Progress in analytical methods to predict and control azimuthal combustion instability modes in annular chambers

Longitudinal low-frequency thermoacoustic unstable modes in combustion chambers have been intensively studied experimentally, numerically, and theoretically, leading to significant progress in both understanding and controlling these acoustic modes. However, modern annular gas turbines may also exhibit azimuthal modes, which are much less studied and feature specific mode structures and dynamic behaviors, leading to more complex situations. Moreover, dealing with 10–20 burners mounted in the same chamber limits the use of high fidelity simulations or annular experiments to investigate these modes because of their complexity and costs. Consequently, for such circumferential acoustic modes, theoretical tools have been developed to uncover underlying phenomena controlling their stability, nature, and dynamics. This review presents recent progress in this field. First, Galerkin and network models are described with their pros and cons in both the temporal and frequency framework. Then, key features of such acoustic modes are unveiled, focusing on their specificities such as symmetry breaking, non-linear modal coupling, forcing by turbulence. Finally, recent works on uncertainty quantifications, guided by theoretical studies and applied to annular combustors, are presented. The objective is to provide a global view of theoretical research on azimuthal modes to highlight their complexities and potential

Theoretical and numerical study of symmetry breaking effects on azimuthal thermoacoustic modes in annular combustors

A large range of physical problems, from molecules to giant stars, contains rotating symmetry and can exhibit azimuthal waves or vibrations. When this symmetry is broken, the system can become unstable with chaotic behaviors. Symmetry breaking is investigated in annular combustors prone to azimuthal thermo-acoustic instabilities. First, theories reveal that two types of symmetry breaking exist : due to different burner types distributed along the chamber or due to the flow itself . It leads to frequency splitting, fixes the mode structure and can destabilize the configuration. A UQ analysis is also performed to quantify the symmetry breaking effect due to uncertainties of flame descriptions or behaviors. To complete theory, Large Eddy Simulations are performed on a single-sector as well as on a complete 360° configuration of the annular experiment of Cambridge. Numerical results are compared to experimental data showing a good agreement. In particular, an unstable azimuthal mode at 1800 Hz grows in both LES and experiment. However, LES cannot investigate the limit cycle because of its extreme cost. To tackle this problem, a new methodology is developed, called AMT, where theory or Helmholtz solver predictions are injected into LES or DNS. This method allows to study symmetry breaking, mode nature and dynamics as well as evaluating damping in realistic annular configurations

Étude théorique et numérique des effets de brisures de symétrie sur les modes thermo-acoustiques azimutaux dans les chambres annulaires

Une large gamme de problèmes physiques, des petites molécules aux étoiles géantes, contiennent des symétries de rotation et sont sujets à des oscillations azimutales ou transverses. Quand cette symétrie est rompue, le système peut devenir instable. Dans cette thèse, les brisures de symétries sont étudiées dans les chambres de combustion annulaires, sujettes à des instabilités thermo-acoustiques azimutales. En premier lieu, deux types de brisures sont obtenus analytiquement : la première en répartissant des bruleurs différents le long de la chambre et la seconde provoquée par le champ moyen lui-même. Ces ruptures de symétries entraînent une séparation des fréquences, fixe la structure du mode et peut déstabiliser le système. De plus, une approche Quantification d’Incertitudes (UQ) permet d’évaluer l’effet de la rupture de symétries provoquée par les incertitudes sur la description ou le comportement des flammes. Pour compléter cette théorie, des Simulations aux Grandes Echelles (SGE) sont réalisées sur un mono-secteur ainsi que sur une configuration complète 360° de l’expérience annulaire de Cambridge. Les résultats numériques sont comparés aux données expérimentales et montrent un bon accord. En particulier, un mode instable à 1800 Hz croît dans les deux cas. Cependant, la SGE, limitée par son coût important, ne permet pas l’étude du cycle limite s’établissant après plusieurs centaines de millisecondes. Pour pallier à ce problème, une nouvelle approche, appelée AMT, est développée : les résultats d’une théorie ou d’un solveur acoustique sont injectés dans une simulation SGE. Cette approche permet d’étudier les brisures de symétries, la nature et la dynamique des modes acoustiques, ainsi que d’évaluer l’amortissement dans des configurations réalistes. ABSTRACT : A large range of physical problems, from molecules to giant stars, contains rotating symmetry and can exhibit azimuthal waves or vibrations. When this symmetry is broken, the system can become unstable with chaotic behaviors. Symmetry breaking is investigated in annular combustors prone to azimuthal thermo-acoustic instabilities. First, theories reveal that two types of symmetry breaking exist : due to different burner types distributed along the chamber or due to the flow itself . It leads to frequency splitting, fixes the mode structure and can destabilize the configuration. A UQ analysis is also performed to quantify the symmetry breaking effect due to uncertainties of flame descriptions or behaviors. To complete theory, Large Eddy Simulations are performed on a single-sector as well as on a complete 360° configuration of the annular experiment of Cambridge. Numerical results are compared to experimental data showing a good agreement. In particular, an unstable azimuthal mode at 1800 Hz grows in both LES and experiment. However, LES cannot investigate the limit cycle because of its extreme cost. To tackle this problem, a new methodology is developed, called AMT, where theory or Helmholtz solver predictions are injected into LES or DNS. This method allows to study symmetry breaking, mode nature and dynamics as well as evaluating damping in realistic annular configurations

