A Spectrally Reduced Dynamic Turbulence Subgrid Model

对充分发展的湍流的理解仍然是科学和工程研究人员难以捉摸的挑战。虽然纳维-斯托克斯方程提供了典型流体的准确描述，但当它在高速下变得混乱时，我们的理解是基本的。事实上，在理解这个方程及其解方面取得的实质性进展构成了克莱数学研究所提出的七项千年挑战之一。现代计算机缺乏精确模拟地球大气或运动物体周围湍流所必需的存储空间和速度，即使是这个问题的近似解决方案也无法满足现代计算机的要求。子网格模型经常被用来在今天的计算机上进行模拟，以模拟即使在下个世纪的计算机上也很可能无法直接实现的计算。然而，在实践中使用的大多数子网格模型都是在模拟中保留的最小尺度上从系统中去除能量的特别装置。然而，为了使子网格模型忠实于底层物理，它还必须在其他尺度上注入能量，以描述小尺度波浪的跳动。这个重要的影响被目前使用的许多子网格模型所忽略。该项目的目标是为下一代湍流模拟开发可靠的子网格模型，该模型基于一种称为频谱还原的技术，该技术允许人们准确地估计在模拟中没有直接保留的尺度上的速度。最近，在一篇硕士论文中开发了一种实用的计算技术，用于将谱约简应用于一维模拟的Navier-Stokes方程（它与Navier-Stokes方程具有许多特性）。该方法是尝试用几个可变分辨率的网格数值逼近解。电网之间的数据通信旨在节约能源。将该方法推广到完整的Navier-Stokes方程将得到一个具有物理基础且不受任意参数化影响的高雷诺数湍流动力学子网格模型。这样的子网格模型将大大减少紊流阻力和输运模拟的计算成本。

The understanding of fully developed turbulence remains an elusive challenge to researchers in science and engineering. Although the Navier-Stokes equation provides an accurate description of typical fluids, when it becomes chaotic at high speeds, our understanding is rudimentary. Indeed, substantial progress towards understanding this equation and its solutions constitutes one of the seven millennium challenges proposed by the Clay Mathematics Institute. Even approximate solutions of the problem defy modern computers, which lack the storage and speed necessary to accurately simulate turbulence in the Earth's atmosphere or around a moving body. Subgrid models are often used to allow a simulation on today's computer to mimic computations that will very likely not be directly possible even on the computers of the next century. However, most of the subgrid models used in practice are ad hoc devices that remove energy from the system at the smallest scales retained in the simulation. However, for a subgrid model to be faithful to the underlying physics, it must also inject energy at other scales, to describe the beating of small-scale waves. This important effect is neglected by many subgrid models in use today. The goal of this project is to develop reliable subgrid models for the next generation of turbulence simulations, based on a technique known as spectral reduction, which allows one to accurately estimate velocities at scales that are not directly retained in the simulation. Recently, a practical computational technique for applying spectral reduction to a one-dimensional analogue of the Navier-Stokes equation (with which it shares many properties) was developed in a master's thesis. The method is an attempt to numerically approximate the solution using several grids of variable resolution. The communication of data between the grids is designed to conserve energy. The generalization of the method to the full Navier-Stokes equation would lead to a dynamical subgrid model for high-Reynolds number turbulence that has a physical basis and is free of arbitrary parametrization. Such a subgrid model would greatly reduce the computational cost of simulations of turbulent drag and transport.