Adaptive space-time analysis of three-dimensional flow instabilitites

我们的研究的目的是开发数学模型，用于精确计算适合于流动不稳定性分析的流动结构，以揭示在平稳或突然膨胀的对称性破缺开始时起作用的机制。为此，我们确定了两个目标： 1. 开发用于分析PIV流体流量测量数据的工具; 2. 使用新的理论和数值方法结合最先进的测量技术来阐明在通过轴对称膨胀的流动中发生的分叉现象的开始。 第一个目的是由于缺乏一个普遍接受的涡结构的数学定义。这导致了大量欧拉准则的使用。这样的标准的替代方案是在有限时间范围内计算的李雅普诺夫指数（所谓的FTLE）。在这个项目中，我们将研究公式的准确计算的FTLE字段的基础上的误差估计，这将允许一个最佳的网格点分布在计算域。 第二个目标是提供轴对称管道内瞬态动力学的广泛的定量研究。我们记得，在平面几何形状与突然膨胀的情况下，这是众所周知的，雷诺数高于临界值，流动的对称性丢失。 在轴对称的情况下，现有的实验研究表明，对称性破缺分岔的存在，这是不预测的对称流的线性稳定性计算。因此，在这样的流动中向湍流的转变不能准确地预测。我们建议调查的物理机制，负责这些差异，并解释为什么现有的理论未能预测临界雷诺数以上的流动成为湍流。

The aim of our research is to develop mathematical models for the accurate computation of flow structures suited to the analysis of flow instabilities, in order to reveal the mechanism at play at the onset of symmetry breaking in smooth or sudden expansions. To achieve this, we have identified two objectives: 1. To develop tools for the analysis of PIV-based fluid flow measurement data; 2. To elucidate the onset of bifurcation phenomena occurring in flows through axisymmetric expansions using new theoretical and numerical approaches in conjunction with state-of-the-art measurement techniques. The first objective stems from the lack of a universally accepted mathematical definition of a vortex structure. This has led to the use of a considerable number of Eulerian criteria. An alternative to such criteria is the Lyapunov exponent computed over a finite-time horizon (the so called FTLE). In this project, we will study formulations for the accurate computation of the FTLE field based on error estimation that will allow an optimal distribution of grid points within the computational domain. The second objective is to provide extensive quantitative investigations of the transient dynamics within axisymmetric conduit. We recall that in the case of planar geometries with abrupt expansions, it is well known that, for Reynolds numbers higher than a critical value, the symmetry of the flow is lost. In the axisymmetric case, available experimental studies indicate the existence of a symmetry breaking bifurcation that is not predicted by the linear stability computation of the symmetric flow. Hence the transition to turbulence in such flows cannot be accurately predicted. We propose to investigate the physical mechanisms responsible for these discrepancies, and to explain why existing theories fail to predict critical Reynolds numbers above which flow becomes turbulent.