Asymptotic approximation of the large-scale structure of turbulence in axisymmetric jets: a first principle jet noise prediction method

Lead Research Organisation: University of York
Department Name: Mathematics

Abstract

Ever since the jet age began in the 1950s, governments, scientists, and engineers have been acutely aware of the health effects created by aircraft noise--the prolonged exposure of which is highly damaging to human health. Increased noise pollution, for example, has been linked to cognitive impairment and behavioural issues in children, sleep disturbance (and consequent health issues therefrom) as well as the obvious hearing damage caused by the repeated intrusion of high levels of noise. The World Health Organization estimates that 1-million healthy life years are lost in Europe due to noise; this is mainly by cardiovascular disease via the persistent increase in stress level-with aviation noise being the largest contributor here. Moreover, the Aviation Environment Federation found that these issues place a £540M/year burden on UK government expenditure. While there has been tremendous progress in understanding aircraft noise, the doubling of flights in the past 20 years to a staggering 40 million (in the pre-Covid year 2019) has heightened the need for research into the physics of jet noise to uncover new reduced-order turbulence models. This proposal develops a novel mathematical model for jet flow turbulence using asymptotic analysis. The re-constructed turbulence structure will be used within a numerical code for fast noise prediction of a high-speed axisymmetric jet flow.

Fundamentally, a jet flow breaking down into turbulence creates pressure fluctuations that propagate away as sound. In 1952, Lighthill showed that the Navier-Stokes equations can be exactly re-arranged into a form where a wave operator acting on the pressure fluctuation, is equal to the double-divergence of the jet's Reynolds stress. When the auto-covariance of the Reynolds stress was assumed to be known for a fluid at rest, scaling properties of the acoustic spectrum were obtained such as the celebrated 8th power law. The generalized acoustic analogy formulated by Goldstein in 2003 advanced this idea by dividing the fluid mechanical variables into a steady base flow and its perturbation. The acoustic spectrum per unit volume is a tensor product of a propagator and the auto-covariance of the purely fluctuating Reynolds stress tensor. The propagator can be calculated by determining the Green's function of the Linearized Euler operator for an appropriate jet base flow however, as in Lighthill's theory, the auto-covariance tensor is assumed to be known, which invariably requires the use of Large-Eddy Simulation (LES) and experiments to obtain an approximate functional form for it. But LES data still uses immense computational resources and computing time when different nozzle operating points are needed for design optimization or when complex jets are considered.

What makes any alternative to modelling so complex is that the turbulence closure problem precludes a closed-form theory for the auto-covariance tensor. However, our recent work revealed that the peak noise can be accurately predicted when the propagator is determined at low frequencies that are of the same order as the jet spread rate (that is lesser than unity). This proposal, therefore, sets out an alternative, first-of-its-kind, analytical approach to determine the fluctuating Reynolds stress for a given mean flow solution. By solving the governing equations at this asymptotic scaling where the jet evolves temporally at the same rate it spreads in space, we determine the Large-Scale Turbulence (LST) structure in the jet. This approach is defined by a 2-dimensional system of equations for an axisymmetric jet and the computational time is expected to be an order-of-magnitude faster than LES. The LST-based solution of the Reynolds stress auto-covariance for peak jet noise will be compared to LES data provided by our project partners at several jet operating conditions. We aim to show that the LST model of turbulence provides accurate noise predictions and is a viable alternative to LES.

Publications

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Description Collaboration with Prof. Christophe Bogey (Ecolé Centralé de Lyon) 
Organisation Ecole Centrale de Lyon
Country France 
Sector Academic/University 
PI Contribution The collaboration with Prof. Bogey is an exciting opportunity bringing together applied mathematical modelling of jet noise that we have pioneered in my group with his expert Large-Eddy Simulation database. The Bogey datasets were obtained over the past 15 years and this represents an exciting opportunity to prove the robustness of our current approaches and also how the new project on turbulence reconstruction compares over a wide range of inflow conditions that the Bogey dataset has. Currently we have a plan in place on the order of calculations (i.e. which database to start from). My PhD student (Fay Oglethorpe) will do this as part of her doctoral work. She will begin her PhD in September 2024.
Collaborator Contribution Prof. Bogey has provided initial test data and code to begin the post-processing for the first jet in our plan for analysis of Bogey datasets. Fay Oglethorpe will use the initial data and code to familiarise herself with the Bogey database. This is scheduled to begin in September/October 2024.
Impact Currently underway. We are aiming for the November 2024 abstract submission for the American Institute of Aeronautics & Astronautics (2025) Conference.
Start Year 2024
 
Description Collaboration with Prof. Dani Rodriguez (Polytechnic University of Madrid) 
Organisation Technical University of Madrid
Country Spain 
Sector Academic/University 
PI Contribution The potential collaboration between PI (Koshuriyan) was established during the American Institute of Aeronautics & Astronautics (AIAA) Aeroacoustics meeting in June 2022. The low order asymptotic model of the turbulence velocity field (and subsequent jet noise source term reconstruction) that we will construct in this project will be compared to the so-called parabolized stability equations (PSE) that Prof. Rodriguez has focused a large part of his career on. Prof. Rodriguez has agreed to send us the codes he uses for the parabolized stability calculation. This is important because it will shows two things: (a). how our first-principles asymptotic theory compares against an ad-hoc engineering approach (b). where our approach fits within the realm of fully numerical turbulence simulations on the one hand and ad-hoc approaches like PSE.
Collaborator Contribution So far there has only been discussion between myself and Prof. Rodriguez. Bearing in mind that my PDRA (Dr. Sadaf Arabi) started on Jan 2nd 2024. We hope that in the middle of this year the collaboration will begin with more earnestness.
Impact The project is still underway.
Start Year 2024
 
Description Industrial Engagement with BAE Systems PLC 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Industry/Business
Results and Impact 2 members of BAE Systems PLC came to visit my group at the University of York. The BAE Engineers were from various branches of the company in different parts of England. The aim of the visit was to develop a deeper of understanding of BAE's Engineering challenges in Acoustics as well as to present the mathematical modelling we have on in the areas of jet turbulence and aeroacoustics. The meeting was held on 27/11/23.

We also discussed the potential of the currrent project on turbulence reconstruction via first principle asymptotic analysis. We have planned future meetings to take place this year and early 2025. We hope that BAE will provide support for EPSRC CASE PhD studentships to complement Koshuriyan's current EPSRC NIA award.
Year(s) Of Engagement Activity 2023