A low-order model is presented to study the propagation and interaction of acoustic and entropic perturbations through a convergent-divergent nozzle. The calculations deal with choked, unchoked, as well as compact and noncompact nozzles. In the choked case, a normal shock exists in the divergent section of the nozzle. First, for circumferential waves and for a compact choked nozzle, it is shown that the pressure, entropy, and vorticity perturbations at the nozzle outlet can be obtained directly from the perturbations at the nozzle inlet. Thus, for the choked case, there is no need to model either the linear waves or the mean flow within the nozzle. Then, to validate the models developed, cylindrical configurations corresponding to the so-called Entropy Wave Generator and Hot Acoustic Testrig are studied. For the Entropy Wave Generator, an entropy wave is generated upstream of a nozzle by an electrical heating device, and for the Hot Acoustic Testrig, a speaker is used to generate pressure waves. In these two configurations and for the choked case, the supersonic region between the nozzle throat and the normal shock is assumed to be acoustically compact. The results of the low-order model are found to give excellent agreement with the experimental results of the Entropy Wave Generator and Hot Acoustic Testrig. To give insight into the physics, the model is used to undertake a parametric study for a range of nozzle lengths and shock strengths. The low-order model is finally used to calculate the direct to indirect (entropy and vorticity) combustion noise ratio for an idealized thin annular combustor. For this model combustor, the direct acoustic noise is found to dominate within the combustor, whereas the entropy indirect noise is found to be the main source of noise downstream of the choked nozzle. The indirect vorticity noise has a negligible contribution.

The prometheus combustor design system aims to reduce the complexity of evaluating combustor designs by automatically defining preprocessing, simulation, and postprocessing tasks based on the automatic identification of combustor features within the computer-aided design (CAD) environment. This system enables best practice to be codified and topological changes to a combustor's design to be more easily considered within an automated design process. The following paper presents the prometheus combustor design system and its application to the multiobjective isothermal optimization of a combustor prediffuser and the multifidelity isothermal optimization of a fuel injector feed arm in combination with a surrogate modeling strategy accelerated via a high-performance graphical processing unit (GPU).

The following paper presents an overview of the Prometheus design system and its applications to gas turbine combustor design. Unlike a traditional “optimizer-centric” method, Prometheus aims to reduce both the level of workflow complexity and rework by taking a more “geometry-centric” approach to design optimization by shifting the control of script generation away from the optimization program to the computer aided design (CAD) package. Prometheus therefore enables significant geometry changes to be automatically reflected in all subsequent scripts necessary for the analysis of a combustor. Prometheus’ current capabilities include automatic fluid volume generation and aero-thermal and thermo-acoustic network generation as well as automatic mesh and computational fluid dynamics (CFD) script generation.

A Conditional Moment Closure (CMC) approach embedded in an LES CFD framework is presented for simulation of the reactive flow field of an aero-engine combustor operating at altitude relight conditions. Before application to the combustor geometry, the CMC model was validated on the standard lab-scale Sandia flame D. For the combustor simulation, a global mechanism for n-heptane was used along with a Lagrangian approach for the spray, to which a secondary break-up model was applied. The simulation modelled a multi-sector sub-atmospheric rig that was used to verify the altitude relight capability of the combustor. A comprehensive suite of diagnostics was applied to the rig test, including high-speed OH and kerosene PLIF as well as high speed OH* chemiluminescence. The CMC-based CFD simulation was able to predict well the position of the flame front and fuel distribution at the low pressure, low temperature conditions typical of altitude relight. Furthermore, the simulation of the ignition showed strong similarities with OH* chemiluminescence measurements of the event. An EBU-based LES was run too and showed to be unable to capture the flame front as well as the CMC model could. This work demonstrates that CMC LES can be an effective tool to support assessment of the relight capability of aero-engine combustors.

A model that simulates the possible flame trajectories following spark ignition in a generic recirculating flame has been applied to a realistic aero-engine combustor. The model has been previously validated for gaseous and simple spray flames. It uses a CFD solution of the un-ignited flow and estimates the volume of the combustor that could be ignited given a particular flow field, spray distribution, and spark location, shape and size, and also provides a measure of the variability between independent sparking events. From this information, the ease of igniting the combustor can be assessed, and hence the combustor and injector geometry and spark placement decisions can be informed at a very early stage of the design process. Results for igniting a Rolls-Royce test combustor run with kerosene at high-altitude relight conditions for which experimental data are available and for which a RANS CFD solution has been developed, demonstrate the usefulness of the model’s output. The results are consistent with experiment and also reveal that the spark characteristics and location used in the experiments, developed over a number of years by trial-and-error methods, are indeed close to optimum.

