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

自从20世纪50年代喷气式飞机时代开始以来，政府、科学家和工程师们就敏锐地意识到飞机噪音对健康的影响——长时间接触这种噪音对人体健康危害极大。例如，噪音污染的增加与儿童的认知障碍和行为问题、睡眠障碍（以及由此产生的健康问题）以及高水平噪音反复侵入造成的明显听力损害有关。世界卫生组织估计，在欧洲，由于噪音损失了100万健康生命年；这主要是由持续增加的压力水平引起的心血管疾病造成的，其中航空噪音是最大的因素。此外，航空环境联合会发现，这些问题每年给英国政府带来5.4亿英镑的支出负担。虽然在了解飞机噪音方面取得了巨大进展，但过去20年（2019年新冠肺炎前一年）的航班数量翻了一番，达到惊人的4000万架次，这增加了对喷气噪声物理学研究的需求，以发现新的低阶湍流模型。本文利用渐近分析方法建立了一种新的射流湍流数学模型。重建的湍流结构将在数值程序中用于高速轴对称射流的快速噪声预测。从根本上说，射流分解成湍流会产生压力波动，并以声音的形式传播出去。1952年，Lighthill证明，Navier-Stokes方程可以精确地重新排列成一种形式，其中作用于压力波动的波算符等于射流雷诺兹应力的双散度。当假定静止流体的雷诺应力的自协方差已知时，声谱的标度特性就得到了，例如著名的第八次幂定律。Goldstein在2003年提出的广义声学类比将流体力学变量划分为稳定基流及其扰动，从而推进了这一思想。单位体积的声谱是传播量和纯波动雷诺应力张量的自协方差的张量积。传播量可以通过确定一个适当的喷射基流的线性化欧拉算子的格林函数来计算，然而，正如在Lighthill的理论中，假设自协方差张量是已知的，这总是需要使用大涡模拟（LES）和实验来获得它的近似函数形式。但是当设计优化需要不同喷嘴工作点或考虑复杂射流时，LES数据仍然需要大量的计算资源和计算时间。使建模的任何替代方案如此复杂的是，湍流闭合问题排除了自协方差张量的闭合形式理论。然而，我们最近的工作表明，当传播器在与射流传播速率（小于单位）相同阶的低频下确定时，可以准确地预测峰值噪声。因此，这一建议提出了一种替代的、首创的分析方法来确定给定平均流解的波动雷诺应力。通过在这个渐近尺度上求解控制方程，其中射流以与空间传播相同的速率在时间上演化，我们确定了射流中的大尺度湍流（LST）结构。该方法由轴对称射流的二维方程组定义，计算时间预计比LES快一个数量级。基于lst的峰值射流噪声的雷诺应力自协方差解将与我们的项目合作伙伴在几种射流运行条件下提供的LES数据进行比较。我们的目标是证明湍流的LST模型提供了准确的噪声预测，是一个可行的替代LES。

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.