EAGER: A Non-Boussinesq, Non-Incompressible Framework for Studying Atmospheric Turbulence

本文介绍了一种新的研究大气湍流的理论框架，它挑战了两个最广泛使用的假设：空气的不可压缩性和Boussinesq近似。不可压缩的假设在数学上由空气团之后的密度守恒表示，并且意味着流场是非发散的。在Boussinesq近似下，除了重力项之外，动量方程的所有项都可以用一个恒定的统一值来代替空气密度。这两种假设在大气流动中一般都是有效的，并且有很大的好处，即它们显著地简化了方程组及其封闭。然而，真实的大气既不是不可压缩的，也不是均匀的。压缩性效应在风力涡轮机附近可能特别重要，风力涡轮机的转子尖端以接近马赫数~0.3的速度移动，马赫数为不可压缩假设的极限。此外，大多数用于日常天气预报的数值中尺度模式是可压缩的。考虑到当今强大而廉价的计算机系统，似乎是时候摆脱不必要的假设，并试图解决“真实的”湍流空气流，这是可压缩的，而不是均匀的。在这个新的框架中，空气不可压缩性和Boussinesq近似被抛弃，取而代之的是一个简单的，更少的限制，新的假设：空气密度不是湍流。而风速，温度和压力是湍流，因此它们可以被分解为平均（或分解）和波动（或涡流）分量，空气密度被假设为在时间上变化足够缓慢，其涡流分量为零。这仅仅意味着空气密度对气流变化的响应比其他气象变量慢，以这种方式，它在模拟的网格分辨率和时间步长下完全解析，并且在这些时间和空间尺度下没有混乱的湍流波动。因为在这样的假设下，密度将保持四维（与Boussinesq近似的0维相反），这样的流动在设计上不是不可压缩的，因为空气密度随着包裹的时间而变化，并且Boussinesq近似变得毫无意义。相关方程需要完全重写，但令人惊讶的是，它们与不可压缩方程和Boussinesq方程非常相似，只是在动量方程和连续性方程中各增加了一项。这些项很容易用数值计算，因为它们仅取决于平均流属性，并且不需要任何新的封闭或参数化。大涡模拟（LES）模式WiTTS（风力涡轮机和湍流模拟器），最近在特拉华州大学内部开发的，将修改和测试与新的框架，以评估可压缩效应的大小。影响可能发生在风力发电行业，该方法目前依赖于不可压缩和Boussinesq LES代码来了解涡轮机尾流对风力性能的有害影响涡轮机位于更下游。即使不可压缩性误差仅为~5- 10%，了解湍流尾流中的可压缩性也将使他们能够减少尾流损失并优化风力涡轮机布局，这可以在风力项目的生命周期内节省数百万美元，从而降低能源价格。最后，随着风力发电场变得越来越普遍，越来越多的人住在风力发电场附近，了解由涡轮机尾流引起的温度、湿度、污染物和灰尘的混合以及它们的声波，可以对住在风力涡轮机附近的人的健康和福祉产生积极影响。因为这个框架是变革性的，并且挑战了湍流理论的“现状”，所以它很适合EAGER。

A new theoretical framework for studying atmospheric turbulence is introduced, which challenges two of the most widely-used assumptions: the incompressibility of air and the Boussinesq approximation. The incompressible assumption is represented mathematically by the conservation of density following an air parcel and implies that the flow field is non-divergent. With the Boussinesq approximation, air density can be replaced by a constant, uniform value in all terms of the momentum equation, except for the gravity term. Both assumptions are generally valid in atmospheric flows and have the great benefit that they simplify noticeably the system of equations and their closures. However, the real atmosphere is neither incompressible nor homogeneous. Compressibility effects may be especially important in the proximity of wind turbines, whose rotor tips move at speeds that are close to Mach number of ~0.3, the limit for the incompressible assumption. Also, the majority of numerical mesoscale models used for routine weather forecasting are compressible. Given today's powerful and inexpensive computer systems, it seems time to step away from un-necessary assumptions and attempt to resolve "real" turbulent air flows, which are compressible and not homogeneous.In this new framework, the air incompressibility and the Boussinesq approximation are tossed and replaced by a simple, less-restrictive, new assumption: that air density is not turbulent. Whereas wind velocity, temperature, and pressure are turbulent and therefore they can be decomposed into a mean (or resolved) and a fluctuating (or eddy) component, air density is assumed to be varying slowly enough in time that its eddy component is zero. This simply means that air density responds to flow changes slower than the other meteorological variables, in such a way that it is fully resolved at the grid resolution and time step of the simulations and without chaotic, turbulent fluctuations below those temporal and spatial scales. Because under such an assumption density would remain 4-D (as opposed to 0-D as with the Boussinesq approximation), such a flow is not incompressible by design, because air density changes with time following a parcel, and the Boussinesq approximation becomes moot. The relevant equations need to be rewritten entirely, but, surprisingly, they remain very similar to their incompressible and Boussinesq counterparts, except for one additional term each in the momentum and in the continuity equations. These terms are easy to calculate numerically because they depend only on mean-flow properties and do not require any new closure or parameterization. The Large-Eddy Simulation (LES) model WiTTS (Wind Turbine and Turbulence Simulator), recently developed in-house at the University of Delaware, will be modified and tested with and without the new framework to evaluate the magnitude of the compressible effects.Impacts can occur in the wind industry, which currently relies on incompressible and Boussinesq LES codes to understand the deleterious effects of turbine wakes on the performance of wind turbines located further downstream. Even if incompressibility errors are just ~5-10%, understanding compressibility in turbulent wakes will allow them to reduce wake losses and optimize wind turbine layout, which could introduce savings of the order of millions of dollars over the lifetime of a wind project and therefore lower energy prices. Lastly, as wind farms are becoming increasingly widespread and more people live near a wind farm, understanding mixing of temperature, humidity, pollutants and dust induced by turbine wakes, as well as their sound waves, can positively impact the health and well-being of people who live in the proximity of wind turbines. Because this framework is transformative and challenges the "status-quo" of turbulence theory, it fits well with EAGER.