Mathematical Methods for Turbulent Flow

由于纳维尔和斯托克斯的贡献，普通流体行为的基本方程在科学和工程中起着重要作用。然而，当流体表现混乱时，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。我们最近发现了经典四阶龙格-库塔积分器的指数版本，它的阶数为4，这在以前被认为是不可能的！我们将推广这种技术来寻找任意龙格-库塔积分器的指数版本，并探索自适应指数积分器对，湍流壳模型的基本工具，是否可以用于模拟真实流体。指数积分器还可以处理在使用惩罚方法实现复杂几何的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被称为别名的非基谐波。隐式处理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，隐式去锯齿化的并行化版本可以处理任意数量的输入和输出向量，这是一个重要的进步，它开启了许多新的应用，例如，由于Basdevant的一个聪明的二维对流公式，每个龙格-库塔阶段只需要4个fft，而不是通常的5个。我们计划将隐式去噪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地震成像和湍流通量剖面中出现的部分fft。计算出的通量分布可以用来计算训练湍流子网格模型所需的阻尼率，以便从每个保留尺度中去除正确的能量。鉴于最近对绘制不同强迫情景下二维强迫耗散湍流的吸引子的兴趣，我们正试图饱和已知的动力学函数解析约束。目标是学习紊流的不变测度。我们还希望使用最优输运理论来开发一种尊重重排（卡西米尔）不变量的二维湍流模拟技术。湍流可以用最先进的2D和3D矢量图形语言渐近线可视化。提议的WebGL输出格式将为平板电脑和智能手机带来渐近线的强大功能。我们还提出了一种可移植的压缩二进制格式（v3d），用于支持顶点着色的3D矢量图形和一种实现顺序无关透明度的技术。

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.