Mathematical Methods for Turbulent Flow

湍流的数学方法

• 批准号: RGPIN-2019-06127
• 负责人: Bowman, John
• 金额: $ 1.53万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758332
• 关键词: Mathematical; Methods; Turbulent; Flow

The fundamental equations underlying the behaviour of ordinary fluids, due to Navier and Stokes, play an important role in science and engineering. However, when fluids behave chaotically, our ability to solve these equations is limited: the extreme range of interacting space and time scales makes mathematical analysis and direct numerical simulation difficult, even on massively parallel computers. We have recently discovered an exponential version of the classical four-stage Runge-Kutta integrator with stiff order 4, a feat that was previously thought to be impossible! We will generalize this technique to find exponential versions of arbitrary Runge-Kutta integrators and explore whether adaptive exponential integrators pairs, essential tools for turbulent shell models, could be useful in simulations of real fluids. Exponential integrators could also handle the stiff linearity that arises when implementing the pseudospectral method for complex geometries using a penalty method. Convolutions are at the heart of the pseudospectral method for simulating turbulent flow: they can be computed efficiently using the discrete fast Fourier transform (FFT). The cyclic nature of the FFT requires that non-fundamental harmonics, called aliases, be removed. Implicit dealiasing speeds up these computations by a factor of two and in 2D uses 2/3 (in 3D: 4/9) of the memory required by conventional zero padding. A recent parallelized version of implicit dealiasing handles an arbitrary number of input and output vectors, a crucial advance that is opening up many new applications, such as a clever formulation, due to Basdevant, of 2D convection that requires only 4 FFTs per Runge-Kutta stage, instead of the usual 5. We plan to apply implicit dealiasing to dissipationless regularizations of the Euler equations, to signal and image denoising, sparse FFTs, and more general boundary conditions. We also propose to apply it to compute partial FFTs, which arise in seismic imaging and turbulent flux profiles. The computed flux profiles could be used to calculate the damping rates needed to train turbulence subgrid models to remove the correct amount of energy from each retained scale. In view of recent interest in mapping out the attractor for 2D forced-dissipative turbulence under different forcing scenarios, we are attempting to saturate known function analytic constraints on the dynamics. The goal is to learn about invariant measures for turbulent flows. We would also like to use optimal transport theory to develop a simulation technique for 2D turbulence that respects rearrangement (Casimir) invariants. Turbulence can be visualized with the state-of-the-art 2D and 3D vector graphics language Asymptote. A proposed WebGL output format would bring the power of Asymptote to tablets and smart phones. We also propose a portable compressed binary format (v3d) for 3D vector graphics that supports vertex shading and a technique for implementing order-independent transparency.

纳维尔和斯托克斯提出的描述普通流体行为的基本方程在科学和工程中起着重要的作用。然而，当流体行为混乱时，我们解决这些方程的能力是有限的：相互作用的空间和时间尺度的极端范围使得数学分析和直接数值模拟变得困难，即使在大规模并行计算机上也是如此。我们最近发现了一个指数版本的经典四级龙格库塔积分器与刚性阶4，一个壮举，以前被认为是不可能的！我们将推广这种技术，找到指数版本的任意龙格库塔积分，并探讨是否自适应指数积分对，湍流壳模型的基本工具，可能是有用的模拟真实的流体。指数积分器还可以处理当使用罚方法对复杂几何形状实施伪谱方法时出现的刚性线性。卷积是模拟湍流的伪谱方法的核心：它们可以使用离散快速傅里叶变换（FFT）有效地计算。FFT的循环特性要求去除称为混叠的非基波谐波。隐式去混叠将这些计算速度提高了两倍，并且在2D中使用传统零填充所需内存的2/3（在3D中：4/9）。最近的一个并行化版本的隐式去混叠处理任意数量的输入和输出向量，这是一个关键的进步，开辟了许多新的应用，例如一个聪明的公式，由于Basdevant，2D对流，每个龙格库塔阶段只需要4个FFT，而不是通常的5个。我们计划将隐式去混叠应用于欧拉方程的无耗散正则化、信号和图像去噪、稀疏FFT和更一般的边界条件。我们还建议将其应用于计算部分FFT，这出现在地震成像和湍流通量剖面。计算出的通量分布可用于计算训练湍流子网格模型所需的阻尼率，以从每个保留的尺度中去除正确的能量。鉴于最近的兴趣映射出的吸引子的二维强迫耗散湍流在不同的强迫情况下，我们正试图饱和已知的功能解析约束的动态。我们的目标是了解湍流的不变措施。我们还想使用最优输运理论来开发一种尊重重排（Casimir）不变量的二维湍流模拟技术。湍流可以用最先进的2D和3D矢量图形语言Asymptote可视化。一个提议的WebGL输出格式将把Asymptote的力量带到平板电脑和智能手机上。我们还提出了一个可移植的压缩二进制格式（v3 d）的三维矢量图形，支持顶点着色和实现顺序无关的透明度的技术。