Influence of viscosity on entropy noise generation through a nozzle

Abstract

Entropy (temperature) fluctuations produced by turbulent flames generate noise when they are accelerated by the flow. This so-called entropy noise is an important contributor to core noise in modern aeroengines and several semi-analytical models exist in the literature for its prediction. All these models assume the flow to be inviscid. In the present paper, contribution of viscosity on entropy noise generation and scattering through a nozzle is investigated numerically with URANS simulations and analytically through the extension of the 2D inviscid model of Emmanuelli et al. (2020). Simulations indicate noise generation and scattering is slightly reduced in the medium-frequency range in the presence of viscosity with variations below 3 dB in comparison to reference inviscid data. This noise variation is qualitatively well reproduced by the low-order model. The major effect of viscosity on noise generation and propagation lies in the presence of boundary layers. Viscous entropy noise sources and viscous diffusion of acoustic perturbations have a negligible impact on noise. Discrepancies between simulations and analytical solutions are found to come from the radial evolution of the acoustic waves in thick boundary layers, not accounted for in the model, and which impact noise scattering.

Introduction

The important efforts carried out during the last decades to reduce jet and fan noise in turbojet engines led to the emergence of additional noise sources, previously masked, among which combustion noise stands out [1], [2]. Combustion noise issues are even more important for turboshaft engines where jet and fan noises are absent [3], [4]. Two different mechanisms are involved in combustion noise. Direct combustion noise is associated with the acoustic fluctuations generated by the turbulent flame through heat release fluctuations [5], [6], whereas indirect combustion noise is produced when flow heterogeneities, such as entropy (cold and hot spots), vorticity (turbulent spots) and compositional (mixture heterogeneities) fluctuations are accelerated by the mean flow through nozzles and turbine stages [7], [8], [9], [10]. Such perturbations lead to a loss of balance, to which the flow reacts by emitting acoustic waves [11], [12]. Relative contributions of direct and indirect combustion noise to global sound emission of aeroengines is still an open question [1], [13] but recent analytical investigations suggest indirect noise may dominate direct noise in several practical situations [1], [14], hence the need for its modelling and reduction. Concerning indirect noise sources, entropy noise is thought to dominate vorticity noise because of the large entropy fluctuations produced by the turbulent flames inside combustion chambers, as well as large dissipation of turbulent structures by viscosity, particularly important at high temperature [15]. As for compositional noise, it was shown analytically to be a possible contributor in lean mixtures and supercritical nozzle flow regimes only [16], [17].

The observations above outline the need for modelling of entropy noise. The present paper focuses on its generation in a nozzle. Nozzle flow is a simple configuration very well suited to the investigation of entropy noise through analytical, numerical and experimental approaches, before possible application to more realistic turbine geometries. Analytical studies dedicated to entropy noise in nozzles essentially build on the seminal work of Marble and Candel [18] assuming one-dimensional flow (flow variables do not vary along radial and azimuthal directions). Noise produced and scattered through the nozzle is expressed in the form of transfer functions evaluated algebraically in the compact limit, i.e. acoustic and entropic wavelengths are assumed large compared to the nozzle dimensions. Non-compact transfer functions are also evaluated in the simplified configuration of a linear velocity profile, an hypothesis later relaxed by Moase et al. [19] and Giauque et al. [20] considering a piecewise-linear velocity profile. Using a different approach, Bohn [21] solves a system of partial differential equations over pressure, velocity and entropy to determine the transfer functions of generic nozzles and provides asymptotic solutions in the large frequency limit in the case of a linear velocity profile, whereas Duran and Moreau [22] consider a system of partial differential equations over mass flow rate, stagnation temperature and entropy solved with the Magnus expansion. Additionally, Mahmoudi et al. [23] describe the nozzle geometry as a succession of ducts of constant radii and apply the compact solutions of Marble and Candel between ducts to reconstruct frequency-dependent transfer functions. The one-dimensional hypothesis was recently relaxed by Duran and Morgans [24], Dowling and Mahmoudi [1] and Mahmoudi et al. [23] by considering azimuthal flow fluctuations in annular nozzles and by Emmanuelli et al. [25] and Huet et al. [26] to account for the radial evolution of the flow. The former studies allow to investigate the contribution of the different azimuthal modes in indirect noise, whereas the latter addresses the problem of shear dispersion of entropy fluctuations by the mean flow and subsequent noise modifications. The role of entropy dispersion in indirect noise production is also addressed in a different way by Mahmoudi et al. [27] in a combustion chamber. The authors convect entropy fluctuations along streamlines and reconstruct a radially averaged entropy fluctuation at combustor outlet, used to feed a one-dimensional model for the nozzle. To end, the isentropic nozzle assumption is considered by De Domenico et al. [28]. Marble and Candel’s [18] model is adapted to non isentropic flows, corresponding for instance to the presence of recirculation, and compact transfer functions are expressed as a function of a pressure loss parameter.