Lean premixed prevaporised (LPP) combustion can reduce NOx emissions from gas turbines, but often leads to combustion instability. Acoustic waves produce fluctuations in heat release, for instance by perturbing the fuel-air ratio. These heat fluctuations will in turn generate more acoustic waves and in some situations linear oscillations grow into large amplitude self-sustained oscillations. The resulting limit cycles can cause structural damage. Thermoacoustic oscillations will have a low amplitude initially. Thus linear models can describe the initial growth and hence give stability predictions. An unstable linear mode will grow in amplitude until nonlinear effects become sufficiently important to achieve a limit cycle. While the frequency of the linear mode can often provide a good approximation to that of the resulting limit cycle, linear theories give no prediction of its resulting amplitude. In previous work, we developed a low-order frequency-domain method to model thermoacoustic limit cycles in LPP combustors. This was based on a ‘describing function’ approach and is only applicable when there is a dominant mode and the main nonlinearity is in the combustion response to flow perturbations. In this paper that method is extended into the time domain. The main advantage of the time-domain approach is that limit-cycle stability, the influence of harmonics, and the interaction between different modes can be simulated. In LPP combustion, fluctuations in the inlet fuel-air ratio have been shown to be the dominant cause of unsteady combustion: these occur because velocity perturbations in the premix ducts cause a time-varying fuel-air ratio, which then convects downstream. If the velocity perturbation becomes comparable to the mean flow, there will be an amplitude-dependent effect on the equivalence ratio fluctuations entering the combustor and hence on the rate of heat release. Since the Mach number is low, the velocity perturbation can be comparable to the mean flow, with even reverse flow occurring, while the disturbances are still acoustically linear in that the pressure perturbation is still much smaller than the mean. Hence while the combustion response to flow velocity and equivalence ratio fluctuations must be modelled nonlinearly, the flow perturbations generated as a result of the unsteady combustion can be treated as linear. In developing a time-domain network model for nonlinear thermoacoustic oscillations an initial frequency-domain calculation is performed. The linear network model, LOTAN, is used to categorise the combustor geometry by finding the transfer function for the response of flow perturbations (at the fuel injectors, say) to heat-release oscillations. This transfer function is then converted into the time domain through an inverse Fourier transform to obtain the Green’s function, which thus relates unsteady flow to heat release at previous times. By combining this with a nonlinear flame model (relating heat release to unsteady flow at previous times) a complete time-domain solution can be found by stepping forward in time. If an unstable mode is present, its amplitude will initially grow exponentially (in accordance with linear theory) until saturation effects in the flame model become significant, and eventually a stable limit cycle will be attained. The time-domain approach enables determination of the limit-cycle. In addition, the influence of harmonics and the interaction and exchange of energy between different modes can be simulated. These effects are investigated for longitudinal and circumferential instabilities in an example combustor system and results are compared to frequency-domain limit-cycle predictions.

Combustion oscillations that arise in gas turbines can lead to plant damage. One method used to predict these oscillations is to analyse the acoustics using a simple linear model. This model requires a transfer function to describe the response of the heat release to flow perturbations. A transfer function has been obtained for a swirled premixed combustion system using experiments under atmospheric conditions and CFD. These results have been compared with analytical models. The experimental and computational transfer functions both indicate a low frequency zero. A time-delay spread model gives a good representation of the computational transfer function. The experimental transfer function is described well by a model that combines a time-delay spread with a constant gain.

