Turbulence, Coherent Structures, Dynamical Systems and Symmetry

Abstract

For turbulent flows at relatively low speeds there exists an excellent mathematical model in the incompressible Navier–Stokes equations. Why then is the 'problem of turbulence' so difficult? One reason is that these nonlinear partial differential equations appear to be insoluble, except through numerical simulations, which offer useful approximations but little direct understanding. Three recent developments offer new hope. First, the discovery by experimentalists of coherent structures in certain turbulent flows. Secondly, the suggestion that strange attractors and other ideas from finite-dimensional dynamical systems theory might play a role in the analysis of the governing equations. And, finally, the introduction of the Karhunen-Loève or proper orthogonal decomposition. This book introduces these developments and describes how they may be combined to create low-dimensional models of turbulence, resolving only the coherent structures. This book will interest engineers, especially in the aerospace, chemical, civil, environmental and geophysical areas, as well as physicists and applied mathematicians concerned with turbulence.

Structure-preserving model reduction for mechanical systems

“…The Karhunen-Loève method is perhaps best known in fluid computations, as described in [11]. The literature is huge in this area and we cite only the recent work of [34,35] as examples.…”

“…The method has been extensively analyzed in the literature, although the original concept goes back to Pearson [32]. We give here a brief outline of the method; for details, see [11].…”

Reduced-order representation of near-wall structures in the late transitional boundary layer

“…There has been a great deal of effort to extract coherent structures from experimental and numerical data; however, many of the identification methods rely on a priori knowledge of the specific form of the structure. One method for the extraction of spatial modes while still avoiding subjectivity is proper orthogonal decomposition (POD) (Bakewell & Lumley 1967;Sirovich 1987;Holmes, Lumley & Berkooz 1996). Since it is based on a sequence of flow snapshots, POD is equally applicable to experiments and numerical simulations.…”

Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

“…A second level of model reduction then becomes necessary: after the reduction of the infinite-dimensional system to a ("large") finite dimensional one, we seek to exploit the dissipativity of the original PDE to construct (or approximate) accurate, dynamic, "small" finite dimensional models that can be used in controller design. Such "further reduced" models are often based on modal representations of the dynamics, the modes coming from the leading part of the linearized problem ( [8]), from their Krylov subspace approximations ( [44]), from empirically determined eigenfunctions ( [43], and references therein), or from an appropriate (dissipative) part of the linear problem operator, such as the eigenfunctions of the Stokes operator in the case of the Navier-Stokes equations ( [17], [73], [74]). Beyond the qualitative similarity with singular perturbation methods that construct invariant manifolds and exploit them in controller design for finite dimensional systems ( [57]), there is an extensive interest in the use of inertial manifolds and approximate inertial manifolds for closed loop dynamics analysis and controller design (see, for example, [12,15,69,70]).…”