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 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.

Acoustic and Entropy Waves in Nozzles in Combustion Noise Framework

“…For an axisymmetric configuration, Leyko et al [16] using the compact nozzle approximation of Marble & Candel [6] and considering reflection from EWG rig outlet found excellent agreement between their analytical solution and experimental measurements. In addition to the choked isentropic nozzle studied by Leyko et al [16] a subsonic configuration has been studied by Giauque et al [25] and Duràn et al [34]. The analytical results of Duràn et al [34] based on a compact nozzle hypothesis for unchoked subsonic nozzle showed that the strength of the indirect noise is two orders of magnitude smaller than the direct noise.…”

“…The compact solution of Marble & Candel [6] has been extended to non-compact nozzles in the linear regime for choked nozzles with a shock [18,22,23], or with an isentropic supersonic expansion [24], and for subcritical nozzles [25]. Stow et al [18] and Goh & Morgans [22] used an asymptotic expansion of the linearized Euler equations to show that slightly non-compact nozzles can be modelled through use of an effective nozzle length that is slightly longer than the physical length.…”

“…Goh & Morgans [22] used an asymptotic analysis of linearized Euler equations to determine analytical solutions to the nozzle response to incoming acoustic and entropic disturbances for the various flow configurations. For non-compact choked nozzles with a supersonic diffuser [24] and for subcritical non-compact nozzles [25], the linear transfer functions have been calculated numerically and analytically for the upstream and downstream acoustic waves generated by an incident entropy disturbance. Duran & Morgans [26] studied numerically using high resolution 1-D and 2-D computations the propagation of circumferential waves (acoustic, entropy and vorticity waves) through an annular non-compact nozzle with isentropic mean flow throughout the divergent section of the nozzle (i.e.…”

Solution of the quasi-one-dimensional linearized Euler equations using flow invariants and the Magnus expansion

“…The transfer functions of the realistic nozzle geometry calculated with the Magnus expansion are also compared with a numerical solver (Silva 2010;Giauque et al 2012), solving the linearized Euler equations in the frequency domain with a second order centred spatial scheme. The discretization is such as to have at least 100 points per acoustic wavelength for Ω = 5 at the inlet.…”

Theory and application of reverberated direct and indirect noise

“…Direct and indirect noise in a reflective environment 437 Giauque, Huet & Cléro 2012;Durán, Moreau & Poinsot 2013;Lourier et al 2014). In the case where the nozzle is operated in the supersonic regime, Leyko et al (2011) indicated that the measured pressure signal was primarily due to indirect noise.…”