Lean premixed prevaporised (LPP) combustion can reduce NOx emissions from gas turbines, but often leads to combustion instability. Acoustic waves produce fluctuations in heat release, for instance by perturbing the fuel-air ratio. These heat fluctuations will in turn generate more acoustic waves and in some situations self-sustained oscillations can form. The resulting limit cycles can have large amplitude causing structural damage. Thermoacoustic oscillations will have a low amplitude initially. Thus linear models can give stability predictions. An unstable linear mode will grow in amplitude until nonlinear effects become important and a limit cycle is achieved. While the frequency of the linear mode can provide a good approximation to that of the resulting limit cycle, linear theories give no prediction of its amplitude. A low-order model for thermoacoustic limit cycles in LPP combustors is described. The approach is based on the fact that the main nonlinearity is in the combustion response to flow perturbations. In LPP combustion, fluctuations in the inlet fuel-air ratio have been shown to be the dominant cause of unsteady combustion: these occur because velocity perturbations in the premix ducts cause a time-varying fuel-air ratio, which then convects downstream. If the velocity perturbation becomes comparable to the mean flow, there will be an amplitude-dependent effect on the equivalence ratio fluctuations entering the combustor and hence on the rate of heat release. A simple nonlinear flame model for this dependence is developed and is assumed to be the major non-linear effect on the limit cycle. Since the Mach number is low, the velocity perturbation can be comparable to the mean flow, with even reverse flow occurring, while the disturbances are still acoustically linear in that the pressure perturbation is still much smaller than the mean. Hence elsewhere the perturbations are treated as linear. In this nonlinear flame model, the flame transfer function describing the combustion response to changes in inlet flow is a function of both frequency and amplitude. The nonlinear flame transfer function is incorporated into a linear thermoacoustic network model for plane waves. Frequency, amplitude and modeshape predictions are compared with results from an atmospheric test rig. The approach is extended to circumferential waves in a thin annular geometry, where the nonlinearity leads to modal coupling.

Combustion instability has become a major issue for gas turbine manufacturers. Stricter emission regulations, particularly on nitrogen oxides, have led to the development of new combustion methods, such as lean premixed prevaporized(LPP)combustion,to replacethetraditionaldiffusion e ame.However,LPPcombustionismuchmore liable to generate strong oscillations, which can damage equipment and limit operating conditions. As a tutorial, methods to investigate combustion instabilities are reviewed. Theemphasis is on gas turbine applications and LPP combustion. The e ow is modeled as a one-dimensional mean with linear perturbations. Calculations are typically done in the frequency domain. The techniques described lead to predictions for the frequencies of oscillations and the susceptibility to instabilities for which linear disturbances grow expotentially in time. Appropriate boundary conditions are discussed, as is the change in the linearized e ow across zones of heat addition and/or area change. Many of the key concepts are e rst introduced by considering one-dimensional perturbations. Later higher-order modes, particularly circumferential waves, are introduced, and modal coupling is discussed. The modeling of a simplie ed combustion system, from compressor outlet to turbine inlet, is described. The approaches are simple and fast enough to be used at the design stage.

W. S. Cheung, G. J. M. Sims, R. W. Copplestone, J. R. Tilston and C. W. Wilson QinetiQ Limited Farnborough, Hampshire GU14 0LX United Kingdom Tel: +44 1252 397970 Fax: +44 1252 397722 E-mail: swcheung@qinetiq.com Simon R. Stow and Ann P. Dowling Department of Engineering University of Cambridge Cambridge CB2 1PZ United Kingdom Tel: +44 1223 332600 Fax: +44 1223 332662 E-mail: srs31@eng.cam.ac.uk June 16–19, 2003, Atlanta, Georgia, USA

