Distinguished material surfaces and coherent structures in three-dimensional fluid flows

Abstract

We prove analytic criteria for the existence of finite-time attracting and repelling material surfaces and lines in three-dimensional unsteady flows. The longest lived such structures define coherent structures in a Lagrangian sense. Our existence criteria involve the invariants of the velocity gradient tensor along fluid trajectories. An alternative approach to coherent structures is shown to lead to their characterization as local maximizers of the largest finite-time Lyapunov exponent field computed directly from particle paths. Both approaches provide effective tools for extracting distinguished Lagrangian structures from three-dimensional velocity data. We illustrate the results on steady and unsteady ABC-type flows.

Introduction

The mixing of passive tracers in three-dimensional fluid flows is intimately linked to the presence of coherent structures. Without being specific about their definition for now, we shall picture these structures in the extended phase space of space and time, and refer to them as Lagrangian coherent structures. A fundamental question in the study of both turbulent and laminar mixing is the location and nature of these structures, as well as their interaction with each other.

For steady velocity fields, Lagrangian coherent structures are typically delineated by stable and unstable manifolds of invariant sets that are responsible for the stretching and folding of blobs of initial conditions. The “skeleton” formed by these manifolds has been recognized to have an important role in particle transport. Its existence in certain near-integrable cases has been studied numerically (see, e.g., [4], [33], [38]), analytically (see, e.g., [28], [29], [31]), and even experimentally (see [15]). Other structures, such as invariant tori or cylinders, are known to prevent mixing by providing closed two-dimensional invariant boundaries that trajectories cannot cross (see, e.g., [7], [12], [14], [23]). If the velocity field is given analytically, one can locate hyperbolic fixed points and study their global influence through their stable and unstable manifolds (see, e.g., [8], [12], [25], [43]). Some analytic techniques for locating stable and unstable manifolds also carry over to near-integrable three-dimensional flows with periodic time dependence (see, e.g., [24]). In such flows hyperbolic periodic orbits take over the role of hyperbolic stagnation points as basic organizing centers for mixing. They can be targeted with special numerical methods even in flows that are far from integrable (see, e.g., [2], [27]). Finally, long-term statistical properties of material stretching (associated with hyperbolic foliations) have been analyzed in several studies of three-dimensional steady or time-periodic flows (see, e.g., [18]).

With the advent of computational fluid dynamics, there is an increasing need to understand and evaluate mixing in numerical velocity data with general (turbulent) time dependence. These data sets are discrete and hence velocities are not available at all points. This fact makes it numerically challenging to locate even fixed points, let alone more general dynamical structures. The main difficulty is, however, the fact the velocity data is only available on a finite-time interval. Unlike for flows with periodic or quasiperiodic time dependence, the asymptotic behavior of particles now cannot be recovered from a finite-time velocity sample. As a result, Poincaré maps are no longer available, classical stable and unstable manifolds become undefined, and the tools to locate them either fail or provide results that are difficult to interpret. At the same time, statistics-based approaches, such as long-term plots of passive scalar gradient fields, do suggest the presence of distinct Lagrangian coherent structures that resemble stable and unstable manifolds (see, e.g., [16]).

The same problem already arises in the study of two-dimensional turbulent flows. A large body of literature has been developed on the statistical evaluation of Lagrangian dynamics from two-dimensional velocity data (see [37] for a survey). More recently, several numerical studies have been aimed at understanding the geometry of Lagrangian coherent structures in specific two-dimensional data sets (see, e.g., [9], [30], [35], [34]). A theoretical approach to finite-time invariant manifolds (material lines) has been developed in [19], [20], [21]. It involves simple analytic conditions on the invariants of the velocity gradient tensor along particles paths. The conditions can be used to associate a time to each fluid particle during which it travels in a linearly stable or unstable material line. Lagrangian coherent structures are then identified as local minimizers or maximizers of the stability or instability time field. This result provides a connection between Eulerian approaches to coherent structures, notably the Okubo–Weiss criterion (see, e.g., [41]), and actual Lagrangian structures.

In the present paper, we extend the two-dimensional analysis of Haller [20] to three-dimensional velocity fields of the form $\text{u}\text{(}\text{x}\text{,t),t∈I}$, where I is a finite time interval. Our goal is to extract dynamically important material lines and surfaces from a data set representing $\text{u}$. As in the two-dimensional case, we want to achieve this by deriving conditions under which sample particle paths $\text{x}\text{(t)}$ generated by $\text{u}$ lie on finite-time hyperbolic material lines or surfaces (to be defined below) on some subset of the time interval I. The conditions turn out to involve the eigenvalues of $\text{∇}\text{u}\text{(}\text{x}\text{(t),t)}$, as well as an invariant β of $\text{∇}\text{u}\text{(}\text{x}\text{(t),t)}$ that measures how fast its eigenvectors change in time. Lagrangian coherent structures will then be identified as material lines or surfaces along which our conditions hold for locally the longest time in the flow. A sample result of such an analysis is shown in Fig. 1, which was obtained from a finite-time stability analysis of the well-known ABC flow (see Section 5 for details and notation).

The extension we complete here from two- to three-dimensional flows is technically challenging and requires customized treatments of the different eigenvalue configurations of $\text{∇}\text{u}\text{(}\text{x}\text{(t),t)}$. Accordingly, the resulting attracting and repelling material surfaces and lines fall into ten different categories (cf. Fig. 3), as opposed to two categories in the two-dimensional case. Surprisingly, the formulae obtained in this fashion are still simple, but their numerical implementation requires more thought than in two dimensions.

To compare our analytic results on distinguished Lagrangian structures with actual “bulk” particle dynamics, we use an alternative, purely geometric approach. Again, we seek material lines and surfaces that repel or attract infinitesimally close particles for locally the longest time. However, we now approach such structures by searching for material lines and surfaces transverse to which unit vectors grow by the largest amount over the time interval considered. As we show, such structures are local maximizers of the largest finite-time Lyapunov exponent field, computed directly from trajectories. This second approach to locating Lagrangian coherent structures converges more slowly initially, but provides well-resolved images for long-lived structures. While the direct connection between these images and actual structures is lost for longer times, on intermediate time scales they do offer a good comparison with our analytic results. This can be seen by comparing Fig. 1, Fig. 2.

This paper is organized as follows. In Section 2, we define attracting and repelling material lines and surfaces for three-dimensional finite-time velocity fields. We first give a hyperbolicity time definition of coherent structures and then describe our alternative geometric approach for locating them numerically from available particle data. In Section 3, we make some basic assumptions on the velocity field, then identify different eigenvalue configurations of interest for the matrix $\text{∇}\text{u}\text{(}\text{x}\text{(t),t)}$. We then proceed to the statement of Theorem 1, our main result, on the existence of attracting and repelling material lines and surfaces. A detailed proof of this theorem is given in Appendix A. Section 4 is devoted to the numerical implementation of the main theorem and Section 5 describes our numerical experiments with steady and forced ABC flows. Section 6 offers conclusions and outlines some open questions.