of bluff body stabilized flames: Scaling and dynamics

The interactions between acoustic waves and a premixed combustion process can play an important role in the characteristicunsteadinessofcombustiondevices.Inparticular,theyareoftenresponsiblefortheoccurrenceofselfexcited, combustion-driven oscillations that are detrimental to combustor life and performance. A tutorial review is provided of current understanding of these interactions. First, the mutual interaction mechanisms between the combustion process and acoustic, vorticity, and entropy waves are described. Then, the acoustic‐ e ame interaction literatureisreviewed,primarily focusingon modeling issues.Thisliteratureisessentially organized into fourparts, depending on its treatment of 1) linear or 2) nonlinear analyses of 1) e amelets or 2) distributed reaction zones. A sizeable theoretical literature has accumulated to model the unsteady response of the laminar e ame structure, for example, the burning rate response to pressure perturbations. However, essentially no serious experimental effort has been performed to critically assess these predictions. As such, it is dife cult to determine the state of understanding in this area. On the other hand, good agreement has been achieved between well-coordinated experiments and theory describing the interactions between inherent e ame instabilities and acoustically induced e ow oscillations. Similarly, both the linear and nonlinear kinematic response of simple laminar e ames to acoustic velocity disturbances appear to be well understood, as evidenced by the agreement between surprisingly simple theory and experiment. Other than kinematic nonlinearities, additional potential mechanisms that introduce heat release‐ acoustic nonlinearities, such as e ame holding, or extinction, have been analyzed theoretically, but lack experimentalverie cation.Unsteadyreactormodelshavebeenusedextensivelytomodelcombustionprocessesinthe distributed reaction zone regimes. None of thesepredictionsappears to have been subjected to direct experimental scrutiny. Itisunlikelythatthismodeling approach willbeusefulforquantitativecombustion responsecalculations, due to their largely heuristic nature and the dife culty in rationally modeling the key interactions between reaction rate and the global characteristics of the combustion region, such as its volume. Several areas in need of work are particularly highlighted. These include e nite amplitude effects, modeling approaches for interactions outside of the e amelet regime, turbulent e ame wrinkling effects, and unsteady vortex‐ e ame interactions.

Modeling Premixed Combustion-Acoustic Wave Interactions: A Review

Laminar flame speeds of H2/CO mixtures : Effect of CO2 dilution, preheat temperature, and pressure

This paper presents a theoretical investigation of combustion instabilities in low NOX gas turbines (LNGT) that burn fuel in a lean premixed mode. It is shown that these instabilities may be caused by interactions of combustor pressure oscillations with the reactants' supply rates, producing equivalence ratio perturbations in the inlet duct. These perturbations are convected by the mean flow to the combustor where they produce large-amplitude heat-release oscillations that drive combustor pressure oscillations. It is shown in this study that in contrast to earlier analyses, which assumed a uniform instantaneous heat release throughout the flame region, the heat release within the flame may exhibit strong spatial dependence that can significantly affect the combustor stability. The proposed instability mechanism is incorporated into a model that is used to predict LNGT stability limits. The model results show that LNGT are highly prone to combustion instabilities, especially under lean operating conditions, and that the regions of instability can be approximately described in terms of a ratio of the reactants' convective time from the fuel injector to the combustor and the period of the oscillations (with some modifications that account for the structure of the combustion region). Significantly, the developed model's predictions are in good agreement with available experimental data, strongly suggesting that the proposed mechanism and the developed model properly account for the essential physics of the problem.

Tim Lieuwen

Papers

Combustion Instabilities In Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling

Lean blowoff of bluff body stabilized flames: Scaling and dynamics

Modeling Premixed Combustion-Acoustic Wave Interactions: A Review

Laminar flame speeds of H2/CO mixtures : Effect of CO2 dilution, preheat temperature, and pressure

The role of equivalence ratio oscillations in driving combustion instabilities in low NOx gas turbines