Analytical Analysis of Indirect Combustion Noise in Subcritical Nozzles

Abstract

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.

Thermoacoustic Shape Optimization of a Subsonic Nozzle

“…In this article, they also propose a methodology to extend its capabilities to arbitrarily shaped nozzles. The present authors have recently worked on this idea and shown that the model can indeed be extended and applied to arbitrarily shaped nozzles and provide their acoustic transfer functions both for direct and indirect combustion noise [12].…”

A nonlinear model for indirect combustion noise through a compact nozzle

“…Finally, to ensure the nozzle compactness for all simulated configurations, the compactness criterion Ω 0 = ω 0 × L 0 dξ/u(ξ ), where ω 0 = 2πf 0 is the angular frequency of the forcing signal and L the axial length of the nozzle, has been calculated and is reported in table 5. This criterion was initially proposed by Giauque et al (2012) and represents the ratio between the time needed by the entropy wave to pass through the nozzle and its period τ 0 = 1/f 0 . Its low value indicates that the fluctuation can indeed be considered constant through the entire nozzle, so that the nozzle appears to be compact even for frequencies several times larger than f 0 .…”

“…These conditions allow acoustic and entropy waves to leave the domain without noticeable reflections and are modified to inject pressure and temperature fluctuations (Kaufmann, Nicoud & Poinsot 2002). The reader may refer to Giauque et al (2012) for additional details on the solver and its validation.…”

“…Moase, Brear & Manzie (2007), Giauque, Huet & Clero (2012) and Durán et al generalized the approach of Marble and Candel to any non-compact nozzle by solving the linearized Euler equations in the frequency domain. It allowed them to compute the linear transfer functions for the upstream and downstream acoustic waves generated by an entropy forcing.…”

“…The shocked configuration was investigated numerically and analytically by Leyko et al (2011) using large eddy simulation (LES) and Mühlbauer, Noll & Aigner (2009) with an unsteady Reynoldsaveraged Navier-Stokes (URANS) approach. The subsonic configuration has been reproduced numerically by Mühlbauer et al (2009), Giauque et al (2012) and Durán, Moreau & Poinsot (2013b) and has also been investigated analytically by Howe (2010). Dealing with the turbine configuration, Leyko et al (2010), Durán et al (2013a) and Mishra & Bodony (2013) numerically investigated the actuator disk theory by comparing two-dimensional simulations with the analytical model of Cumpsty & Marble (1977) in the presence of small amplitude entropy perturbations.…”