Lean premixed prevaporised (LPP) combustion can reduce NOx emissions from gas turbines, but often leads to combustion instability. A flame transfer function describes the change in the rate of heat release in response to perturbations in the inlet flow as a function of frequency. It is a quantitative assessment of the susceptibility of combustion to disturbances. The resulting fluctuations will in turn generate more acoustic waves and in some situations self-sustained oscillations can result. Flame transfer functions for LPP combustion are poorly understood at present but are crucial for predicting combustion oscillations. This paper describes an experiment designed to measure the flame transfer function of a simple combustor incorporating realistic components. Tests were conducted initially on this combustor at atmospheric pressure (1.2 bar and 550 K) to make an early demonstration of the combustion system. The test rig consisted of a plenum chamber with an inline siren, followed by a single LPP premixer/duct and a combustion chamber with a silencer to prevent natural instabilities. The siren was used to induce variable frequency pressure/acoustic signals into the air approaching the combustor. Both unsteady pressure and heat release measurements were undertaken. There was good coherence between the pressure and heat release signals. At each test frequency, two unsteady pressure measurements in the plenum were used to calculate the acoustic waves in this chamber and hence estimate the mass-flow perturbation at the fuel injection point inside the LPP duct. The flame transfer function relating the heat release perturbation to this mass flow was found as a function of frequency. The same combustor hardware and associated instrumentation were then used for the high pressure (15 bar and 800 K) tests. Flame transfer function measurements were taken at three combustion conditions that simulated the staging point conditions (Idle, Approach and Take-off) of a large turbofan gas turbine. There was good coherence between pressure and heat release signals at Idle, indicating a close relationship between acoustic and heat release processes. Problems were encountered at high frequencies for the Approach and Take-off conditions, but the flame transfer function for the Idle case had very good qualitative agreement with the atmospheric-pressure tests. The flame transfer functions calculated here could be used directly for predicting combustion oscillations in gas turbine using the same LPP duct at the same operating conditions. More importantly they can guide work to produce a general analytical model.

Lean premixed prevaporised (LPP) combustion can reduce NOx emissions from gas turbines, but often leads to combustion instability. Acoustic waves produce fluctuations in heat release, for instance by perturbing the fuel-air ratio or flame shape. These heat fluctuations will in turn generate more acoustic waves and in some situations self-sustained oscillations can result. A linear model for thermoacoustic oscillations in LPP combustors is described. A thin annular geometry is assumed and so circumferential modes are included but radial dependence is ignored. The formulation is in terms of a network of modules such as straight ducts and area changes. At certain operating conditions, the flow is predicted to be unstable, with linear waves growing in amplitude. Helmholtz resonators can be used to absorb acoustic energy and, when carefully designed and installed at appropriate locations, can stabilise the flow. Helmholtz resonators are included in the model. Connecting a Helmholtz resonator to an annular duct destroys the axisymmetry of the geometry. This results in coupling of the circumferential modes which must be calculated. The model is used to investigate the best arrangement of resonators around the circumference of an annular duct to achieve maximum damping of a circumferential oscillation.

Small perturbations of a choked flow through a thin annular nozzle are investigated. Two cases are considered, corresponding to a ‘choked outlet’ and a ‘choked inlet’ respectively. For the first case, either an acoustic or entropy or vorticity wave is assumed to be travelling downstream towards the nozzle contraction. An asymptotic analysis for low frequency is used to find the reflected acoustic wave that is created. The boundary condition found by Marble & Candel (1977) for a compact choked nozzle is shown to apply to first order, even for circumferentially varying waves. The next-order correction can be expressed as an ‘effective length’ dependent on the mean flow (and hence the particular geometry of the nozzle) in a quantifiable way. For the second case, an acoustic wave propagates upstream and is reflected from a convergent–divergent nozzle. A normal shock is assumed to be present. By considering the interaction of the shock's position and flow perturbations, the reflected propagating waves are found for a compact nozzle. It is shown that a significant entropy disturbance is produced even when the shock is weak, and that for circumferential modes a vorticity wave is also present. Numerical calculations are conducted using a sample geometry and good agreement with the analysis is found at low frequency in both cases, and the range of validity of the asymptotic theory is determined.

We consider two viscous flows, both of which are in a class of three-dimensional flow states that are closely related to the classical Jeffery–Hamel solutions. In the first configuration, we consider a flow between two planes, intersecting at an angle α, and driven by a line-source-like solution in the neighbourhood of the apex of intersection (just as in classical, two-dimensional, Jeffery–Hamel flow). However, in addition we allow for a flow in the direction of the line of intersection of the planes (in order to capture the broader class of three-dimensional solutions). In this flow, two solution scenarios are possible; the first of these originates as a bifurcation from Jeffery–Hamel flow, whilst the second scenario describes a radial velocity of the classical Jeffery–Hamel form (also with a zero azimuthal velocity component), but with an axial velocity determined from the radial flow. Both of these solutions are exact within the Navier–Stokes framework. In the second configuration, we consider the high Reynolds number, three-dimensional flow in a diverging channel, with (generally) non-straight walls close to a plane of symmetry, and driven by a pressure gradient. Similarity solutions are found, and a connection with Jeffery–Hamel flows is established for the particular case of a flow through straight (but non-parallel) channel walls, and again, additional three-dimensional solutions are found. One member of this general class (corresponding to the flow through a straight-walled channel, driven by linearly increasing pressure in both the axial and cross-channel directions), leads to a further family of exact Navier–Stokes solutions.

