A physical model for indirect noise in non-isentropic nozzles: transfer functions and stability

Abstract

We propose a mathematical model from physical principles to predict the sound generated in nozzles with dissipation. The focus is on the sound generated from the acceleration of temperature inhomogeneities (also known as entropy waves), which is referred to as indirect noise. First, we model the dissipation caused by flow recirculation and wall friction with a friction factor, which enables us to derive quasi-one-dimensional equations from conservation laws. The model is valid for both compact nozzles and nozzles with a spatial extent. Second, the predictions from the proposed model are compared against the experimental data available in the literature. Third, we compute the nozzle transfer functions for a range of Helmholtz numbers and friction factors. It is found that the friction and the Helmholtz number have a significant effect on the gain/phase of the reflected and transmitted waves. The analysis is performed from subsonic to supersonic regimes (with and without shock waves). The acoustic transfer functions vary significantly because of non-isentropic effects and the Helmholtz number, in particular, in the subsonic-choked regime. Finally, we calculate the effect that the friction of a nozzle guide vane has on thermoacoustic stability. It is found that the friction and the Helmholtz number can change thermoacoustic stability from a linearly stable regime to a linearly unstable regime. The study opens up new possibilities for the accurate prediction of indirect noise in realistic nozzles with implications on both noise emissions and thermoacoustic stability.

Compositional noise in nozzles with dissipation

“…Because we assume that the flow is chemically frozen with no mass generation, Ṡm = 0 and ṠZ = 0. As shown in Jain & Magri (2022a), the momentum and entropy source terms are…”

“…where the two-/three-dimensional dissipation effects, such as recirculation and wall friction, are averaged across the cross-section and parametrised with a friction factor, f (Jain & Magri 2022a), M is the Mach number, D is the diameter of the nozzle and R is the gas constant. The compressibility factor, Λ, and competition factor, ζ , are defined, respectively, as…”

“…Linearising (2.1)-(2.4) around the mean flow and collecting the mean-flow terms yields (Jain & Magri 2022a)…”

“…As a solution strategy, first, the primitive variables are decomposed in travelling waves (figure 1) et al 2000), as detailed in Jain & Magri (2022a). The boundary conditions are specified according to the transfer function that is being calculated.…”

“…Additionally, Yang, Guzmán-Iñigo & Morgans (2020) and Guzmán-Iñigo et al (2022) modelled the effect of non-isentropicity in entropy-noise generation in a sudden area expansion, which is a canonical flow with separation and mean pressure losses. Recently, Jain & Magri (2022a) derived the governing differential equations from first principles to model the non-isentropicity of nozzles with a spatial extent, which provided physical interpretation of the indirect sound-generation process in non-isentropic nozzles. A detailed parametric study of the nozzle and flow parameters was carried out by Jain & Magri (2022b).…”