Wavelet transforms are recent mathematical techniques, based on group theory and square integrable representations, which allows one to unfold a signal, or a field, into both space and scale, and possibly directions. They use analyzing functions, called wavelets, which are localized in space. The scale decomposition is obtained by dilating or contracting the chosen analyzing wavelet before convolving it with the signal. The limited spatial support of wavelets is important because then the behavior of the signal at infinity does not play any role. Therefore the wavelet analysis or syn­ thesis can be performed locally on the signal, as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions. Wavelet transforms have been applied mostly to signal processing, image coding, and numerical analysis, and they are still evolving. So far there are only two complete presentations of this topic, both written in French, one for engineers (Gasquet & Witomski 1 990) and the other for mathematicians (Meyer 1 990a), and two conference proceedings, the first in English (Combes et al 1 989), the second in French (Lemarie 1 990a). In preparation are a textbook (Holschneider 199 1 ), a course (Dau­ bee hies 1 99 1), three conference procecdings (Mcyer & Paul 199 1 , Beylkin et al 199 1b, Farge et al 1 99 1), and a special issue of IEEE Transactions

Wavelet Transforms and their Applications to Turbulence

▪ Abstract We review the direct numerical simulation (DNS) of turbulent flows. We stress that DNS is a research tool, and not a brute-force solution to the Navier-Stokes equations for engineering problems. The wide range of scales in turbulent flows requires that care be taken in their numerical solution. We discuss related numerical issues such as boundary conditions and spatial and temporal discretization. Significant insight into turbulence physics has been gained from DNS of certain idealized flows that cannot be easily attained in the laboratory. We discuss some examples. Further, we illustrate the complementary nature of experiments and computations in turbulence research. Examples are provided where DNS data has been used to evaluate measurement accuracy. Finally, we consider how DNS has impacted turbulence modeling and provided further insight into the structure of turbulent boundary layers.

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DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

In this review we will focus on a topic of fundamental importance for both plasma physics and astrophysics, namely the occurrence of large-amplitude low-frequency fluctuations of the fields that describe the plasma state. This subject will be treated within the context of the expanding solar wind and the most meaningful advances in this research field will be reported emphasizing the results obtained in the past decade or so. As a matter of fact, Ulysses’ high latitude observations and new numerical approaches to the problem, based on the dynamics of complex systems, brought new important insights which helped to better understand how turbulent fluctuations behave in the solar wind. In particular, numerical simulations within the realm of magnetohydrodynamic (MHD) turbulence theory unraveled what kind of physical mechanisms are at the basis of turbulence generation and energy transfer across the spectral domain of the fluctuations. In other words, the advances reached in these past years in the investigation of solar wind turbulence now offer a rather complete picture of the phenomenological aspect of the problem to be tentatively presented in a rather organic way.

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The Solar Wind as a Turbulence Laboratory

Small-scale turbulence has been an area of especially active research in the recent past, and several useful research directions have been pursued. Here, we selectively review this work. The emphasis is on scaling phenomenology and kinematics of small-scale structure. After providing a brief introduction to the classical notions of universality due to Kolmogorov and others, we survey the existing work on intermittency, refined similarity hypotheses, anomalous scaling exponents, derivative statistics, intermittency models, and the structure and kinematics of small-scale structure—the latter aspect coming largely from the direct numerical simulation of homogeneous turbulence in a periodic box.

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The phenomenology of small-scale turbulence

A large-Reynolds-number asymptotic theory is presented for the problem of a vortex tube of finite circulation [Gcy ] subjected to uniform non-axisymmetric irrotational strain, and aligned along an axis of positive rate of strain. It is shown that at leading order the vorticity field is determined by a solvability condition at first-order in e = 1/R[Gcy ] where R[gcy ] = [gcy ]/ν. The first-order problem is solved completely, and contours of constant rate of energy dissipation are obtained and compared with the family of contour maps obtained in a previous numerical study of the problem. It is found that the region of large dissipation does not overlap the region of large enstrophy; in fact, the dissipation rate is maximal at a distance from the vortex axis at which the enstrophy has fallen to only 2.8% of its maximum value. The correlation between enstrophy and dissipation fields is found to be 0.19 + O(e2). The solution reveals that the stretched vortex can survive for a long time even when two of the principal rates of strain are positive, provided R[gcy ] is large enough. The manner in which the theory may be extended to higher orders in e is indicated. The results are discussed in relation to the high-vorticity regions (here described as ‘sinews’) observed in many direct numerical simulations of turbulence.

Stretched vortices - the sinews of turbulence; large-Reynolds-number asymptotics

Le processus de cascade d'energie est etudie numeriquement sur un modele scalaire de la turbulence tridimensionnelle entierement developpee. L'energie se propage a travers l'intervalle inerte par intermittence, comme des explosions alternant avec des periodes de repos (on revoit le point de vue de Siggia). Pendant la phase activee, le premier exposant de Lyapunov local oscille fortement et le support du premier vecteur de Lyapunov s'etend sur le sous-intervalle inerte

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Temporal Intermittency in the Energy Cascade Process and Local Lyapunov Analysis in Fully-Developed Model Turbulence

A model equation of fully developed three-dimensional turbulence is proposed which exhibits the Kolmogorov's similarity law in its chaotic state The structure of the chaotic attractor is investigated by examining the Lyapunov spectrum for several values of viscosity The Lyapunov spectrum has a scaling property in the interior of the attractor It appears that the distribution function of the Lyapunov exponents has a singularity at a null Lyapunov exponent in the inviscid limit

Lyapunov Spectrum of a Chaotic Model of Three-Dimensional Turbulence

The growth of the gradient of a scalar temperature in a quasigeostrophic flow is studied numerically in detail. We use a flow evolving from a simple initial condition which was regarded by Constantin et al. as a candidate for a singularity formation in a finite time. For the inviscid problem, we propose a completely different interpretation of the growth, that is, the temperature gradient can be fitted equally well by a double-exponential function of time rather than an algebraic blowup. It seems impossible to distinguish whether the flow blows up or not on the basis of the inviscid computations at hand. In the viscous case, a comparison is made between a series of computations with different Reynolds numbers. The critical time at which the temperature gradient attains the first local maximum is found to depend double logarithmically on the Reynolds number, which suggests the global regularity of the inviscid flow.

Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow

An analysis of the data of a direct numerical simulation of a forced incompressible isotropic turbulence at a high Reynolds number (Rλ≊180) is made to investigate the interaction among three Fourier modes of wave numbers that form a triangle. The triad interaction is classified into six types according to the direction of the energy transfer to each Fourier mode: (+,+,−), (+,−,+), (+,−,−), (−,+,+), (−,+,−), (−,−,+), where the + (or −) denotes the energy gain (or loss) of the modes of the largest, the intermediate, and the smallest wave numbers in this order. The last three types of the interaction are very few. In the first type of the interaction, a comparable amount of energy is exchanged typically among three modes of comparable magnitude of wave numbers. In the second and third types, the magnitudes of the larger two wave numbers are comparable and much larger than the smallest one, and a great amount of energy is exchanged between the former two. This behavior of the triad interaction agrees very wel...

Koji Ohkitani

Papers

Stretched vortices - the sinews of turbulence; large-Reynolds-number asymptotics

Temporal Intermittency in the Energy Cascade Process and Local Lyapunov Analysis in Fully-Developed Model Turbulence

Lyapunov Spectrum of a Chaotic Model of Three-Dimensional Turbulence

Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow

Triad interactions in a forced turbulence