Gal Berkooz

Low-dimensional models of coherent structures in turbulence

Abstract For fluid flow one has a well-accepted mathematical model: the Navier-Stokes equations. Why, then, is the problem of turbulence so intractable? One major difficulty is that the equations appear insoluble in any reasonable sense. (A direct numerical simulation certainly yields a “solution”, but it provides little understanding of the process per se .) However, three developments are beginning to bear fruit: (1) The discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows; (2) the suggestion, by Ruelle, Takens and others, that strange attractors and other ideas from dynamical systems theory might play a role in the analysis of the governing equations, and (3) the introduction of the statistical technique of Karhunen-Loeve or proper orthogonal decomposition, by Lumley in the case of turbulence. Drawing on work on modeling the dynamics of coherent structures in turbulent flows done over the past ten years, and concentrating on the near-wall region of the fully developed boundary layer, we describe how these three threads can be drawn together to weave low-dimensional models which yield new qualitative understanding. We focus on low wave number phenomena of turbulence generation, appealing to simple, conventional modeling of inertial range transport and energy dissipation.

Galerkin projections and the proper orthogonal decomposition for equivariant equations

Abstract In this paper we are interested in Galerkin projections of certain partial differential evolution equations, that retain the symmetries of the original nontruncated equations. In particular, we extend several results concerning the Galerkin projections, based on the proper orthogonal decomposition (which is also known as the Karhunen-Loeve expansion) for equations which are equivariant under the action of compact Abelian symmetry groups. We provide sufficient conditions for numerical schemes, based on the linear and the nonlinear Galerkin method, to retain the symmetries of the original system. We perform a rational analysis of methods which use symmetry to enlarge the ensemble size of a data set. Our analysis has the potential for computational savings.

Observations on the Proper Orthogonal Decomposition

The Proper Orthogonal Decomposition (P.O.D.), also known as the Karhunen-Loeve expansion, is a procedure for decomposing a stochastic field in an L2 optimal sense. It is used in diverse disciplines from image processing to turbulence. Recently the P.O.D. is receiving much attention as a tool for studying dynamics of systems in infinite dimensional space. This paper reviews the mathematical fundamentals of this theory. Also included are results on the span of the eigenfunction basis, a geometric corollary due to Chebyshev’s inequality and a relation between the P.O.D. symmetry and ergodicity.

On the relation between low-dimensional models and the dynamics of coherent structures in the turbulent wall layer

In this paper we establish some rigorous connections between the dynamics of coherent structures in the wall region of the turbulent boundary layer and the low-dimensional models of the type studied by Aubry et al. (1988). An important first step is to determine what sort of connection is feasible. We choose to study the energy budget of the models in comparison with the energy budget of the real flow. This is done by comparing the respective kinetic energy equations. In the process we reexamine some of the assumptions and approximations of Aubry et al. (1988) and perform order of magnitude analyses to determine when they hold. We find that, for the models developed in that paper, involving modes which do not vary in the streamwise direction, the energy production lies within positive, experimentally determined, bounds. Moreover, the dissipation due to neglected modes may be reproduced correctly with an order 1 value of the Heisenberg parameter α, as assumed by Aubry et al.

Coherent structures in random media and wavelets

Abstract We construct low-dimensional dynamical models for the motion of coherent structures such as those frequently observed in extended turbulent flows. These models are derived from the wavelet based Galerkin projection of the PDE describing the flow and account for the (local) interaction of a small number of coherent structures. We show that, under rather general assumptions, the wavelet projection is close to the proper orthogonal decomposition, in the average energy sense. In the specific case of the 1D Kuramoto-Sivashinsky equation, we show (numerically) that the dynamics of our model agrees qualitatively with that of the original KS equation.

