Phase prediction of the response of choked nozzles to entropy and acoustic disturbances

Abstract

The development and transmission of sound through the exit of an aeroengine combustor is often investigated by modelling the complex geometry as a convergent-divergent nozzle. However, these analytical acoustic predictions are usually limited to the compact case, where the length of the nozzle is insignificant compared to the wavelength of the flow perturbations, or to cases where the variation of the mean velocity through the nozzle may be treated as linear, or piece-wise linear. Considering terms up to first order in frequency for the conservation of mass, momentum and energy, this paper investigates an alternative approach by deriving effective lengths for the passage of the flow perturbations through a supercritical convergent-divergent nozzle. The effects due to the presence of a normal shock wave are also studied using a linearised form of the Rankine-Hugoniot relations. The analyses lead to predictions for the phase and magnitude of the transmitted acoustic waves from finite-length nozzles, and are valid for low non-dimensional frequencies. It has been found that these predictions agree well with the numerical results from inviscid simulations.

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

“…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 rewritten using the reduced frequency (or Helmholtz number), Ω = f L n /c 0 , which compares the nozzle length with a characteristic acoustic wavelength. In this form the equations read (2002) and Goh & Morgans (2011a) studied the non-compact response of the choked nozzle performing an asymptotic expansion in Ω, considering ϕ(x, Ω) = ϕ (0) (x) + ϕ (1) (x)Ω + O(Ω 2 ) and similarly for ν and ρ /ρ, where ( ) (0) stands for the zeroth-order solution of the asymptotic expansion, obtained using the compact theory of Marble & Candel (1977) explained in § 2. Using the fact that the zeroth-order terms are constant through the nozzle in the choked case only, they obtained an analytical expression for the first-order correction and used it to deduce an equivalent nozzle length that corrects the phase of the reflection and transmission coefficients.…”

“…This case will be treated separately in § 5.2. Stow et al (2002) and Goh & Morgans (2011a) performed an asymptotic expansion in variables ϕ, ν and ρ /ρ using the fact that the zero-order terms are constant through the nozzle for choked flows. Writing the equation for the first order correction in the choked case they found a method to correct the phase of the reflection coefficient through an equivalent nozzle length.…”

“…The shock wave acts as an interface between the two regions. The response of the shock wave to acoustic and entropy perturbations has been studied in depth (Marble & Candel 1977;Kuo & Dowling 1996;Stow et al 2002;Moase et al 2007;Goh & Morgans 2011a;Leyko et al 2011). The main steps of the method are summarized here, the reader is referred to Moase et al (2007) and Goh & Morgans (2011a) for more details.…”

“…They used an asymptotic analysis to calculate the first order correction to the low-frequency hypothesis in the case of a choked nozzle with no specific assumption on the nozzle geometry and used this analysis to obtain an effective nozzle length that corrects the phase prediction of the reflection coefficient. Goh & Morgans (2011a) used this result to extend the effective nozzle length to compute the transmission coefficients of a choked flow.…”