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This article revisits the problem of indirect combustion noise in nozzles of finite length. The analytical model proposed by Moase for indirect combustion noise is rederived and applied to subcritical nozzles having shapes of increasing complexity. This model is based on the equations formulated by Marble and Candel for which an explicit solution is obtained in the subsonic framework. The discretization of the nozzle into n elementary units of finite length implies the determination of 2n integration constants for which a set of linear equations is provided in this article. The analytical method is applied to configurations of increasing complexity. Analytical solutions are compared to numerical results obtained using SUNDAY (a 1D nonlinear Euler solver in temporal space) and CEDRE (3D Navier-Stokes flow solver). Excellent agreement is found for all configurations thereby showing that acceleration discontinuities at the boundaries between adjacent elements do not influence the actual acoustic transfer functions. The issue of nozzle compactness is addressed. It is found that in the subcritical domain, spectral results should be nondimensionalized using the flow-through-time of the entire nozzle. Doing so, transfer functions of nozzles of different lengths are successfully compared and a compactness criterion is proposed that writes x à РL 0 df=uðfÞ < 1 where L is the axial length of the nozzle. Finally, the entropy wave generator (EWG) experimental setup is considered. Analytical results are compared to the results reported by Howe. Both models give similar trends and show the important role of the rising time of the fluctuating temperature front on the amplitude of the indirect acoustic emission. The experimental temperature profile and the impedance coefficients at the inlet and outlet are introduced into the analytical formulation. Results show that the indirect combustion noise mechanism is not alone responsible for the acoustic emission in the subcritical case.

This paper presents an analysis of the energy transported by disturbances in gaseous combustion. It extends the previous work of Myers (J. Fluid Mech., vol. 226, 1991, 383–400) and so includes non-zero mean-flow quantities, large-amplitude disturbances, varying specific heats and chemical non-equilibrium. This extended form of Myers’ ‘disturbance energy’ then enables complete identification of the conditions under which the famous Rayleigh source term can be derived from the equations governing combusting gas motion. These are: small disturbances in an irrotational, homentropic, non-diffusive (in terms of species, momentum and energy) and stationary mean flow at chemical equilibrium. Under these assumptions, the Rayleigh source term becomes the sole source term in a conservation equation for the classical acoustic energy. It is also argued that the exact disturbance energy flux should become an acoustic energy flux in the far-field surrounding a (reacting or non-reacting) jet. In this case, the volume integral of the disturbance energy source terms are then directly related to the area-averaged far-field sound produced by the jet. This is demonstrated by closing the disturbance energy budget over a set of aeroacoustic, direct numerical simulations of a forced, low-Mach-number, laminar, premixed flame. These budgets show that several source terms are significant, including those involving the mean-flow and entropy fields. This demonstrates that the energetics of sound generation cannot be examined by considering the Rayleigh source term alone.

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The present paper deals with the generation of sound by the passage of acoustic or entropy perturbations through a nozzle in the nonlinear regime and in the lowfrequency limit. The analytical model of Marble and Candel for compact nozzles (J. Sound Vib., vol. 55, 1977, pp. 225-243), initially developed for excitations in the linear regime, is rederived and extended to the nonlinear domain. Full nonlinear and second-order models are written for both subcritical and supercritical nozzles in the absence of shock and a detailed methodology is provided for the resolution of the second-order system. The accuracy of the second-order model is assessed for entropy forcings. It is shown to be accurate for all waves, with the exception of the upstream generated wave for subcritical diverging geometries where higher-order nonlinear contributions cannot be neglected. In the context of indirect combustion noise, the phenomenon of regime change of the nozzle due to an incoming entropy fluctuation is also addressed. Regime change is related to a Mach number modification induced by temperature and velocity fluctuations. In the present study, it translates into a limitation of the maximum amplitude of the incoming entropy forcing. Such limitations are to be considered for subcritical nozzles with significant inlet or outlet Mach numbers, where the flow transition is observed even for very low-amplitude entropy excitations. With the constraint of those limitations, the analytical extended nozzle describing functions representing the full nonlinear response for indirect combustion noise are validated through detailed comparisons with numerical simulations.

This article revisits the problem of indirect combustion noise in nozzles of finite length. The analytical model proposed by Moase et al. (JFM 2007) for indirect combustion noise is red-erived and applied to subcritical nozzles having shapes of increasing complexity. This model is based on the equations formulated by Marble & Candel (JSV 1977) for which an explicit solution is obtained in the subsonic framework. The discretization of the nozzle into n elementary units of finite length implies the determination of 2n integration constants for which a set of linear equations is provided in this article. The analytical method is applied to configurations of increasing complexity. Analytical solutions are compared to numerical results obtained using SUNDAY (a 1D non linear Euler solver in temporal space) and CEDRE (3D Navier-Stokes flow solver). An excellent agreement is found for all configurations thereby showing that acceleration discontinuities at the boundaries between adjacent elements do not influence the actual acoustic transfer functions. The issue of nozzle compactness is addressed. It is found that in the subcritical domain, spectral results should be nondimensionalized using the flow-through-time of the entire nozzle. Doing so, transfer functions of nozzles of different lengths are successfully compared and a compactness criterion is proposed that writes ω*∫0Ldζ/uζ<1 where L is the axial length of the nozzle. Finally, the EWG experimental setup of Bake et al. (JSV 2009) is considered. Analytical results are compared to the results reported by Howe (JFM 2010). Both models give similar trends and show the important role of the rising time of the fluctuating temperature front on the amplitude of the indirect acoustic emission. The experimental temperature profile and the impedance coefficients at the inlet and outlet provided by Bake et al. (JSV 2009) and Leyko et al. (JSV 2011) are introduced into the analytical formulation. Results show that the indirect combustion noise mechanism cannot be held responsible alone for the acoustic emission in the subcritical case.

Pilot flames, created by additional injectors of pure fuel, are often used in turbulent burners to enhance flame stabilisation and reduce combustion instabilities. The exact mechanisms through which these additional rich zones modify the flame anchoring location and the combustion dynamics are often difficult to identify, especially when they include unsteady hydrodynamic motion. This study presents Large Eddy Simulations (LES) of the reacting flow within a large-scale gas turbine burner for two different cases of piloting, where either 2 or 6 percent of the total methane used in the burner is injected through additional pilot flame lines. For each case, LES shows how the pilot fuel injection affects both flame stabilisation and flame stability. The 6 percent case leads to a stable flame and limited hydrodynamic perturbations in the initial flame zone. The 2 percent case is less stable, with a small-lift-off of the flame and a Precessing Vortex Core (PVC) in the cold stabilisation zone. This PVC traps some of the lean cold gases issuing from the pilot passage stream, changes the flame stabilisation point and induces instability.

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