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

自20世纪50年代喷气式飞机时代开始以来，各国政府、科学家和工程师一直敏锐地意识到飞机噪音对健康的影响--飞机噪音的长期暴露对人类健康造成了极大的损害。例如，噪声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)以及高水平噪声反复侵入造成的明显听力损害有关。世界卫生组织估计，欧洲因噪音而损失了100万年的健康寿命；这主要是由于压力水平的持续增加而导致的心血管疾病--其中航空噪音是最大的贡献者。此外，航空环境联合会发现，这些问题给英国政府支出带来了5.4亿GB/年的负担。虽然在理解飞机噪声方面取得了巨大的进展，但在过去20年中，航班翻了一番，达到惊人的4000万架次(2019年冰冻前)，这加剧了对喷气噪声物理的研究，以发现新的降阶湍流模型。这一建议利用渐近分析建立了一种新的射流湍流数学模型。重建的湍流结构将被用于高速轴对称射流的快速噪声预测的数值代码中。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证明了Navier-Stokes方程可以精确地重新排列成这样一种形式，即作用于压力涨落的波算符等于喷流雷诺应力的双重发散。当假定静止流体的雷诺应力自协方差已知时，得到了声谱的标度性质，如著名的8次方定律。Goldstein在2003年提出的广义声学类比将流体力学变量分解为稳定的基流及其扰动，从而提出了这一思想。单位体积的声谱是传播子和纯波动雷诺应力张量的自协方差的张量积。对于适当的喷流基流，可以通过确定线性化欧拉算符的格林函数来计算传播子。然而，在莱特希尔理论中，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，当需要不同的喷管工作点进行优化设计或考虑复杂喷流时，大涡模拟数据仍然需要大量的计算资源和计算时间。使建模变得如此复杂的任何替代方案是，湍流闭合问题排除了自协方差张量的闭合理论。然而，我们最近的工作表明，当传播子被确定为与喷流扩散率相同量级(即小于1)的低频时，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脉动雷诺应力。通过求解这个渐近标度下的控制方程，我们确定了射流中的大尺度湍流(LST)结构。这种方法是由轴对称射流的二维方程组定义的，计算时间预计将比大涡模拟快一个数量级。基于LST的峰值喷气噪声雷诺应力自协方差的解决方案将与我们的项目合作伙伴在几种喷气操作条件下提供的大涡模拟数据进行比较。我们的目标是表明，湍流的LST模型提供了准确的噪声预测，是一种可行的大涡模拟的替代方案。

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.