Discharge coefficient of an orifice jet in cross flow: influence of inlet conditions and optimum velocity ratio

International audienceThe present work aims to characterize the discharge performance of aircraft door vent flaps. For this purpose, three different configurations with increasing complexity are studied with a RANS and a LES solver. The first configuration consists of an orifice plate in a duct for which experimental pressure loss data are available in the literature. This configuration is used as a reference for the validation of the RANS and LES setups. The duct placed downstream of the orifice is then removed to produce an unconfined geometry in which the orifice jet discharges either into an open atmosphere or a transverse flow. Finally, a classic jet in cross flow is also studied. The main objective is to analyze the discharge coefficient variations depending on three key parameters: (i) the jet Reynolds number, (ii) the inlet velocity profile, and (iii) the velocity ratio between the jet and the cross flow. Results show that for cases without cross flow, the jet Reynolds number has no influence on the discharge performance whereas a steady decrease of the orifice pressure loss is observed as the duct inlet velocity profile is deformed from that of a flat profile. The Poiseuille profile is found to minimize the pressure loss. In addition, numerical data of the reference configuration compare well with experimental values when such a profile is prescribed. Finally, simulations with a cross flow evidence an optimal velocity ratio for which the discharge coefficient is maximum and exceeds the freejet value

Sensitivity analysis of thermo-acoustic eigenproblems with adjoint methods

International audienceThis paper outlines two new applications of adjoint methods in the study of thermoacoustic instability. The first is to calculate gradients for the active subspace method, which is used in uncertainty quantification. The second is to calculate gradients in a nonlinear thermo-acoustic Helmholtz solver. Two methods are presented. The first, which uses the discrete adjoint approach, is specifically for nonlinear Helmholtz eigenvalue problems that are solved iteratively. The second, which uses a hybrid adjoint approach, is more general and can be applied to both problems

Analytical methods for azimuthal thermo-acoustic modes in annular combustion chambers

Large power densities in gas turbines can be accompanied by combustion instabilities (Culick & Kuentzmann 2006; Lieuwen & Yang 2005) due to a coupling between the flames and acoustics, creating high pressure and heat release oscillations in the chamber. Such oscillations may destroy the whole propulsion system and combustion instabilities have been a key issue for aeronautics and propulsion systems (Candel 2002; Culick & Kuentzmann 2006; Lieuwen & Yang 2005) especially in high-performance engines (Harrje & Reardon 1972; Culick 1987) for a long time. Full-scale experiments in this field are difficult (Poinsot et al. 1987; Lee & Lieuwen 2003; Lee & Anderson 1999) and numerical simulations have been used heavily to replicate the complex mechanisms involved in combustion instabilities in full-scale geometries (Wolf et al. 2009; Staffelbach et al. 2009; Wolf et al. 2010). These simulations are not sufficient to understand or control unstable modes: low-order models and theory on simplified geometries (Dowling 1995, 1997; Kopitz et al. 2005; Nicoud et al. 2007) are needed to guide both large-eddy simulations (LES) and experiments. Annular chambers used in gas turbines sometimes exhibit a specific class of unstable modes: azimuthal modes (Figure 1) propagating along the azimuthal direction eθ and not only in the longitudinal direction ez (Candel 1992; Crighton et al. 1992; Lieuwen & Yang 2005; O’Connor et al. 2015). Mechanisms leading to azimuthal instabilities are more complex than those encountered in longitudinal configurations