Lean premixed prevaporised (LPP) combustion can reduce NOx emissions from gas turbines, but often leads to combustion instability. Acoustic waves produce fluctuations in heat release, for instance by perturbing the fuel–air ratio or flame shape. These heat fluctuations will in turn generate more acoustic waves and in some situations self-sustained oscillations can result. A linear model for thermoacoustic oscillations in LPP combustors is described. A thin annular combustor is assumed and so circumferential modes are included but radial dependence is ignored. The geometry consists of straight ducts joined by short regions of area change. Perturbations to the flow can be thought of as a combination of acoustic, entropy and vorticity waves. The development of these waves along the straight ducts is found using a propagation matrix approach. At the entrance to the combustion chamber, a flame model is used in which the unsteady heat release is related to fluctuations in fuel–air ratio. Various possible inlet and outlet conditions are described. The model is then applied to a simplified example based on a sector rig. The resonant modes are found numerically and compared with the frequencies that occurred in experiments.

We consider two viscous #ows, both of which are in a class of three-dimensional #ow states that are closely related to the classical Je!ery–Hamel solutions. In the 4rst con4guration, we consider a #ow between two planes, intersecting at an angle , and driven by a line-source-like solution in the neighbourhood of the apex of intersection (just as in classical, two-dimensional, Je!ery–Hamel #ow). However, in addition we allow for a #ow in the direction of the line of intersection of the planes (in order to capture the broader class of three-dimensional solutions). In this #ow, two solution scenarios are possible; the 4rst of these originates as a bifurcation from Je!ery–Hamel #ow, whilst the second scenario describes a radial velocity of the classical Je!ery–Hamel form (also with a zero azimuthal velocity component), but with an axial velocity determined from the radial #ow. Both of these solutions are exact within the Navier–Stokes framework. In the second con4guration, we consider the high Reynolds number, three-dimensional #ow in a diverging channel, with (generally) non-straight walls close to a plane of symmetry, and driven by a pressure gradient. Similarity solutions are found, and a connection with Je!ery–Hamel #ows is established for the particular case of a #ow through straight (but non-parallel) channel walls, and again, additional three-dimensional solutions are found. One member of this general class (corresponding to the #ow through a straight-walled channel, driven by linearly increasing pressure in both the axial and cross-channel directions), leads to a further family of exact Navier–Stokes solutions. c © 2001 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.

The incompressible boundary layer in the corner formed by two intersecting, semi-infinite planes is investigated, when the free-stream flow, aligned with the corner, is taken to be of the form U∞F(x), x representing the non-dimensional streamwise distance from the leading edge. In Dhanak & Duck (1997) similarity solutions for F(x) = xn were considered, and it was found that solutions exist for only a range of values of n, whilst for ∞ > n > −0.018, approximately, two solutions exist. In this paper, we extend the work of Dhanak & Duck to the case of non-90° corner angles and allow for streamwise development of solutions. In addition, the effect of transpiration at the walls of the corner is investigated. The governing equations are of boundary-layer type and as such are parabolic in nature. Crucially, although the leading-order pressure term is known a priori, the third-order pressure term is not, but this is nonetheless present in the leading-order governing equations, together with the transverse and crossflow viscous terms.Particular attention is paid to flows which develop spatially from similarity solutions. It turns out that two scenarios are possible. In some cases the problem may be treated in the usual parabolic sense, with standard numerical marching procedures being entirely appropriate. In other cases standard marching procedures lead to numerically inconsistent solutions. The source of this difficulty is linked to the existence of eigensolutions emanating from the leading edge (which are not present in flows appropriate to the first scenario), analogous to those found in the computation of some two-dimensional hypersonic boundary layers (Neiland 1970; Mikhailov et al. 1971; Brown & Stewartson 1975). In order to circumvent this difficulty, a different numerical solution strategy is adopted, based on a global Newton iteration procedure.A number of numerical solutions for the entire corner flow region are presented.