Wavelet projections of the Kuramoto-Sivashinsky equation I. heteroclinic cycles and modulated traveling waves for short systems

Abstract We study fairly low dimensional projections of the Kuramoto-Sivashinsky partial differential equation, with periodic boundary conditions on a short interval, onto bases spanned by periodic wavelets. Such projections break the translation-reflection symmetry ( O (2)), replacing it by a finite dihedral group Dk. However, we show that, for the Perrier-Basdevant wavelets used here, the loss of symmetry is sufficiently mild that key global features of the dynamics are preserved. In particular, we observe heteroclinic cycles and modulated travelling waves arising from interactions of unstable modes on a four dimensional subspace spanned by appropriate combinations of wavelets. We use invariant manifold reductions in our analysis and pay particular attention to symmetries and the relation between periodic wavelets and Fourier modes, which preserve full ( O (2)) symmetry and are also optimal in that Fourier truncations maximise the energy (L2 norm) among all finite dimensional models. This study provides a foundation for current and future work in which we use wavelet bases to extract local models of evolution equations in large space domains.

Local models of spatio-temporally complex fields

Abstract We investigate the ability of local models of the one space dimensional Kuramoto-Sivashinsky partial differential equation with periodic boundary conditions, obtained by projection on a small set of Fourier modes on a short subinterval, to reproduce coherent events typical of solutions of the same equation on a much longer interval. We find that systems containing as few as two linearly unstable modes can produce realistic local events in the short term, but that for more reliable short time tracking and long term statistics, three or four interacting modes are required, and that the length of the short interval plays a subtle role, certain “resonant” lengths giving superior results.

Use of Analytical Flow Sensitivities in Static Aeroelasticity

Modern computational fluid dynamics tools can be used to predict sensitivities for flows encountered by aircraft and missiles throughout the flight regime. In this study, we describe one important multidisciplinary application of aerodynamic sensitivity analysis, namely, the coupling of the aerodynamic sensitivities with a structural dynamics analysis and optimization code

Lagrangian and Eulerian view of the bursting period

Low-dimensional models for the turbulent wall layer display an intermittent phenomenon with an ejection phase and a sweep phase that strongly resembles the bursting phenomenon observed in experimental flows. The probability distribution of inter-burst times has the observed shape [E. Stone and P. J. Holmes, Physica D 37, 20 (1989); SIAM J. Appl. Math. 50, 726 (1990); Phys. Lett. A 5, 29 (1991); P. J. Holmes and E. Stone, in Studies in Turbulence, edited by T. B. Gatski, S. Sarkar, and C. G. Speziale (Springer, Heidelberg, 1992)]. However, the time scales both for bursts and interburst durations are unrealistically long, a fact that was not appreciated until recently. We believe that the long time scales are due to the model’s inclusion of only a single coherent structure, when in fact a succession of quasi-independent structures are being swept past the sensor in an experiment. A simple statistical model of this situation restores the magnitude of the observed bursting period, although there is a great de...

The proper orthogonal decomposition, wavelets and modal approaches to the dynamics of coherent structures

We present brief precis of three related investigations. Fuller accounts can be found elsewhere. The investigations bear on the identification and prediction of coherent structures in turbulent shear flows. A second unifying thread is the Proper Orthogonal Decomposition (POD), or Karhunen-Loeve expansion, which appears in all three investigations described. The first investigation demonstrates a close connection between the coherent structures obtained using linear stochastic estimation, and those obtained from the POD. Linear stochastic estimation is often used for the identification of coherent structures. The second investigation explores the use (in homogeneous directions) of wavelets instead of Fourier modes, in the construction of dynamical models; the particular problem considered here is the Kuramoto-Sivashinsky equation. The POD eigenfunctions, of course, reduce to Fourier modes in homogeneous situations, and either can be shown to converge optimally fast; we address the question of how rapidly (by comparison) a wavelet representation converges, and how the wavelet-wavelet interactions can be handled to construct a simple model. The third investigation deals with the prediction of POD eigenfunctions in a turbulent shear flow. We show that energy-method stability theory, combined with an anisotropic eddy viscosity, and erosion of the mean velocity profile by the growing eigenfunctions, produces eigenfunctions very close to those of the POD, and the same eigenvalue spectrum at low wavenumbers.