An analytical model for azimuthal thermoacoustic modes in an annular chamber fed by an annular plenum

This study describes an analytical method for computing azimuthal modes due to flame/acoustics coupling in annular combustors. It is based on a quasi-one-dimensional zero-Mach-number formulation where N burners are connected to an upstream annular plenum and a downstream chamber. Flames are assumed to be compact and are modeled using identical flame transfer function for all burners, characterized by an amplitude and a phase shift. Manipulation of the corresponding acoustic equations leads to a simple methodology called ANR (annular network reduction). It makes it possible to retain only the useful information related to the azimuthal modes of the annular cavities. It yields a simple dispersion relation that can be solved numerically and makes it possible to construct coupling factors between the different cavities of the combustor. A fully analytical resolution can be performed in specific situations where coupling factors are small (weak coupling). A bifurcation appears at high coupling factors, leading to a frequency lock-in of the two annular cavities (strong coupling). This tool is applied to an academic case where four burners connect an annular plenum to a chamber. For this configuration, analytical results are compared with a full three-dimensional Helmholtz solver to validate the analytical model in both weak and strong coupling regimes. Results show that this simple analytical tool can predict modes in annular combustors and investigate strategies for controlling them

Route to chaos on a dragonfly wing cross section in gliding flight

The route from linear towards nonlinear and chaotic aerodynamic regimes of a fixed dragonfly wing cross section in gliding flight is investigated numerically using direct Navier-Stokes simulations (DNSs). The dragonfly wing consists of two corrugations combined with a rear arc, which is known to provide overall good aerodynamic mean performance at low Reynolds numbers. First, the three regimes (linear, nonlinear, and chaotic) are characterized, and validated using two different fluid solvers. In particular, a peculiar transition to chaos when changing the angle of attack is observed for both solvers: The system undergoes a sudden transition to chaos in less than 0.1 degree. Second, a physical insight is given on the flow interaction between the corrugations and the rear arc, which is shown as the key phenomenon controlling the unsteady vortex dynamics and the sudden transition to chaos. Additionally, aerodynamic performances in the three regimes are given, showing that optimal performances are closely connected to the transition to chaos

Stabilization of a premixed laminar flame on a rotating cylinder

This paper investigates the stabilization of a laminar premixed flame on a rotating cylinder. Experiments and DNS are combined to analyze the effects of rotation on the flow topology and flame stabilization. Both experiment and simulation reveal that the usual stabilization pattern (two symmetric flame roots on both sides of the cylinder) is strongly affected by rotation. The flame roots positions on the upper and lower sides of the cylinder are modified with increasing rotation speeds. One of the two flame roots is quenched over a long region. The distance of the flame roots to the flameholder changes with the rotation speed until a bifurcation takes place: at a critical rotation speed, the flame roots merge, and the flame stabilizes upstream of the cylinder. DNS results are used to explain the flame topologies observed experimentally

Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part I. Sensitivity

We present an adjoint-based method for the calculation of eigenvalue perturbations in nonlinear, degenerate and non self-adjoint eigenproblems. This method is applied to a thermo-acoustic annular combustor network, the stability of which is governed by a nonlinear eigenproblem. We calculate the first- and second-order sensitivities of the growth rate and frequency to geometric, flow and flame parameters. Three different configurations are analysed. The benchmark sensitivities are obtained by finite difference, which involves solving the nonlinear eigenproblem at least as many times as the number of parameters. By solving only one adjoint eigenproblem, we obtain the sensitivities to any thermo-acoustic parameter, which match the finite-difference solutions at much lower computational cost.The authors are grateful to the 2014 Center for Turbulence Research Summer Program (Stanford University) where the ideas of this work were born. L.M. and M.P.J acknowledge the European Research Council – Project ALORS 2590620 for financial support. L.M gratefully acknowledges the financial support received from the Royal Academy of Engineering Research Fellowships scheme. The authors thank Prof. Franck Nicoud for fruitful discussions. Fig. 1 was adapted from the article of S.R. Stow and A.P. Dowling, A time-domain network model for nonlinear thermoacoustic oscillations, ASME Turbo Expo, GT2008-50770 [9] with permission of the original publisher ASME