Computing Lagrangian means in multi-timescale fluid flows

计算多时间尺度流体流动中的拉格朗日均值

• 批准号: EP/Y021479/1
• 负责人: Hossein Kafiabad
• 金额: $ 47.88万
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY021479%2F1
• 关键词: Computing; Lagrangian; means; multi; timescale

Time averaging is one of the most essential tools in analysing fluid flows with multiple time scales, which are ubiquitous in nature and industries. A prominent use of time averaging is the flow decomposition into fast and slow parts to understand different phenomena associated with each time scale. For instance, in geophysical flows the wave dynamics is associated with the fast part, and the slow dynamics can be reduced to a balance between a few forces (geostrophic and hydrostatic balance). Another application of time averaging is to filter out the fast variations that are not fully captured in numerical simulations or observations to make meaningful inferences from the remaining slow dynamics. Similarly, the noise and error inherent in measurements or simulations are removed by averaging. For fluids, time-averaging can be performed in two different ways. The most straightforward approach is to average time series of flow variables at fixed spatial points, to obtain the so-called Eulerian mean. Another approach is to average flow variables along particle trajectories instead of fixed positions, which gives the Lagrangian mean. Lagrangian averaging has several pivotal advantages over its Eulerian counterpart as illustrated by a growing number of studies. For instance, it removes the Doppler shift that eclipses the separation of time scales between the background flow and waves. However, the widespread adoption of Lagrangian averaging has been hindered by computational complications. To compute the Lagrangian mean in numerical models usually particles are tracked using interpolated velocities at particle positions at every time steps. This is a computational challenge that requires extra memory space (considering time series of variables are stored at all grid points) and is ill-suited for efficient computational parallelisation. In this project, we develop a numerical approach that circumvents these difficulties. This new approach is based on the evolution of partial means instead of particle tracking. Partial means can be viewed as means over shorter intervals than the total averaging period. In this approach, we compute the Lagrangian means as solutions to a set of Partial Differential Equations (PDEs) that describe the evolution of these partial means. This paradigm could be a breakthrough in computing Lagrangian means as these PDEs can be discretised in a variety of ways and solved on-the-fly (i.e. simultaneously with the dynamical governing equations). Hence, they do not require storing any time series and substantially reduce the memory footprint compared to particle tracking. The basic idea of this numerical technique is put forward by the PI. The goals of this proposal are to 1) expand the theoretical underpinnings of this novel method, 2) develop and implement a set of numerical schemes to solve the associated PDEs, and 3) apply them to 3D fluid models and laboratory data.

时间平均是分析具有多个时间尺度的流体流动的最重要的工具之一，这些时间尺度无处不在。时间平均的显着用途是流动分解为快速和缓慢的部分，以了解与每个时间尺度相关的不同现象。例如，在地球物理流中，波动力学与快速部分相关，并且可以将缓慢的动力学降低至几种力（地球和静水平衡）之间的平衡。时间平均的另一个应用是滤除未在数值模拟或观察值中未完全捕获的快速变化，以从剩余的慢速动态中进行有意义的推论。同样，测量或模拟中固有的噪声和误差通过平均消除。对于流体，可以通过两种不同的方式进行时间平衡。最直接的方法是在固定空间点处流量变量的平均时间序列，以获得所谓的Eulerian平均值。另一种方法是沿粒子轨迹而不是固定位置的平均流量变量，这给出了拉格朗日平均值。 Lagrangian的平均值比越来越多的研究所说明的是其欧拉的相比，具有多个关键优势。例如，它消除了多普勒的偏移，从而黯然失色的时间尺度在背景流和波之间。但是，计算并发症阻碍了拉格朗日平均的广泛采用。为了计算数值模型中的拉格朗日平均值，通常在每个时间步骤的粒子位置都使用插值速度跟踪粒子。这是一个计算挑战，需要额外的记忆空间（考虑变量的时间序列在所有网格点存储），并且不适合有效的计算并行化。在这个项目中，我们开发了一种数值方法来规避这些困难。这种新方法基于部分均值而不是粒子跟踪的演变。与总平均期相比，部分手段可以视为均值的均值。在这种方法中，我们计算了Lagrangian作为描述这些部分手段演变的一组部分微分方程（PDE）的解决方案。这种范式可能是计算拉格朗日手段的突破，因为这些PDE可以通过多种方式离散并在fly上解决（即与动态的管理方程式同时解决）。因此，与粒子跟踪相比，它们不需要存储任何时间序列，并且大大减少了记忆足迹。 PI提出了这种数值技术的基本思想。该提案的目标是1）扩大这种新颖方法的理论基础，2）开发和实施一组数值方案以求解相关的PDE，3）将它们应用于3D流体模型和实验室数据。