Wavelets: Application to Turbulence

Abstract

Turbulence is a highly nonlinear regime encountered in fluid flows. Such flows are described by continuous fields, for example, velocity or pressure, assuming that the characteristic scale of the fluid motions is much larger than the mean free path of the molecular motions. The prediction of the spacetime evolution of fluid flows from first principles is given by the solutions of the Navier– Stokes equations. The turbulent regime develops when the nonlinear term of Navier–Stokes equations strongly dominates the linear term; the ratio of the norms of both terms is the Reynolds number Re, which characterizes the level of turbulence. In this regime nonlinear instabilities dominate, which leads to the flow sensitivity to initial conditions and unpredictability. The corresponding turbulent fields are highly fluctuating and their detailed motions cannot be predicted. However, if one assumes some statistical stability of the turbulence regime, averaged quantities, such as mean and variance, or other related quantities, for example, diffusion coefficients, lift or drag, may still be predicted. When turbulent flows are statistically stationary (in time) or homogeneous (in space), as it is classically supposed, one studies their energy spectrum, given by the modulus of the Fourier transform of the velocity autocorrelation. Unfortunately, since the Fourier representation spreads the information in physical space among the phases of all Fourier coefficients, the energy spectrum loses all structural information in time or space. This is a major limitation of the classical way of analyzing turbulent flows. This is why we have proposed to use the wavelet representation instead and define new analysis tools that are able to preserve time and space locality. The same is true for computing turbulent flows. Indeed, the Fourier representation is well suited to study linear motions, for which the superposition principle holds and whose generic behavior is, either to persist at a given scale, or to spread to larger ones. In contrast, the superposition principle does not hold for nonlinear motions, their archetype being the turbulent regime, which therefore cannot be decomposed into a sum of independent motions that can be separately studied. Generically, their evolution involves a wide range of scales, exciting smaller and smaller ones, even leading to finite-time singularities, e.g., shocks. The ‘‘art’’ of predicting the evolution of such nonlinear phenomena consists of disentangling the active from the passive elements: the former should be deterministically computed, while the latter could either be discarded or their effect statistically modeled. The wavelet representation allows to analyze the dynamics in both space and scale, retaining only those degrees of freedom which are essential to predict the flow evolution. Our goal is to perform a kind of ‘‘distillation’’ and retain only the elements which are essential to compute the nonlinear dynamics.