The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows

TLDR

Turbulence remains a critical yet poorly understood phenomenon, despite the Navier‑Stokes equations, because its nonlinear dynamics at high Reynolds numbers involve many interacting degrees of freedom across a wide range of scales and unresolved mathematical questions. The study aims to derive the complex turbulent behavior directly from the fundamental mass and momentum conservation laws embodied in the Navier‑Stokes equations. Ruelle and Takens proposed that turbulence could be a manifestation of chaotic dynamics arising from the Navier‑Stokes equations.

Abstract

It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical