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

Abstract

The acoustic and entropy transfer functions of quasi-one-dimensional nozzles are studied analytically for both subsonic and choked flows with and without shock waves. The present analytical study extends both the compact nozzle solution obtained by Marble & Candel (1977) and the effective nozzle length proposed by Stow et al. (2002) and by Goh & Morgans (2011a) to non-zero frequencies for both modulus and phase through an asymptotic expansion of the linearized Euler equations. It also extends the piecewiselinear approximation of the velocity profile in the nozzle proposed by Moase et al. (2007) to any arbitrary profile or equivalently any nozzle geometry. The equations are written as a function of three variables, namely the dimensionless mass, total temperature and entropy fluctuations, yielding a first order linear system of differential equations with varying coefficients, which is solved using the Magnus expansion. The solution shows that both the modulus and the phase of the transfer functions of the nozzle have a strong dependence on the frequency. This holds for both choked flows and subsonic converging diverging nozzles. The method is used to compare two different nozzle geometries with the same inlet and outlet Mach numbers showing that, even if the compact solution predicts no differences between the transfer functions of the two nozzles, significant differences are found at non-zero frequencies. A parametric study is finally performed to calculate the indirect to direct noise ratio for a model combustor, showing that this ratio decreases at higher frequencies.

Entropy noise: A review of theory, progress and challenges

“…Figure 8. The acoustic admittance, Y of three different nozzle geometries, as measured experimentally 52 (, Á and «) and predicted 57 (_______, ---and Á Á Á). 0 is a non-dimensional frequency.…”

On the dissipation and dispersion of entropy waves in heat transferring channel flows

“…For more than three decades the formulations of Marble and Candel [12] were the main tools for modeling of acoustic effects of entropy waves in combustion dynamics. Over the last ten years, a number of attempts were made to release the assumptions made by Marble and Candel [12] about the linearity of the system and compactness of the nozzle [13,14,15,16,17,18]. These resulted in improving the predictability of the acoustic conversion of entropy waves.…”

Acoustic and Entropy Waves in Nozzles in Combustion Noise Framework

“…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.…”

“…It was also found that [15] the ratio of indirect to direct noise is small for laboratory experiments but large in most real aeroengines, with the indirect noise exceeding the direct noise by one order of magnitude in real gas turbines but being negligible in most laboratory rigs [15]. Duran & Moreau [23] studied numerically the ratio of indirect to direct noise for both subsonic and choked nozzles for different frequencies and again neglected any mean heat input. They studied the axisymmetric nozzle geometry of Bell et al [10] and only considered the converging section, for simplicity.…”