The hypotheses used in the models may appear restrictive in comparison with the complex phenomena occurring in real flows. Concerning acoustics, it is often argued that waves cannot exhibit radial variations because corresponding radial modes are cut-off for the low frequencies considered and viscosity can be neglected due to the short propagation distance. Limited contribution of acoustic-boundary layer interactions also plays in favour of the one-dimensional flow assumption. Duran and Moreau [22] and Huet [29] showed that such one-dimensional models indeed capture the acoustic reflection and transmission coefficients of subsonic and choked nozzle flows with a good accuracy. The compact model is also suitable to capture entropy noise in the compact limit, as illustrated by Leyko et al. [30] with the reproduction of the experimental Entropy Wave Generator test case in choked configuration [31]. When larger frequencies are considered, Morgans et al. [32] and Giusti et al. [33] numerically demonstrated that turbulent dissipation of entropy is negligible for constant section duct flows. This finding was also observed through a nozzle flow by Becerril [34] and Moreau et al. [35]. Common to these studies is the importance of shear dispersion of entropy perturbations by the mean flow and possible contribution of turbulent mixing in entropy dispersion, the latter being restricted to large frequencies because of the limited diameter of the duct where only small-scale turbulence can develop. Inside the combustion chamber, however, large turbulent structures may grow and contribute to the dispersion of entropy fluctuations before they reach the nozle, as demonstrated by Xia et al. [36] in a realistic gas turbine combustor flow-field. Additionally, thermal diffusion may play a role in the attenuation of entropy fluctuations. It was found numerically to be negligible by Xia et al. [36] in the Siemens SGT-100 gas turbine combustor. From a theoretical point of view, thermal diffusion drops with pressure and is expected to be negligible in industrial, pressurized combustors but may contribute to entropy damping in ambient pressure, lab scale experiments [37]. Finally, most of the models use the strong assumption of negligible vorticity. Entropy-induced vorticity originates from a baroclinic torque when density (entropy) fluctuations are misaligned with the mean pressure gradient [24] and was observed numerically by several authors. Its contribution was shown to be minor for inviscid nozzle flows by Duran and Morgans [24] and Emmanuelli et al. [25] but a potential noise generator when viscosity is considered [34], [35], [38]. In addition, viscosity influences the mean flow development, in particular through the presence of boundary layers, and may affect entropy dispersion and noise generation in comparison to inviscid configurations.

The objective of the present paper is to assess the role of viscosity in entropy noise generation within a nozzle through analytical modelling and numerical simulations with the URANS approach. This numerical modelling is sufficient to capture the influence of viscosity on the mean flow, in particular boundary layers, as well as on the propagation of acoustics and convection of entropy and vorticity perturbations. These phenomena are expected to be the dominant ones in the frequency bandwidth of indirect combustion noise considered here, below 1 kHz. The interaction between these perturbations and small-scale turbulence is not directly computed and cannot be accounted for in the present simulations, but it is expected to be of second order following literature results [29], [39]. In addition, URANS already proved successful in simulating entropy convection and related noise production through nozzles and turbine stages [31], [39], [40], [41]. The paper is organized as follows. Derivation of the Riemann invariants in duct flows with boundary layers is first detailed in Section 2 under simplifying assumptions. The inclusion of viscous terms in a two-dimensional low-order model is presented in Section 3. This modelling allows for an analytical discussion on the role of viscosity in entropy noise generation. The geometry considered to evaluate the transfer functions is detailed in Section 4, along with the description of the mean flow fields and the numerical procedure to compute the transfer functions with Euler and URANS equations. Modelled and simulated transfer functions are then compared in Section 5. The role of viscosity in scattering and generation of acoustic waves is discussed for both approaches and the accuracy and limitations of the model are outlined, before providing conclusions in Section 6.